final

CEVAP 1.a Ortalama Matematik Başarısı – Okul Türü Bazında

data_env <- new.env()
load("D:/OLC_733/final/PISA_STU_2022 (1).rda", envir = data_env)
veri <- data_env[[ls(data_env)[1]]]


# Gerekli paketler
library(dplyr)

## Warning: package 'dplyr' was built under R version 4.4.3

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(ggplot2)
library(haven)   # as_factor için

## Warning: package 'haven' was built under R version 4.4.3

library(forcats) # fct_reorder için

# STRATUM değişkenini etiket isimlerine dönüştür
veri <- veri %>%
  mutate(STRATUM = haven::as_factor(STRATUM))

# Ortalama matematik puanını her okul türü için hesapla
ortalama_math <- veri %>%
  group_by(STRATUM) %>%
  summarise(ortalama_puan = mean(PV1MATH, na.rm = TRUE)) %>%
  arrange(desc(ortalama_puan))

# Tabloyu yazdır
print(ortalama_math)

## # A tibble: 36 × 2
##    STRATUM                                                       ortalama_puan
##    <fct>                                                                 <dbl>
##  1 TUR - stratum 02: Science High School - A                              635.
##  2 TUR - stratum 03: Science High School - B                              599.
##  3 TUR - stratum 08: Anatolian High School- A                             593.
##  4 TUR - stratum 19: Anatolian Imam and Preacher High School - A          575.
##  5 TUR - stratum 04: Science High School - C                              573.
##  6 TUR - stratum 09: Anatolian High School- B                             564.
##  7 TUR - stratum 06: Social Sciences High School - B                      545.
##  8 TUR - stratum 20: Anatolian Imam and Preacher High School - B          539.
##  9 TUR - stratum 05: Social Sciences High School - A                      538.
## 10 TUR - stratum 10: Anatolian High School- C                             533.
## # ℹ 26 more rows

# Grafikle göster (okul türü adlarıyla)
ggplot(ortalama_math, aes(x = fct_reorder(STRATUM, ortalama_puan), y = ortalama_puan)) +
  geom_col(fill = "steelblue") +
  coord_flip() +
  labs(title = "Okul Turune Göre Ortalama Matematik Başarısı (PV1MATH)",
       x = "Okul Turu (STRATUM Etiketi)",
       y = "Ortalama Matematik Puanı") +
  theme_minimal()

Matematik başarısı yönünden bazı aykırılıklar olsa da genel olarak okulların en başarılıdan en az başarılıya doğru sıralaması şu şekilde gözükmektedir.Fen liseleri, anadolu liseleri,anadolu imam hatip liseleri ve mesleki ve teknik anadolu liseleleri.

CEVAP 1.b Cinsiyete Göre Matematik Başarısı – Okul Türü Bazında

# Cinsiyet değişkenini anlamlı hale getirelim
veri <- veri %>%
  mutate(CINSIYET = case_when(
    ST004D01T == 1 ~ "Erkek",
    ST004D01T == 2 ~ "Kız",
    TRUE ~ NA_character_
  )) %>%
  filter(!is.na(CINSIYET))

# Okul türü ve cinsiyet bazında ortalama puan hesapla
ortalama_cinsiyet <- veri %>%
  group_by(STRATUM, CINSIYET) %>%
  summarise(ortalama_puan = mean(PV1MATH, na.rm = TRUE)) %>%
  arrange(STRATUM, CINSIYET)

## `summarise()` has grouped output by 'STRATUM'. You can override using the
## `.groups` argument.

# Grafik
ggplot(ortalama_cinsiyet, aes(x = fct_reorder(STRATUM, ortalama_puan, .fun = mean),
                              y = ortalama_puan, fill = CINSIYET)) +
  geom_col(position = "dodge") +
  coord_flip() +
  labs(title = "Okul Turune Göre Cinsiyete Göre Ortalama Matematik Puanı",
       x = "Okul TUru",
       y = "Ortalama Matematik Puanı",
       fill = "Cinsiyet") +
  theme_minimal()

Hemen hemen her okul türünde kızlar erkeklerden daha yüksek bir matematik başarısı göstermiştir diyebiliriz. Buna istisna olarak spor liseleri ve güzel sanatlar liselerinde erkekler daha başarılıdır.

