/* Genel yazı tipi ve sayfa ayarları */
body {
  font-family: 'Libre Baskerville', 'Times New Roman', serif; /* Akademik serif yazı tipi */
  font-size: 19px; /* Daha büyük yazı boyutu */
  font-weight: 500; /* Yazılar daha belirgin */
  background-color: #fffde7; /* Tereyağı sarısına yakın açık pastel sarı */
  color: #4e342e; /* Daha koyu kahverengi tonu (yüksek kontrast için) */
}
/* Başlıklar */
h1, h2, h3 {
  color: #f57f17; /* Doygun altın sarısı ton */
  font-weight: 700; /* Daha kalın başlıklar */
  text-transform: uppercase;
  font-family: 'Lora', 'Georgia', serif; /* Akademik ve zarif serif başlık fontu */
}
/* İçindekiler tablosu */
.tocify {
  background-color: #fff8e1; /* Tereyağı sarısı tonu */
  border-radius: 6px;
  padding: 12px;
  font-family: 'Lato', sans-serif; /* Temiz sans-serif font */
  font-size: 17px;
  color: #3e2723; /* Koyu kahverengi metin */
  font-weight: 600;
}
/* Kod bloklarının (chunk) arka planı */
pre {
  background-color: #fff9c4 !important; /* Açık sarı arka plan */
  color: #e65100 !important; /* Koyu kehribar kod yazısı */
  padding: 14px;
  border-radius: 8px;
  font-family: 'Fira Code', 'Courier New', Courier, monospace;
  font-size: 17px;
  font-weight: 600;
}
/* Inline kodlar */
code {
  background-color: #fff59d; /* Yumuşak sarı ton */
  color: #bf360c; /* Derin kehribar */
  padding: 4px 8px;
  border-radius: 4px;
  font-size: 17px;
  font-weight: bold;
}
/* Grafik ve görseller */
img, .figure {
  border: 2px solid #fbc02d; /* Doygun tereyağı sarısı çerçeve */
  border-radius: 10px;
  padding: 6px;
}

0.1 R KODLAR

library(dplyr)
library(psych)
library(ggplot2)
library(tidyr)
library(tidyverse)
library(QuantPsyc)
library(EFAtools)
library(EFA.MRFA)
library(EFA.dimensions)
library(knitr)
library(data.table)
library(lavaan)
library(lavaanPlot)
library(devtools)
library(DT)
library(tibble)
library(EFAfactors)
library(outliers)
library(foreign)
library(haven)
library(highr)
library(corrplot)
library(semptools)
library(semTools)
library(sur)
library(gtools)
library(irtoys)
library(kableExtra)
library(lattice)
library(ggfortify)
library(latticeExtra)
library(plotly)
library(flextable)
library(purrr)
library(rgl)
library(olsrr)
library(scatterplot3d)
library(broom)
library(GGally)
library(stargazer)
library(effectsize)
library(rockchalk)
library(quartets)
library(flextable)
library(apaTables)
library(officer)
library("MPsychoR")
library("Gifi")
library("mirt")
library("EGAnet")
library("eRm")
library("TAM")
library("ltm")
library("WrightMap")
library("cowplot")
library(reshape2)
library(gridExtra)

Veri seti olusturma:

set.seed(123)  
n_items <- 20
n_persons <- 500
a <- runif(n_items, 0.5, 2) 
b <- rnorm(n_items, 0, 1) 
c <- runif(n_items, 0, 0.25)
theta <- rnorm(n_persons, 0, 1)
p <- matrix(nrow = n_persons, ncol = n_items)
for(i in 1:n_items){
  for(j in 1:n_persons){
    p[j,i] <- c[i] + (1 - c[i]) / (1 + exp(-a[i] * (theta[j] - b[i])))}}
df <- data.frame(theta, p)
colnames(df) <- c("theta", paste("madde", 1:n_items, sep = "_"))
head(df)

Elde edilen verinin tablo gosterimi:

library(knitr)
library(kableExtra)
df[1:21, ] %>% kable("html", digits = 3, caption = "Madde Olasılıkları Tablosu (İlk 10 Birey)") %>% kable_styling(full_width = F, bootstrap_options = c("striped", "hover", "condensed", "responsive"))

Madde Olasılıkları Tablosu (İlk 10 Birey)
theta	madde_1	madde_2	madde_3	madde_4	madde_5	madde_6	madde_7	madde_8	madde_9	madde_10	madde_11	madde_12	madde_13	madde_14	madde_15	madde_16	madde_17	madde_18	madde_19	madde_20
-0.695	0.286	0.165	0.302	0.243	0.549	0.286	0.343	0.930	0.307	0.497	0.735	0.463	0.690	0.512	0.549	0.870	0.284	0.477	0.643	0.050
-0.208	0.340	0.295	0.400	0.403	0.730	0.328	0.431	0.970	0.383	0.624	0.871	0.581	0.815	0.670	0.619	0.942	0.355	0.534	0.741	0.082
-1.265	0.241	0.083	0.218	0.139	0.367	0.245	0.277	0.828	0.253	0.360	0.518	0.347	0.515	0.326	0.468	0.703	0.220	0.416	0.515	0.035
2.169	0.756	0.956	0.889	0.979	0.996	0.604	0.917	1.000	0.900	0.963	0.998	0.952	0.994	0.981	0.878	0.999	0.783	0.794	0.967	0.858
1.208	0.580	0.811	0.738	0.889	0.974	0.484	0.772	0.998	0.729	0.893	0.990	0.868	0.973	0.933	0.796	0.996	0.620	0.699	0.919	0.492
-1.123	0.250	0.098	0.236	0.157	0.405	0.255	0.290	0.861	0.264	0.392	0.573	0.373	0.559	0.369	0.488	0.754	0.234	0.431	0.547	0.038
-0.403	0.316	0.236	0.358	0.331	0.660	0.311	0.392	0.957	0.349	0.573	0.824	0.533	0.770	0.609	0.591	0.920	0.325	0.511	0.703	0.066
-0.467	0.309	0.218	0.345	0.309	0.636	0.305	0.381	0.952	0.339	0.557	0.807	0.517	0.753	0.588	0.582	0.910	0.315	0.503	0.690	0.062
0.780	0.498	0.678	0.642	0.788	0.942	0.432	0.673	0.995	0.620	0.836	0.978	0.802	0.950	0.886	0.749	0.990	0.536	0.651	0.882	0.306
-0.083	0.357	0.338	0.429	0.453	0.770	0.340	0.458	0.976	0.408	0.656	0.895	0.612	0.841	0.706	0.637	0.954	0.375	0.548	0.763	0.096
0.253	0.407	0.468	0.511	0.595	0.860	0.374	0.539	0.987	0.484	0.735	0.942	0.693	0.897	0.792	0.683	0.975	0.435	0.589	0.817	0.151
-0.029	0.364	0.358	0.442	0.475	0.787	0.345	0.471	0.978	0.419	0.669	0.904	0.626	0.851	0.722	0.644	0.958	0.385	0.555	0.772	0.103
-0.043	0.362	0.353	0.439	0.469	0.783	0.344	0.467	0.977	0.416	0.666	0.902	0.622	0.849	0.718	0.642	0.957	0.382	0.553	0.770	0.101
1.369	0.611	0.849	0.770	0.915	0.980	0.504	0.805	0.998	0.766	0.910	0.993	0.888	0.979	0.945	0.812	0.997	0.650	0.716	0.930	0.568
-0.226	0.338	0.289	0.396	0.396	0.723	0.327	0.427	0.969	0.380	0.620	0.867	0.577	0.812	0.665	0.617	0.941	0.352	0.532	0.737	0.081
1.516	0.640	0.878	0.797	0.933	0.985	0.522	0.831	0.999	0.797	0.923	0.995	0.904	0.983	0.955	0.826	0.997	0.677	0.732	0.939	0.635
-1.549	0.225	0.062	0.189	0.112	0.308	0.228	0.255	0.748	0.237	0.304	0.418	0.302	0.434	0.247	0.430	0.587	0.196	0.388	0.452	0.032
0.585	0.463	0.603	0.594	0.724	0.919	0.410	0.624	0.993	0.568	0.802	0.968	0.764	0.934	0.856	0.725	0.986	0.498	0.628	0.860	0.237
0.124	0.386	0.416	0.479	0.540	0.829	0.361	0.507	0.983	0.453	0.706	0.926	0.662	0.878	0.761	0.665	0.968	0.411	0.573	0.797	0.126
0.216	0.401	0.453	0.502	0.579	0.852	0.370	0.529	0.986	0.475	0.727	0.938	0.684	0.891	0.783	0.678	0.973	0.428	0.584	0.811	0.143
0.380	0.427	0.520	0.543	0.646	0.886	0.387	0.571	0.989	0.515	0.762	0.954	0.721	0.913	0.819	0.699	0.980	0.459	0.604	0.834	0.179

Bu calismada olusturulan veri seti, 3 parametreli lojistik model (3PL) temel alinarak simele edilmis olup, her biri farkli ayiricilik (a), gucluk (b) ve tahmin (c) parametrelerine sahip 20 maddeye iliskin, 500 birey icin hesaplanan dogru yanit verme olasiliklarini icermektedir.
Veri setinde yer alan her bir satir, bireylerin yetenek duzeyi (θ) goz onunde bulundurularak, ilgili bireyin her bir maddeyi dogru yanitlamasina iliskin tahmin edilen olasiliklari gostermektedir.
- Bu baglamda, madde_1’den madde_20’ye kadar olan sutunlar, her bir maddenin, bireyin yetenek duzeyine karsilik gelen dogru yanit olasiliklarini ifade etmektedir.
- Olasilik degerleri 0 ile 1 arasinda degismekte olup, bireyin ilgili maddeyi dogru yanitlamasi beklenen olasiliginn buyuklugunu yansitir; bu deger ne kadar 1’e YAKIN ise, bireyin dogru yanit verme ihtimali o denli YUKSEK kabul edilmektedir.
- Bu yapi sayesinde, hem bireylerin hem de maddelerin ozelliklerine bagli olarak test performansinin daha duyarli ve ayrintili bicimde analiz edilmesi mumkun hale gelmektedir (Hambleton, Swaminathan & Rogers, 1991).

1 3PL MTK

3 PL model icin madde karakteristik egrileri formulu:

\[P_i(\theta) = c_i + (1 - c_i) * \frac{\exp[Da_i(\theta - b_i)]}{1 + \exp[Da_i(\theta - b_i)]} = c_i + \frac{1 - c_i}{1 + \exp(-[Da_i(\theta - b_i)])}\]

Burada:
- \(P_i(\theta)\): Yetenek duzeyi \(\theta\) olan bir bireyin i maddesine dogru yanit verme olasiligi
- \(a_i\): i maddesinin ayirt edicilik parametresi
- \(b_i\): i maddesinin gucluk parametresi
- \(c_i\): i maddesinin sans parametresi
- \(D\): Olcekleme faktoru (genellikle 1.7 olarak alinir)
Olasilik Egrileri Grafigi:

library(tidyr)
library(ggplot2)
df_plot <- df[, c("theta", "madde_3", "madde_7", "madde_12", "madde_16")]
df_long <- df_plot %>% pivot_longer( cols = !theta, names_to = "madde", values_to = "olasilik")
ggplot(df_long, aes(x = theta, y = olasilik, colour = madde)) +
  geom_line() +
  geom_hline(aes(yintercept = 0.5), linetype = 3) +
  ggtitle("Olasılık Eğrileri") +
  scale_x_continuous(breaks = seq(-4, 4, by = 1)) +
  labs(
    colour = "Maddeler",
    x = expression(theta),
    y = expression(P(theta))) +
  theme_bw() + theme(
    text = element_text(size = 12),
    axis.text.x = element_text(colour = "black"),
    axis.text.y = element_text(colour = "black"))

Bu grafik, 4 farkli maddeye (madde_3, madde_7, madde_12 ve madde_16) ait dogru yanit verme olasiliklarini, bireylerin gizil yetenek duzeyine (theta) gore gorsellestirmektedir.
Grafik yatay eksende theta degerlerini, dikey eksende ise ilgili maddenin dogru yanitlanma olasiligini (P(theta)) gostermektedir.
Egrilerin sekli, 3 Parametreli Lojistik Model (3PL) kapsaminda tahmin edilen ayirt edicilik (a), gucluk (b) ve sans (c) parametrelerine baglidir.
Orn., madde_16’nin egrisi digerlerine kiyasla daha dik sekilde yukselmekte olup, bu durum ilgili maddenin daha yuksek ayirt edicilige sahip oldugunu ve bireylerin yetenek duzeylerini daha hassas ayirdigini gosterir.
Eger bir egri yatay eksende daha sagda konumlanmissa, bu maddeye dogru yanit verebilmek icin daha yuksek theta degerine sahip olunmasi gerektigini yani maddenin daha zor oldugunu ifade eder.
Bu tur gorsel analizler, testteki her bir maddenin hangi yetenek duzeyinde en etkili oldugunu ortaya koyarak olcme aracinin yapisal gecerligine iliskin kanit sunmaktadir.

set.seed(123)
df_binary <- df 
for(i in 2:ncol(df)) {
  df_binary[,i] <- ifelse(runif(nrow(df)) <= df[,i], 1, 0)}
head(df_binary)

veri <- df_binary[, -1] 
head(veri)

3PL model icin test edilecek model kodu:

ucpl_model <- "F=1-20"

ucpl_uyum <- mirt(data = veri, model = ucpl_model, itemtype = "3PL", SE = T)

## Iteration: 1, Log-Lik: -5577.245, Max-Change: 1.75719Iteration: 2, Log-Lik: -5469.059, Max-Change: 0.41358Iteration: 3, Log-Lik: -5448.025, Max-Change: 0.33291Iteration: 4, Log-Lik: -5441.084, Max-Change: 0.29521Iteration: 5, Log-Lik: -5437.920, Max-Change: 0.32005Iteration: 6, Log-Lik: -5436.239, Max-Change: 0.23165Iteration: 7, Log-Lik: -5434.669, Max-Change: 0.15783Iteration: 8, Log-Lik: -5434.304, Max-Change: 0.60333Iteration: 9, Log-Lik: -5434.167, Max-Change: 0.98978Iteration: 10, Log-Lik: -5434.106, Max-Change: 0.02446Iteration: 11, Log-Lik: -5434.003, Max-Change: 0.01469Iteration: 12, Log-Lik: -5433.984, Max-Change: 0.01026Iteration: 13, Log-Lik: -5433.964, Max-Change: 0.00310Iteration: 14, Log-Lik: -5433.960, Max-Change: 0.00211Iteration: 15, Log-Lik: -5433.957, Max-Change: 0.00180Iteration: 16, Log-Lik: -5433.953, Max-Change: 0.00141Iteration: 17, Log-Lik: -5433.953, Max-Change: 0.00102Iteration: 18, Log-Lik: -5433.953, Max-Change: 0.00093Iteration: 19, Log-Lik: -5433.952, Max-Change: 0.00017Iteration: 20, Log-Lik: -5433.952, Max-Change: 0.00070Iteration: 21, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 22, Log-Lik: -5433.952, Max-Change: 0.00056Iteration: 23, Log-Lik: -5433.952, Max-Change: 0.00023Iteration: 24, Log-Lik: -5433.952, Max-Change: 0.00048Iteration: 25, Log-Lik: -5433.952, Max-Change: 0.00022Iteration: 26, Log-Lik: -5433.952, Max-Change: 0.00037Iteration: 27, Log-Lik: -5433.952, Max-Change: 0.00016Iteration: 28, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 29, Log-Lik: -5433.952, Max-Change: 0.00039Iteration: 30, Log-Lik: -5433.952, Max-Change: 0.00008
## 
## Calculating information matrix...

ucpl_par <- coef(ucpl_uyum, IRTpars = T, simplify = T)

ucpl_par$items

##                  a          b           g u
## madde_1  0.9405489  0.8588193 0.112263262 1
## madde_2  1.7888304  0.5267211 0.076453641 1
## madde_3  1.0613814  0.5132669 0.143408803 1
## madde_4  2.5285032  0.2080746 0.182462011 1
## madde_5  2.1986842 -0.1911775 0.396648970 1
## madde_6  0.5969278  1.5163780 0.072103470 1
## madde_7  1.9248363  0.5764446 0.281374076 1
## madde_8  1.6876074 -2.1107879 0.171791935 1
## madde_9  1.1459505  0.3771295 0.111431597 1
## madde_10 1.0362589 -0.8488462 0.002582630 1
## madde_11 1.7798288 -1.3279202 0.047657150 1
## madde_12 1.3668452 -0.1238128 0.194582796 1
## madde_13 1.3364282 -0.6511787 0.451559239 1
## madde_14 1.4244116 -0.6979000 0.001253227 1
## madde_15 0.5105399 -1.1980455 0.054692523 1
## madde_16 2.2764261 -1.6927067 0.003702569 1
## madde_17 1.2643324  0.8925815 0.222323806 1
## madde_18 0.9724285  0.9861186 0.346005098 1
## madde_19 1.0850333 -1.1925337 0.001892854 1
## madde_20 2.4323381  1.2748461 0.053789032 1

library(knitr)
library(kableExtra)
ucpl_par$items %>% kable("html", digits = 3, caption = "3PL Madde Parametreleri Tablosu") %>% kable_styling(full_width = F, bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% add_header_above(c(" " = 1, "Ayırıcılık (a)" = 1, "Gucluk (b)" = 1, "Tahmin (c)" = 1, "Tekrar Sayisi (u)" = 1))

3PL Madde Parametreleri Tablosu
	Ayırıcılık (a)	Gucluk (b)	Tahmin (c)	Tekrar Sayisi (u)
	a	b	g	u
madde_1	0.941	0.859	0.112	1
madde_2	1.789	0.527	0.076	1
madde_3	1.061	0.513	0.143	1
madde_4	2.529	0.208	0.182	1
madde_5	2.199	-0.191	0.397	1
madde_6	0.597	1.516	0.072	1
madde_7	1.925	0.576	0.281	1
madde_8	1.688	-2.111	0.172	1
madde_9	1.146	0.377	0.111	1
madde_10	1.036	-0.849	0.003	1
madde_11	1.780	-1.328	0.048	1
madde_12	1.367	-0.124	0.195	1
madde_13	1.336	-0.651	0.452	1
madde_14	1.424	-0.698	0.001	1
madde_15	0.511	-1.198	0.055	1
madde_16	2.276	-1.693	0.004	1
madde_17	1.264	0.893	0.222	1
madde_18	0.972	0.986	0.346	1
madde_19	1.085	-1.193	0.002	1
madde_20	2.432	1.275	0.054	1

Bu tabloda yer alan degerler, 3 parametreli lojistik model (3PL) kapsaminda madde duzeyinde tahmin edilen parametreleri gostermektedir.
Modeldeki “a” parametresi, her bir maddenin ayirt edicilik duzeyini temsil eder; bu parametre ne kadar YUKSEK ise, ilgili madde bireyler arasinda yetenek duzeyine gore DAHA IYI bir AYIRIM yapabilmektedir.
“b” parametresi, maddenin gucluk duzeyini gosterir ve bireyin o maddeyi dogru yanitlamasi icin sahip olmasi gereken yetenek duzeyini yansitir; bu deger theta (θ) olceginde yorumlanir.
“c” parametresi, ozellikle DUSUK yetenek duzeyindeki bireylerin maddeyi dogru yanitlayabilme olasiligini temsil eder ve genellikle “tahmin (sans) parametresi” olarak adlandirilir; coktan secmeli testlerde bu parametrenin degeri genellikle 0 ile 0.25 arasinda degismektedir.
Son olarak, “u” sutununda yer alan degerler, ilgili parametrelerin tahmin edilmesinde kullanilan veri sikligini ya da yanitlayan birey sayisini simgelemektetir.
Bu parametreler araciligiyla her bir maddenin olcme gucu ve olcme dogrulugu ayrintili sekilde analiz edilebilmekte ve testin genel gecerligi artirilabilmektedir (Embretson & Reise, 2000; Hambleton, Swaminathan, & Rogers, 1991).

plot(ucpl_uyum, type = "trace", which.items = 1:20)

Bu grafik, 3 Parametreli Lojistik Model (3PL) kullanilarak elde edilen madde yanit egri fonksiyonlarini gostermektedir.
Her bir panel, bir maddeye ait egriyi temsil etmekte olup, bireylerin yetenek duzeyi (theta) ile dogru yanit verme olasiligi arasindaki iliskiyi yansitmaktadir.
Yatay eksen theta degerini yani bireyin gizil yetenegini, dikey eksen ise ilgili maddeyi dogru yanitlama olasiligini ifade etmektedir.
Egri sekilleri, her bir maddenin ayirt edicilik (a), gucluk (b) ve sans (c) parametrelerine bagli olarak sekillenmektedir.
Egri ne kadar dik ise madde o kadar ayirt edici kabul edilir; egri saga dogru kaydikca madde daha zor hale gelir; alt sinirin yukarida baslamasi ise dusuk yetenekli bireylerin bile SANS ile dogru yanit verebilecegini gosterir.
Bu grafik, testte yer alan tum maddelerin olcme islevine iliskin genel bir degerlendirme saglar ve madde kalitesinin analiz edilmesine olanak tanir.

plot(ucpl_uyum, type = "trace", which.items = 1:4, facet_items = F)

Bu grafik, 3 Parametreli Lojistik Model (3PL) kapsaminda degerlendirilen ilk dort maddeye ait olasilik fonksiyonlarini gostermektedir.
Her bir egri, ilgili maddenin bireyin gizil yetenek duzeyi (theta) temelinde dogru yanitlanma olasiligini temsil eder.
Yatay eksende theta yer almakta olup bireylerin yetenek duzeyini ifade ederken, dikey eksende ise dogru yanit verme olasiligi (P(theta)) gosterilmektedir.
Egri sekilleri, maddenin ayirt edicilik (a), gucluk (b) ve sans (c) parametrelerine bagli olarak degisir.
Daha dik egriye sahip olan maddeler, bireylerin yetenek duzeylerine karsi daha hassas bir ayrim yapar.
Egri sola ya da saga kaydiginda, maddenin kolay ya da zor oldugu yorumu yapilabilir.
Bu tur grafikler, testin madde duzeyindeki olcme niteligini degerlendirmek ve hangi maddelerin hangi yetenek duzeylerinde daha etkili oldugunu analiz etmek icin kullanilir.

fit3pl <- tpm(veri)
coef(fit3pl)

##                Gussng     Dffclt    Dscrmn
## madde_1  1.111881e-01  0.8546614 0.9401635
## madde_2  7.358496e-02  0.5214554 1.7720172
## madde_3  1.430746e-01  0.5119956 1.0622246
## madde_4  1.813544e-01  0.2060510 2.5131647
## madde_5  4.018852e-01 -0.1772559 2.2411697
## madde_6  3.026510e-02  1.3811175 0.5404138
## madde_7  2.823694e-01  0.5782851 1.9355488
## madde_8  1.981718e-01 -2.0711955 1.7078674
## madde_9  1.085069e-01  0.3701222 1.1384512
## madde_10 2.460911e-04 -0.8532327 1.0350438
## madde_11 6.783189e-02 -1.2989022 1.8039900
## madde_12 1.949483e-01 -0.1227380 1.3684696
## madde_13 4.529876e-01 -0.6462657 1.3386986
## madde_14 1.183904e-04 -0.6990382 1.4246560
## madde_15 1.692018e-02 -1.3510183 0.4987451
## madde_16 1.730458e-04 -1.6957463 2.2704352
## madde_17 2.233717e-01  0.8933010 1.2732651
## madde_18 3.469721e-01  0.9882459 0.9773798
## madde_19 7.918629e-05 -1.1964440 1.0828713
## madde_20 5.371835e-02  1.2748783 2.4284157

library(knitr)
library(kableExtra)
parametreler <- as.data.frame(coef(fit3pl))
parametreler %>% kable("html", digits = 3, caption = "3PL Modeline Ait Madde Parametreleri") %>% kable_styling(full_width = F, bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% add_header_above(c("Madde Parametreleri" = 4))

3PL Modeline Ait Madde Parametreleri
Madde Parametreleri
	Gussng	Dffclt	Dscrmn
madde_1	0.111	0.855	0.940
madde_2	0.074	0.521	1.772
madde_3	0.143	0.512	1.062
madde_4	0.181	0.206	2.513
madde_5	0.402	-0.177	2.241
madde_6	0.030	1.381	0.540
madde_7	0.282	0.578	1.936
madde_8	0.198	-2.071	1.708
madde_9	0.109	0.370	1.138
madde_10	0.000	-0.853	1.035
madde_11	0.068	-1.299	1.804
madde_12	0.195	-0.123	1.368
madde_13	0.453	-0.646	1.339
madde_14	0.000	-0.699	1.425
madde_15	0.017	-1.351	0.499
madde_16	0.000	-1.696	2.270
madde_17	0.223	0.893	1.273
madde_18	0.347	0.988	0.977
madde_19	0.000	-1.196	1.083
madde_20	0.054	1.275	2.428

Bu tabloda yer alan degerler, 3 parametreli lojistik model (3PL) cercevesinde tahmin edilen madde duzeyinde istatistiksel parametreleri gostermektedir.
“Dscrmn” basligi altinda sunulan degerler, her bir maddenin ayirt edicilik katsayisini ifade eder ve bireylerin gizil yetenek duzeylerini birbirinden ayirma gucunu temsil eder; bu deger ne kadar YUKSEK ise, madde o kadar ayirt edicidir.
“Dffclt” basligi ise her bir maddenin zorluk duzeyini yansitir; bu katsayi bireyin ilgili maddeyi dogru yanitlamasi icin sahip olmasi gereken theta (yetkinlik) duzeyini belirtir.
- Negatif degerler daha kolay, pozitif degerler ise daha zor maddeleri temsil eder.
“Gussng” ise tahmin parametresidir ve ozellikle dusuk yetenek duzeyindeki bireylerin dogru yanit verme olasiligini yani SANS ile dogru bilme ihtimalini gosterir; bu deger genellikle 0 ile 0.25 arasinda degisir.

plot(fit3pl, item = 1:6, legend = T)

Bu grafik, 3PLM cercevesinde tahmin edilen ilk 6 maddeye ait madde karakteristik egrilerini gostermektedir.
Yatay eksende bireylerin gizil yetenek duzeyi (Ability ya da theta), dikey eksende ise her bir maddenin dogru yanitlanma olasiligi (Probability) yer almaktadir.
Her bir egri, ilgili maddenin farkli theta degerlerindeki bireyler icin dogru yanit verme ihtimalini modellemektedir.
Egrilerin sekli, maddenin ayirt edicilik (Discrimination), gucluk (Difficulty) ve sans (Guessing) parametrelerinden etkilenmektedir.
Orn., daha dik bir egriye sahip olan madde, bireylerin yetenek duzeylerini daha hassas sekilde ayirt ederken, daha saga kaymis bir egri ise daha zor bir maddeyi temsil eder.
Ayrica, egri alt sinirda sifirdan yukarda basliyorsa, bu durum bireyin DUSUK yetenegine ragmen SANS ile dogru yanit verebilecegi anlamina gelir.
Kavramsal olarak genel bir faktor tarafindan hesaplanan olcek puanlarındaki varyans yuzdesini yansitan McDonald’s hiyerarsik Omega’si boyutlulugu degerlendirmek amaci ile kullanilabilir.

sonuc <- omega(veri)

sonuc$alpha  # 0.7717837

## [1] 0.7717837

sonuc$omega_h  # 0.7004879

## [1] 0.7004879

birpl_model <- "F=1-20
                CONSTRAIN=(1-20, a1)"
birpl_uyum <- mirt(data = veri, model = birpl_model, SE = T)

## Iteration: 1, Log-Lik: -5513.426, Max-Change: 0.08895Iteration: 2, Log-Lik: -5509.335, Max-Change: 0.03307Iteration: 3, Log-Lik: -5508.089, Max-Change: 0.02065Iteration: 4, Log-Lik: -5507.471, Max-Change: 0.00588Iteration: 5, Log-Lik: -5507.426, Max-Change: 0.00427Iteration: 6, Log-Lik: -5507.405, Max-Change: 0.00197Iteration: 7, Log-Lik: -5507.399, Max-Change: 0.00171Iteration: 8, Log-Lik: -5507.395, Max-Change: 0.00115Iteration: 9, Log-Lik: -5507.393, Max-Change: 0.00078Iteration: 10, Log-Lik: -5507.391, Max-Change: 0.00025Iteration: 11, Log-Lik: -5507.391, Max-Change: 0.00022Iteration: 12, Log-Lik: -5507.391, Max-Change: 0.00013Iteration: 13, Log-Lik: -5507.391, Max-Change: 0.00009
## 
## Calculating information matrix...