CEVAP 1.c Matematik Kaygısı ile Başarı Arasındaki İlişki (Genel)

library(ggplot2)
library(ggpubr)  # Korelasyon ve regresyon çizgisi için

## Warning: package 'ggpubr' was built under R version 4.4.3

# Korelasyon hesapla
correlation <- cor(veri$ANXMAT, veri$PV1MATH, use = "complete.obs")

# Scatter plot + regresyon çizgisi + korelasyon katsayısı
ggscatter(veri, x = "ANXMAT", y = "PV1MATH",
          add = "reg.line",  # Regresyon çizgisi ekle
          conf.int = TRUE,   # Güven aralığı
          cor.coef = TRUE,   # Korelasyon katsayısını göster
          cor.method = "pearson",
          xlab = "Matematik Kaygısı (ANXMAT)",
          ylab = "Matematik Basarısı (PV1MATH)",
          title = "Matematik Kaygısı ile Matematik Başarısı Arasındaki İlişki") +
  theme_minimal()

## Warning: Removed 191 rows containing non-finite outside the scale range
## (`stat_smooth()`).

## Warning: Removed 191 rows containing non-finite outside the scale range
## (`stat_cor()`).

## Warning: Removed 191 rows containing missing values or values outside the scale range
## (`geom_point()`).

Korelasyon Katsayısı (R): -0.12 Negatif yönlü zayıf bir ilişki olduğunu söyleyebiliriz. Bu, matematik kaygısı arttıkça matematik başarısının hafifçe azalma eğiliminde olduğunu gösterir.

Anlamlılık Düzeyi (p) < 2.2e- 16 : Korelasyonun istatistiksel olarak son derece anlamlı olduğunu söyleyebiliriz (p değeri çok küçük). Ancak, korelasyonun gücü zayıf olduğundan kaygının başarı üzerindeki etkisi sınırlı olabilir.

CEVAP 1.d Matematik Kaygısı ile Başarı Arasındaki İlişki – Okul Türüne Göre

library(ggplot2)
library(dplyr)

# STRATUM'u faktör yapalım (eğer değilse)
veri <- veri %>%
  mutate(STRATUM = as.factor(STRATUM))

# Her okul türü için scatter + regresyon çizgisi, facet_wrap ile
ggplot(veri, aes(x = ANXMAT, y = PV1MATH)) +
  geom_point(alpha = 0.4, color = "blue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  facet_wrap(~ STRATUM, scales = "free") +   # Her okul türü için ayrı grafik
  labs(title = "Okul Turune Göre Matematik Kaygısı ile Matematik Başarısı İlişkisi",
       x = "Matematik Kaygısı (ANXMAT)",
       y = "Matematik Basarısı (PV1MATH)") +
  theme_minimal() +
  theme(strip.text = element_text(size = 8))  # Facet etiketleri okunaklı olsun

## `geom_smooth()` using formula = 'y ~ x'

## Warning: Removed 191 rows containing non-finite outside the scale range
## (`stat_smooth()`).

## Warning: Removed 191 rows containing missing values or values outside the scale range
## (`geom_point()`).

Çoğu okul türünde kırmızı çizgi negatif eğimli, bu da matematik kaygısı arttıkça matematik başarısının azaldığını gösteriyor.Bununla birlikte bu negatif ilişki özellikle bazı okul türlerinde daha belirgin. Fen ve Anadolu liseleri gibi akademik başarı odaklı okullarda bu etkinin daha güçlü olduğunu gözlemliyoruz. Bu okullarda yüksek başarıya rağmen kaygının başarıyı düşürdüğünü söyleyebiliriz. Meslek liseleri gibi görece daha az başarılı okullarda ise daha yatay bir çizgi var. Bu da bu okullardaki öğrencilerin başarısızlık nedenlerinin kaygıdan daha çok başka faktörlere bağlı olduğunu gösteriyor.

CEVAP 1.e Matematik Kaygısı ile Başarı Arasındaki İlişkide okul öncesi/sonrası ders çalışma süresinin rolü (STUDYHMW)

library(tidyr)

## Warning: package 'tidyr' was built under R version 4.4.3

library(dplyr)
library(stringr)

alt_veri <- veri %>%
  filter(str_detect(as.character(STRATUM), "stratum 02|stratum 15|stratum 35")) %>%
  select(STRATUM, PV1MATH, ANXMAT, STUDYHMW) %>%
  drop_na()

library(lavaan)

## Warning: package 'lavaan' was built under R version 4.4.3

## This is lavaan 0.6-19
## lavaan is FREE software! Please report any bugs.

# Modeli tanımlayalım
model_mediation <- '
  STUDYHMW ~ a*ANXMAT
  PV1MATH ~ b*STUDYHMW + c_prime*ANXMAT

  # Dolaylı ve toplam etkiler
  indirect := a*b
  total := c_prime + (a*b)
'

# Modeli fit edelim
fit_mediation <- sem(model_mediation, data=alt_veri, estimator="MLR")

# Sonuçları gösterelim
summary(fit_mediation, standardized=TRUE, fit.measures=TRUE)

## lavaan 0.6-19 ended normally after 8 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         5
## 
##   Number of observations                          1131
## 
## Model Test User Model:
##                                               Standard      Scaled
##   Test Statistic                                 0.000       0.000
##   Degrees of freedom                                 0           0
## 
## Model Test Baseline Model:
## 
##   Test statistic                                 5.592       5.382
##   Degrees of freedom                                 3           3
##   P-value                                        0.133       0.146
##   Scaling correction factor                                  1.039
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.000       1.000
##                                                                   
##   Robust Comparative Fit Index (CFI)                            NA
##   Robust Tucker-Lewis Index (TLI)                               NA
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -9623.568   -9623.568
##   Loglikelihood unrestricted model (H1)             NA          NA
##                                                                   
##   Akaike (AIC)                               19257.135   19257.135
##   Bayesian (BIC)                             19282.290   19282.290
##   Sample-size adjusted Bayesian (SABIC)      19266.408   19266.408
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000          NA
##   90 Percent confidence interval - lower         0.000          NA
##   90 Percent confidence interval - upper         0.000          NA
##   P-value H_0: RMSEA <= 0.050                       NA          NA
##   P-value H_0: RMSEA >= 0.080                       NA          NA
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.000
##   P-value H_0: Robust RMSEA <= 0.050                            NA
##   P-value H_0: Robust RMSEA >= 0.080                            NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.000       0.000
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   STUDYHMW ~                                                            
##     ANXMAT     (a)   -0.085    0.090   -0.944    0.345   -0.085   -0.032
##   PV1MATH ~                                                             
##     STUDYHM    (b)    0.411    0.732    0.561    0.575    0.411    0.016
##     ANXMAT  (c_pr)   -4.276    2.100   -2.037    0.042   -4.276   -0.060
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .STUDYHMW         10.970    0.310   35.403    0.000   10.970    0.999
##    .PV1MATH        7683.680  378.391   20.306    0.000 7683.680    0.996
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     indirect         -0.035    0.070   -0.499    0.618   -0.035   -0.000
##     total            -4.311    2.098   -2.055    0.040   -4.311   -0.061

CEVAP 2.a Belirtilen “orneklem1” fonksiyonunu iterasyon sayısını da ekleyecek şekilde for döngüsü kullanarak “orneklem2” adıyla güncelleyiniz.

orneklem2 <- function(evren, size = 20, iterasyon = 100) {
  ortalamalar <- numeric(iterasyon)  # iterasyon sayısı kadar boş vektör oluştur
  for (i in 1:iterasyon) {
    ortalamalar[i] <- mean(sample(evren, size))
  }
  return(ortalamalar)
}

CEVAP 2.b “orneklem2” fonksiyonunda oluşacak olan örneklem dağılımlarının ortalamasını ve standart sapmasını fonksiyona çıktı olarak ekleyiniz.

orneklem2 <- function(evren, size = 20, iterasyon = 100) {
  ortalamalar <- numeric(iterasyon)  # iterasyon sayısı kadar boş vektör oluştur
  for (i in 1:iterasyon) {
    ortalamalar[i] <- mean(sample(evren, size))
  }
  # Örneklem dağılımının ortalaması ve standart sapması
  liste <- list(
    orneklem_ortalamalari = ortalamalar,
    ortalama = mean(ortalamalar),
    standart_sapma = sd(ortalamalar)
  )
  return(liste)
}

CEVAP 2.c Elde ettiğiniz fonksiyonu kullanarak oluşturduğunuz 10 örneklem büyüklüğünde 5, 30 ve 100 tekrarlı örneklemler seçiniz. Örneklem ortalamaları dağılımının grafiğini çiziniz. Örneklem ortalamaları dağılımın ortalamasını ve standart hatasını hesaplayınız.

library(mlmRev)

## Warning: package 'mlmRev' was built under R version 4.4.3

## Zorunlu paket yükleniyor: lme4

## Zorunlu paket yükleniyor: Matrix

## 
## Attaching package: 'Matrix'

## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

attach(Exam)

# Daha önce tanımlanan orneklem2 fonksiyonu burada tekrar kullanılıyor
orneklem2 <- function(evren, size = 10, iterasyon = 100) {
  ortalamalar <- numeric(iterasyon)
  for (i in 1:iterasyon) {
    ortalamalar[i] <- mean(sample(evren, size))
  }
  list(
    orneklem_ortalamalari = ortalamalar,
    ortalama = mean(ortalamalar),
    standart_hata = sd(ortalamalar)
  )
}