Q3 <- residuals(birpl_uyum, type = 'Q3', method = 'ML')

## Q3 summary statistics:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -0.156  -0.082  -0.055  -0.051  -0.023   0.107 
## 
##          madde_1 madde_2 madde_3 madde_4 madde_5 madde_6 madde_7 madde_8
## madde_1    1.000  -0.010  -0.098  -0.081  -0.062  -0.104  -0.071  -0.065
## madde_2   -0.010   1.000  -0.045  -0.143  -0.053  -0.028  -0.141   0.016
## madde_3   -0.098  -0.045   1.000  -0.140  -0.027  -0.130  -0.027  -0.042
## madde_4   -0.081  -0.143  -0.140   1.000  -0.005  -0.001   0.007  -0.022
## madde_5   -0.062  -0.053  -0.027  -0.005   1.000  -0.087  -0.092  -0.070
## madde_6   -0.104  -0.028  -0.130  -0.001  -0.087   1.000  -0.025  -0.019
## madde_7   -0.071  -0.141  -0.027   0.007  -0.092  -0.025   1.000  -0.064
## madde_8   -0.065   0.016  -0.042  -0.022  -0.070  -0.019  -0.064   1.000
## madde_9   -0.074  -0.016  -0.018   0.005  -0.134  -0.086  -0.134  -0.055
## madde_10  -0.085  -0.036  -0.055  -0.156  -0.048  -0.100   0.023   0.010
## madde_11  -0.036   0.005  -0.044  -0.053  -0.054  -0.078  -0.053  -0.020
## madde_12  -0.103   0.001  -0.035  -0.041  -0.046   0.098  -0.066  -0.030
## madde_13  -0.054  -0.058  -0.057  -0.047   0.031  -0.114  -0.082  -0.064
## madde_14  -0.048  -0.065  -0.009   0.047  -0.107  -0.115  -0.053   0.056
## madde_15  -0.044  -0.085  -0.106  -0.116  -0.110  -0.006  -0.053  -0.056
## madde_16  -0.065  -0.013  -0.054  -0.065   0.107  -0.082  -0.067   0.022
## madde_17  -0.037  -0.120   0.000  -0.008   0.028  -0.097  -0.097   0.013
## madde_18  -0.056  -0.096  -0.061  -0.061  -0.122  -0.028  -0.076  -0.055
## madde_19   0.024  -0.056  -0.082  -0.131   0.024  -0.089  -0.047  -0.056
## madde_20  -0.065   0.007  -0.107   0.000  -0.085  -0.077   0.003   0.005
##          madde_9 madde_10 madde_11 madde_12 madde_13 madde_14 madde_15 madde_16
## madde_1   -0.074   -0.085   -0.036   -0.103   -0.054   -0.048   -0.044   -0.065
## madde_2   -0.016   -0.036    0.005    0.001   -0.058   -0.065   -0.085   -0.013
## madde_3   -0.018   -0.055   -0.044   -0.035   -0.057   -0.009   -0.106   -0.054
## madde_4    0.005   -0.156   -0.053   -0.041   -0.047    0.047   -0.116   -0.065
## madde_5   -0.134   -0.048   -0.054   -0.046    0.031   -0.107   -0.110    0.107
## madde_6   -0.086   -0.100   -0.078    0.098   -0.114   -0.115   -0.006   -0.082
## madde_7   -0.134    0.023   -0.053   -0.066   -0.082   -0.053   -0.053   -0.067
## madde_8   -0.055    0.010   -0.020   -0.030   -0.064    0.056   -0.056    0.022
## madde_9    1.000   -0.103   -0.060   -0.020   -0.023   -0.093   -0.063   -0.051
## madde_10  -0.103    1.000    0.023   -0.110   -0.049   -0.070   -0.030    0.031
## madde_11  -0.060    0.023    1.000   -0.116   -0.061   -0.031   -0.069    0.062
## madde_12  -0.020   -0.110   -0.116    1.000   -0.037   -0.125   -0.058   -0.039
## madde_13  -0.023   -0.049   -0.061   -0.037    1.000   -0.076    0.050   -0.072
## madde_14  -0.093   -0.070   -0.031   -0.125   -0.076    1.000   -0.095    0.083
## madde_15  -0.063   -0.030   -0.069   -0.058    0.050   -0.095    1.000   -0.121
## madde_16  -0.051    0.031    0.062   -0.039   -0.072    0.083   -0.121    1.000
## madde_17  -0.044   -0.072   -0.042   -0.101   -0.109   -0.082   -0.067   -0.078
## madde_18  -0.085   -0.027    0.017   -0.156   -0.041   -0.037   -0.035   -0.125
## madde_19  -0.052   -0.077   -0.066   -0.032   -0.062    0.005   -0.057    0.021
## madde_20   0.009   -0.030   -0.059   -0.037    0.020   -0.073   -0.044   -0.008
##          madde_17 madde_18 madde_19 madde_20
## madde_1    -0.037   -0.056    0.024   -0.065
## madde_2    -0.120   -0.096   -0.056    0.007
## madde_3     0.000   -0.061   -0.082   -0.107
## madde_4    -0.008   -0.061   -0.131    0.000
## madde_5     0.028   -0.122    0.024   -0.085
## madde_6    -0.097   -0.028   -0.089   -0.077
## madde_7    -0.097   -0.076   -0.047    0.003
## madde_8     0.013   -0.055   -0.056    0.005
## madde_9    -0.044   -0.085   -0.052    0.009
## madde_10   -0.072   -0.027   -0.077   -0.030
## madde_11   -0.042    0.017   -0.066   -0.059
## madde_12   -0.101   -0.156   -0.032   -0.037
## madde_13   -0.109   -0.041   -0.062    0.020
## madde_14   -0.082   -0.037    0.005   -0.073
## madde_15   -0.067   -0.035   -0.057   -0.044
## madde_16   -0.078   -0.125    0.021   -0.008
## madde_17    1.000   -0.020   -0.103   -0.112
## madde_18   -0.020    1.000   -0.073   -0.083
## madde_19   -0.103   -0.073    1.000   -0.032
## madde_20   -0.112   -0.083   -0.032    1.000

library(knitr)
library(kableExtra)
Q3 %>% round(3) %>% kable("html", caption = "Q3 Artik Korelasyon Matrisi") %>% kable_styling(full_width = T, bootstrap_options = c("striped", "hover", "condensed", "responsive"), font_size = 12) %>% scroll_box(width = "100%", height = "500px")

Q3 Artik Korelasyon Matrisi
	madde_1	madde_2	madde_3	madde_4	madde_5	madde_6	madde_7	madde_8	madde_9	madde_10	madde_11	madde_12	madde_13	madde_14	madde_15	madde_16	madde_17	madde_18	madde_19	madde_20
madde_1	1.000	-0.010	-0.098	-0.081	-0.062	-0.104	-0.071	-0.065	-0.074	-0.085	-0.036	-0.103	-0.054	-0.048	-0.044	-0.065	-0.037	-0.056	0.024	-0.065
madde_2	-0.010	1.000	-0.045	-0.143	-0.053	-0.028	-0.141	0.016	-0.016	-0.036	0.005	0.001	-0.058	-0.065	-0.085	-0.013	-0.120	-0.096	-0.056	0.007
madde_3	-0.098	-0.045	1.000	-0.140	-0.027	-0.130	-0.027	-0.042	-0.018	-0.055	-0.044	-0.035	-0.057	-0.009	-0.106	-0.054	0.000	-0.061	-0.082	-0.107
madde_4	-0.081	-0.143	-0.140	1.000	-0.005	-0.001	0.007	-0.022	0.005	-0.156	-0.053	-0.041	-0.047	0.047	-0.116	-0.065	-0.008	-0.061	-0.131	0.000
madde_5	-0.062	-0.053	-0.027	-0.005	1.000	-0.087	-0.092	-0.070	-0.134	-0.048	-0.054	-0.046	0.031	-0.107	-0.110	0.107	0.028	-0.122	0.024	-0.085
madde_6	-0.104	-0.028	-0.130	-0.001	-0.087	1.000	-0.025	-0.019	-0.086	-0.100	-0.078	0.098	-0.114	-0.115	-0.006	-0.082	-0.097	-0.028	-0.089	-0.077
madde_7	-0.071	-0.141	-0.027	0.007	-0.092	-0.025	1.000	-0.064	-0.134	0.023	-0.053	-0.066	-0.082	-0.053	-0.053	-0.067	-0.097	-0.076	-0.047	0.003
madde_8	-0.065	0.016	-0.042	-0.022	-0.070	-0.019	-0.064	1.000	-0.055	0.010	-0.020	-0.030	-0.064	0.056	-0.056	0.022	0.013	-0.055	-0.056	0.005
madde_9	-0.074	-0.016	-0.018	0.005	-0.134	-0.086	-0.134	-0.055	1.000	-0.103	-0.060	-0.020	-0.023	-0.093	-0.063	-0.051	-0.044	-0.085	-0.052	0.009
madde_10	-0.085	-0.036	-0.055	-0.156	-0.048	-0.100	0.023	0.010	-0.103	1.000	0.023	-0.110	-0.049	-0.070	-0.030	0.031	-0.072	-0.027	-0.077	-0.030
madde_11	-0.036	0.005	-0.044	-0.053	-0.054	-0.078	-0.053	-0.020	-0.060	0.023	1.000	-0.116	-0.061	-0.031	-0.069	0.062	-0.042	0.017	-0.066	-0.059
madde_12	-0.103	0.001	-0.035	-0.041	-0.046	0.098	-0.066	-0.030	-0.020	-0.110	-0.116	1.000	-0.037	-0.125	-0.058	-0.039	-0.101	-0.156	-0.032	-0.037
madde_13	-0.054	-0.058	-0.057	-0.047	0.031	-0.114	-0.082	-0.064	-0.023	-0.049	-0.061	-0.037	1.000	-0.076	0.050	-0.072	-0.109	-0.041	-0.062	0.020
madde_14	-0.048	-0.065	-0.009	0.047	-0.107	-0.115	-0.053	0.056	-0.093	-0.070	-0.031	-0.125	-0.076	1.000	-0.095	0.083	-0.082	-0.037	0.005	-0.073
madde_15	-0.044	-0.085	-0.106	-0.116	-0.110	-0.006	-0.053	-0.056	-0.063	-0.030	-0.069	-0.058	0.050	-0.095	1.000	-0.121	-0.067	-0.035	-0.057	-0.044
madde_16	-0.065	-0.013	-0.054	-0.065	0.107	-0.082	-0.067	0.022	-0.051	0.031	0.062	-0.039	-0.072	0.083	-0.121	1.000	-0.078	-0.125	0.021	-0.008
madde_17	-0.037	-0.120	0.000	-0.008	0.028	-0.097	-0.097	0.013	-0.044	-0.072	-0.042	-0.101	-0.109	-0.082	-0.067	-0.078	1.000	-0.020	-0.103	-0.112
madde_18	-0.056	-0.096	-0.061	-0.061	-0.122	-0.028	-0.076	-0.055	-0.085	-0.027	0.017	-0.156	-0.041	-0.037	-0.035	-0.125	-0.020	1.000	-0.073	-0.083
madde_19	0.024	-0.056	-0.082	-0.131	0.024	-0.089	-0.047	-0.056	-0.052	-0.077	-0.066	-0.032	-0.062	0.005	-0.057	0.021	-0.103	-0.073	1.000	-0.032
madde_20	-0.065	0.007	-0.107	0.000	-0.085	-0.077	0.003	0.005	0.009	-0.030	-0.059	-0.037	0.020	-0.073	-0.044	-0.008	-0.112	-0.083	-0.032	1.000

Bu tabloda sunulan degerler, madde tepkisi kurami cercevesinde uygulanan model uyumuna iliskin artik korelasyon (Q3) istatistiklerini temsil etmektedir.
Q3 matrisi, her bir madde ciftinin model tarafindan aciklanamayan ortak varyansini yansitmakta olup, model uyumu degerlendirmelerinde siklikla kullanilmaktadir.
- Ideal bir modelde, maddeler arasindaki artik korelasyon degerlerinin sifira yakin olmasi beklenir; bu durum, ilgili modelin gozlemlenen madde yanitlarindaki iliskileri yeterli duzeyde acikladigini gosterir.
- POZITIF sapmalar, olculmek istenen gizil ozellik disinda ortak baska bir faktore isaret edebilirken, NEGATIF degerler modelin fazla uyum sagladigini (overfitting) gosterebilir.
- Bu nedenle, Q3 degerleri madde bagimliligini, lokal bagimsizlik varsayiminin ihlali durumlarini ve modelin yapisal uygunlugunu anlamada onemli bir tanilama araci olarak kullanilmaktadir (Yen, 1984; Reeve et al., 2007).

Q3[lower.tri(Q3, diag = TRUE)] <- NA
sum(abs(Q3) > 0.2, na.rm = TRUE)  # 0

## [1] 0

Bu sonuc, madde tepkisi kurami cercevesinde uygulanan modelin lokal bagimsizlik varsayimini buyuk olcude sagladigini gostermektedir.
Q3 matrisi, her bir madde ciftinin model tarafindan aciklanamayan artik korelasyonlarini yansitir.
Kodda yer alan \(abs(Q3) > 0.2\) ifadesi, mutlak degeri \(0.2\)’den buyuk olan artik korelasyonlarin sayisini hesaplamaktadir.
Elde edilen degerin sifir (0) olmasi, tum madde ciftlerinin artik korelasyonlarinin bu esik degerin altinda kaldigini ve bu nedenle modelin gozlemlenen veri uzerinde yeterli yapisal uyum sagladigini ortaya koymaktadir.
- Bu durum, testte yer alan maddelerin, gizil ozellik disinda bagimsiz sekilde calistigini ve herhangi bir lokal bagimlilik sorununun bulunmadigini desteklemektedir.
- Bu bulgu, olcum aracinin yapisal gecerligine iliskin guclu bir gosterge olarak yorumlanabilir (Yen, 1984; Chen & Thissen, 1997).
- YEREL BAGIMLILIGA karar vermek icin bunu kullandik!!

ikipl_model <- "F=1-20"
ikipl_uyum <- mirt(data = veri, model = ikipl_model, itemtype = "2PL", SE = T)

## Iteration: 1, Log-Lik: -5513.426, Max-Change: 0.65652Iteration: 2, Log-Lik: -5454.963, Max-Change: 0.32085Iteration: 3, Log-Lik: -5447.197, Max-Change: 0.19128Iteration: 4, Log-Lik: -5445.766, Max-Change: 0.11326Iteration: 5, Log-Lik: -5445.258, Max-Change: 0.06665Iteration: 6, Log-Lik: -5445.081, Max-Change: 0.03558Iteration: 7, Log-Lik: -5445.006, Max-Change: 0.02236Iteration: 8, Log-Lik: -5444.974, Max-Change: 0.01689Iteration: 9, Log-Lik: -5444.958, Max-Change: 0.00532Iteration: 10, Log-Lik: -5444.954, Max-Change: 0.00452Iteration: 11, Log-Lik: -5444.949, Max-Change: 0.00326Iteration: 12, Log-Lik: -5444.946, Max-Change: 0.00179Iteration: 13, Log-Lik: -5444.945, Max-Change: 0.00128Iteration: 14, Log-Lik: -5444.944, Max-Change: 0.00105Iteration: 15, Log-Lik: -5444.944, Max-Change: 0.00087Iteration: 16, Log-Lik: -5444.943, Max-Change: 0.00012Iteration: 17, Log-Lik: -5444.943, Max-Change: 0.00013Iteration: 18, Log-Lik: -5444.943, Max-Change: 0.00012Iteration: 19, Log-Lik: -5444.943, Max-Change: 0.00038Iteration: 20, Log-Lik: -5444.943, Max-Change: 0.00006
## 
## Calculating information matrix...

ikipl_par <- coef(ikipl_uyum, IRTpars = T, simplify = T)

library(knitr)
uyumdegerleri <- rbind(M2(birpl_uyum), 
                       M2(ikipl_uyum), 
                       M2(ucpl_uyum))

library(kableExtra)
library(knitr)
cbind(data.frame(Modeller = c("1PL", "2PL", "3PL")), uyumdegerleri) %>% kable("html", caption = "Uyum Degerleri Tablosu", digits = 2) %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

Uyum Degerleri Tablosu
	Modeller	M2	df	p	RMSEA	RMSEA_5	RMSEA_95	SRMSR	TLI	CFI
stats	1PL	287.74	189	0.00	0.03	0.02	0.04	0.07	0.95	0.95
stats1	2PL	172.06	170	0.44	0.00	0.00	0.02	0.04	1.00	1.00
stats2	3PL	150.29	150	0.48	0.00	0.00	0.02	0.04	1.00	1.00

Bu tablo, 1PL, 2PL ve 3PL modellerine ait yapisal uyum istatistiklerini karsilastirmali olarak sunmaktadir.
Ilk sutunlarda yer alan M2, model uyumuna iliskin genel uyum testi istatistigini ifade etmekte olup, bu degerin KUCUK olmasi modelin gozlemlenen veriye DAHA IYI uyduguna isaret eder.
df (degrees of freedom) serbestlik derecelerini, p ise M2 istatistiginin anlamlilik duzeyini gostermektedir.
RMSEA ve SRMSR gibi degerler mutlak uyumu, TLI ve CFI gibi indeksler ise karsilastirmali uyumu degerlendirmektedir.
RMSEA ve SRMSR degerlerinin 0.08’in, TLI ve CFI degerlerinin ise 0.90’in uzerinde olmasi, kabul edilebilir uyum sinirlari icinde oldugunu gosterir.
Tablo incelendiginde, 3PL modelinin diger modellere kiyasla DAHA DUSUK M2 ve RMSEA degerlerine ve daha yuksek TLI ve CFI puanlarina sahip oldugu gorulmektedir.
Bu durum, 3PL modelinin veriye daha iyi uyum sagladigini ve olcme aracinin yapisal gecerligini DAHA IYI temsil ettigini ortaya koymaktadir (Maydeu-Olivares & Joe, 2006; Reise et al., 2005).

uyum1 <- anova(birpl_uyum, ikipl_uyum)

library(kableExtra)
library(knitr)
cbind(data.frame(Modeller = c("1PL", "2PL")), uyum1[, 1:6]) %>% kable("html", caption = "1PL ve 2PL Modellerine Ait Uyum Degerleri", digits = 2) %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

1PL ve 2PL Modellerine Ait Uyum Degerleri
	Modeller	AIC	SABIC	HQ	BIC	logLik	X2
birpl_uyum	1PL	11056.78	11078.63	11091.51	11145.29	-5507.39	NA
ikipl_uyum	2PL	10969.89	11011.51	11036.04	11138.47	-5444.94	124.9

Bu tabloda, 1PL (Rasch modeli) ve 2PL (iki parametreli lojistik model) iceriklerinde hesaplanan temel uyum istatistikleri karsilastirmali olarak sunulmaktadir.
AIC (Akaike Information Criterion) ve BIC (Bayesian Information Criterion) gibi bilgi kriterleri, modellerin veri uzerindeki uyumunu ve parsimony (sadeligini) birlikte dikkate alarak degerlendirir; bu degerlerin dusuk olmasi daha iyi bir model secimini yansitir.
logLik (log-likelihood), modelin gozlemlenen veri uzerinden hesaplanan logaritmik olasiligini gostermekte olup, daha yuksek degerler modelin veriye daha iyi uyduguna isaret eder.
X2 testi ise, 1PL ve 2PL modelleri arasinda hiyerarsik karsilastirma yaparak, daha genis parametre yapisina sahip olan modelin (burada 2PL) manidar derecede daha iyi uyum saglayip saglamadigini kontrol eder.
Tabloda, 2PL modelinin AIC, BIC ve logLik degerleri acisindan 1PL modeline kiyasla daha avantajli oldugu, X2 farklilik testinin sonucunda da manidar bir iyilesme gosterdigi gorulmektedir.
Bu bulgular, olcum aracinin daha karmasik yapili 2PL modeli ile daha DOGRU sekilde temsil edilebilecegini ortaya koymaktadir (Burnham & Anderson, 2002).

uyum2 <- anova(ikipl_uyum, ucpl_uyum)

library(kableExtra)
library(knitr)
cbind(data.frame(Modeller = c("2PL", "3PL")), uyum2[, 1:6]) %>% kable("html", caption = "2PL ve 3PL Modellerine Ait Uyum Degerleri", digits = 2) %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

2PL ve 3PL Modellerine Ait Uyum Degerleri
	Modeller	AIC	SABIC	HQ	BIC	logLik	X2
ikipl_uyum	2PL	10969.89	11011.51	11036.04	11138.47	-5444.94	NA
ucpl_uyum	3PL	10987.90	11050.34	11087.13	11240.78	-5433.95	21.98

Bu tablo, 2PL (iki parametreli lojistik) ve 3PL (uc parametreli lojistik) modellerine ait temel uyum istatistiklerini karsilastirmali olarak sunmaktadir.
AIC (Akaike Information Criterion), SABIC (Sample-size Adjusted BIC), HQ (Hannan-Quinn Criterion) ve BIC (Bayesian Information Criterion) gibi bilgi kriterleri, modelin veri uzerindeki uyumunu ve model karmasikligini birlikte degerlendiren istatistiklerdir; bu degerlerin DAHA DUSUK olmasi, modelin hem veriye iyi uydugunu hem de gereksiz parametre kullanmadigini gosterir.
logLik (log-likelihood), modelin gozlemlenen veriye gore ne kadar olasi oldugunu ifade eder; daha YUKSEK logLik degeri DAHA IYI UYUMU temsil eder.
X2 testi ise 2PL ve 3PL modelleri arasinda manidar bir fark olup olmadigini ortaya koyar; burada elde edilen deger (\(21.90\)), 3PL modelinin eklenen tahmin (guessing) parametresi sayesinde veri uzerinde manidar bir iyilesme sagladigini gostermektedir.
Bu bulgular, 3PL modelinin DAHA IYI MODEL UYUM sundugunu ve testteki maddelerin cevaplanma davranisini daha isabetli sekilde yansittigini ortaya koymaktadir (Burnham & Anderson, 2002; Embretson & Reise, 2000).