# 3 farklı iterasyonla örneklemler
set.seed(123)  # Aynı sonuçlar için
orneklem_5 <- orneklem2(evren = normexam, size = 10, iterasyon = 5)
orneklem_30 <- orneklem2(evren = normexam, size = 10, iterasyon = 30)
orneklem_100 <- orneklem2(evren = normexam, size = 10, iterasyon = 100)

# Grafik için
par(mfrow = c(1, 3))  # 3 grafiği yan yana çiz
hist(orneklem_5$orneklem_ortalamalari, main = "n = 10, iterasyon = 5", xlab = "Örneklem Ortalamaları", col = "lightblue")
hist(orneklem_30$orneklem_ortalamalari, main = "n = 10, iterasyon = 30", xlab = "Örneklem Ortalamaları", col = "lightgreen")
hist(orneklem_100$orneklem_ortalamalari, main = "n = 10, iterasyon = 100", xlab = "Örneklem Ortalamaları", col = "lightcoral")

orneklem_5$ortalama

## [1] 0.03052185

orneklem_5$standart_hata

## [1] 0.1660067

orneklem_30$ortalama

## [1] -0.03890927

orneklem_30$standart_hata

## [1] 0.2703031

orneklem_100$ortalama

## [1] 0.04972833

orneklem_100$standart_hata

## [1] 0.3072292

CEVAP 2.d Elde ettiğiniz fonksiyonu kullanarak oluşturduğunuz 50 örneklem büyüklüğünde 5, 30 ve 100 tekrarlı örneklemler seçiniz. Örneklem ortalamaları dağılımının grafiğini çiziniz. Örneklem ortalamaları dağılımın ortalamasını ve standart hatasını hesaplayınız.

library(mlmRev)
attach(Exam)

## The following objects are masked from Exam (pos = 3):
## 
##     intake, normexam, schavg, schgend, school, sex, standLRT, student,
##     type, vr

# Fonksiyon tanımı (önceden verilmişti)
orneklem2 <- function(evren, size = 50, iterasyon = 100) {
  ortalamalar <- numeric(iterasyon)
  for (i in 1:iterasyon) {
    ortalamalar[i] <- mean(sample(evren, size))
  }
  list(
    orneklem_ortalamalari = ortalamalar,
    ortalama = mean(ortalamalar),
    standart_hata = sd(ortalamalar)
  )
}

# 3 farklı iterasyonla örneklem
set.seed(42)
orneklem_5 <- orneklem2(evren = normexam, size = 50, iterasyon = 5)
orneklem_30 <- orneklem2(evren = normexam, size = 50, iterasyon = 30)
orneklem_100 <- orneklem2(evren = normexam, size = 50, iterasyon = 100)

# Grafikler
par(mfrow = c(1, 3))  # 3 grafiği yan yana
hist(orneklem_5$orneklem_ortalamalari, main = "n = 50, iterasyon = 5", xlab = "Örneklem Ortalamaları", col = "skyblue")
hist(orneklem_30$orneklem_ortalamalari, main = "n = 50, iterasyon = 30", xlab = "Örneklem Ortalamaları", col = "lightgreen")
hist(orneklem_100$orneklem_ortalamalari, main = "n = 50, iterasyon = 100", xlab = "Örneklem Ortalamaları", col = "salmon")

# Ortalamalar ve standart hatalar
orneklem_5$ortalama

## [1] 0.1147119

orneklem_5$standart_hata

## [1] 0.2479696

orneklem_30$ortalama

## [1] 0.01763558

orneklem_30$standart_hata

## [1] 0.1277681

orneklem_100$ortalama

## [1] -0.007014406

orneklem_100$standart_hata

## [1] 0.1382155

CEVAP 3.a MTK sayıltılarını (tek boyutluluk ve yerel bağımsızlık) test ederek verinin hangi modele daha iyi uyum sağlandığını raporlayınız.