library(TAM)
model_1PL <- tam.mml(resp = veri)

## ....................................................
## Processing Data      2025-05-19 21:33:23.130221 
##     * Response Data: 500 Persons and  20 Items 
##     * Numerical integration with 21 nodes
##     * Created Design Matrices   ( 2025-05-19 21:33:23.133011 )
##     * Calculated Sufficient Statistics   ( 2025-05-19 21:33:23.137344 )
## ....................................................
## Iteration 1     2025-05-19 21:33:23.141452
## E Step
## M Step Intercepts   |----
##   Deviance = 11049.3001
##   Maximum item intercept parameter change: 0.319324
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.065942
## ....................................................
## Iteration 2     2025-05-19 21:33:23.14658
## E Step
## M Step Intercepts   |---
##   Deviance = 11015.9534 | Absolute change: 33.3467 | Relative change: 0.00302712
##   Maximum item intercept parameter change: 0.018871
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.01788
## ....................................................
## Iteration 3     2025-05-19 21:33:23.147929
## E Step
## M Step Intercepts   |--
##   Deviance = 11015.4098 | Absolute change: 0.5436 | Relative change: 4.935e-05
##   Maximum item intercept parameter change: 0.011642
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.010737
## ....................................................
## Iteration 4     2025-05-19 21:33:23.149567
## E Step
## M Step Intercepts   |--
##   Deviance = 11015.1344 | Absolute change: 0.2754 | Relative change: 2.5e-05
##   Maximum item intercept parameter change: 0.00835
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.006298
## ....................................................
## Iteration 5     2025-05-19 21:33:23.150988
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.9852 | Absolute change: 0.1492 | Relative change: 1.355e-05
##   Maximum item intercept parameter change: 0.006111
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.003742
## ....................................................
## Iteration 6     2025-05-19 21:33:23.15216
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.9007 | Absolute change: 0.0845 | Relative change: 7.67e-06
##   Maximum item intercept parameter change: 0.004539
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.002268
## ....................................................
## Iteration 7     2025-05-19 21:33:23.153154
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.8517 | Absolute change: 0.049 | Relative change: 4.45e-06
##   Maximum item intercept parameter change: 0.00341
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.001412
## ....................................................
## Iteration 8     2025-05-19 21:33:23.154121
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.8228 | Absolute change: 0.0289 | Relative change: 2.62e-06
##   Maximum item intercept parameter change: 0.002583
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000905
## ....................................................
## Iteration 9     2025-05-19 21:33:23.155272
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.8057 | Absolute change: 0.0171 | Relative change: 1.56e-06
##   Maximum item intercept parameter change: 0.001969
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000598
## ....................................................
## Iteration 10     2025-05-19 21:33:23.156773
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7955 | Absolute change: 0.0102 | Relative change: 9.3e-07
##   Maximum item intercept parameter change: 0.001508
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000408
## ....................................................
## Iteration 11     2025-05-19 21:33:23.158025
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7894 | Absolute change: 0.0061 | Relative change: 5.5e-07
##   Maximum item intercept parameter change: 0.001158
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000286
## ....................................................
## Iteration 12     2025-05-19 21:33:23.159565
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7857 | Absolute change: 0.0036 | Relative change: 3.3e-07
##   Maximum item intercept parameter change: 0.000892
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000205
## ....................................................
## Iteration 13     2025-05-19 21:33:23.160616
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7835 | Absolute change: 0.0022 | Relative change: 2e-07
##   Maximum item intercept parameter change: 0.000687
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.00015
## ....................................................
## Iteration 14     2025-05-19 21:33:23.161545
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7822 | Absolute change: 0.0013 | Relative change: 1.2e-07
##   Maximum item intercept parameter change: 0.000531
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0.000111
## ....................................................
## Iteration 15     2025-05-19 21:33:23.162392
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7814 | Absolute change: 8e-04 | Relative change: 7e-08
##   Maximum item intercept parameter change: 0.00041
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 8.4e-05
## ....................................................
## Iteration 16     2025-05-19 21:33:23.163211
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.781 | Absolute change: 5e-04 | Relative change: 4e-08
##   Maximum item intercept parameter change: 0.000317
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 6.3e-05
## ....................................................
## Iteration 17     2025-05-19 21:33:23.164127
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7807 | Absolute change: 3e-04 | Relative change: 3e-08
##   Maximum item intercept parameter change: 0.000245
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 4.8e-05
## ....................................................
## Iteration 18     2025-05-19 21:33:23.165435
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7805 | Absolute change: 2e-04 | Relative change: 2e-08
##   Maximum item intercept parameter change: 0.00019
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 3.7e-05
## ....................................................
## Iteration 19     2025-05-19 21:33:23.167163
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7804 | Absolute change: 1e-04 | Relative change: 1e-08
##   Maximum item intercept parameter change: 0.000147
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 2.8e-05
## ....................................................
## Iteration 20     2025-05-19 21:33:23.172159
## E Step
## M Step Intercepts   |--
##   Deviance = 11014.7804 | Absolute change: 1e-04 | Relative change: 1e-08
##   Maximum item intercept parameter change: 0.000113
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 2.2e-05
## ....................................................
## Iteration 21     2025-05-19 21:33:23.175433
## E Step
## M Step Intercepts   |-
##   Deviance = 11014.7803 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 8.8e-05
##   Maximum item slope parameter change: 0
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 1.7e-05
## ....................................................
## Item Parameters
##    xsi.index xsi.label     est
## 1          1   madde_1  0.4338
## 2          2   madde_2  0.4837
## 3          3   madde_3  0.1013
## 4          4   madde_4 -0.1690
## 5          5   madde_5 -1.2142
## 6          6   madde_6  0.7295
## 7          7   madde_7 -0.1013
## 8          8   madde_8 -3.1504
## 9          9   madde_9  0.0916
## 10        10  madde_10 -0.8745
## 11        11  madde_11 -1.9362
## 12        12  madde_12 -0.6030
## 13        13  madde_13 -1.7316
## 14        14  madde_14 -0.8745
## 15        15  madde_15 -0.7998
## 16        16  madde_16 -2.6789
## 17        17  madde_17  0.2273
## 18        18  madde_18 -0.2368
## 19        19  madde_19 -1.2625
## 20        20  madde_20  1.6738
## ...................................
## Regression Coefficients
##      [,1]
## [1,]    0
## 
## Variance:
##        [,1]
## [1,] 0.9794
## 
## 
## EAP Reliability:
## [1] 0.766
## 
## -----------------------------
## Start:  2025-05-19 21:33:23.129197
## End:  2025-05-19 21:33:23.188063 
## Time difference of 0.05886602 secs

birpl_if <- tam.fit(model_1PL)

## Item fit calculation based on 40 simulations
## |**********|
## |----------|

str(birpl_if)

## List of 3
##  $ itemfit:'data.frame': 20 obs. of  9 variables:
##   ..$ parameter   : chr [1:20] "madde_1" "madde_2" "madde_3" "madde_4" ...
##   ..$ Outfit      : num [1:20] 1.101 0.895 1.058 0.857 0.913 ...
##   ..$ Outfit_t    : num [1:20] 2.55 -2.79 1.61 -4.24 -1.72 ...
##   ..$ Outfit_p    : num [1:20] 1.09e-02 5.26e-03 1.07e-01 2.24e-05 8.59e-02 ...
##   ..$ Outfit_pholm: num [1:20] 0.141689 0.073686 0.751529 0.000402 0.687285 ...
##   ..$ Infit       : num [1:20] 1.069 0.921 1.039 0.899 0.985 ...
##   ..$ Infit_t     : num [1:20] 1.774 -2.079 1.081 -2.93 -0.268 ...
##   ..$ Infit_p     : num [1:20] 0.07606 0.0376 0.27982 0.00339 0.78851 ...
##   ..$ Infit_pholm : num [1:20] 1 0.6015 1 0.0644 1 ...
##  $ time   : POSIXct[1:2], format: "2025-05-19 21:33:23" "2025-05-19 21:33:23"
##  $ CALL   : language tam.fit(tamobj = model_1PL)
##  - attr(*, "class")= chr "tam.fit"

library(kableExtra)
library(knitr)
birpl_if_df <- birpl_if$itemfit
birpl_if_clean <- birpl_if_df[, c("parameter", "Outfit", "Outfit_t", "Outfit_p")]
colnames(birpl_if_clean) <- c("item", "X2", "t", "p")
birpl_if_clean[, c("X2", "t", "p")] <- round(birpl_if_clean[, c("X2", "t", "p")], 3)
kable(birpl_if_clean, format = "html", caption = "1PL Modeline Ait Madde Uyum Degerleri (Outfit Temelli)") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = F, font_size = 13)

1PL Modeline Ait Madde Uyum Degerleri (Outfit Temelli)
item	X2	t	p
madde_1	1.101	2.546	0.011
madde_2	0.895	-2.790	0.005
madde_3	1.058	1.610	0.107
madde_4	0.857	-4.240	0.000
madde_5	0.913	-1.717	0.086
madde_6	1.185	4.064	0.000
madde_7	1.007	0.192	0.848
madde_8	0.709	-2.055	0.040
madde_9	1.019	0.533	0.594
madde_10	0.989	-0.264	0.791
madde_11	0.760	-3.327	0.001
madde_12	0.963	-0.963	0.336
madde_13	0.985	-0.207	0.836
madde_14	0.904	-2.251	0.024
madde_15	1.194	4.258	0.000
madde_16	0.689	-2.902	0.004
madde_17	1.079	2.112	0.035
madde_18	1.176	4.640	0.000
madde_19	0.979	-0.397	0.691
madde_20	0.875	-1.884	0.060

library(TAM)
library(knitr)
library(kableExtra)
model_2PL <- tam.mml.2pl(resp = veri)

## ....................................................
## Processing Data      2025-05-19 21:33:23.492715 
##     * Response Data: 500 Persons and  20 Items 
##     * Numerical integration with 21 nodes
##     * Created Design Matrices   ( 2025-05-19 21:33:23.494405 )
##     * Calculated Sufficient Statistics   ( 2025-05-19 21:33:23.495479 )
## ....................................................
## Iteration 1     2025-05-19 21:33:23.497722
## E Step
## M Step Intercepts   |----
## M Step Slopes       |----
##   Deviance = 11049.3001
##   Maximum item intercept parameter change: 0.319324
##   Maximum item slope parameter change: 0.430084
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 2     2025-05-19 21:33:23.502066
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10923.9273 | Absolute change: 125.3727 | Relative change: 0.01147689
##   Maximum item intercept parameter change: 0.245791
##   Maximum item slope parameter change: 0.220955
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 3     2025-05-19 21:33:23.503906
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10901.7344 | Absolute change: 22.1929 | Relative change: 0.00203572
##   Maximum item intercept parameter change: 0.183075
##   Maximum item slope parameter change: 0.154486
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 4     2025-05-19 21:33:23.505735
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10895.9753 | Absolute change: 5.7592 | Relative change: 0.00052856
##   Maximum item intercept parameter change: 0.140239
##   Maximum item slope parameter change: 0.113417
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 5     2025-05-19 21:33:23.508058
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10893.5533 | Absolute change: 2.4219 | Relative change: 0.00022233
##   Maximum item intercept parameter change: 0.109945
##   Maximum item slope parameter change: 0.086189
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 6     2025-05-19 21:33:23.510094
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10892.2558 | Absolute change: 1.2975 | Relative change: 0.00011912
##   Maximum item intercept parameter change: 0.087986
##   Maximum item slope parameter change: 0.067178
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 7     2025-05-19 21:33:23.512161
## E Step
## M Step Intercepts   |---
## M Step Slopes       |---
##   Deviance = 10891.4767 | Absolute change: 0.7791 | Relative change: 7.153e-05
##   Maximum item intercept parameter change: 0.071586
##   Maximum item slope parameter change: 0.053359
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 8     2025-05-19 21:33:23.51504
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.9779 | Absolute change: 0.4988 | Relative change: 4.58e-05
##   Maximum item intercept parameter change: 0.058971
##   Maximum item slope parameter change: 0.042992
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 9     2025-05-19 21:33:23.518415
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.6456 | Absolute change: 0.3323 | Relative change: 3.051e-05
##   Maximum item intercept parameter change: 0.049021
##   Maximum item slope parameter change: 0.03502
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 10     2025-05-19 21:33:23.521025
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.4186 | Absolute change: 0.227 | Relative change: 2.085e-05
##   Maximum item intercept parameter change: 0.041016
##   Maximum item slope parameter change: 0.02877
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 11     2025-05-19 21:33:23.523585
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.2609 | Absolute change: 0.1577 | Relative change: 1.448e-05
##   Maximum item intercept parameter change: 0.034478
##   Maximum item slope parameter change: 0.023796
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 12     2025-05-19 21:33:23.525465
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.1503 | Absolute change: 0.1107 | Relative change: 1.016e-05
##   Maximum item intercept parameter change: 0.029079
##   Maximum item slope parameter change: 0.019787
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 13     2025-05-19 21:33:23.527003
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.0721 | Absolute change: 0.0782 | Relative change: 7.18e-06
##   Maximum item intercept parameter change: 0.024584
##   Maximum item slope parameter change: 0.016524
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 14     2025-05-19 21:33:23.528551
## E Step
## M Step Intercepts   |---
## M Step Slopes       |--
##   Deviance = 10890.0165 | Absolute change: 0.0555 | Relative change: 5.1e-06
##   Maximum item intercept parameter change: 0.020818
##   Maximum item slope parameter change: 0.013848
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 15     2025-05-19 21:33:23.530083
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.977 | Absolute change: 0.0396 | Relative change: 3.63e-06
##   Maximum item intercept parameter change: 0.01765
##   Maximum item slope parameter change: 0.011638
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 16     2025-05-19 21:33:23.531666
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.9487 | Absolute change: 0.0282 | Relative change: 2.59e-06
##   Maximum item intercept parameter change: 0.014976
##   Maximum item slope parameter change: 0.009803
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 17     2025-05-19 21:33:23.533611
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.9286 | Absolute change: 0.0202 | Relative change: 1.85e-06
##   Maximum item intercept parameter change: 0.012715
##   Maximum item slope parameter change: 0.008274
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 18     2025-05-19 21:33:23.535042
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.9141 | Absolute change: 0.0144 | Relative change: 1.32e-06
##   Maximum item intercept parameter change: 0.010799
##   Maximum item slope parameter change: 0.006994
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 19     2025-05-19 21:33:23.536502
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.9038 | Absolute change: 0.0103 | Relative change: 9.5e-07
##   Maximum item intercept parameter change: 0.009175
##   Maximum item slope parameter change: 0.00592
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 20     2025-05-19 21:33:23.537907
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8964 | Absolute change: 0.0074 | Relative change: 6.8e-07
##   Maximum item intercept parameter change: 0.007796
##   Maximum item slope parameter change: 0.005016
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 21     2025-05-19 21:33:23.539436
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8911 | Absolute change: 0.0053 | Relative change: 4.9e-07
##   Maximum item intercept parameter change: 0.006625
##   Maximum item slope parameter change: 0.004254
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 22     2025-05-19 21:33:23.542685
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8873 | Absolute change: 0.0038 | Relative change: 3.5e-07
##   Maximum item intercept parameter change: 0.005631
##   Maximum item slope parameter change: 0.003611
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 23     2025-05-19 21:33:23.545203
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8846 | Absolute change: 0.0027 | Relative change: 2.5e-07
##   Maximum item intercept parameter change: 0.004786
##   Maximum item slope parameter change: 0.003067
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 24     2025-05-19 21:33:23.547758
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8827 | Absolute change: 0.0019 | Relative change: 1.8e-07
##   Maximum item intercept parameter change: 0.004068
##   Maximum item slope parameter change: 0.002606
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 25     2025-05-19 21:33:23.549813
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8813 | Absolute change: 0.0014 | Relative change: 1.3e-07
##   Maximum item intercept parameter change: 0.003457
##   Maximum item slope parameter change: 0.002216
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 26     2025-05-19 21:33:23.551304
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8803 | Absolute change: 0.001 | Relative change: 9e-08
##   Maximum item intercept parameter change: 0.002939
##   Maximum item slope parameter change: 0.001884
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 27     2025-05-19 21:33:23.552699
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8795 | Absolute change: 7e-04 | Relative change: 7e-08
##   Maximum item intercept parameter change: 0.002498
##   Maximum item slope parameter change: 0.001603
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 28     2025-05-19 21:33:23.554046
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.879 | Absolute change: 5e-04 | Relative change: 5e-08
##   Maximum item intercept parameter change: 0.002124
##   Maximum item slope parameter change: 0.001364
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 29     2025-05-19 21:33:23.556591
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8787 | Absolute change: 4e-04 | Relative change: 3e-08
##   Maximum item intercept parameter change: 0.001805
##   Maximum item slope parameter change: 0.001161
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 30     2025-05-19 21:33:23.559045
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8784 | Absolute change: 3e-04 | Relative change: 2e-08
##   Maximum item intercept parameter change: 0.001535
##   Maximum item slope parameter change: 0.000989
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 31     2025-05-19 21:33:23.560971
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8782 | Absolute change: 2e-04 | Relative change: 2e-08
##   Maximum item intercept parameter change: 0.001305
##   Maximum item slope parameter change: 0.000842
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 32     2025-05-19 21:33:23.562496
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8781 | Absolute change: 1e-04 | Relative change: 1e-08
##   Maximum item intercept parameter change: 0.001109
##   Maximum item slope parameter change: 0.000717
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 33     2025-05-19 21:33:23.564523
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.878 | Absolute change: 1e-04 | Relative change: 1e-08
##   Maximum item intercept parameter change: 0.000943
##   Maximum item slope parameter change: 0.000611
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 34     2025-05-19 21:33:23.566462
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8779 | Absolute change: 1e-04 | Relative change: 1e-08
##   Maximum item intercept parameter change: 0.000802
##   Maximum item slope parameter change: 0.000521
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 35     2025-05-19 21:33:23.567906
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8779 | Absolute change: 1e-04 | Relative change: 0
##   Maximum item intercept parameter change: 0.000682
##   Maximum item slope parameter change: 0.000444
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 36     2025-05-19 21:33:23.569335
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8778 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.00058
##   Maximum item slope parameter change: 0.000378
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 37     2025-05-19 21:33:23.57086
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8778 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000494
##   Maximum item slope parameter change: 0.000322
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 38     2025-05-19 21:33:23.572667
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8778 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.00042
##   Maximum item slope parameter change: 0.000275
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 39     2025-05-19 21:33:23.574852
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8778 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000357
##   Maximum item slope parameter change: 0.000234
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 40     2025-05-19 21:33:23.576528
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000304
##   Maximum item slope parameter change: 0.000199
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 41     2025-05-19 21:33:23.578135
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000259
##   Maximum item slope parameter change: 0.00017
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 42     2025-05-19 21:33:23.579689
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.00022
##   Maximum item slope parameter change: 0.000145
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 43     2025-05-19 21:33:23.581294
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000187
##   Maximum item slope parameter change: 0.000124
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 44     2025-05-19 21:33:23.583724
## E Step
## M Step Intercepts   |--
## M Step Slopes       |--
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000159
##   Maximum item slope parameter change: 0.000105
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 45     2025-05-19 21:33:23.594682
## E Step
## M Step Intercepts   |--
## M Step Slopes       |-
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000136
##   Maximum item slope parameter change: 9e-05
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 46     2025-05-19 21:33:23.595996
## E Step
## M Step Intercepts   |--
## M Step Slopes       |-
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 0.000116
##   Maximum item slope parameter change: 7.7e-05
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Iteration 47     2025-05-19 21:33:23.597259
## E Step
## M Step Intercepts   |-
## M Step Slopes       |-
##   Deviance = 10889.8777 | Absolute change: 0 | Relative change: 0
##   Maximum item intercept parameter change: 9.8e-05
##   Maximum item slope parameter change: 6.5e-05
##   Maximum regression parameter change: 0
##   Maximum variance parameter change: 0
## ....................................................
## Item Parameters
##    xsi.index xsi.label     est
## 1          1   madde_1  0.4020
## 2          2   madde_2  0.5364
## 3          3   madde_3  0.0943
## 4          4   madde_4 -0.2264
## 5          5   madde_5 -1.2966
## 6          6   madde_6  0.6471
## 7          7   madde_7 -0.1047
## 8          8   madde_8 -3.7737
## 9          9   madde_9  0.0857
## 10        10  madde_10 -0.8954
## 11        11  madde_11 -2.4450
## 12        12  madde_12 -0.6282
## 13        13  madde_13 -1.7122
## 14        14  madde_14 -1.0223
## 15        15  madde_15 -0.7028
## 16        16  madde_16 -3.8302
## 17        17  madde_17  0.2110
## 18        18  madde_18 -0.2059
## 19        19  madde_19 -1.3130
## 20        20  madde_20  1.9374
## ...................................
## Regression Coefficients
##      [,1]
## [1,]    0
## 
## Variance:
##      [,1]
## [1,]    1
## 
## 
## EAP Reliability:
## [1] 0.791
## 
## -----------------------------
## Start:  2025-05-19 21:33:23.492228
## End:  2025-05-19 21:33:23.602874 
## Time difference of 0.110646 secs

ikipl_if <- tam.fit(model_2PL)

## Item fit calculation based on 40 simulations
## |**********|
## |----------|

ikipl_if_df <- ikipl_if$itemfit
ikipl_if_clean <- ikipl_if_df[, c("parameter", "Outfit", "Outfit_t", "Outfit_p")]
colnames(ikipl_if_clean) <- c("item", "X2", "t", "p")
ikipl_if_clean[, c("X2", "t", "p")] <- round(ikipl_if_clean[, c("X2", "t", "p")], 3)
kable(ikipl_if_clean, format = "html", caption = "2PL Modeline Ait Madde Uyum Degerleri (Outfit Temelli)") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = F, font_size = 16)

2PL Modeline Ait Madde Uyum Degerleri (Outfit Temelli)
item	X2	t	p
madde_1	1.007	0.232	0.817
madde_2	0.995	-0.118	0.906
madde_3	1.011	0.351	0.726
madde_4	0.951	-1.058	0.290
madde_5	0.936	-1.212	0.226
madde_6	0.997	-0.068	0.946
madde_7	0.994	-0.175	0.861
madde_8	0.788	-1.538	0.124
madde_9	0.998	-0.058	0.954
madde_10	1.001	0.016	0.988
madde_11	0.921	-1.050	0.294
madde_12	1.005	0.098	0.922
madde_13	0.958	-0.602	0.547
madde_14	1.018	0.323	0.747
madde_15	1.001	0.038	0.970
madde_16	1.086	0.326	0.744
madde_17	0.998	-0.072	0.942
madde_18	0.998	-0.072	0.943
madde_19	1.019	0.339	0.735
madde_20	1.080	1.017	0.309

library(mirt)
library(knitr)
library(kableExtra)
model_1PL <- mirt(data = veri, model = 1, itemtype = "Rasch", SE = T)

## Iteration: 1, Log-Lik: -5507.654, Max-Change: 0.09565Iteration: 2, Log-Lik: -5507.400, Max-Change: 0.00497Iteration: 3, Log-Lik: -5507.394, Max-Change: 0.00265Iteration: 4, Log-Lik: -5507.392, Max-Change: 0.00145Iteration: 5, Log-Lik: -5507.391, Max-Change: 0.00074Iteration: 6, Log-Lik: -5507.391, Max-Change: 0.00036Iteration: 7, Log-Lik: -5507.391, Max-Change: 0.00021Iteration: 8, Log-Lik: -5507.391, Max-Change: 0.00010Iteration: 9, Log-Lik: -5507.391, Max-Change: 0.00007
## 
## Calculating information matrix...

birpl_if <- mirt::itemfit(model_1PL)
birpl_if_df <- as.data.frame(birpl_if)
birpl_if_df[, 2:5] <- round(birpl_if_df[, 2:5], 2)
colnames(birpl_if_df) <- c("item", "X2", "sd", "RMSEA", "p")
kable(birpl_if_df, format = "html", caption = "1PL Modeline Ait Madde Uyum Degerleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 14)

1PL Modeline Ait Madde Uyum Degerleri
item	X2	sd	RMSEA	p
madde_1	14.04	13	0.01	0.37
madde_2	19.43	13	0.03	0.11
madde_3	15.90	12	0.03	0.20
madde_4	18.22	13	0.03	0.15
madde_5	27.93	12	0.05	0.01
madde_6	31.35	12	0.06	0.00
madde_7	15.00	13	0.02	0.31
madde_8	14.76	10	0.03	0.14
madde_9	5.97	12	0.00	0.92
madde_10	12.32	13	0.00	0.50
madde_11	19.13	12	0.03	0.09
madde_12	7.12	13	0.00	0.90
madde_13	12.87	13	0.00	0.46
madde_14	16.89	13	0.02	0.20
madde_15	32.91	13	0.06	0.00
madde_16	21.05	12	0.04	0.05
madde_17	13.78	12	0.02	0.32
madde_18	33.88	13	0.06	0.00
madde_19	8.91	12	0.00	0.71
madde_20	15.58	12	0.02	0.21

library(kableExtra)
library(knitr)
kable(birpl_if_df, format = "html", caption = "1PL Modeline Ait Madde Uyum Degerleri (Outfit & Infit)") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

1PL Modeline Ait Madde Uyum Degerleri (Outfit & Infit)
item	X2	sd	RMSEA	p
madde_1	14.04	13	0.01	0.37
madde_2	19.43	13	0.03	0.11
madde_3	15.90	12	0.03	0.20
madde_4	18.22	13	0.03	0.15
madde_5	27.93	12	0.05	0.01
madde_6	31.35	12	0.06	0.00
madde_7	15.00	13	0.02	0.31
madde_8	14.76	10	0.03	0.14
madde_9	5.97	12	0.00	0.92
madde_10	12.32	13	0.00	0.50
madde_11	19.13	12	0.03	0.09
madde_12	7.12	13	0.00	0.90
madde_13	12.87	13	0.00	0.46
madde_14	16.89	13	0.02	0.20
madde_15	32.91	13	0.06	0.00
madde_16	21.05	12	0.04	0.05
madde_17	13.78	12	0.02	0.32
madde_18	33.88	13	0.06	0.00
madde_19	8.91	12	0.00	0.71
madde_20	15.58	12	0.02	0.21

Bu tablo, 1PL (Rasch) modeline dayali olarak her bir maddeye iliskin uyum istatistiklerini sunmaktadir.
- “Outfit” ve “Infit” sutunlari, ham ortalama kare uyum degerlerini temsil eder.
- Outfit, yetenek duzeyi madde zorlugu ile cok farkli olan bireylerin beklenmedik tepkilerine daha duyarlidir.
- Infit ise bireyin yetenegi ile madde zorlugu birbirine yakin oldugunda modele uyumu daha hassas sekilde olcer.
- Her iki istatistik icin ideal deger \(1.00\) civaridir ve \(0.5\) ile \(1.5\) arasindaki degerler genellikle kabul edilebilir olarak yorumlanir.
“Outfit_t” ve “Infit_t”, bu istatistiklerin standartlastirilmis t-degerleridir.
- Bu degerlerin \(-2\) ile \(+2\) araliginda olmasi, istatistiksel olarak manidar bir uyumsuzluk olmadigina isaret ederken, bu sinirlarin disina cikan degerler potansiyel model-uyum sorunu oldugunu gosterir.
  - Buna karsilik “Outfit_p” ve “Infit_p”, ilgili uyum degerlerinin p-degerleridir ve \(0.05\)’ten kucuk olmalari halinde madde ile model arasinda manidar farklilik oldugunu ifade eder.
- “Outfit_pholm” ve “Infit_pholm” ise coklu test duzeltmesi icin Holm yontemi ile ayarlanmis p-degerleridir.
- Bu duzenleme, birden fazla hipotez testi yapildiginda ortaya cikabilecek Tip I hata riskini AZALTMAYI amaclar.
Bu istatistikler, modelin yapisal gecerligini degerlendirmek, problemli maddeleri saptamak ve olcegin ozel amacina uygunlugu acisindan son derece onemlidir.
- Modelden ciddi sekilde sapan maddeler (ozellikle YUKSEK Outfit/Infit ve DUSUK p-degeri tasiyanlar), iceriksel olarak gozatilmali ve gerekirse duzenlenmelidir.
- Bu analizler, olcegin gecerligi ve olcme kalitesini arttirmaya yoneliktir (Wright & Linacre, 1994; Bond & Fox, 2015).

ikipl_model <- "F=1-20"
ikipl_uyum <- mirt(data = veri, model = ikipl_model, itemtype = "2PL", SE = T)

## Iteration: 1, Log-Lik: -5513.426, Max-Change: 0.65652Iteration: 2, Log-Lik: -5454.963, Max-Change: 0.32085Iteration: 3, Log-Lik: -5447.197, Max-Change: 0.19128Iteration: 4, Log-Lik: -5445.766, Max-Change: 0.11326Iteration: 5, Log-Lik: -5445.258, Max-Change: 0.06665Iteration: 6, Log-Lik: -5445.081, Max-Change: 0.03558Iteration: 7, Log-Lik: -5445.006, Max-Change: 0.02236Iteration: 8, Log-Lik: -5444.974, Max-Change: 0.01689Iteration: 9, Log-Lik: -5444.958, Max-Change: 0.00532Iteration: 10, Log-Lik: -5444.954, Max-Change: 0.00452Iteration: 11, Log-Lik: -5444.949, Max-Change: 0.00326Iteration: 12, Log-Lik: -5444.946, Max-Change: 0.00179Iteration: 13, Log-Lik: -5444.945, Max-Change: 0.00128Iteration: 14, Log-Lik: -5444.944, Max-Change: 0.00105Iteration: 15, Log-Lik: -5444.944, Max-Change: 0.00087Iteration: 16, Log-Lik: -5444.943, Max-Change: 0.00012Iteration: 17, Log-Lik: -5444.943, Max-Change: 0.00013Iteration: 18, Log-Lik: -5444.943, Max-Change: 0.00012Iteration: 19, Log-Lik: -5444.943, Max-Change: 0.00038Iteration: 20, Log-Lik: -5444.943, Max-Change: 0.00006
## 
## Calculating information matrix...

ikipl_par <- coef(ikipl_uyum, IRTpars = T, simplify = T)
ikipl_if <- mirt::itemfit(ikipl_uyum)
ikipl_if[,2:5] <- round(ikipl_if[,2:5], 2)
names(ikipl_if ) <- c("item", "X2", "sd", "RMSEA", "p")
head(ikipl_if)

library(kableExtra)
library(knitr)
numeric_cols <- sapply(ikipl_if_df, is.numeric)
ikipl_if_df[numeric_cols] <- round(ikipl_if_df[numeric_cols], 2)
kable(ikipl_if_df, format = "html", caption = "2PL Modeline Ait Madde Uyum Degerleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 14)

2PL Modeline Ait Madde Uyum Degerleri
parameter	Outfit	Outfit_t	Outfit_p	Outfit_pholm	Infit	Infit_t	Infit_p	Infit_pholm
madde_1	1.01	0.23	0.82	1	1.00	0.12	0.90	1
madde_2	0.99	-0.12	0.91	1	1.00	0.01	1.00	1
madde_3	1.01	0.35	0.73	1	1.00	0.11	0.91	1
madde_4	0.95	-1.06	0.29	1	1.00	0.05	0.96	1
madde_5	0.94	-1.21	0.23	1	1.00	0.05	0.96	1
madde_6	1.00	-0.07	0.95	1	1.00	-0.02	0.99	1
madde_7	0.99	-0.18	0.86	1	1.00	-0.02	0.99	1
madde_8	0.79	-1.54	0.12	1	1.01	0.11	0.91	1
madde_9	1.00	-0.06	0.95	1	1.00	-0.02	0.99	1
madde_10	1.00	0.02	0.99	1	1.00	0.04	0.96	1
madde_11	0.92	-1.05	0.29	1	0.99	-0.08	0.94	1
madde_12	1.00	0.10	0.92	1	1.00	0.10	0.92	1
madde_13	0.96	-0.60	0.55	1	1.00	0.06	0.95	1
madde_14	1.02	0.32	0.75	1	0.99	-0.19	0.85	1
madde_15	1.00	0.04	0.97	1	1.00	-0.02	0.98	1
madde_16	1.09	0.33	0.74	1	0.97	-0.19	0.85	1
madde_17	1.00	-0.07	0.94	1	1.00	-0.03	0.98	1
madde_18	1.00	-0.07	0.94	1	1.00	-0.02	0.99	1
madde_19	1.02	0.34	0.73	1	1.00	0.06	0.95	1
madde_20	1.08	1.02	0.31	1	0.98	-0.28	0.78	1

Bu tabloda sunulan degerler, 2 Parametreli Lojistik Model (2PL) baglaminda her bir maddeye iliskin madde uyum istatistiklerini ortaya koymaktadir.
“Outfit” ve “Infit” degerleri, maddelerin modele ne derece uyum sagladigini gosteren istatistiklerdir; burada “Outfit” istatistigi ekstrem (beklenmedik) yanitlara duyarlidirken, “Infit” istatistigi ise bireyin yetenek seviyesine yakin maddelere verdigi yanitlara daha duyarlidir.
Bu istatistiklere ait t-test (“Outfit_t”, “Infit_t”), p-degeri (“Outfit_p”, “Infit_p”) ve p-holm (“Outfit_pholm”, “Infit_pholm”) degerleri ise her bir madde icin bu uyum istatistiklerinin istatistiksel olarak manidar olup olmadigini test etmektedir.
Genelde p-degerinin \(0.05\)’ten buyuk olmasi, maddenin modele uygunlugu icin yeterli kabul edilirken; p-holm degeri, coklu test duzeltmesini hesaba katarak anlamlilik degerlendirmesi yapmaktadir.
Tabloya bakildiginda, madde bazinda hem Outfit hem de Infit istatistiklerinin buyuk cogunlugunun 1’e YAKIN oldugu, bu nedenle madde uyumunun kabul edilebilir duzeyde oldugu gorulmektedir.
Bu uyum, olcegin genellenebilirligini ve madde kalitesini destekleyici bir bulgu olarak degerlendirilebilir.

mod_3PL <- mirt(data = veri, model = 1, itemtype = "3PL")

## Iteration: 1, Log-Lik: -5577.245, Max-Change: 1.75719Iteration: 2, Log-Lik: -5469.059, Max-Change: 0.41358Iteration: 3, Log-Lik: -5448.025, Max-Change: 0.33291Iteration: 4, Log-Lik: -5441.084, Max-Change: 0.29521Iteration: 5, Log-Lik: -5437.920, Max-Change: 0.32005Iteration: 6, Log-Lik: -5436.239, Max-Change: 0.23165Iteration: 7, Log-Lik: -5434.669, Max-Change: 0.15783Iteration: 8, Log-Lik: -5434.304, Max-Change: 0.60333Iteration: 9, Log-Lik: -5434.167, Max-Change: 0.98978Iteration: 10, Log-Lik: -5434.106, Max-Change: 0.02446Iteration: 11, Log-Lik: -5434.003, Max-Change: 0.01469Iteration: 12, Log-Lik: -5433.984, Max-Change: 0.01026Iteration: 13, Log-Lik: -5433.964, Max-Change: 0.00310Iteration: 14, Log-Lik: -5433.960, Max-Change: 0.00211Iteration: 15, Log-Lik: -5433.957, Max-Change: 0.00180Iteration: 16, Log-Lik: -5433.953, Max-Change: 0.00141Iteration: 17, Log-Lik: -5433.953, Max-Change: 0.00102Iteration: 18, Log-Lik: -5433.953, Max-Change: 0.00093Iteration: 19, Log-Lik: -5433.952, Max-Change: 0.00017Iteration: 20, Log-Lik: -5433.952, Max-Change: 0.00070Iteration: 21, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 22, Log-Lik: -5433.952, Max-Change: 0.00056Iteration: 23, Log-Lik: -5433.952, Max-Change: 0.00023Iteration: 24, Log-Lik: -5433.952, Max-Change: 0.00048Iteration: 25, Log-Lik: -5433.952, Max-Change: 0.00022Iteration: 26, Log-Lik: -5433.952, Max-Change: 0.00037Iteration: 27, Log-Lik: -5433.952, Max-Change: 0.00016Iteration: 28, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 29, Log-Lik: -5433.952, Max-Change: 0.00039Iteration: 30, Log-Lik: -5433.952, Max-Change: 0.00008

res <- M2(mod_3PL)
print(res)

##            M2  df         p       RMSEA RMSEA_5   RMSEA_95      SRMSR       TLI
## stats 150.294 150 0.4778863 0.001981898       0 0.02102183 0.03519982 0.9998282
##             CFI
## stats 0.9998644

library(mirt)
ucpl_model <- "F = 1-20"
ucpl_uyum <- mirt(data = veri, model = ucpl_model, itemtype = "3PL", SE = T)

## Iteration: 1, Log-Lik: -5577.245, Max-Change: 1.75719Iteration: 2, Log-Lik: -5469.059, Max-Change: 0.41358Iteration: 3, Log-Lik: -5448.025, Max-Change: 0.33291Iteration: 4, Log-Lik: -5441.084, Max-Change: 0.29521Iteration: 5, Log-Lik: -5437.920, Max-Change: 0.32005Iteration: 6, Log-Lik: -5436.239, Max-Change: 0.23165Iteration: 7, Log-Lik: -5434.669, Max-Change: 0.15783Iteration: 8, Log-Lik: -5434.304, Max-Change: 0.60333Iteration: 9, Log-Lik: -5434.167, Max-Change: 0.98978Iteration: 10, Log-Lik: -5434.106, Max-Change: 0.02446Iteration: 11, Log-Lik: -5434.003, Max-Change: 0.01469Iteration: 12, Log-Lik: -5433.984, Max-Change: 0.01026Iteration: 13, Log-Lik: -5433.964, Max-Change: 0.00310Iteration: 14, Log-Lik: -5433.960, Max-Change: 0.00211Iteration: 15, Log-Lik: -5433.957, Max-Change: 0.00180Iteration: 16, Log-Lik: -5433.953, Max-Change: 0.00141Iteration: 17, Log-Lik: -5433.953, Max-Change: 0.00102Iteration: 18, Log-Lik: -5433.953, Max-Change: 0.00093Iteration: 19, Log-Lik: -5433.952, Max-Change: 0.00017Iteration: 20, Log-Lik: -5433.952, Max-Change: 0.00070Iteration: 21, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 22, Log-Lik: -5433.952, Max-Change: 0.00056Iteration: 23, Log-Lik: -5433.952, Max-Change: 0.00023Iteration: 24, Log-Lik: -5433.952, Max-Change: 0.00048Iteration: 25, Log-Lik: -5433.952, Max-Change: 0.00022Iteration: 26, Log-Lik: -5433.952, Max-Change: 0.00037Iteration: 27, Log-Lik: -5433.952, Max-Change: 0.00016Iteration: 28, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 29, Log-Lik: -5433.952, Max-Change: 0.00039Iteration: 30, Log-Lik: -5433.952, Max-Change: 0.00008
## 
## Calculating information matrix...

ucpl_par <- coef(ucpl_uyum, IRTpars = T, simplify = T)
print(ucpl_par$items)

##                  a          b           g u
## madde_1  0.9405489  0.8588193 0.112263262 1
## madde_2  1.7888304  0.5267211 0.076453641 1
## madde_3  1.0613814  0.5132669 0.143408803 1
## madde_4  2.5285032  0.2080746 0.182462011 1
## madde_5  2.1986842 -0.1911775 0.396648970 1
## madde_6  0.5969278  1.5163780 0.072103470 1
## madde_7  1.9248363  0.5764446 0.281374076 1
## madde_8  1.6876074 -2.1107879 0.171791935 1
## madde_9  1.1459505  0.3771295 0.111431597 1
## madde_10 1.0362589 -0.8488462 0.002582630 1
## madde_11 1.7798288 -1.3279202 0.047657150 1
## madde_12 1.3668452 -0.1238128 0.194582796 1
## madde_13 1.3364282 -0.6511787 0.451559239 1
## madde_14 1.4244116 -0.6979000 0.001253227 1
## madde_15 0.5105399 -1.1980455 0.054692523 1
## madde_16 2.2764261 -1.6927067 0.003702569 1
## madde_17 1.2643324  0.8925815 0.222323806 1
## madde_18 0.9724285  0.9861186 0.346005098 1
## madde_19 1.0850333 -1.1925337 0.001892854 1
## madde_20 2.4323381  1.2748461 0.053789032 1

library(mirt)
library(knitr)
library(kableExtra)
ucpl_model <- "F = 1-20"
ucpl_uyum <- mirt(data = veri, model = ucpl_model, itemtype = "3PL", SE = T)

## Iteration: 1, Log-Lik: -5577.245, Max-Change: 1.75719Iteration: 2, Log-Lik: -5469.059, Max-Change: 0.41358Iteration: 3, Log-Lik: -5448.025, Max-Change: 0.33291Iteration: 4, Log-Lik: -5441.084, Max-Change: 0.29521Iteration: 5, Log-Lik: -5437.920, Max-Change: 0.32005Iteration: 6, Log-Lik: -5436.239, Max-Change: 0.23165Iteration: 7, Log-Lik: -5434.669, Max-Change: 0.15783Iteration: 8, Log-Lik: -5434.304, Max-Change: 0.60333Iteration: 9, Log-Lik: -5434.167, Max-Change: 0.98978Iteration: 10, Log-Lik: -5434.106, Max-Change: 0.02446Iteration: 11, Log-Lik: -5434.003, Max-Change: 0.01469Iteration: 12, Log-Lik: -5433.984, Max-Change: 0.01026Iteration: 13, Log-Lik: -5433.964, Max-Change: 0.00310Iteration: 14, Log-Lik: -5433.960, Max-Change: 0.00211Iteration: 15, Log-Lik: -5433.957, Max-Change: 0.00180Iteration: 16, Log-Lik: -5433.953, Max-Change: 0.00141Iteration: 17, Log-Lik: -5433.953, Max-Change: 0.00102Iteration: 18, Log-Lik: -5433.953, Max-Change: 0.00093Iteration: 19, Log-Lik: -5433.952, Max-Change: 0.00017Iteration: 20, Log-Lik: -5433.952, Max-Change: 0.00070Iteration: 21, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 22, Log-Lik: -5433.952, Max-Change: 0.00056Iteration: 23, Log-Lik: -5433.952, Max-Change: 0.00023Iteration: 24, Log-Lik: -5433.952, Max-Change: 0.00048Iteration: 25, Log-Lik: -5433.952, Max-Change: 0.00022Iteration: 26, Log-Lik: -5433.952, Max-Change: 0.00037Iteration: 27, Log-Lik: -5433.952, Max-Change: 0.00016Iteration: 28, Log-Lik: -5433.952, Max-Change: 0.00012Iteration: 29, Log-Lik: -5433.952, Max-Change: 0.00039Iteration: 30, Log-Lik: -5433.952, Max-Change: 0.00008
## 
## Calculating information matrix...

ucpl_par <- coef(ucpl_uyum, IRTpars = T, simplify = T)$items
ucpl_par_round <- round(ucpl_par, 2)
ucpl_par_df <- data.frame(item = rownames(ucpl_par_round), ucpl_par_round, row.names = NULL)
kable(ucpl_par_df, format = "html", caption = "3PL Modeline Ait Madde Parametreleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 14)

3PL Modeline Ait Madde Parametreleri
item	a	b	g	u
madde_1	0.94	0.86	0.11	1
madde_2	1.79	0.53	0.08	1
madde_3	1.06	0.51	0.14	1
madde_4	2.53	0.21	0.18	1
madde_5	2.20	-0.19	0.40	1
madde_6	0.60	1.52	0.07	1
madde_7	1.92	0.58	0.28	1
madde_8	1.69	-2.11	0.17	1
madde_9	1.15	0.38	0.11	1
madde_10	1.04	-0.85	0.00	1
madde_11	1.78	-1.33	0.05	1
madde_12	1.37	-0.12	0.19	1
madde_13	1.34	-0.65	0.45	1
madde_14	1.42	-0.70	0.00	1
madde_15	0.51	-1.20	0.05	1
madde_16	2.28	-1.69	0.00	1
madde_17	1.26	0.89	0.22	1
madde_18	0.97	0.99	0.35	1
madde_19	1.09	-1.19	0.00	1
madde_20	2.43	1.27	0.05	1

Bu tabloda sunulan degerler, 3-Parametreli Lojistik Model’e (3PL) dayali olarak her bir madde icin tahmin edilen madde parametrelerini temsil etmektedir.
Bu modelde, “a” parametresi ayirma (discrimination), “b” parametresi gucluk (difficulty) ve “c” parametresi alt duzey yanitlama (guessing) olasiligini ifade ederken, “u” parametresi genellikle sabit bir deger alarak modelde teknik bir kontrol degiskeni olarak kullanilir.
- “a” parametresi, bireylerin yetenek duzeyindeki degisikliklerin madde yanitlari uzerindeki etkisini belirtir; bu deger ne kadar YUKSEK ise, madde bireylerin yeteneklerini o kadar IYI AYIRT eder.
- “b” parametresi, maddeyi basarili sekilde yanitlamak icin gereken yetenek duzeyini ifade eder; daha yuksek “b” degerleri daha zor maddeleri isaret eder.
- “c” parametresi, dusuk yetenekli bireylerin dogru yanit verme olasiligini yani SANS faktoru etkisini yansitir; bu deger \(0\) ile \(1\) arasindadir ve genellikle \(0.00\) ile \(0.35\) arasinda olmasi beklenir.
Bu tabloda yer alan degerler madde analizinde kullanilarak olcegin olculen yapilari ne kadar dogru yansittigi ve hangi maddelerin olcegin genel yapisina daha uygun oldugu konusunda bilgi verir.
- Madde parametrelerinin model beklentilerine uygun dagilim gostermesi, model uyumunun yeterli oldugunu ve maddelerin istatistiksel olarak gecerli oldugunu gosterir.

1.1 Yetenek Parametresi Kestirimi

MTK modellerinde baslica 3 yontem İle yetenek kestirilir:
- Maksimum Likelihood (ML)
- Maksimum a Posteriori (MAP)
- Expected/estimated a Posteriori (EAP)

ML <- fscores(ikipl_uyum, method = "ML", full.scores.SE = T)
MAP <- fscores(ikipl_uyum, method = "MAP", full.scores.SE = T)
EAP <- fscores(ikipl_uyum, method = "EAP", full.scores.SE = T)

head(ML)

##               F      SE_F
## [1,] -0.5011763 0.4571663
## [2,] -1.1825774 0.4312117
## [3,] -1.5392975 0.4316295
## [4,]  3.2511342 1.3516795
## [5,]  1.3242297 0.6384556
## [6,] -1.4525457 0.4297604

library(kableExtra)
library(knitr)
ML_df <- as.data.frame(ML)
colnames(ML_df) <- c("F", "SE_F") 
kable(ML_df, format = "html", caption = "Maximum Likelihood Yontemine Gore Birey Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