# Gerekli paketler
library(mirt)

## Warning: package 'mirt' was built under R version 4.4.3

## Zorunlu paket yükleniyor: stats4

## Zorunlu paket yükleniyor: lattice

## 
## Attaching package: 'mirt'

## The following object is masked from 'package:lme4':
## 
##     fixef

library(psych)

## Warning: package 'psych' was built under R version 4.4.3

## 
## Attaching package: 'psych'

## The following object is masked from 'package:lavaan':
## 
##     cor2cov

## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha

# Veri setini yükle
binary_data <- readRDS("D:/OLC_733/final/binary.Rds")




# 1 faktörlü model kur
mod1 <- mirt(binary_data, 1)

## Iteration: 1, Log-Lik: -23301.605, Max-Change: 0.36415Iteration: 2, Log-Lik: -23195.704, Max-Change: 0.12903Iteration: 3, Log-Lik: -23187.443, Max-Change: 0.05170Iteration: 4, Log-Lik: -23186.110, Max-Change: 0.02777Iteration: 5, Log-Lik: -23185.667, Max-Change: 0.01436Iteration: 6, Log-Lik: -23185.488, Max-Change: 0.00798Iteration: 7, Log-Lik: -23185.364, Max-Change: 0.00208Iteration: 8, Log-Lik: -23185.346, Max-Change: 0.00136Iteration: 9, Log-Lik: -23185.337, Max-Change: 0.00109Iteration: 10, Log-Lik: -23185.327, Max-Change: 0.00053Iteration: 11, Log-Lik: -23185.326, Max-Change: 0.00042Iteration: 12, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 13, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 14, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 15, Log-Lik: -23185.324, Max-Change: 0.00016Iteration: 16, Log-Lik: -23185.323, Max-Change: 0.00010

# Faktör uygunluğunu test et (basit karşılaştırma)
mod2 <- mirt(binary_data, 2)

## Iteration: 1, Log-Lik: -23972.798, Max-Change: 0.37715Iteration: 2, Log-Lik: -23335.709, Max-Change: 0.21477Iteration: 3, Log-Lik: -23215.916, Max-Change: 0.13098Iteration: 4, Log-Lik: -23180.097, Max-Change: 0.08397Iteration: 5, Log-Lik: -23167.091, Max-Change: 0.05065Iteration: 6, Log-Lik: -23162.020, Max-Change: 0.03130Iteration: 7, Log-Lik: -23159.924, Max-Change: 0.02088Iteration: 8, Log-Lik: -23159.004, Max-Change: 0.01372Iteration: 9, Log-Lik: -23158.556, Max-Change: 0.01105Iteration: 10, Log-Lik: -23158.182, Max-Change: 0.00776Iteration: 11, Log-Lik: -23158.131, Max-Change: 0.00685Iteration: 12, Log-Lik: -23158.093, Max-Change: 0.00642Iteration: 13, Log-Lik: -23157.963, Max-Change: 0.00245Iteration: 14, Log-Lik: -23157.952, Max-Change: 0.00239Iteration: 15, Log-Lik: -23157.944, Max-Change: 0.00233Iteration: 16, Log-Lik: -23157.912, Max-Change: 0.00209Iteration: 17, Log-Lik: -23157.908, Max-Change: 0.00197Iteration: 18, Log-Lik: -23157.904, Max-Change: 0.00191Iteration: 19, Log-Lik: -23157.885, Max-Change: 0.00136Iteration: 20, Log-Lik: -23157.883, Max-Change: 0.00129Iteration: 21, Log-Lik: -23157.881, Max-Change: 0.00124Iteration: 22, Log-Lik: -23157.870, Max-Change: 0.00102Iteration: 23, Log-Lik: -23157.868, Max-Change: 0.00102Iteration: 24, Log-Lik: -23157.867, Max-Change: 0.00101Iteration: 25, Log-Lik: -23157.857, Max-Change: 0.00090Iteration: 26, Log-Lik: -23157.856, Max-Change: 0.00088Iteration: 27, Log-Lik: -23157.854, Max-Change: 0.00086Iteration: 28, Log-Lik: -23157.846, Max-Change: 0.00076Iteration: 29, Log-Lik: -23157.845, Max-Change: 0.00075Iteration: 30, Log-Lik: -23157.844, Max-Change: 0.00074Iteration: 31, Log-Lik: -23157.837, Max-Change: 0.00067Iteration: 32, Log-Lik: -23157.836, Max-Change: 0.00066Iteration: 33, Log-Lik: -23157.835, Max-Change: 0.00066Iteration: 34, Log-Lik: -23157.828, Max-Change: 0.00060Iteration: 35, Log-Lik: -23157.827, Max-Change: 0.00060Iteration: 36, Log-Lik: -23157.826, Max-Change: 0.00060Iteration: 37, Log-Lik: 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-23157.672, Max-Change: 0.00013Iteration: 323, Log-Lik: -23157.672, Max-Change: 0.00013Iteration: 324, Log-Lik: -23157.672, Max-Change: 0.00013Iteration: 325, Log-Lik: -23157.672, Max-Change: 0.00013Iteration: 326, Log-Lik: -23157.672, Max-Change: 0.00012Iteration: 327, Log-Lik: -23157.672, Max-Change: 0.00012Iteration: 328, Log-Lik: -23157.672, Max-Change: 0.00012Iteration: 329, Log-Lik: -23157.672, Max-Change: 0.00012Iteration: 330, Log-Lik: -23157.672, Max-Change: 0.00012Iteration: 331, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 332, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 333, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 334, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 335, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 336, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 337, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 338, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 339, Log-Lik: -23157.671, Max-Change: 0.00012Iteration: 340, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 341, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 342, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 343, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 344, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 345, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 346, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 347, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 348, Log-Lik: -23157.670, Max-Change: 0.00012Iteration: 349, Log-Lik: -23157.669, Max-Change: 0.00012Iteration: 350, Log-Lik: -23157.669, Max-Change: 0.00012Iteration: 351, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 352, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 353, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 354, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 355, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 356, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 357, Log-Lik: -23157.669, Max-Change: 0.00011Iteration: 358, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 359, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 360, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 361, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 362, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 363, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 364, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 365, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 366, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 367, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 368, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 369, Log-Lik: -23157.668, Max-Change: 0.00011Iteration: 370, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 371, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 372, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 373, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 374, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 375, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 376, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 377, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 378, Log-Lik: -23157.667, Max-Change: 0.00011Iteration: 379, Log-Lik: -23157.667, Max-Change: 0.00010Iteration: 380, Log-Lik: -23157.667, Max-Change: 0.00010Iteration: 381, Log-Lik: -23157.667, Max-Change: 0.00010Iteration: 382, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 383, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 384, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 385, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 386, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 387, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 388, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 389, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 390, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 391, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 392, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 393, Log-Lik: -23157.666, Max-Change: 0.00010Iteration: 394, Log-Lik: -23157.665, Max-Change: 0.00010