Maximum Likelihood Yontemine Gore Birey Yetenek Tahminleri
F	SE_F
-0.5011763	0.4571663
-1.1825774	0.4312117
-1.5392975	0.4316295
3.2511342	1.3516795
1.3242297	0.6384556
-1.4525457	0.4297604
-0.7927151	0.4442352
-0.4048674	0.4618439
0.2972389	0.5074801
-0.5380680	0.4554332
-0.0556245	0.4814198
-0.3222478	0.4660617
-0.4836769	0.4579992
3.7454263	1.6634029
-0.4896868	0.4577123
1.8517894	0.7569643
-1.4476672	0.4296946
0.0599327	0.4891350
0.3173421	0.5092030
0.7051471	0.5481905
0.0681906	0.4897149
-1.6043239	0.4339646
-0.7728846	0.4450581
-0.7596045	0.4456144
-1.0445432	0.4348961
-0.1880175	0.4734160
0.3551667	0.5125200
-0.0161439	0.4839740
1.2161433	0.6195764
1.9830485	0.7945029
-0.8845888	0.4405588
-Inf	NA
0.8010902	0.5596681
-0.8969731	0.4400828
-0.3487912	0.4646837
0.9705517	0.5819660
-0.5174147	0.4563997
-1.1520231	0.4318998
-0.1078152	0.4781654
-1.0656336	0.4342459
0.2174653	0.5009114
0.5801811	0.5343801
-0.2145161	0.4719087
1.0706548	0.5964778
-0.8130826	0.4434001
0.4226678	0.5186893
3.7454263	1.6634029
0.6998465	0.5475791
-0.1260735	0.4770583
0.5192622	0.5280913
0.1072154	0.4925100
1.8133156	0.7466223
-0.6007356	0.4525557
-0.5545894	0.4546667
3.6878572	1.6244663
-0.4865740	0.4578608
2.9015130	1.1625334
2.5717754	1.0087307
0.0547446	0.4887726
-0.6932857	0.4484498
0.1707602	0.4972590
0.5921277	0.5356468
-0.1675445	0.4746009
0.3629462	0.5132147
-0.4333035	0.4604381
-0.3121539	0.4665918
0.2252489	0.5015338
-1.4967396	0.4305431
-0.2511721	0.4698706
0.7296594	0.5510484
-0.7927151	0.4442352
-0.5157831	0.4564765
-2.3374132	0.5280407
-0.1975849	0.4728684
-0.0435929	0.4821895
-0.4564470	0.4593097
-0.6473136	0.4504682
-1.0210922	0.4356499
-1.6244428	0.4348608
-0.5444304	0.4551374
-0.3286122	0.4657292
0.0391478	0.4876926
-0.8688306	0.4411718
-0.8061329	0.4436839
2.7707255	1.0986339
-0.2200324	0.4715986
-0.1783505	0.4739732
0.1764959	0.4977000
-1.3501592	0.4291704
-0.0686083	0.4805974
1.1446769	0.6079166
0.6582632	0.5428623
-0.1090257	0.4780916
0.2947016	0.5072646
-1.8952709	0.4557552
1.3365076	0.6407007
-1.4406501	0.4296070
1.2611418	0.6272475
2.7951585	1.1102838
-0.4508538	0.4595811
1.6659357	0.7096497
-0.6066568	0.4522879
-2.1288820	0.4879703
-1.5912472	0.4334269
-1.8247653	0.4486618
-0.7349621	0.4466570
-1.4483641	0.4297038
-0.3169087	0.4663417
2.4713666	0.9667284
-1.4517732	0.4297498
0.6623233	0.5433167
1.2301590	0.6219377
0.6618942	0.5432686
-1.3991889	0.4292517
-0.5486313	0.4549425
-0.1888837	0.4733663
1.1170555	0.6035746
-1.2715687	0.4297238
1.2778647	0.6301660
-0.9075189	0.4396816
0.9350467	0.5770672
-1.7684516	0.4438519
-0.8600881	0.4415153
Inf	NA
-0.6499631	0.4503507
0.1766815	0.4977143
0.9942135	0.5853012
-0.5333324	0.4556540
0.6707536	0.5442643
0.0559703	0.4888581
-0.2786353	0.4683774
-0.4502648	0.4596098
-0.2947196	0.4675156
2.9015130	1.1625334
-0.8871192	0.4404611
-1.4189157	0.4293866
0.3263708	0.5099858
0.1856417	0.4984076
0.0761476	0.4902775
-0.5187894	0.4563351
-1.5639067	0.4324146
3.6878572	1.6244663
0.0662854	0.4895808
-0.4127515	0.4614519
0.1629635	0.4966628
-0.4125013	0.4614643
2.5406708	0.9954768
0.1700903	0.4972076
0.9440989	0.5783043
-0.3918559	0.4624946
-0.2227246	0.4714477
-0.8865530	0.4404830
-0.6940595	0.4484162
-0.8000519	0.4439332
-1.3666549	0.4291587
1.9628417	0.7884925
1.1658234	0.6113018
-1.2198505	0.4304891
-0.3739479	0.4633981
-0.9758177	0.4371875
Inf	NA
3.7348411	1.6561918
0.0924716	0.4914434
0.4499261	0.5212731
-0.3587662	0.4641716
-0.0293264	0.4831120
-1.0057906	0.4361579
-0.1565142	0.4752469
1.9932160	0.7975596
1.1507244	0.6088792
0.8176695	0.5617323
0.9329949	0.5767879
1.5221174	0.6773733
-0.5480515	0.4549693
-1.3604305	0.4291586
1.6504867	0.7060084
-0.6017765	0.4525085
-0.9787699	0.4370842
-1.4087947	0.4293098
-0.6719883	0.4493796
0.1764959	0.4977000
0.5723156	0.5335522
0.5195036	0.5281157
0.5978287	0.5362552
-0.6379473	0.4508845
-0.3187486	0.4662451
-1.5044159	0.4307143
-1.1184767	0.4327452
1.1479473	0.6084367
-1.7011307	0.4390716
3.7454263	1.6634029
0.2218515	0.5012616
1.1912196	0.6154388
-1.0714219	0.4340723
-1.0345261	0.4352143
-1.5858076	0.4332135
-0.6447286	0.4505829
0.4377362	0.5201110
0.4014561	0.5167157
-0.8734980	0.4409894
-1.7452787	0.4420889
-1.0957846	0.4333658
2.5619783	1.0045326
-0.6487619	0.4504039
-0.8103111	0.4435132
1.4226339	0.6570622
-0.1786477	0.4739560
-1.6712017	0.4372755
-0.6160480	0.4518647
0.7467136	0.5530667
-0.3571580	0.4642540
-0.5332977	0.4556556
-0.4756361	0.4583843
-0.9207806	0.4391828
Inf	NA
-0.8964522	0.4401027
1.4806145	0.6687085
1.7960700	0.7420813
-0.4554004	0.4593604
-0.9911346	0.4366559
-0.3857221	0.4628030
-0.2043394	0.4724841
-0.5260341	0.4559952
0.9814332	0.5834926
2.4713666	0.9667284
1.6815976	0.7133854
0.1855112	0.4983975
-1.2341556	0.4302485
-0.3656767	0.4638187
0.1027989	0.4921891
1.1389638	0.6070112
0.8934590	0.5714860
0.2302898	0.5019390
-1.0338690	0.4352353
0.5108551	0.5272456
-1.1172126	0.4327788
-0.7551819	0.4458005
-0.3376348	0.4652601
0.6328502	0.5400487
0.1248594	0.4938037
-2.3937900	0.5406860
0.5958544	0.5360443
1.0673898	0.5959875
0.0319035	0.4871957
-0.2637865	0.4691812
0.1767758	0.4977216
-0.8573998	0.4416213
2.5717754	1.0087307
0.0007608	0.4850931
-0.3121539	0.4665918
0.5701727	0.5333274
2.0100620	0.8026720
1.8439843	0.7548425
-0.3501094	0.4646158
1.2003098	0.6169388
-0.1090375	0.4780908
Inf	NA
-1.7386532	0.4416077
0.0355047	0.4874424
0.2895770	0.5068307
-0.5329778	0.4556706
-1.2240813	0.4304157
0.1138038	0.4929909
-1.2364948	0.4302112
-0.4281382	0.4606918
0.3936447	0.5159971
-1.2840310	0.4295845
0.0700810	0.4898483
1.9628417	0.7884925
1.6901103	0.7154347
0.9305024	0.5764492
-0.0223837	0.4835648
-2.4179291	0.5463351
-0.6136844	0.4519710
-0.0606841	0.4810983
0.0160403	0.4861179
-0.2644736	0.4691439
-1.6224131	0.4347665
3.2267076	1.3376100
0.7686535	0.5556994
1.0976088	0.6005707
0.9067459	0.5732510
-1.6519571	0.4362247
0.2409281	0.5027996
-0.0993378	0.4786849
0.7924599	0.5586031
-0.4036671	0.4619037
1.7221477	0.7232676
1.2094201	0.6184526
-0.0514724	0.4816846
0.8337208	0.5637542
-1.1169529	0.4327857
0.0873491	0.4910758
1.4432552	0.6611443
0.4033224	0.5168881
-1.8480458	0.4508711
-1.5782702	0.4329277
0.0001184	0.4850502
2.2869481	0.8954850
-0.8337717	0.4425629
1.1841592	0.6142807
0.3526668	0.5122978
2.1911408	0.8614707
-0.4243938	0.4608762
-0.2997546	0.4672477
-1.1584119	0.4317492
-0.2563829	0.4695851
-0.1075568	0.4781812
-1.7214780	0.4404071
-0.5609853	0.4543715
3.6878572	1.6244663
-2.4729087	0.5597274
-0.5808506	0.4534600
0.4582459	0.5220725
-0.9177982	0.4392944
0.6001381	0.5365024
1.0721955	0.5967096
0.7285116	0.5509135
-2.4622294	0.5570688
2.5541368	1.0011881
-0.9828422	0.4369423
1.5749273	0.6888101
0.2399567	0.5027206
0.2319249	0.5020707
0.8348822	0.5639014
0.2231389	0.5013647
0.7250724	0.5505098
-0.5987412	0.4526460
1.4436641	0.6612259
-0.1043969	0.4783745
2.3376331	0.9143057
-1.4339281	0.4295306
-0.8244879	0.4429372
0.8367345	0.5641364
0.7671105	0.5555129
-0.8111053	0.4434807
-0.9288806	0.4388814
-0.5649733	0.4541879
-0.7920229	0.4442638
-1.9129735	0.4577259
-0.2543982	0.4696937
0.2065732	0.5000471
-0.0347344	0.4827611
-0.7436523	0.4462878
0.8737215	0.5688948
2.0748530	0.8228967
-0.0775379	0.4800368
-0.8621743	0.4414331
-0.5176474	0.4563888
1.0224355	0.5893550
-1.3505677	0.4291697
-1.4306758	0.4294963
1.0090816	0.5874264
-0.8190348	0.4431581
-0.5880243	0.4531328
1.5831838	0.6906410
-0.3592472	0.4641470
1.5330261	0.6796975
-1.1317244	0.4324007
-1.1341278	0.4323397
-0.7953814	0.4441253
-1.4426701	0.4296314
-0.5106048	0.4567205
-0.2377277	0.4706120
1.1701189	0.6119960
-1.4021388	0.4292681
-0.3390197	0.4651883
0.0486735	0.4883506
-0.2545287	0.4696866
0.0351451	0.4874177
1.0578589	0.5945631
0.3946204	0.5160866
1.1912196	0.6154388
-0.3966977	0.4622519
0.0959497	0.4916938
-2.4876094	0.5634322
0.5576658	0.5320228
0.4127516	0.5177627
0.6044942	0.5369698
-0.7924998	0.4442441
2.5581071	1.0028798
0.8355117	0.5639812
-0.1169804	0.4776077
1.3496536	0.6431282
-0.2585436	0.4694670
-0.6559154	0.4500873
2.3500594	0.9190076
-0.1435749	0.4760118
2.2168073	0.8703837
-1.5518635	0.4320157
-1.0357591	0.4351748
0.0384442	0.4876443
0.6185890	0.5384923
-0.4095338	0.4616117
-0.4594899	0.4591623
-1.1400286	0.4321918
0.8439047	0.5650491
1.6573110	0.7076116
-2.2907559	0.5181570
1.8439843	0.7548425
-1.9124448	0.4576660
1.0574595	0.5945036
0.4656737	0.5227905
-0.6070244	0.4522713
-0.6094441	0.4521621
0.0074121	0.4855376
0.5402903	0.5302301
0.3851903	0.5152241
-1.8190278	0.4481373
-2.0367957	0.4736647
2.2360641	0.8771665
1.8091433	0.7455183
0.2755672	0.5056536
0.5823568	0.5346100
0.8541618	0.5663628
-2.3830374	0.5382149
1.1765594	0.6130410
-0.5098102	0.4567580
-0.2401945	0.4704754
-0.4559082	0.4593358
0.9873891	0.5843333
-0.7244293	0.4471067
-0.3633724	0.4639362
-0.6452169	0.4505612
-0.7882405	0.4444201
1.0022078	0.5864410
0.1432231	0.4951704
1.2234146	0.6207984
0.5756027	0.5338976
0.7914513	0.5584791
1.2473951	0.6248762
0.3201170	0.5094430
-0.9457764	0.4382610
0.0396142	0.4877248
0.2920123	0.5070367
0.4790529	0.5240940
1.2600746	0.6270626
-0.7007906	0.4481243
0.6115177	0.5377265
0.0001184	0.4850502
0.0534697	0.4886838
-0.1384295	0.4763179
2.1911408	0.8614707
-0.3954910	0.4623123
0.4756334	0.5237596
0.4874660	0.5249204
0.9229361	0.5754247
-1.1131981	0.4328862
Inf	NA
0.0060955	0.4854494
-0.6269931	0.4513736
2.1245810	0.8390316
0.1984013	0.4994037
-0.8632806	0.4413896
-0.2424215	0.4703523
-1.8084255	0.4471888
0.3369763	0.5109124
-0.4307373	0.4605641
-0.0588002	0.4812178
1.3528819	0.6437281
-0.8094981	0.4435463
-1.3568572	0.4291610
0.5548215	0.5317277
2.2168073	0.8703837
-1.9187436	0.4583849
-0.2032492	0.4725460
-0.4713394	0.4585907
-1.2677284	0.4297704
0.4004009	0.5166184
-0.8334845	0.4425744
0.3901412	0.5156761
-0.2090973	0.4722146
0.1971644	0.4993067
0.2447379	0.5031096
-1.4281372	0.4294708
-1.4512715	0.4297429
-0.0242068	0.4834457
0.7950395	0.5589208
-1.2367363	0.4302074
0.1526462	0.4958798
0.7463427	0.5530225
-0.4844830	0.4579606
-0.4911809	0.4576412
-0.3526045	0.4644876
1.1679118	0.6116390
0.4520349	0.5214753
1.1561076	0.6097398
0.2386277	0.5026128
1.4531583	0.6631281
-0.6686052	0.4495282
1.0515345	0.5936233
0.3218954	0.5095971
-2.1379655	0.4894945
-0.2060116	0.4723893
3.6878572	1.6244663
-0.0646295	0.4808485
0.4732090	0.5235230
0.7150706	0.5493414
-1.4250662	0.4294412
-0.7878973	0.4444343

Bu tabloda yer alan degerler, bireylerin Maximum Likelihood (ML) yontemine gore tahmin edilen yetenek duzeylerini (F) ve bu tahminlere iliskin standart hatalari (SE_F) gostermektedir.
“F” sutununda bulunan degerler, bireylerin olcekteki gizil ozelliklerine (genellikle “yetkinlik”, “basari” ya da “yetkinlik seviyesi” gibi adlandirilir) iliskin tahminlerdir.
- Bu degerler, testte verilen yanitlara gore bireyin konumlandigi yetenek seviyesini yansitir.
- Pozitif degerler ortalamanin uzerinde bir yetenek seviyesini, negatif degerler ise ortalamanin altinda bir performansi ifade eder.
- Orn., \(F = 3.3\) olan birey oldukca yuksek bir yetenek seviyesine sahiptir, \(F = -1.5\) olan birey ise dusuk performans gostermistir.
“SE_F” sutunu ise bu tahminlerin ne kadar guvenilir oldugunu gosterir.
- Daha kucuk standart hata degerleri, yetenek tahminlerinin daha duyarlilikli ve guvenilir oldugunu belirtir.
- Orn., \(SE_F = 0.43\) olan bir tahmin daha az belirsizlik icerirken, \(SE_F = 1.35\) olan bir deger daha yuksek bir belirsizlikle birlikte gelir.
- Bu bilgiler olcegin birey duzeyinde ayiriciligi ve guvenilirligi hakkinda bilgi verir.
- Yani, hem bireylerin yetenek duzeyleri hem de bu degerlerin ne kadar guvenilir oldugu ayni anda degerlendirilmelidir.

head(MAP)

##               F      SE_F
## [1,] -0.4138823 0.4189514
## [2,] -0.9953929 0.4000572
## [3,] -1.2993110 0.3945927
## [4,]  1.7167424 0.5853362
## [5,]  0.9668267 0.5026532
## [6,] -1.2263351 0.3953203

library(kableExtra)
library(knitr)
MAP_df <- as.data.frame(MAP)
colnames(MAP_df) <- c("F", "SE_F")
kable(MAP_df, format = "html", caption = "MAP Yontemine Gore Birey Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

MAP Yontemine Gore Birey Yetenek Tahminleri
F	SE_F
-0.4138823	0.4189514
-0.9953929	0.4000572
-1.2993110	0.3945927
1.7167424	0.5853362
0.9668267	0.5026532
-1.2263351	0.3953203
-0.6606955	0.4102709
-0.3332336	0.4220083
0.2368484	0.4489774
-0.4448975	0.4178089
-0.0451463	0.4342603
-0.2644332	0.4247278
-0.3991937	0.4194986
1.7938351	0.5956198
-0.4042366	0.4193103
1.2613533	0.5313897
-1.2222059	0.3953750
0.0483772	0.4388048
0.2525620	0.4498903
0.5462796	0.4688765
0.0550203	0.4391398
-1.3533453	0.3943802
-0.6437893	0.4108382
-0.6324764	0.4112199
-0.8765602	0.4034056
-0.1534759	0.4293706
0.2820130	0.4516288
-0.0130792	0.4357832
0.8985926	0.4966213
1.3238074	0.5380839
-0.7392184	0.4076869
-2.6556942	0.5204696
0.6157793	0.4739128
-0.7498262	0.4073446
-0.2864968	0.4238435
0.7349559	0.4830427
-0.4275259	0.4184467
-0.9690893	0.4007607
-0.0877084	0.4322936
-0.8947101	0.4028685
0.1740838	0.4454323
0.4537480	0.4624967
-0.1752950	0.4284302
0.8030257	0.4885421
-0.6780759	0.4096916
0.3341711	0.4547971
1.7938351	0.5956198
0.5423997	0.4686016
-0.1026446	0.4316177
0.4078719	0.4594705
0.0863356	0.4407413
1.2422006	0.5293805
-0.4977321	0.4159003
-0.4588085	0.4173020
1.7862877	0.5945995
-0.4016244	0.4194078
1.6415075	0.5756072
1.5498510	0.5641836
0.0442007	0.4385951
-0.5760929	0.4131478
0.1370478	0.4434153
0.4626872	0.4630969
-0.1366484	0.4301056
0.2880504	0.4519897
-0.3569969	0.4210938
-0.2560532	0.4250669
0.1802357	0.4457727
-1.2636242	0.3948898
-0.2055484	0.4271492
0.5641678	0.4701527
-0.6606955	0.4102709
-0.4261544	0.4184973
-1.8904901	0.4143238
-0.1613487	0.4290296
-0.0353618	0.4347212
-0.3763679	0.4203571
-0.5371206	0.4145057
-0.8563869	0.4040115
-1.3699263	0.3943770
-0.4502530	0.4176134
-0.2697199	0.4245148
0.0316320	0.4379677
-0.7257283	0.4081247
-0.6721436	0.4098889
1.6078062	0.5713512
-0.1798426	0.4282360
-0.1455269	0.4297167
0.1416070	0.4436606
-1.1392373	0.3967402
-0.0557169	0.4337662
0.8521633	0.4926433
0.5118214	0.4664575
-0.0886978	0.4322486
0.2348621	0.4488627
-1.5849369	0.3974889
0.9744210	0.5033385
-1.2162625	0.3954561
0.9272966	0.4991316
1.6143556	0.5721733
-0.3716838	0.4205345
1.1653357	0.5215196
-0.5027338	0.4157219
-1.7544453	0.4048606
-1.3425317	0.3943985
-1.5306371	0.3960977
-0.6115038	0.4119321
-1.2227960	0.3953671
-0.2599999	0.4249070
1.5172891	0.5602413
-1.2256814	0.3953288
0.5148179	0.4666658
0.9075777	0.4974029
0.5145013	0.4666437
-1.1810534	0.3959917
-0.4537902	0.4174845
-0.1541884	0.4293396
0.8339494	0.4911101
-1.0719077	0.3981669
0.9378570	0.5000650
-0.7588632	0.4070544
0.7103853	0.4811089
-1.4863554	0.3952823
-0.7182479	0.4083685
1.9795098	0.6215586
-0.5393641	0.4144269
0.1417546	0.4436686
0.7512084	0.4843368
-0.4409125	0.4179548
0.5210323	0.4670990
0.0451876	0.4386445
-0.2282668	0.4262041
-0.3711907	0.4205532
-0.2415925	0.4256562
1.6415075	0.5756072
-0.7413855	0.4076169
-1.1978236	0.3957251
0.2596074	0.4503029
0.1488707	0.4440532
0.0614161	0.4394638
-0.4286815	0.4184041
-1.3198352	0.3944769
1.7862877	0.5945995
0.0534881	0.4390624
-0.3398179	0.4217536
0.1308453	0.4430828
-0.3396089	0.4217617
1.5400105	0.5629857
0.1365150	0.4433867
0.7166704	0.4816010
-0.3223743	0.4224302
-0.1820627	0.4281414
-0.7409006	0.4076325
-0.5767497	0.4131251
-0.6669544	0.4100618
-1.1533252	0.3964752
1.3144849	0.5370707
0.8660071	0.4938188
-1.0274634	0.3992345
-0.3074429	0.4230147
-0.8174710	0.4052047
1.9795098	0.6215586
1.7924721	0.5954353
0.0745206	0.4401327
0.3550838	0.4560997
-0.2947981	0.4235138
-0.0237736	0.4352713
-0.8432294	0.4044115
-0.1275932	0.4305047
1.3284580	0.5385912
0.8561313	0.4929793
0.6276479	0.4747939
0.7089587	0.4809974
1.0851243	0.5136532
-0.4533020	0.4175022
-1.1480115	0.3965737
1.1569619	0.5206825
-0.4986113	0.4158689
-0.8200071	0.4051260
-1.1892244	0.3958593
-0.5580266	0.4137746
0.1416070	0.4436606
0.4478521	0.4621027
0.4080546	0.4594824
0.4669463	0.4633841
-0.5291920	0.4147845
-0.2615275	0.4248452
-1.2700782	0.3948275
-0.9402041	0.4015600
0.8543100	0.4928249
-1.4324557	0.3946515
1.7938351	0.5956198
0.1775513	0.4456240
0.8825162	0.4952325
-0.8996922	0.4027225
-0.8679420	0.4036633
-1.3380253	0.3944098
-0.5349320	0.4145826
0.3457425	0.4555156
0.3178369	0.4537926
-0.7297230	0.4079948
-1.4679153	0.3950232
-0.9206656	0.4021150
1.5467750	0.5638085
-0.5383470	0.4144626
-0.6757099	0.4097702
1.0267615	0.5081385
-0.1457712	0.4297060
-1.4081869	0.3944893
-0.5106700	0.4154396
0.5765607	0.4710450
-0.2934594	0.4235668
-0.4408833	0.4179558
-0.3924495	0.4197512
-0.7702324	0.4066911
1.9795098	0.6215586
-0.7493800	0.4073590
1.0610554	0.5113580
1.2334918	0.5284737
-0.3754913	0.4203902
-0.8306316	0.4047978
-0.3172581	0.4226299
-0.1669103	0.4287899
-0.4347732	0.4181800
0.7424423	0.4836374
1.5172891	0.5602413
1.1737642	0.5223659
0.1487671	0.4440476
-1.0397642	0.3989301
-0.3005524	0.4232862
0.0827995	0.4405586
0.8484081	0.4923259
0.6813313	0.4788568
0.1842168	0.4459938
-0.8673768	0.4036803
0.4015032	0.4590575
-0.9391157	0.4015906
-0.6287106	0.4113474
-0.2772185	0.4242139
0.4930129	0.4651587
0.1004532	0.4414757
-1.9246273	0.4172043
0.4654718	0.4632846
0.8008343	0.4883618
0.0257876	0.4376781
-0.2159779	0.4267136
0.1418295	0.4436726
-0.7159482	0.4084437
1.5498510	0.5641836
0.0006158	0.4364446
-0.2560532	0.4250669
0.4462444	0.4619956
1.3361040	0.5394279
1.2574988	0.5309837
-0.2875935	0.4237998
0.8883941	0.4957388
-0.0887092	0.4322481
1.9795098	0.6215586
-1.4626207	0.3949572
0.0286934	0.4378219
0.2308483	0.4486315
-0.4406141	0.4179657
-1.0311019	0.3991438
0.0916108	0.4410148
-1.0417752	0.3988810
-0.3526773	0.4212591
0.3118086	0.4534247
-1.0826038	0.3979239
0.0565402	0.4392166
1.3144849	0.5370707
1.1783197	0.5228250
0.7072248	0.4808621
-0.0181396	0.4355405
-1.9388939	0.4184678
-0.5086723	0.4155106
-0.0492641	0.4340674
0.0129752	0.4370473
-0.2165463	0.4266899
-1.3682566	0.3943759
1.7121173	0.5847292
0.5924413	0.4721981
0.8210382	0.4900326
0.6906455	0.4795747
-1.3924886	0.3944225
0.1926102	0.4464620
-0.0807821	0.4326096
0.6095853	0.4734553
-0.3322314	0.4220471
1.1953012	0.5245460
0.8942683	0.4962466
-0.0417685	0.4344191
0.6390926	0.4756494
-0.9388921	0.4015969
0.0704107	0.4399222
1.0390467	0.5092850
0.3192761	0.4538807
-1.5487121	0.3965117
-1.3317733	0.3944290
0.0000957	0.4364194
1.4512627	0.5524363
-0.6957471	0.4091068
0.8779394	0.4948393
0.2800714	0.4515131
1.4136397	0.5481021
-0.3495468	0.4213792
-0.2457669	0.4254855
-0.9745899	0.4006116
-0.2098555	0.4269690
-0.0874973	0.4323032
-1.4488506	0.3948031
-0.4641973	0.4171064
1.7862877	0.5945995
-1.9706020	0.4214010
-0.4809471	0.4165017
0.3614491	0.4564999
-0.7676752	0.4067726
0.4686704	0.4635006
0.8040590	0.4886272
0.5633322	0.4700928
-1.9645285	0.4208259
1.5442975	0.5635069
-0.8235059	0.4050177
1.1151605	0.5165594
0.1918442	0.4464192
0.1855075	0.4460656
0.6399216	0.4757116
0.1785686	0.4456803
0.5608272	0.4699133
-0.4960478	0.4159604
1.0392893	0.5093077
-0.0849166	0.4324208
1.4702355	0.5546531
-1.2105644	0.3955365
-0.6878155	0.4093688
0.6412395	0.4758106
0.5913250	0.4721167
-0.6763878	0.4097477
-0.7771793	0.4064701
-0.4675583	0.4169847
-0.6601052	0.4102906
-1.5983581	0.3979021
-0.2082149	0.4270376
0.1654651	0.4449581
-0.0281646	0.4350623
-0.6188969	0.4116804
0.6674410	0.4777934
1.3648225	0.5426005
-0.0629937	0.4334282
-0.7200326	0.4083103
-0.4277215	0.4184395
0.7704629	0.4858852
-1.1395864	0.3967335
-1.2078065	0.3955763
0.7613699	0.4851519
-0.6831581	0.4095230
-0.4870004	0.4162843
1.1197960	0.5170120
-0.2951985	0.4234979
1.0913833	0.5142549
-0.9516113	0.4012412
-0.9536808	0.4011838
-0.6629698	0.4101948
-1.2179739	0.3954325
-0.4218026	0.4186580
-0.1944430	0.4276164
0.8688084	0.4940578
-1.1835635	0.3959505
-0.2783699	0.4241678
0.0393106	0.4383503
-0.2083227	0.4270330
0.0284033	0.4378075
0.7944262	0.4878358
0.3125619	0.4534706
0.8825162	0.4952325
-0.3264142	0.4222729
0.0773098	0.4402759
-1.9788951	0.4221964
0.4368489	0.4613715
0.3265416	0.4543265
0.4719204	0.4637205
-0.6605118	0.4102770
1.5455537	0.5636598
0.6403695	0.4757452
-0.0952022	0.4319536
0.9825159	0.5040720
-0.2116420	0.4268943
-0.5444055	0.4142502
1.4747901	0.5551884
-0.1169785	0.4309759
1.4239468	0.5492814
-1.3098019	0.3945283
-0.8690028	0.4036315
0.0310646	0.4379396
0.4824186	0.4644339
-0.3371303	0.4218574
-0.3789167	0.4202607
-0.9587618	0.4010434
0.6463362	0.4761939
1.1606681	0.5210525
-1.8613773	0.4120273
1.2574988	0.5309837
-1.5979586	0.3978895
0.7941573	0.4878138
0.3671250	0.4568582
-0.5030445	0.4157108
-0.5050889	0.4156380
0.0059982	0.4367064
0.4237620	0.4605085
0.3052762	0.4530278
-1.5261612	0.3960025
-1.6896613	0.4014797
1.4315699	0.5501574
1.2401006	0.5291615
0.2198613	0.4480020
0.4553774	0.4626059
0.6536102	0.4767429
-1.9182046	0.4166469
0.8730018	0.4944162
-0.4211349	0.4186827
-0.1964797	0.4275305
-0.3759166	0.4203741
0.7465310	0.4839632
-0.6025474	0.4122380
-0.2986334	0.4233620
-0.5353454	0.4145681
-0.6568793	0.4103986
0.7566770	0.4847749
0.1151166	0.4422465
0.9032590	0.4970268
0.4503170	0.4622673
0.6088628	0.4734021
0.9185721	0.4983644
0.2547276	0.4500169
-0.7916760	0.4060115
0.0320082	0.4379864
0.2327560	0.4487413
0.3773314	0.4575059
0.9266207	0.4990720
-0.5824640	0.4129278
0.4771550	0.4640756
0.0000957	0.4364194
0.0431741	0.4385436
-0.1127616	0.4311640
1.4136397	0.5481021
-0.3254072	0.4223121
0.3747249	0.4573400
0.3837382	0.4579147
0.7019548	0.4804516
-0.9356590	0.4016881
1.9795098	0.6215586
0.0049330	0.4366545
-0.5199244	0.4151117
1.3861251	0.5449844
0.1589914	0.4446037
-0.7209792	0.4082794
-0.1983189	0.4274530
-1.5178685	0.3958339
0.2678699	0.4507894
-0.3548506	0.4211759
-0.0477307	0.4341392
0.9844981	0.5042521
-0.6750159	0.4097933
-1.1449598	0.3966311
0.4347093	0.4612299
1.4239468	0.5492814
-1.6027137	0.3980423
-0.1660125	0.4288285
-0.3888470	0.4198865
-1.0686104	0.3982429
0.3170229	0.4537428
-0.6955016	0.4091149
0.3091026	0.4532601
-0.1708296	0.4286215
0.1580110	0.4445502
0.1956133	0.4466302
-1.2056523	0.3956078
-1.2252569	0.3953344
-0.0196187	0.4354697
0.6114379	0.4735920
-1.0419828	0.3988759
0.1226291	0.4426448
0.5762917	0.4710255
-0.3998700	0.4194733
-0.4054905	0.4192635
-0.2896696	0.4237173
0.8673695	0.4939350
0.3566980	0.4562010
0.8596574	0.4932785
0.1907962	0.4463606
1.0449120	0.5098351
-0.5551586	0.4138746
0.7901648	0.4874870
0.2561150	0.4500981
-1.7606840	0.4052241
-0.1682876	0.4287307
1.7862877	0.5945995
-0.0524763	0.4339173
0.3728760	0.4572226
0.5535321	0.4693922
-1.2030459	0.3956463
-0.6565867	0.4104084