anova(mod1, mod2)

##           AIC    SABIC       HQ      BIC    logLik     X2 df p
## mod1 46470.65 46576.50 46569.35 46735.33 -23185.32            
## mod2 46463.33 46619.99 46609.41 46855.06 -23157.67 55.316 24 0

Veride tek boyutluluk varsayımının yaklaşık olarak sağlandığını söyleyebiliriz, çünkü 1 faktörlü model daha sade ve bilgi kriterlerine göre tercih edilebilir durumdadır. Ancak istatistiksel olarak 2 faktörlü modelin anlamlı fark yaratması, sınırlı çok boyutluluk olabileceğini düşünmemizi sağlıyor.

# Q3 artık korelasyonları (yerel bağımsızlık testi)
res <- residuals(mod1, type = "Q3")

## Q3 summary statistics:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -0.161  -0.057  -0.036  -0.034  -0.010   0.098 
## 
##          madde_1 madde_2 madde_3 madde_4 madde_5 madde_6 madde_7 madde_8
## madde_1    1.000  -0.061  -0.027   0.002  -0.023  -0.107  -0.161  -0.048
## madde_2   -0.061   1.000   0.041  -0.050  -0.073  -0.036  -0.044  -0.026
## madde_3   -0.027   0.041   1.000   0.024  -0.054  -0.036  -0.057   0.006
## madde_4    0.002  -0.050   0.024   1.000   0.004   0.001  -0.038  -0.074
## madde_5   -0.023  -0.073  -0.054   0.004   1.000   0.014  -0.024  -0.056
## madde_6   -0.107  -0.036  -0.036   0.001   0.014   1.000   0.098  -0.052
## madde_7   -0.161  -0.044  -0.057  -0.038  -0.024   0.098   1.000   0.018
## madde_8   -0.048  -0.026   0.006  -0.074  -0.056  -0.052   0.018   1.000
## madde_9   -0.064  -0.072  -0.021   0.003  -0.075  -0.032  -0.066  -0.090
## madde_10   0.008  -0.055  -0.003  -0.057  -0.094  -0.079  -0.008   0.030
## madde_11  -0.056   0.022  -0.062  -0.052  -0.052  -0.026  -0.047  -0.030
## madde_12  -0.111   0.009  -0.035  -0.038  -0.054  -0.094  -0.016   0.002
## madde_13  -0.045  -0.010   0.003  -0.052  -0.061   0.045  -0.034  -0.006
## madde_14  -0.031   0.022   0.035  -0.023  -0.008  -0.008  -0.040  -0.066
## madde_15  -0.060  -0.007  -0.030  -0.013  -0.046  -0.065  -0.053  -0.087
## madde_16  -0.037  -0.041  -0.049  -0.074   0.005  -0.117  -0.004  -0.034
## madde_17  -0.047  -0.014   0.003  -0.070  -0.116  -0.104  -0.033  -0.051
## madde_18  -0.040  -0.061  -0.012  -0.070  -0.045  -0.020  -0.041  -0.023
## madde_19  -0.079  -0.020  -0.036  -0.040  -0.073  -0.022  -0.021  -0.005
## madde_20  -0.008  -0.020   0.016  -0.066  -0.098  -0.035  -0.110  -0.048
## madde_21  -0.016  -0.001   0.002  -0.048  -0.046  -0.045  -0.072   0.004
## madde_22  -0.051  -0.044  -0.029  -0.036  -0.048  -0.068  -0.034  -0.071
## madde_23  -0.039   0.009  -0.017  -0.020  -0.060  -0.022  -0.034  -0.086
## madde_24  -0.048  -0.070  -0.014  -0.092   0.029   0.014  -0.008  -0.034
## madde_25  -0.076  -0.059  -0.015  -0.022  -0.021   0.009  -0.029   0.011
##          madde_9 madde_10 madde_11 madde_12 madde_13 madde_14 madde_15 madde_16
## madde_1   -0.064    0.008   -0.056   -0.111   -0.045   -0.031   -0.060   -0.037
## madde_2   -0.072   -0.055    0.022    0.009   -0.010    0.022   -0.007   -0.041
## madde_3   -0.021   -0.003   -0.062   -0.035    0.003    0.035   -0.030   -0.049
## madde_4    0.003   -0.057   -0.052   -0.038   -0.052   -0.023   -0.013   -0.074
## madde_5   -0.075   -0.094   -0.052   -0.054   -0.061   -0.008   -0.046    0.005
## madde_6   -0.032   -0.079   -0.026   -0.094    0.045   -0.008   -0.065   -0.117
## madde_7   -0.066   -0.008   -0.047   -0.016   -0.034   -0.040   -0.053   -0.004
## madde_8   -0.090    0.030   -0.030    0.002   -0.006   -0.066   -0.087   -0.034
## madde_9    1.000   -0.036    0.015    0.009    0.037    0.016   -0.002   -0.050
## madde_10  -0.036    1.000   -0.051   -0.098   -0.035   -0.036   -0.067   -0.114
## madde_11   0.015   -0.051    1.000   -0.046    0.047   -0.045    0.034    0.024
## madde_12   0.009   -0.098   -0.046    1.000    0.019   -0.045   -0.029   -0.069
## madde_13   