MAP cikti tablosunda sunulan degerler, MTK kapsaminda *Maksimum A Posteriori (MAP)** tahmin yontemi kullanilarak elde edilen bireysel yetenek tahminlerini ve bu tahminlere iliskin standart hatalari yansitmaktadir.
- Tabloya ait “F” sutunu, her bireyin gozlemledigi yanit desenine ve secilen onsel dagilima dayali olarak hesaplanan gizil ozellik (genellikle yetenek veya yeterlik) degerini ifade eder.
Bu sutundaki yuksek degerler bireyin daha yuksek bir gizil ozellige sahip oldugunu, NEGATIF degerler ise evren ortalamasina gore DAHA DUSUK bir yetenek seviyesini gosterir.
“SE_F” sutunu ise her yetenek tahminine iliskin standart hatayi belirtmekte olup, tahminin ne kadar guvenilir oldugunu ortaya koyar.
- DAHA DUSUK hata degerleri daha yuksek guvenirligi, daha yuksek hata degerleri ise artan belirsizligi temsil eder; bu durum genellikle daha az ayirt edici maddeler veya yetersiz yanit sayisi gibi faktorlerden kaynaklanabilir.
- MAP yontemi, gozlemlenen verilerle onsel bilgiyi dengede tutarak, ozellikle sinirli veya uclarda kalan yanit desenlerinde bile kararli tahminler uretme yetisine sahiptir.
Bu nedenle, genis capli sinavlar ve bilgisayar uyumlu test uygulamalari gibi ortamlarda hem dogruluk hem de hesaplama verimliligi acisindan tercih edilen bir yontemdir.

head(EAP)

##               F      SE_F
## [1,] -0.3978568 0.4213053
## [2,] -0.9868642 0.4040602
## [3,] -1.2995100 0.4021196
## [4,]  1.7923834 0.5972560
## [5,]  1.0130331 0.5109648
## [6,] -1.2239083 0.4018864

library(kableExtra)
library(knitr)
EAP_df <- as.data.frame(EAP)
colnames(EAP_df) <- c("F", "SE_F")
kable(EAP_df, format = "html", caption = "EAP Yontemiyle Hesaplanan Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

EAP Yontemiyle Hesaplanan Yetenek Tahminleri
F	SE_F
-0.3978568	0.4213053
-0.9868642	0.4040602
-1.2995100	0.4021196
1.7923834	0.5972560
1.0130331	0.5109648
-1.2239083	0.4018864
-0.6473217	0.4125101
-0.3162627	0.4245719
0.2633563	0.4537345
-0.4292177	0.4201001
-0.0240259	0.4378846
-0.2465901	0.4275123
-0.3830014	0.4218860
1.8726161	0.6074358
-0.3881018	0.4216859
1.3182456	0.5416437
-1.2196414	0.4018883
0.0711410	0.4428080
0.2794069	0.4547096
0.5800954	0.4749046
0.0779064	0.4431699
-1.3557285	0.4026421
-0.6302314	0.4130525
-0.6187965	0.4134207
-0.8659374	0.4064953
-0.1340741	0.4325637
0.3094996	0.4565648
0.0085881	0.4395373
0.9425675	0.5045038
1.3831674	0.5487213
-0.7267411	0.4101178
-2.7477146	0.5380610
0.6514336	0.4802597
-0.7374769	0.4098114
-0.2689406	0.4265537
0.7739423	0.4899869
-0.4116535	0.4207716
-0.9600470	0.4045348
-0.0672870	0.4357466
-0.8843722	0.4060773
0.1992829	0.4499403
0.4852305	0.4681264
-0.1562155	0.4315394
0.8440270	0.4958598
-0.6648941	0.4119624
0.3628256	0.4599403
1.8726161	0.6074358
0.5761151	0.4746124
-0.0824612	0.4350110
0.4382447	0.4649108
0.1098079	0.4448979
1.2983486	0.5395123
-0.4826248	0.4181120
-0.4432813	0.4195686
1.8647652	0.6064321
-0.3854598	0.4217894
1.7140224	0.5874949
1.6185233	0.5758731
0.0668881	0.4425814
-0.5618141	0.4153161
0.1615031	0.4477754
0.4943896	0.4687640
-0.1169927	0.4333644
0.3156702	0.4569496
-0.3403124	0.4235894
-0.2380991	0.4278803
0.2055604	0.4503052
-1.2624931	0.4019410
-0.1869027	0.4301445
0.5984496	0.4762610
-0.6473217	0.4125101
-0.4102668	0.4208250
-1.9248313	0.4291725
-0.1420642	0.4321923
-0.0140764	0.4383850
-0.3599118	0.4228009
-0.5224311	0.4166836
-0.8454602	0.4069777
-1.3730212	0.4028668
-0.4346322	0.4198948
-0.2519462	0.4272812
0.0540908	0.4419032
-0.7130909	0.4105134
-0.6588959	0.4121482
1.6789099	0.5831851
-0.1608293	0.4313278
-0.1260058	0.4329408
0.1661528	0.4480390
-1.1341389	0.4022374
-0.0347729	0.4373478
0.8946718	0.5002456
0.5447533	0.4723341
-0.0682923	0.4356976
0.2613276	0.4536120
-1.5990006	0.4088970
1.0208815	0.5116989
-1.2135018	0.4018937
0.9721991	0.5071922
1.6857339	0.5840194
-0.3551730	0.4229905
1.2185618	0.5311469
-0.4876799	0.4179280
-1.7792081	0.4183205
-1.3444613	0.4025122
-1.5416345	0.4067889
-0.5975997	0.4141141
-1.2202510	0.4018879
-0.2420982	0.4277067
1.5845972	0.5718233
-1.2232327	0.4018866
0.5478259	0.4725554
0.9518412	0.5053407
0.5475013	0.4725320
-1.1771765	0.4019894
-0.4382081	0.4197597
-0.1347973	0.4325300
0.8758940	0.4986055
-1.0650636	0.4029167
0.9831047	0.5081920
-0.7466246	0.4095537
0.7486647	0.4879241
-1.4949964	0.4053747
-0.7055230	0.4107356
2.0653673	0.6325009
-0.5246983	0.4166036
0.1663032	0.4480476
0.7906683	0.4913679
-0.4251888	0.4202533
0.5541987	0.4730158
0.0678930	0.4426349
-0.2099371	0.4291161
-0.3546741	0.4230105
-0.2234444	0.4285204
1.7140224	0.5874949
-0.7289341	0.4100549
-1.1944693	0.4019304
0.2866047	0.4551501
0.1735612	0.4484607
0.0844206	0.4435198
-0.4128220	0.4207267
-1.3208395	0.4022812
1.8647652	0.6064321
0.0763460	0.4430863
-0.3229271	0.4242980
0.1551782	0.4474182
-0.3227156	0.4243067
1.6082702	0.5746447
0.1609598	0.4477447
0.7551297	0.4884489
-0.3052701	0.4250265
-0.1630815	0.4312248
-0.7284434	0.4100690
-0.5624778	0.4152935
-0.6536494	0.4123117
-1.1486261	0.4021382
1.3734727	0.5476526
0.9089485	0.5015037
-1.0196047	0.4035363
-0.2901528	0.4256574
-0.8059920	0.4079588
2.0653673	0.6325009
1.8711983	0.6072544
0.0977699	0.4442416
0.3842179	0.4613266
-0.2773482	0.4261969
-0.0022906	0.4389821
-0.8321112	0.4073021
-0.1077989	0.4337991
1.3880041	0.5492560
0.8987635	0.5006052
0.6636215	0.4811971
0.7471974	0.4878053
1.1354199	0.5227435
-0.4377146	0.4197783
-1.1431604	0.4021738
1.2098763	0.5302539
-0.4835134	0.4180796
-0.8085629	0.4078929
-1.1855991	0.4019577
-0.5435574	0.4159443
0.1661528	0.4480390
0.4791903	0.4677078
0.4384319	0.4649234
0.4987539	0.4690691
-0.5144188	0.4169674
-0.2436459	0.4276397
-1.2691810	0.4019640
-0.9306326	0.4050999
0.8968854	0.5004401
-1.4384104	0.4040041
1.8726161	0.6074358
0.2028210	0.4501458
0.9259787	0.5030168
-0.8894345	0.4059653
-0.8571878	0.4066991
-1.3397683	0.4024619
-0.5202194	0.4167617
0.3746616	0.4607050
0.3461214	0.4588708
-0.7171327	0.4103956
-1.4756146	0.4048635
-0.9107554	0.4055070
1.6153184	0.5754887
-0.5236704	0.4166398
-0.6625017	0.4120363
1.0750046	0.5168407
-0.1262538	0.4329292
-1.4129980	0.4035069
-0.4957005	0.4176377
0.6111683	0.4772096
-0.2759924	0.4262543
-0.4251593	0.4202545
-0.3761799	0.4221547
-0.7581353	0.4092338
2.0653673	0.6325009
-0.7370252	0.4098242
1.1104963	0.5202875
1.2893034	0.5385493
-0.3590250	0.4228363
-0.8193347	0.4076198
-0.3000905	0.4252419
-0.1477079	0.4319312
-0.4189816	0.4204904
0.7816463	0.4906214
1.5845972	0.5718233
1.2273055	0.5320493
0.1734555	0.4484546
-1.0321760	0.4033520
-0.2831754	0.4259508
0.1062048	0.4447009
0.8907999	0.4999061
0.7187879	0.4855236
0.2096231	0.4505421
-0.8566140	0.4067126
0.4317245	0.4644718
-0.9295249	0.4051221
-0.6149902	0.4135442
-0.2595426	0.4269550
0.5254699	0.4709543
0.1241951	0.4456890
-1.9614755	0.4323626
0.4972430	0.4689634
0.8417694	0.4956671
0.0481411	0.4415898
-0.1974783	0.4296703
0.1663796	0.4480519
-0.7031966	0.4108043
1.6185233	0.5758731
0.0225220	0.4402543
-0.2380991	0.4278803
0.4775433	0.4675939
1.3959567	0.5501375
1.3142407	0.5412132
-0.2700514	0.4265064
0.9320433	0.5035589
-0.0683039	0.4356970
2.0653673	0.6325009
-1.4700538	0.4047249
0.0510992	0.4417454
0.2572285	0.4533648
-0.4248871	0.4202648
-1.0233224	0.4034808
0.1151835	0.4451926
-1.0342320	0.4033228
-0.3359413	0.4237667
0.3399575	0.4584789
-1.0760197	0.4027871
0.0794544	0.4432529
1.3734727	0.5476526
1.2320319	0.5325386
0.7454140	0.4876609
0.0034402	0.4392741
-1.9767998	0.4337508
-0.4936815	0.4177106
-0.0282127	0.4376750
0.0350997	0.4409071
-0.1980547	0.4296446
-1.3712789	0.4028427
1.7875676	0.5966507
0.6274677	0.4784356
0.8625867	0.4974532
0.7283644	0.4862886
-1.3965831	0.4032233
0.2181893	0.4510437
-0.0602491	0.4360902
0.6450726	0.4797731
-0.3152483	0.4246137
1.2496537	0.5343720
0.9381049	0.5041025
-0.0205914	0.4380570
0.6753808	0.4821079
-0.9292974	0.4051266
0.0935830	0.4440145
1.0877161	0.5180683
0.3475931	0.4589645
-1.5607089	0.4074440
-1.3332597	0.4023958
0.0219928	0.4402269
1.5158200	0.5637488
-0.6827637	0.4114159
0.9212569	0.5025959
0.3075153	0.4564414
1.4766442	0.5592341
-0.3327733	0.4238956
-0.2276751	0.4283350
-0.9656525	0.4044323
-0.1912703	0.4299482
-0.0670724	0.4357570
-1.4556010	0.4043815
-0.4487288	0.4193643
1.8647652	0.6064321
-2.0108776	0.4369516
-0.4656598	0.4187347
0.3907304	0.4617523
-0.7555460	0.4093053
0.5005206	0.4691928
0.8450916	0.4959508
0.5975921	0.4761973
-2.0043483	0.4363263
1.6127370	0.5751794
-0.8121098	0.4078025
1.1665388	0.5258509
0.2174075	0.4509978
0.2109403	0.4506190
0.6762301	0.4821739
0.2038592	0.4502061
0.5950215	0.4760065
-0.4809225	0.4181741
1.0879673	0.5180927
-0.0644503	0.4358849
1.5355804	0.5660497
-1.2076179	0.4019019
-0.6747426	0.4116598
0.6775841	0.4822792
0.6263237	0.4783492
-0.6631872	0.4120151
-0.7651699	0.4090408
-0.4521263	0.4192373
-0.6467249	0.4125289
-1.6132087	0.4094838
-0.1896067	0.4300229
0.1904892	0.4494317
-0.0067567	0.4387553
-0.6050716	0.4138681
0.7045094	0.4843906
1.4258341	0.5534735
-0.0421701	0.4369804
-0.7073285	0.4106824
-0.4118513	0.4207640
0.8104899	0.4930212
-1.1344977	0.4022347
-1.2047708	0.4019069
0.8011284	0.4922381
-0.6700330	0.4118041
-0.4717781	0.4185091
1.1713430	0.5263347
-0.2777537	0.4261797
1.1419030	0.5233871
-0.9422446	0.4048714
-0.9443518	0.4048307
-0.6496210	0.4124379
-1.2152694	0.4018918
-0.4058662	0.4209948
-0.1756397	0.4306531
0.9118378	0.5017594
-1.1797634	0.4019790
-0.2607089	0.4269051
0.0619087	0.4423168
-0.1897160	0.4300180
0.0508038	0.4417299
0.8351684	0.4951050
0.3407278	0.4585278
0.9259787	0.5030168
-0.3093598	0.4248570
0.1006116	0.4443960
-2.0197944	0.4378148
0.4679191	0.4669308
0.3550228	0.4594393
0.5038511	0.4694264
-0.6471360	0.4125160
1.6140458	0.5753362
0.6766903	0.4822097
-0.0749007	0.4353766
1.0292485	0.5124847
-0.1930819	0.4298670
-0.5297928	0.4164245
1.5403245	0.5666044
-0.0970199	0.4343122
1.4873754	0.5604645
-1.3104088	0.4021967
-0.8582647	0.4066739
0.0535131	0.4418727
0.5146104	0.4701844
-0.3202069	0.4244097
-0.3624904	0.4226980
-0.9495262	0.4047317
0.6828207	0.4826873
1.2137204	0.5306487
-1.8936095	0.4265995
1.3142407	0.5412132
-1.6127856	0.4094659
0.8348914	0.4950814
0.3965380	0.4621334
-0.4879938	0.4179166
-0.4900600	0.4178417
0.0279991	0.4405379
0.4545158	0.4660138
0.3332788	0.4580560
-1.5369144	0.4066338
-1.7101494	0.4141939
1.4953128	0.5613777
1.2961674	0.5392797
0.2460091	0.4526918
0.4868999	0.4682423
0.6902976	0.4832721
-1.9545785	0.4317482
0.9161633	0.5021430
-0.4051911	0.4210209
-0.1777056	0.4305595
-0.3594553	0.4228191
0.7858543	0.4909691
-0.5885482	0.4144144
-0.2812321	0.4260327
-0.5206372	0.4167470
-0.6434638	0.4126317
0.7962974	0.4918356
0.1391418	0.4465188
0.9473836	0.5049379
0.4817155	0.4678826
0.6443285	0.4797162
0.9631909	0.5063705
0.2816193	0.4548448
-0.7798530	0.4086440
0.0544738	0.4419234
0.2591767	0.4534822
0.4069826	0.4628222
0.9715011	0.5071284
-0.5682524	0.4150970
0.5092157	0.4698037
0.0219928	0.4402269
0.0658427	0.4425258
-0.0927373	0.4345170
1.4766442	0.5592341
-0.3083404	0.4248992
0.4043152	0.4626458
0.4135397	0.4632569
0.7399940	0.4872232
-0.9260073	0.4051929
2.0653673	0.6325009
0.0269152	0.4404817
-0.5050531	0.4173015
1.4480029	0.5559735
0.1838848	0.4490516
-0.7082861	0.4106543
-0.1795709	0.4304751
-1.5281728	0.4063536
0.2950467	0.4556694
-0.3381406	0.4236774
-0.0266536	0.4377530
1.0312975	0.5126777
-0.6618000	0.4120581
-1.1400221	0.4021952
0.4657276	0.4667804
1.4873754	0.5604645
-1.6178220	0.4096799
-0.1467968	0.4319733
-0.3725359	0.4222988
-1.0616874	0.4029582
0.3452891	0.4588178
-0.6825155	0.4114234
0.3371908	0.4583035
-0.1516848	0.4317478
0.1828847	0.4489942
0.2212545	0.4512238
-1.2025473	0.4019112
-1.2227940	0.4018868
0.0019357	0.4391973
0.6469751	0.4799184
-1.0344443	0.4033198
0.1468008	0.4469472
0.6108921	0.4771889
-0.3836854	0.4218591
-0.3893700	0.4216363
-0.2721542	0.4264171
0.9103537	0.5016280
0.3858694	0.4614344
0.9023998	0.5009254
0.2163378	0.4509350
1.0937861	0.5186573
-0.5406593	0.4160449
0.8307789	0.4947323
0.2830367	0.4549315
-1.7858693	0.4187531
-0.1491054	0.4318667
1.8647652	0.6064321
-0.0314785	0.4375119
0.4024231	0.4625209
0.5875362	0.4754527
-1.1998575	0.4019170
-0.6431679	0.4126411

EAP cikti tablosunda yer alan degerler, MTK cercevesinde Empirical Bayes (Beklenen a Posteriori - EAP) yontemi ile elde edilen bireysel yetenek tahminlerini ve bu tahminlere ait standart hatalari gostermektedir.
“F” sutunu, her bireyin yanit desenine gore hesaplanan gizil ozellik (latent trait) yani yetenek degerini yansitir.
Bu degerler, bireylerin testte olcmek istenen yetenek boyutundaki konumunu belirtir; POZITIF degerler ortalamanin uzerinde, NEGATIF degerler ise ortalamanin altinda bir yetenek duzeyine isaret eder.
“SE_F” sutunu ise bu tahminlere ait standart hatayi ifade eder ve tahminin ne kadar guvenilir oldugunu gosterir.
DUSUK standart hata degerleri daha kesin tahminleri, YUKSEK hata degerleri ise tahmindeki belirsizligi ifade eder. ben
EAP yontemi, posterior dagilimin matematiksel beklenen degerini temel alir ve genellikle dagilim uzerinde daha dengeli ve istikrarli tahminler sunar.
Bu nedenle, ozellikle aykiri veya uclarda yer alan yanitlar icin daha kararli sonuclar uretme kapasitesine sahiptir ve uygulamalarda siklikla tercih edilmektedir.

1.1.0.1 3PL icin:

ML_ucPL <- fscores(ucpl_uyum, method = "ML", full.scores.SE = T)
MAP_ucPL <- fscores(ucpl_uyum, method = "MAP", full.scores.SE = T)
EAP_ucPL <- fscores(ucpl_uyum, method = "EAP", full.scores.SE = T)

head(ML_ucPL)

##               F      SE_F
## [1,] -0.3770755 0.5123649
## [2,] -1.2147417 0.5456406
## [3,] -1.8282694 0.5968482
## [4,]  2.6796206 1.0387485
## [5,]  1.2280019 0.4738721
## [6,] -1.3612614 0.4866118

library(kableExtra)
library(knitr)
ML_df_uc <- as.data.frame(ML_ucPL)
colnames(ML_df_uc) <- c("F", "SE_F") 
kable(ML_df_uc, format = "html", caption = "Maximum Likelihood Yontemine Gore Birey Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

Maximum Likelihood Yontemine Gore Birey Yetenek Tahminleri
F	SE_F
-0.3770755	0.5123649
-1.2147417	0.5456406
-1.8282694	0.5968482
2.6796206	1.0387485
1.2280019	0.4738721
-1.3612614	0.4866118
-0.6555437	0.4889141
-0.2799139	0.4496241
0.3409932	0.4149045
-0.3750438	0.4531702
0.1394424	0.4445953
-0.1789364	0.4440432
-0.3292871	0.4678085
3.3220411	1.4966641
-0.4128241	0.4481628
1.7202307	0.5836619
-1.6795522	0.5216634
0.2183314	0.4479479
0.4216527	0.4176013
0.7296718	0.4429402
-0.0538106	0.5044072
-2.7990450	1.2246530
-0.6734089	0.4634462
-0.6443491	0.5144540
-1.1165729	0.5699324
-0.0641205	0.4635678
0.4065190	0.4180020
0.1656673	0.4438176
1.2600237	0.4899720
1.5192351	0.5198609
-0.7567078	0.4833111
-Inf	NA
0.8232443	0.4680634
-0.8943256	0.5473154
-0.2466863	0.4457056
0.9107721	0.4436044
-0.3477022	0.4436640
-1.0936691	0.4841902
-0.1052631	0.4329285
-1.0694907	0.5341136
0.3022314	0.4139107
0.6117400	0.4682148
-0.0267931	0.5100369
1.0658662	0.4589039
-0.7863212	0.4866781
0.3461441	0.4082617
3.3220411	1.4966641
0.6616317	0.4230019
-0.0199122	0.4207094
0.4347315	0.4408193
0.1911612	0.4355420
1.5972665	0.5428813
-0.7981378	0.5556315
-0.4491153	0.5242656
3.1696932	1.3716605
-0.3355105	0.4732574
2.4737598	0.9202974
2.0559317	0.6934710
0.0846903	0.4172295
-0.5719911	0.4382624
0.2044022	0.4090342
0.6346584	0.4407773
-0.0655983	0.4132419
0.4203498	0.4483356
-0.2897045	0.4423174
-0.1332583	0.4640872
0.3036569	0.4153018
-2.0203881	0.6364744
-0.1708871	0.4489465
0.6601760	0.4363867
-0.6555437	0.4889141
-0.4227848	0.5741925
-2.5691273	0.8236223
0.0143753	0.5079897
0.1673394	0.4795441
-0.3166501	0.4470923
-0.5872130	0.6089240
-1.3338457	0.6320254
-1.4408638	0.4711599
-0.4119555	0.4720770
-0.1873632	0.4184542
0.0434032	0.4428730
-0.8291447	0.5090718
-1.1982893	0.6063760
2.5626148	0.9771152
-0.1755723	0.4402836
-0.0432119	0.4373253
0.2050214	0.4337815
-1.3681641	0.4587231
0.1062217	0.4659606
0.9562366	0.4399416
0.5404370	0.4140627
0.0129689	0.4484341
0.3447325	0.4249518
-1.8074372	0.5024227
1.2809689	0.4883943
-1.4745508	0.5049772
1.1909491	0.4748525
2.2392644	0.7767633
-0.3356482	0.5159638
1.5178889	0.5250323
-0.5212486	0.6218512
-3.1032099	1.7303332
-1.6136721	0.5386048
-1.9763845	0.5618424
-0.7498389	0.5356224
-1.4276277	0.5054436
-0.1629850	0.5133381
2.1476923	0.7467378
-1.3880816	0.4684809
0.7559622	0.4409140
1.0487244	0.4458722
0.8356002	0.4711302
-1.3712110	0.4984154
-0.4816520	0.5013504
-0.0773302	0.4257844
1.1050016	0.4721153
-1.1889987	0.4995509
1.2297552	0.4827303
-1.0581574	0.5772229
0.9852618	0.4661418
-2.0048800	0.5609342
-1.2009523	0.6753838
Inf	NA
-0.5443616	0.4521159
0.1524495	0.4498924
0.8536386	0.4325879
-0.4032812	0.5573842
0.6704002	0.4336991
-0.0492973	0.4483122
-0.1514118	0.4750092
-0.2992695	0.5052504
-0.1573481	0.4705347
2.4737598	0.9202974
-0.8779699	0.5060148
-1.5757503	0.5309842
0.4801762	0.4383551
0.4186240	0.5204096
0.1198014	0.4466687
-0.3860709	0.4874031
-1.7051193	0.5134359
3.1696932	1.3716605
0.1308541	0.4437610
-0.3151427	0.4648228
0.2500452	0.4294204
-0.2749615	0.4753620
2.0577953	0.6994910
0.1991665	0.4721010
0.9184040	0.4454912
-0.2935295	0.4721555
-0.0610084	0.4447578
-0.7205079	0.4417726
-0.7351705	0.6753222
-0.7887386	0.5259011
-1.7039885	0.5548899
1.6015945	0.5478808
1.2581025	0.4969521
-1.4014598	0.5520512
-0.2326244	0.5502733
-0.8318705	0.4623428
Inf	NA
2.6902777	1.0243640
0.2504177	0.4302148
0.5523461	0.4394996
-0.2287781	0.4817920
-0.0566487	0.4041227
-0.8684216	0.4901585
-0.1204282	0.4129589
1.8346444	0.6159931
1.0445292	0.4522706
0.6320267	0.4173617
0.8935502	0.4434993
1.4167222	0.5069203
-0.3937966	0.4433885
-1.3754891	0.4850366
1.3875263	0.5024847
-0.4764886	0.4861793
-1.0330248	0.5190076
-1.4390609	0.5145608
-0.5186657	0.4115190
0.2050214	0.4337815
0.5467602	0.4124940
0.5059000	0.4312100
0.7051909	0.4593510
-0.7132153	0.5748107
-0.1306295	0.4823994
-1.7253101	0.5639970
-1.1330951	0.5165112
1.0231925	0.4489227
-1.4471625	0.4497681
3.3220411	1.4966641
0.2919699	0.4080815
1.0887320	0.4628529
-0.9646252	0.4696609
-1.0011772	0.5135683
-1.7122034	0.5757332
-0.7218380	0.6256105
0.5889789	0.4388905
0.4244812	0.4152074
-1.2529718	0.5970322
-1.5472880	0.4688043
-1.1471102	0.5226154
2.0889236	0.7152624
-0.4809183	0.4545068
-0.8514384	0.5542473
1.3727422	0.4994025
0.0014306	0.4776162
-1.5940509	0.4874664
-0.5794817	0.4745999
0.7059094	0.4292292
-0.4873632	0.7254352
-0.5253430	0.5280885
-0.3207924	0.5442604
-0.7830216	0.4754056
Inf	NA
-0.9680527	0.5566152
1.1640244	0.4619554
1.4729243	0.5171312
-0.3204454	0.4584214
-1.6359251	0.5239067
-0.2354989	0.4373628
-0.1781897	0.4288504
-0.4094135	0.4281065
1.0089202	0.4658462
2.1476923	0.7467378
1.4594809	0.5163469
0.3303332	0.4461086
-1.1186376	0.4694419
-0.2977732	0.4254059
0.2169524	0.4198486
0.9354225	0.4503777
0.9366624	0.4555874
0.4462858	0.4545886
-0.8488440	0.4335760
0.4379312	0.4060807
-1.0886303	0.5119541
-0.8177590	0.5796391
-0.1937770	0.4323306
0.5999279	0.4345501
0.0194219	0.4477393
-3.6051327	1.4869235
0.5440259	0.4246852
0.8629899	0.4263177
0.0640831	0.4293795
-0.1747393	0.4593270
0.1695082	0.4096332
-0.8092270	0.5569556
2.0559317	0.6934710
0.1082051	0.4494837
-0.1332583	0.4640872
0.4308809	0.4644152
1.4850094	0.5228363
1.5066344	0.5242341
-0.2893433	0.4520693
1.1419342	0.4705974
0.0573734	0.4682600
Inf	NA
-1.9387599	0.5662570
0.1918711	0.4319089
0.2314806	0.4246983
-0.7234474	0.5388860
-1.6800723	0.5491057
0.2491025	0.4448527
-1.1999572	0.5191130
-0.5407734	0.4889540
0.3640420	0.4107938
-1.5603966	0.5160633
0.1481933	0.4519063
1.6015945	0.5478808
1.5002467	0.5226369
0.8744803	0.4402989
0.1671011	0.4552772
-Inf	NA
-0.5393488	0.4961241
0.0067506	0.4369308
0.0511167	0.4195943
-0.2085620	0.4764302
-1.7865355	0.5533748
2.4006349	0.8678486
0.7445882	0.4285435
1.0013169	0.4483436
0.9590064	0.4651487
-1.7944711	0.5349472
0.3898354	0.4459276
0.0833056	0.4509134
0.8473737	0.4436258
-0.2483643	0.4636669
1.4459418	0.5128689
1.1388867	0.4687810
0.0447955	0.4375308
0.8047341	0.4439418
-1.3508800	0.5275914
0.1187376	0.4955757
1.2561268	0.4795963
0.5549802	0.4445210
-2.1965722	0.5878547
-2.0246033	0.6336604
-0.0027421	0.4193328
1.8571922	0.6208468
-0.7652598	0.4695135
1.1203311	0.4664277
0.3686552	0.4250974
1.9350416	0.6571914
-0.2722335	0.4537295
-0.1783460	0.4396355
-1.3681326	0.5419201
-0.1262270	0.4303214
-0.1021715	0.4052098
-1.7579530	0.5135994
-0.4369135	0.4922779
3.1696932	1.3716605
-3.0228831	0.9199477
-0.5007767	0.4940972
0.3983168	0.4091858
-0.7500978	0.4445520
0.6697976	0.4308107
0.8970819	0.4345400
0.8225514	0.4452643
-Inf	NA
2.2205282	0.7807025
-0.9018266	0.5211488
1.2545465	0.4684367
0.3702096	0.4380080
0.2746180	0.4320076
0.7366019	0.4255989
0.2235103	0.3994844
0.7688727	0.4334940
-0.4762027	0.4903130
1.2670918	0.4774845
-0.0112936	0.4092862
1.7859325	0.6029301
-1.6403652	0.5119150
-0.6920142	0.4932452
0.8722945	0.4492763
0.7215360	0.4276815
-0.8078585	0.5728948
-0.7926463	0.5085440
-0.5084317	0.4923230
-0.6730720	0.4593208
-1.9077631	0.5609307
-0.0792197	0.4983011
0.3076461	0.4253933
0.0233164	0.4304928
-0.5831206	0.4241870
0.8820797	0.4425002
1.9051635	0.6465036
0.1096899	0.4624580
-0.7577900	0.5021219
-0.3553263	0.4446617
0.9717149	0.4526013
-1.3961492	0.5507914
-1.5082098	0.5183502
0.8888096	0.4434405
-0.8609292	0.5286264
-0.4728806	0.4969466
1.3750769	0.4996511
-0.2489786	0.4580907
1.3324079	0.4915904
-1.3499808	0.5767314
-1.1674652	0.5312197
-0.6215205	0.4735139
-1.9244495	0.6499746
-0.3927528	0.5339337
-0.1799611	0.3955244
1.0269338	0.4511648
-1.3633210	0.5025692
-0.1825863	0.4506813
0.1625150	0.4551526
-0.3823878	0.5144142
0.0967387	0.4132436
0.9441048	0.4428527
0.3713110	0.4083953
1.0887320	0.4628529
-0.2496995	0.5045470
0.2916170	0.4577844
-2.6244066	0.8474998
0.4554800	0.4206751
0.4811459	0.4238497
0.5611548	0.4326384
-0.6295941	0.4369576
1.9270624	0.6368312
0.8334299	0.4398548
-0.0356510	0.4952998
1.1851886	0.4655422
-0.4341929	0.6569045
-0.5581453	0.5087389
1.7714020	0.5954156
0.1052036	0.5111879
2.1199850	0.7397966
-1.4732222	0.4939475
-0.9503540	0.5124349
0.0403440	0.4882157
0.6530249	0.4274648
-0.3302920	0.4192048
-0.7103253	0.5590285
-0.9760218	0.4674539
0.8274753	0.4384998
1.5764504	0.5478874
-2.7395809	1.1103942
1.5066344	0.5242341
-1.9280365	0.5717467
1.0732551	0.4649936
0.6888310	0.4857315
-0.5698705	0.4859902
-0.5420681	0.4716044
0.1619005	0.4503586
0.5832207	0.4565092
0.5124441	0.4374788
-1.9453920	0.5778548
-1.9814163	0.5322683
1.9853602	0.6793011
1.5631800	0.5347959
0.4549725	0.4481857
0.5769875	0.4120040
0.8348187	0.4526859
-2.4488335	0.6329320
0.9797699	0.4419179
-0.4111015	0.5381113
-0.1260068	0.4253982
-0.4148113	0.7485894
0.9706043	0.4574937
-0.6030412	0.4498818
-0.2092595	0.4484217
-0.4642618	0.4441826
-0.6930459	0.4395474
0.9675173	0.4480912
0.2295125	0.4235066
1.0875856	0.4553763
0.4897736	0.4414065
0.6973212	0.4317951
1.2825013	0.4902137
0.3645073	0.4242654
-1.1838654	0.5844084
-0.0434734	0.5088883
0.4508578	0.4428921
0.6247394	0.4459478
1.0054210	0.4390965
-0.5347772	0.4798192
0.6494874	0.4315825
-0.0027421	0.4193328
0.1016241	0.4075567
0.0375193	0.4748182
1.9350416	0.6571914
-0.2372791	0.4301147
0.5742633	0.4351381
0.4704958	0.4231023
0.9331387	0.4477919
-1.0380452	0.5099381
Inf	NA
0.1969503	0.4699696
-1.2153228	0.5917811
1.8408838	0.6170877
0.3445885	0.4456145
-0.9406923	0.5450553
-0.2084097	0.4265643
-1.5608920	0.4591537
0.4353510	0.4320873
-0.3365268	0.4762332
0.1747182	0.4868878
1.1471217	0.4569477
-0.6662317	0.5038364
-1.2327640	0.4911961
0.4979506	0.4299515
2.1199850	0.7397966
-1.9054102	0.5498189
-0.0825989	0.4563414
-0.3119718	0.4500667
-1.8489766	0.6061136
0.3938543	0.4216045
-0.8248727	0.5399789
0.5018644	0.4669646
-0.0880633	0.4684778
0.3036650	0.4195917
0.3746225	0.4260874
-1.4545335	0.5063760
-1.5860075	0.5595243
0.1717172	0.4643257
0.9095571	0.4570915
-1.1769114	0.4993271
0.2471794	0.4115057
0.8239787	0.4493548
-0.3168860	0.4543326
-0.4557786	0.5334467
-0.2253419	0.6166746
1.1288591	0.4726371
0.5373705	0.4422218
1.1217269	0.4727972
0.3585914	0.4226621
1.2058282	0.4709653
-0.6111618	0.4926862
1.0362440	0.4680261
0.4703319	0.4341649
-2.2562492	0.6651932
-0.3029517	0.6157659
3.1696932	1.3716605
0.0024824	0.4150840
0.5271245	0.4248111
0.5871050	0.4437015
-1.3790643	0.4905292
-0.8202978	0.5907477

ML_ucPL tablosunda yer alan degerler, MTK cercevesinde Maximum Likelihood (ML) yontemiyle elde edilen bireysel yetenek tahminlerini ve bu tahminlere ait standart hatalari temsil etmektedir.
“F” sutunu, bireylerin testte olculen gizil ozellige (latent trait) dair tahmin edilen yetenek degerlerini gosterir.
Bu degerler, testte verilen yanitlara gore bireylerin yetenek seviyelerini belirtir; POZITIF degerler ortalamanin UZERINDE yetenegi, NEGATIF degerler ise ortalamanin ALTINDA bir performansi yansitir.
“SE_F” sutunu ise bu tahminlere ait standart hatalari ifade eder; bu degerler tahminin guvenilirligi hakkinda bilgi sunar. Dusuk standart hata degerleri daha kesin tahminleri, yuksek degerler ise tahmindeki belirsizligi yansitir.
ML yontemi, bireyin gozlemlenen yanit desenine en iyi uyan yetenek seviyesini secerek tahminleme yapar.
Bu yontem, prior bilgiye dayanmadan yalnizca veriye odaklandigi icin, verisi YETERLI olan bireylerde oldukca guclu ve dogrudur; ancak uc ya da az bilgilendirici yanit durumlarinda kararsizlik riski tasir.
Bu nedenle, ML yontemi standart test uygulamalarinda yaygin olarak kullanilirken, ozellikle yeterli yanit bilgisine sahip bireyler icin daha anlamli sonuclar sunar.

head(MAP_ucPL)

##               F      SE_F
## [1,] -0.3008162 0.4435508
## [2,] -0.9395421 0.4710788
## [3,] -1.3787281 0.4868404
## [4,]  1.7099922 0.5054370
## [5,]  1.0084702 0.4192223
## [6,] -1.1095768 0.4228089

library(kableExtra)
library(knitr)
MAP_df_uc <- as.data.frame(MAP_ucPL)
colnames(MAP_df_uc) <- c("F", "SE_F")
kable(MAP_df_uc, format = "html", caption = "MAP Yontemine Gore Birey Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

MAP Yontemine Gore Birey Yetenek Tahminleri
F	SE_F
-0.3008162	0.4435508
-0.9395421	0.4710788
-1.3787281	0.4868404
1.7099922	0.5054370
1.0084702	0.4192223
-1.1095768	0.4228089
-0.5327859	0.4260285
-0.2335226	0.4041794
0.2905912	0.3858143
-0.3123821	0.4048111
0.1162480	0.4094581
-0.1497579	0.4018162
-0.2713659	0.4151738
1.8148169	0.5281624
-0.3448175	0.4028190
1.3344910	0.4507342
-1.3340213	0.4492327
0.1815022	0.4126862
0.3585983	0.3882529
0.6100694	0.4051260
-0.0429228	0.4493254
-1.7982221	0.5124349
-0.5569633	0.4111306
-0.5149138	0.4388020
-0.8473031	0.4841599
-0.0528265	0.4187823
0.3457203	0.3879512
0.1381726	0.4091506
1.0229370	0.4298093
1.2206034	0.4296368
-0.6171265	0.4233925
-2.4006819	0.5117880
0.6742613	0.4275495
-0.6926890	0.4679293
-0.2063231	0.4019402
0.7624574	0.4021223
-0.2914911	0.3986602
-0.8903248	0.4262602
-0.0886954	0.3961239
-0.8357258	0.4617184
0.2577365	0.3850317
0.5004100	0.4295822
-0.0212761	0.4531794
0.8837315	0.4105833
-0.6393392	0.4268509
0.2964949	0.3795726
1.8148169	0.5281624
0.5614478	0.3889534
-0.0169199	0.3875147
0.3634862	0.4064373
0.1605078	0.4016013
1.2658573	0.4393598
-0.6106036	0.4822505
-0.3553988	0.4493916
1.7987155	0.5237364
-0.2754135	0.4187882
1.6538973	0.4961531
1.5172787	0.4675084
0.0720977	0.3861362
-0.4819370	0.3922305
0.1749828	0.3802145
0.5311321	0.4047734
-0.0560505	0.3810918
0.3493486	0.4131044
-0.2429447	0.3990300
-0.1098724	0.4168811
0.2586839	0.3862365
-1.4864360	0.5038107
-0.1424915	0.4057349
0.5548680	0.3990784
-0.5327859	0.4260285
-0.3226181	0.4756326
-1.8470147	0.4658136
0.0114224	0.4535280
0.1357105	0.4371737
-0.2647070	0.4019247
-0.4369696	0.4906657
-0.9476665	0.5363259
-1.1869449	0.4138409
-0.3385725	0.4172930
-0.1596447	0.3832469
0.0362730	0.4056622
-0.6613747	0.4448798
-0.8475118	0.5630248
1.6741516	0.5016242
-0.1472645	0.4001243
-0.0362885	0.3998499
0.1723501	0.4004690
-1.1352948	0.4087850
0.0871500	0.4250977
0.8036329	0.3968172
0.4614528	0.3821971
0.0107962	0.4094003
0.2916942	0.3934905
-1.4675390	0.4222309
1.0421151	0.4270633
-1.1832446	0.4390481
0.9784633	0.4172816
1.5914470	0.4794818
-0.2670140	0.4460358
1.2133792	0.4353830
-0.3852040	0.4937250
-1.9329814	0.4861766
-1.2726591	0.4479027
-1.5522746	0.4449425
-0.5864958	0.4600831
-1.1402651	0.4465897
-0.1295114	0.4497316
1.5454428	0.4771153
-1.1429490	0.4166634
0.6328903	0.4039768
0.8780147	0.4004454
0.6826452	0.4303401
-1.1069839	0.4321555
-0.3872284	0.4372813
-0.0654976	0.3905792
0.9060827	0.4230009
-0.9538522	0.4415319
1.0034109	0.4252639
-0.7947567	0.4948487
0.8097085	0.4227386
-1.5757386	0.4455498
-0.8017094	0.5842130
1.9640865	0.5604599
-0.4542383	0.4018971
0.1266535	0.4124256
0.7204649	0.3933363
-0.3118552	0.4656261
0.5641326	0.3987297
-0.0410581	0.4084741
-0.1237844	0.4252648
-0.2399658	0.4394503
-0.1291534	0.4208681
1.6538973	0.4961531
-0.7051571	0.4353252
-1.2424477	0.4528480
0.4021294	0.4051122
0.3267355	0.4760708
0.0997761	0.4098553
-0.3138499	0.4270276
-1.3622540	0.4439264
1.7987155	0.5237364
0.1092353	0.4073160
-0.2600051	0.4150798
0.2108658	0.3971592
-0.2250844	0.4225403
1.5145888	0.4684847
0.1625571	0.4305940
0.7677622	0.4036280
-0.2410460	0.4188046
-0.0509698	0.4048870
-0.6056471	0.3944398
-0.5168766	0.5256944
-0.6215281	0.4543975
-1.3142246	0.4759684
1.2681253	0.4375210
1.0158513	0.4349932
-1.0816251	0.4719884
-0.1799937	0.4693588
-0.6895259	0.4075742
1.9640865	0.5604599
1.7263917	0.5034873
0.2109483	0.3989062
0.4623419	0.4052990
-0.1862776	0.4280257
-0.0487081	0.3741281
-0.7045429	0.4279968
-0.1029515	0.3802071
1.4022409	0.4561146
0.8702094	0.4057531
0.5387061	0.3835604
0.7479343	0.4024636
1.1423484	0.4313959
-0.3303035	0.3978033
-1.1185807	0.4283463
1.1240266	0.4254070
-0.3879112	0.4251460
-0.8188773	0.4492839
-1.1416506	0.4521606
-0.4447343	0.3745786
0.1723501	0.4004690
0.4671810	0.3819002
0.4264398	0.3968536
0.5807243	0.4234144
-0.5399644	0.4855634
-0.1062468	0.4296051
-1.3343770	0.4651524
-0.9008167	0.4459750
0.8544276	0.4034376
-1.2102531	0.3995584
1.8148169	0.5281624
0.2500300	0.3802522
0.9008143	0.4118309
-0.7948563	0.4135231
-0.8000396	0.4391522
-1.3171536	0.4656565
-0.5292714	0.5011136
0.4932405	0.4048192
0.3616921	0.3859118
-0.8961027	0.5550403
-1.2786865	0.4099766
-0.9023266	0.4598878
1.5263306	0.4710872
-0.4003813	0.4047034
-0.6555571	0.4727955
1.1109705	0.4295692
0.0011648	0.4310277
-1.2999317	0.4225269
-0.4750564	0.4202042
0.5963907	0.3936767
-0.3286746	0.5524914
-0.4138089	0.4541519
-0.2501840	0.4605623
-0.6420111	0.4189166
1.9640865	0.5604599
-0.7419845	0.4780060
0.9659459	0.4067999
1.1846968	0.4290601
-0.2657308	0.4097633
-1.2747180	0.4830118
-0.1981032	0.3962937
-0.1506543	0.3920491
-0.3469940	0.3874305
0.8296990	0.4217143
1.5454428	0.4771153
1.1736943	0.4298089
0.2749346	0.4117885
-0.9214611	0.4143776
-0.2525534	0.3879362
0.1841951	0.3900793
0.7798219	0.4055537
0.7772650	0.4110926
0.3689541	0.4190138
-0.7173226	0.3893619
0.3758652	0.3768122
-0.8655873	0.4483202
-0.6184092	0.4839945
-0.1635285	0.3933741
0.5046168	0.3988685
0.0161786	0.4088824
-2.1216172	0.5064869
0.4608195	0.3914684
0.7317886	0.3880497
0.0540825	0.3955134
-0.1446441	0.4126297
0.1450501	0.3806565
-0.6260624	0.4640192
1.5172787	0.4675084
0.0899453	0.4116113
-0.1098724	0.4168811
0.3537334	0.4252191
1.1881458	0.4350234
1.2074356	0.4310423
-0.2409647	0.4059394
0.9387376	0.4192736
0.0470169	0.4256384
1.9640865	0.5604599
-1.5073474	0.4591273
0.1614603	0.3997699
0.1959683	0.3924959
-0.5631525	0.4667425
-1.2964508	0.4802479
0.2075501	0.4104827
-0.9483863	0.4537604
-0.4374582	0.4348488
0.3113212	0.3812210
-1.2349802	0.4571182
0.1228810	0.4147847
1.2681253	0.4375210
1.2009372	0.4344040
0.7336765	0.4000353
0.1381546	0.4180971
-2.2113999	0.5238460
-0.4353262	0.4337478
0.0056679	0.4005156
0.0434486	0.3876906
-0.1704384	0.4249845
-1.4018906	0.4501979
1.6423057	0.4891876
0.6293482	0.3932614
0.8362448	0.4036283
0.7884385	0.4226582
-1.4155775	0.4510509
0.3245149	0.4116234
0.0691603	0.4130683
0.7087301	0.4039753
-0.2051532	0.4136689
1.1652217	0.4282984
0.9374673	0.4180081
0.0375861	0.4014843
0.6728337	0.4043858
-1.0521212	0.4749805
0.0951870	0.4466669
1.0302653	0.4173975
0.4627693	0.4093767
-1.7002823	0.4548410
-1.5093872	0.4879115
-0.0023321	0.3866802
1.4180980	0.4549651
-0.6307873	0.4132202
0.9248200	0.4138957
0.3119644	0.3931862
1.4501561	0.4639156
-0.2265194	0.4064496
-0.1497167	0.3988846
-1.0464249	0.4947190
-0.1066121	0.3931525
-0.0877954	0.3746325
-1.4119807	0.4341449
-0.3540786	0.4293039
1.7987155	0.5237364
-2.0645766	0.4835612
-0.4048347	0.4323610
0.3411113	0.3793366
-0.6293548	0.3961799
0.5646973	0.3969397
0.7563870	0.3939667
0.6867889	0.4062651
-2.3098824	0.5258246
1.5740572	0.4814730
-0.7147435	0.4476581
1.0374685	0.4093293
0.3100465	0.4052178
0.2312079	0.3986584
0.6242928	0.3896544
0.1926391	0.3723460
0.6474247	0.3975474
-0.3869154	0.4259405
1.0409953	0.4157285
-0.0096738	0.3786585
1.3766614	0.4497214
-1.3041809	0.4530769
-0.5606150	0.4288289
0.7267749	0.4075904
0.6104765	0.3917504
-0.6152004	0.4780773
-0.6354107	0.4369308
-0.4118741	0.4297507
-0.5590785	0.4056884
-1.5049560	0.4386143
-0.0635708	0.4428994
0.2601534	0.3944700
0.0196670	0.3958001
-0.4958116	0.3834169
0.7385632	0.4025950
1.4344449	0.4625682
0.0902338	0.4225724
-0.6101214	0.4335834
-0.2976893	0.3992506
0.8085420	0.4079016
-1.0864230	0.4594379
-1.2010603	0.4439301
0.7431646	0.4048186
-0.6756384	0.4590771
-0.3816138	0.4336053
1.1155685	0.4245866
-0.2063166	0.4114374
1.0857347	0.4216938
-1.0096871	0.5020013
-0.9120711	0.4645834
-0.5108913	0.4157416
-1.4126864	0.4982312
-0.3088593	0.4534744
-0.1557269	0.3661228
0.8561908	0.4050150
-1.0905996	0.4453164
-0.1520969	0.4064243
0.1344487	0.4169064
-0.3032600	0.4523081
0.0825938	0.3828447
0.7915584	0.3995627
0.3180771	0.3792892
0.9008143	0.4118309
-0.2001628	0.4404639
0.2404758	0.4214625
-1.8792081	0.4638136
0.3866640	0.3898005
0.4075339	0.3922607
0.4726784	0.3972856
-0.5308983	0.3915008
1.4620612	0.4560547
0.6990259	0.4009764
-0.0286458	0.4426884
0.9802994	0.4108180
-0.3072060	0.5316373
-0.4465904	0.4405292
1.3696701	0.4492460
0.0831737	0.4600639
1.5289247	0.4774650
-1.1940273	0.4284710
-0.7575032	0.4437837
0.0325644	0.4395068
0.5519932	0.3939203
-0.2814542	0.3824699
-0.5435218	0.4808089
-0.8063534	0.4102358
0.6947096	0.3999757
1.2436289	0.4440407
-1.8575258	0.4737206
1.2074356	0.4310423
-1.5108474	0.4435699
0.8860093	0.4147229
0.5546946	0.4465055
-0.4636220	0.4264811
-0.4455701	0.4173209
0.1343872	0.4138662
0.4814885	0.4204281
0.4295091	0.4040753
-1.5115882	0.4528147
-1.5824335	0.4332604
1.4722722	0.4679709
1.2442585	0.4367808
0.3780976	0.4134072
0.4931321	0.3817490
0.6924731	0.4140933
-1.8783639	0.4465381
0.8224532	0.3977873
-0.3218748	0.4580843
-0.1068012	0.3893564
-0.2785338	0.5482746
0.8029264	0.4161412
-0.5040656	0.4000286
-0.1746532	0.4041958
-0.3893043	0.3977230
-0.5824734	0.3963787
0.8075922	0.4048596
0.1943423	0.3929963
0.9048021	0.4063986
0.4093614	0.4067612
0.5880637	0.3955157
1.0416030	0.4290165
0.3085951	0.3928545
-0.8680994	0.5256297
-0.0345550	0.4523026
0.3762045	0.4091414
0.5205097	0.4100301
0.8459565	0.3950640
-0.4377360	0.4194298
0.5473782	0.3970690
-0.0023321	0.3866802
0.0871153	0.3783091
0.0305961	0.4301828
1.4501561	0.4639156
-0.2006305	0.3909203
0.4823095	0.4016333
0.3988444	0.3909847
0.7787998	0.4052981
-0.8280864	0.4439358
1.9640865	0.5604599
0.1609274	0.4300839
-0.8799766	0.5435303
1.4073670	0.4551237
0.2868750	0.4116463
-0.7325829	0.4610911
-0.1765187	0.3900043
-1.2987408	0.4035177
0.3663366	0.3998730
-0.2756536	0.4207541
0.1407902	0.4436514
0.9542303	0.4056044
-0.5363088	0.4331059
-1.0014529	0.4252979
0.4198829	0.3971296
1.5289247	0.4774650
-1.5054504	0.4410358
-0.0684366	0.4129787
-0.2602782	0.4037695
-1.3813969	0.5014765
0.3342178	0.3898722
-0.6426850	0.4634311
0.4106230	0.4300664
-0.0722945	0.4220720
0.2578735	0.3898361
0.3165580	0.3955725
-1.1585858	0.4507736
-1.2205413	0.4744762
0.1409613	0.4253317
0.7533924	0.4136472
-0.9491661	0.4326057
0.2111484	0.3831035
0.6859720	0.4091434
-0.2636414	0.4062523
-0.3578019	0.4564568
-0.1655119	0.5060026
0.9258038	0.4223609
0.4489067	0.4073415
0.9202164	0.4216304
0.3037952	0.3926526
0.9930514	0.4156530
-0.4946702	0.4308207
0.8510283	0.4228107
0.3951089	0.4018494
-1.7216786	0.4448958
-0.2225511	0.5061537
1.7987155	0.5237364
0.0021174	0.3833977
0.4462101	0.3928450
0.4900046	0.4080410
-1.1199551	0.4269563
-0.6159781	0.4874288