0.037   -0.035    0.047    0.019    1.000    0.095    0.002   -0.056
## madde_14   0.016   -0.036   -0.045   -0.045    0.095    1.000   -0.010   -0.022
## madde_15  -0.002   -0.067    0.034   -0.029    0.002   -0.010    1.000    0.009
## madde_16  -0.050   -0.114    0.024   -0.069   -0.056   -0.022    0.009    1.000
## madde_17  -0.023   -0.083   -0.036   -0.032   -0.027    0.012   -0.021   -0.029
## madde_18   0.013    0.041   -0.057   -0.018   -0.032   -0.041   -0.084   -0.086
## madde_19   0.000   -0.028   -0.044   -0.012   -0.074    0.005   -0.057    0.001
## madde_20  -0.039   -0.012   -0.054   -0.001   -0.019   -0.013   -0.065   -0.034
## madde_21  -0.029   -0.047   -0.048   -0.013   -0.076   -0.068   -0.065   -0.071
## madde_22   0.026   -0.008   -0.018   -0.059   -0.027   -0.022   -0.013   -0.052
## madde_23   0.020   -0.061   -0.030   -0.011    0.003   -0.032   -0.027   -0.122
## madde_24  -0.004   -0.036   -0.017   -0.079   -0.069   -0.028   -0.054   -0.031
## madde_25  -0.044   -0.050   -0.054   -0.052   -0.011    0.004    0.004   -0.076
##          madde_17 madde_18 madde_19 madde_20 madde_21 madde_22 madde_23
## madde_1    -0.047   -0.040   -0.079   -0.008   -0.016   -0.051   -0.039
## madde_2    -0.014   -0.061   -0.020   -0.020   -0.001   -0.044    0.009
## madde_3     0.003   -0.012   -0.036    0.016    0.002   -0.029   -0.017
## madde_4    -0.070   -0.070   -0.040   -0.066   -0.048   -0.036   -0.020
## madde_5    -0.116   -0.045   -0.073   -0.098   -0.046   -0.048   -0.060
## madde_6    -0.104   -0.020   -0.022   -0.035   -0.045   -0.068   -0.022
## madde_7    -0.033   -0.041   -0.021   -0.110   -0.072   -0.034   -0.034
## madde_8    -0.051   -0.023   -0.005   -0.048    0.004   -0.071   -0.086
## madde_9    -0.023    0.013    0.000   -0.039   -0.029    0.026    0.020
## madde_10   -0.083    0.041   -0.028   -0.012   -0.047   -0.008   -0.061
## madde_11   -0.036   -0.057   -0.044   -0.054   -0.048   -0.018   -0.030
## madde_12   -0.032   -0.018   -0.012   -0.001   -0.013   -0.059   -0.011
## madde_13   -0.027   -0.032   -0.074   -0.019   -0.076   -0.027    0.003
## madde_14    0.012   -0.041    0.005   -0.013   -0.068   -0.022   -0.032
## madde_15   -0.021   -0.084   -0.057   -0.065   -0.065   -0.013   -0.027
## madde_16   -0.029   -0.086    0.001   -0.034   -0.071   -0.052   -0.122
## madde_17    1.000   -0.061    0.034   -0.068    0.027   -0.044   -0.030
## madde_18   -0.061    1.000   -0.057   -0.048   -0.086    0.035   -0.030
## madde_19    0.034   -0.057    1.000   -0.083   -0.055   -0.062   -0.064
## madde_20   -0.068   -0.048   -0.083    1.000   -0.074   -0.070   -0.008
## madde_21    0.027   -0.086   -0.055   -0.074    1.000   -0.114   -0.048
## madde_22   -0.044    0.035   -0.062   -0.070   -0.114    1.000    0.024
## madde_23   -0.030   -0.030   -0.064   -0.008   -0.048    0.024    1.000
## madde_24   -0.035   -0.044   -0.042    0.010   -0.047   -0.005   -0.043
## madde_25   -0.023   -0.060   -0.004   -0.055   -0.039   -0.046   -0.071
##          madde_24 madde_25
## madde_1    -0.048   -0.076
## madde_2    -0.070   -0.059
## madde_3    -0.014   -0.015
## madde_4    -0.092   -0.022
## madde_5     0.029   -0.021
## madde_6     0.014    0.009
## madde_7    -0.008   -0.029
## madde_8    -0.034    0.011
## madde_9    -0.004   -0.044
## madde_10   -0.036   -0.050
## madde_11   -0.017   -0.054
## madde_12   -0.079   -0.052
## madde_13   -0.069   -0.011
## madde_14   -0.028    0.004
## madde_15   -0.054    0.004
## madde_16   -0.031   -0.076
## madde_17   -0.035   -0.023
## madde_18   -0.044   -0.060
## madde_19   -0.042   -0.004
## madde_20    0.010   -0.055
## madde_21   -0.047   -0.039
## madde_22   -0.005   -0.046
## madde_23   -0.043   -0.071
## madde_24    1.000   -0.114
## madde_25   -0.114    1.000

summary(res)

##     madde_1             madde_2              madde_3         
##  Min.   :-0.161164   Min.   :-0.0727385   Min.   :-0.061608  
##  1st Qu.:-0.061074   1st Qu.:-0.0547764   1st Qu.:-0.035027  
##  Median :-0.047160   Median :-0.0263697   Median :-0.014935  
##  Mean   :-0.009025   Mean   : 0.0135711   Mean   : 0.025352  
##  3rd Qu.:-0.026609   3rd Qu.:-0.0005257   3rd Qu.: 0.003354  
##  Max.   : 1.000000   Max.   : 1.0000000   Max.   : 1.000000  
##     madde_4             madde_5             madde_6          
##  Min.   :-0.092497   Min.   :-0.115529   Min.   :-0.1171432  
##  1st Qu.:-0.056617   1st Qu.:-0.060661   1st Qu.:-0.0653351  
##  Median :-0.038224   Median :-0.048377   Median :-0.0317518  
##  Mean   : 0.003951   Mean   :-0.002959   Mean   : 0.0085484  
##  3rd Qu.:-0.012907   3rd Qu.:-0.021006   3rd Qu.: 0.0008488  
##  Max.   : 1.000000   Max.   : 1.000000   Max.   : 1.0000000  
##     madde_7             madde_8             madde_9        
##  Min.   :-0.161164   Min.   :-0.089871   Min.   :-0.08987  
##  1st Qu.:-0.046954   1st Qu.:-0.056109   1st Qu.:-0.04368  
##  Median :-0.033966   Median :-0.034