MAP_df_uc tablosunda yer alan degerler, MTK cercevesinde Maximum A Posteriori (MAP) yontemi kullanilarak elde edilen bireysel yetenek tahminlerini ve bunlara ait standart hatalari temsil eder.
“F” sutunu, her bireyin testte olculmek istenen gizil ozellige (latent trait) dair tahmin edilen yetenek seviyesini gosterir.
POZITIF degerler, bireyin ortalama UZERINDE bir yetenek seviyesine sahip oldugunu; NEGATIF degerler ise ortalamanin ALTINDA kaldigini ifade eder.
“SE_F” sutunu ise bu tahminlerin guvenilirligini yansitan standart hata degerlerini sunar.
Dusuk SE_F degerleri daha kesin ve guvenilir tahminleri, YUKSEK degerler ise tahmindeki BELIRSIZLIGI gosterir.
MAP yontemi, bireyin gozlemlenen yanitlarindan elde edilen olasilik dagilimi ile secilen prior dagilimi birlestirerek, posterior dagilimin maksimum noktasini baz alir.
Bu nedenle, MAP tahminleri hem veri hem de on bilgi dikkate alinarak hesaplanir ve bu da ozellikle ekstrem yanit desenlerine sahip bireyler icin daha dengeli ve kararli tahminler saglar.

head(EAP_ucPL)

##               F      SE_F
## [1,] -0.3588791 0.4514842
## [2,] -0.9685984 0.4602327
## [3,] -1.4052046 0.4956996
## [4,]  1.7960947 0.5336935
## [5,]  1.0155335 0.4400293
## [6,] -1.1389864 0.4263391

library(kableExtra)
library(knitr)
EAP_df_uc <- as.data.frame(EAP_ucPL)
colnames(EAP_df_uc) <- c("F", "SE_F")
kable(EAP_df_uc, format = "html", caption = "EAP Yontemiyle Hesaplanan Yetenek Tahminleri") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F, font_size = 16)

EAP Yontemiyle Hesaplanan Yetenek Tahminleri
F	SE_F
-0.3588791	0.4514842
-0.9685984	0.4602327
-1.4052046	0.4956996
1.7960947	0.5336935
1.0155335	0.4400293
-1.1389864	0.4263391
-0.5719201	0.4232792
-0.2771617	0.4144571
0.2631932	0.4064685
-0.3550541	0.4119479
0.0497671	0.4455764
-0.1998949	0.4183795
-0.3234626	0.4272639
1.9130475	0.5554722
-0.3762652	0.4076915
1.3750512	0.4770087
-1.3483993	0.4624267
0.1329849	0.4375634
0.3289347	0.4131175
0.5997557	0.4240579
-0.0796070	0.4552199
-1.8446049	0.5558610
-0.5869997	0.4116200
-0.5667021	0.4375938
-0.8824714	0.4698335
-0.1095418	0.4357284
0.3225085	0.4083892
0.0764058	0.4435236
1.0294812	0.4527740
1.2527382	0.4511555
-0.6503938	0.4186721
-2.4956845	0.5352748
0.6549951	0.4477699
-0.7268569	0.4561510
-0.2498737	0.4125807
0.7643636	0.4183601
-0.3325452	0.4064124
-0.9146551	0.4250454
-0.1131505	0.4047754
-0.8661956	0.4481322
0.2284559	0.4058331
0.4708271	0.4497568
-0.1044898	0.4785897
0.8912172	0.4277876
-0.6692246	0.4252217
0.2789143	0.3956972
1.9130475	0.5554722
0.5554447	0.4055264
-0.0513287	0.4029348
0.3412895	0.4214791
0.1265474	0.4210441
1.2984361	0.4636121
-0.6319550	0.4658122
-0.4116543	0.4531283
1.8947211	0.5510777
-0.3278416	0.4299892
1.7338222	0.5248682
1.5788014	0.4945168
0.0391065	0.4023554
-0.5167167	0.3961467
0.1496739	0.3976524
0.5149876	0.4248823
-0.0866212	0.3936416
0.3140842	0.4391837
-0.2821321	0.4072816
-0.1710475	0.4332455
0.2285482	0.4072128
-1.5053974	0.5160734
-0.1883204	0.4172877
0.5518749	0.4127666
-0.5719201	0.4232792
-0.3946842	0.4721454
-1.9148933	0.4964093
-0.0784558	0.4869322
0.0634470	0.4716253
-0.3048010	0.4096289
-0.5075468	0.4782719
-0.9834625	0.5110001
-1.2110099	0.4164527
-0.3827409	0.4225778
-0.1932931	0.3933577
-0.0004138	0.4188202
-0.6903417	0.4332546
-0.8430245	0.5217289
1.7565540	0.5306144
-0.1831568	0.4112588
-0.0844858	0.4213235
0.1393651	0.4180721
-1.1518374	0.4136846
0.0268228	0.4500365
0.8131198	0.4112292
0.4567294	0.3962619
-0.0296327	0.4260380
0.2670639	0.4122447
-1.4968964	0.4335803
1.0520422	0.4488624
-1.2058522	0.4420879
0.9943452	0.4347764
1.6616994	0.5069652
-0.3318770	0.4568467
1.2397912	0.4591205
-0.4723011	0.4901525
-2.0106814	0.5233372
-1.3105882	0.4569734
-1.5882007	0.4619966
-0.6225747	0.4474915
-1.1575871	0.4445312
-0.1998476	0.4610714
1.6116553	0.5053854
-1.1598566	0.4188437
0.6199697	0.4247335
0.8882519	0.4153823
0.6538870	0.4611007
-1.1355724	0.4346835
-0.4268867	0.4352899
-0.1014791	0.4047445
0.9046937	0.4441305
-0.9749961	0.4339315
1.0108428	0.4466025
-0.8266302	0.4751982
0.7986683	0.4448062
-1.6052340	0.4660283
-0.8382073	0.5335427
2.0769142	0.5842827
-0.4928751	0.4069175
0.0925499	0.4259699
0.7245746	0.4082559
-0.3881734	0.4676068
0.5508477	0.4180900
-0.0693019	0.4172652
-0.1741900	0.4391914
-0.3025535	0.4448865
-0.1881496	0.4343949
1.7338222	0.5248682
-0.7437000	0.4322390
-1.2682693	0.4584797
0.3706351	0.4305579
0.2326708	0.5224102
0.0595103	0.4271335
-0.3674753	0.4361963
-1.3761520	0.4540926
1.8947211	0.5510777
0.0763093	0.4240482
-0.2989919	0.4199354
0.1787774	0.4174000
-0.2723008	0.4311986
1.5771763	0.4958290
0.1160464	0.4514254
0.7690490	0.4203804
-0.2935165	0.4275332
-0.1040953	0.4219471
-0.6359837	0.3963132
-0.5879480	0.4997461
-0.6528824	0.4444723
-1.3254234	0.4825750
1.3101656	0.4591562
1.0209811	0.4593145
-1.1105810	0.4702726
-0.2612042	0.4786245
-0.7223982	0.4085821
2.0769142	0.5842827
1.8126472	0.5307947
0.1624258	0.4293789
0.4368228	0.4288108
-0.2366045	0.4391503
-0.0737807	0.3859773
-0.7373002	0.4220844
-0.1335489	0.3930743
1.4500367	0.4829115
0.8786277	0.4216545
0.5392510	0.3964505
0.7489879	0.4183694
1.1613100	0.4544287
-0.3697068	0.4038102
-1.1370895	0.4302780
1.1530314	0.4443201
-0.4347483	0.4289628
-0.8477341	0.4461625
-1.1604950	0.4502056
-0.4715481	0.3786356
0.1393651	0.4180721
0.4562396	0.3989102
0.4153818	0.4119034
0.5476333	0.4488555
-0.5781275	0.4671878
-0.1797902	0.4512261
-1.3656710	0.4742992
-0.9310978	0.4434286
0.8628820	0.4189980
-1.2309733	0.4027712
1.9130475	0.5554722
0.2219039	0.3999329
0.9127810	0.4278560
-0.8241829	0.4111594
-0.8397910	0.4386854
-1.3586507	0.4738941
-0.5916899	0.4890978
0.4668176	0.4303800
0.3369601	0.4074906
-0.8896585	0.5179485
-1.3028755	0.4153109
-0.9235733	0.4514861
1.5906082	0.4987496
-0.4378187	0.4071567
-0.6889944	0.4589706
1.1261132	0.4525471
-0.0646919	0.4523290
-1.3244437	0.4284766
-0.5050785	0.4207171
0.5895677	0.4117820
-0.4046862	0.5242918
-0.4555908	0.4452894
-0.3343450	0.4720323
-0.6720112	0.4138213
2.0769142	0.5842827
-0.7714165	0.4636742
0.9865898	0.4218830
1.2189488	0.4489170
-0.3086836	0.4179127
-1.2374106	0.5025736
-0.2387009	0.4074191
-0.1764419	0.3996390
-0.3805744	0.3951092
0.8209231	0.4432967
1.6116553	0.5053854
1.2064292	0.4498236
0.2333676	0.4372281
-0.9470311	0.4128603
-0.2786090	0.3936250
0.1417967	0.4150797
0.7884338	0.4187570
0.7783588	0.4286183
0.3243978	0.4514121
-0.7417397	0.3895264
0.3662230	0.3916504
-0.8910485	0.4398124
-0.6655293	0.4701430
-0.2033559	0.4056901
0.4957892	0.4144225
-0.0121442	0.4188963
-2.1957411	0.5343860
0.4512568	0.4064558
0.7378973	0.4019538
0.0167704	0.4126649
-0.2000235	0.4271632
0.1182389	0.3967470
-0.6783871	0.4584694
1.5788014	0.4945168
0.0521466	0.4298968
-0.1710475	0.4332455
0.3297711	0.4373681
1.2158969	0.4573886
1.2436716	0.4513447
-0.2858614	0.4194748
0.9428129	0.4397456
-0.0128354	0.4474032
2.0769142	0.5842827
-1.5328109	0.4746301
0.1103806	0.4288396
0.1751374	0.4072603
-0.5866125	0.4631116
-1.2945887	0.4914071
0.1612614	0.4356487
-0.9736387	0.4454626
-0.4555053	0.4328464
0.2963267	0.3973423
-1.2412651	0.4616779
0.0704970	0.4401173
1.3101656	0.4591562
1.2277572	0.4577725
0.7350191	0.4156325
0.0794165	0.4462796
-2.2961711	0.5530162
-0.4714201	0.4303817
-0.0426154	0.4208260
0.0046790	0.4051815
-0.2136678	0.4294552
-1.4371641	0.4668696
1.7205332	0.5171665
0.6234814	0.4095331
0.8432364	0.4193993
0.7756617	0.4448162
-1.4401285	0.4574738
0.2861088	0.4376200
0.0075844	0.4419441
0.7040170	0.4224441
-0.2645789	0.4317364
1.1977001	0.4479462
0.9417029	0.4381109
0.0033443	0.4160107
0.6686473	0.4215287
-1.0535448	0.4702537
0.0503565	0.4583580
1.0510065	0.4347434
0.4355071	0.4341712
-1.7270440	0.4741354
-1.5374992	0.5064098
-0.0287174	0.3985648
1.4694338	0.4812001
-0.6637158	0.4123076
0.9367069	0.4309494
0.2922123	0.4099907
1.5049490	0.4915100
-0.2784151	0.4201447
-0.1936566	0.4122852
-1.0443479	0.4817993
-0.1454046	0.4071466
-0.1097102	0.3845678
-1.4378165	0.4449566
-0.4058117	0.4358488
1.8947211	0.5510777
-2.1273946	0.5082597
-0.4430713	0.4306168
0.3315934	0.3935154
-0.6597857	0.3977472
0.5488419	0.4175147
0.7629299	0.4084732
0.6778305	0.4263140
-2.3962628	0.5525084
1.6435726	0.5098348
-0.7515804	0.4394700
1.0593000	0.4254128
0.2736189	0.4292568
0.2081942	0.4147082
0.6236162	0.4048385
0.1723714	0.3877789
0.6383166	0.4162615
-0.4421036	0.4323041
1.0617250	0.4329694
-0.0388703	0.3922713
1.4303325	0.4735242
-1.3092490	0.4598337
-0.5991746	0.4250723
0.7246969	0.4250792
0.6075135	0.4079282
-0.6608160	0.4666713
-0.6775557	0.4325173
-0.4547319	0.4270943
-0.5956100	0.4076201
-1.5512371	0.4595731
-0.1354008	0.4609984
0.2271031	0.4173885
-0.0206642	0.4127491
-0.5233933	0.3864480
0.7359629	0.4200823
1.4869834	0.4901589
0.0280958	0.4490905
-0.6500092	0.4274891
-0.3387195	0.4067991
0.8124170	0.4246379
-1.1280605	0.4605817
-1.2290127	0.4503078
0.7371623	0.4220793
-0.7051650	0.4460768
-0.4247851	0.4338207
1.1431965	0.4434606
-0.2462228	0.4187486
1.1108907	0.4399196
-1.0337821	0.4845290
-0.9353031	0.4530734
-0.5499093	0.4150383
-1.4525064	0.5111002
-0.3771307	0.4552273
-0.1787971	0.3749565
0.8647465	0.4207545
-1.1080191	0.4411455
-0.2065430	0.4250447
0.0909116	0.4398827
-0.3292071	0.4508701
0.0564090	0.3979824
0.7991089	0.4148650
0.3031090	0.3957428
0.9127810	0.4278560
-0.2666403	0.4495385
0.1865018	0.4525669
-1.9496328	0.4947409
0.3677099	0.4073099
0.3863600	0.4137684
0.4652173	0.4115814
-0.5623599	0.3946722
1.5174264	0.4818702
0.6959878	0.4175957
-0.0912641	0.4616302
0.9964790	0.4271765
-0.3595081	0.5092109
-0.4876713	0.4369677
1.4186021	0.4741901
-0.0347025	0.5195560
1.5931536	0.5062919
-1.2207454	0.4314549
-0.7904739	0.4359887
-0.0047973	0.4462242
0.5392711	0.4121751
-0.3097423	0.3896810
-0.5663924	0.4754546
-0.8388990	0.4108386
0.6912042	0.4171265
1.2739982	0.4694426
-1.9367418	0.5110017
1.2436716	0.4513447
-1.5574410	0.4646710
0.8939098	0.4324742
0.5029642	0.4825568
-0.5003676	0.4283674
-0.4801919	0.4202002
0.0839750	0.4369357
0.4495047	0.4436310
0.4015963	0.4278479
-1.5518743	0.4692911
-1.6142815	0.4455629
1.5299127	0.4959730
1.2756413	0.4605236
0.3405257	0.4414808
0.4799858	0.4003086
0.6766582	0.4360109
-1.9233216	0.4688982
0.8330534	0.4121890
-0.3855746	0.4655303
-0.1453490	0.4028902
-0.3976118	0.5425203
0.7932480	0.4365645
-0.5399406	0.4031989
-0.2267379	0.4205618
-0.4254019	0.4006949
-0.6027509	0.3955709
0.8106008	0.4214950
0.1537083	0.4174418
0.9166587	0.4221583
0.3891376	0.4220280
0.5847755	0.4094994
1.0488228	0.4525631
0.2854940	0.4112561
-0.8789495	0.4938021
-0.0839529	0.4596965
0.3404032	0.4358052
0.4955506	0.4354294
0.8581075	0.4088891
-0.4845366	0.4224354
0.5347493	0.4155211
-0.0287174	0.3985648
0.0627558	0.3926316
-0.0495048	0.4657865
1.5049490	0.4915100
-0.2396339	0.4015686
0.4583133	0.4251561
0.3847095	0.4069480
0.7797563	0.4226800
-0.8576206	0.4354207
2.0769142	0.5842827
0.0982340	0.4607269
-0.8600548	0.5267115
1.4570305	0.4815596
0.2432665	0.4389911
-0.7725215	0.4553132
-0.2013886	0.3970239
-1.3214554	0.4089901
0.3380273	0.4218502
-0.3276045	0.4304625
0.0496354	0.4906972
0.9694969	0.4211568
-0.5837225	0.4320326
-1.0346021	0.4265641
0.4004568	0.4157411
1.5931536	0.5062919
-1.5435917	0.4553309
-0.1219826	0.4279789
-0.3041503	0.4134317
-1.3833056	0.5245487
0.3198996	0.4045723
-0.6750523	0.4512224
0.3647307	0.4587481
-0.1243642	0.4393207
0.2247222	0.4130577
0.2793720	0.4235366
-1.1709249	0.4470516
-1.2481801	0.4777298
0.0780069	0.4551858
0.7487559	0.4339612
-0.9807137	0.4315098
0.1781392	0.4039889
0.6777796	0.4291410
-0.3119940	0.4176945
-0.4079093	0.4514178
-0.2738722	0.5197820
0.9265253	0.4432070
0.4239494	0.4301745
0.9227389	0.4425317
0.2682682	0.4193124
1.0052945	0.4343454
-0.5307083	0.4241967
0.8427091	0.4453150
0.3637628	0.4267093
-1.7829499	0.4781911
-0.3033541	0.5077212
1.8947211	0.5510777
-0.0262528	0.3967808
0.4258021	0.4150116
0.4739816	0.4228941
-1.1473002	0.4302445
-0.6648474	0.4730055

EAP_df_uc tablosunda sunulan degerler, MTK cercevesinde Expected A Posteriori (EAP) yontemi kullanilarak hesaplanan bireysel yetenek tahminlerini ve bu tahminlere iliskin standart hata degerlerini icermektedir.
“F” sutunundaki degerler, her birey icin tahmin edilen gizil ozelligi (latent trait) ifade eder.
Bu degerler, kisilerin testte olculen yetenek duzeylerine dair ortalama posterior dagilimdan elde edilmistir.
POZITIF degerler bireyin ortalama UZERINDE bir yetenek seviyesine sahip oldugunu, NEGATIF degerler ise ortalamanin ALTINDA bir yetenek duzeyine sahip oldugunu gosterir.
“SE_F” sutununda verilen standart hata degerleri ise her bir tahminin belirsizligini yansitir; daha dusuk SE_F degerleri daha guvenilir tahminleri, YUKSEK degerler ise tahmindeki belirsizligi ifade eder.
EAP yontemi, yanitlarin olasilik dagilimi uzerinden ortalama alinarak tahmin yapmasi nedeniyle, hem duzenli hem de ekstrem yanit desenleri icin kararli ve dengeli sonuclar saglar.
Bu yontem, bireylerin yeteneklerini degiskenlik icinde ama sistematik sekilde belirlemek icin tercih edilen Bayesci bir yaklasimdir ve ozellikle genis olcekli test uygulamalarinda siklikla kullanilmaktadir.

calisma_gunlugu_mtk_3pl

Asli_Ece_Kocak

2025-05-19

0.1 R KODLAR

1 3PL MTK

1.1 Yetenek Parametresi Kestirimi

1.1.0.1 3PL icin: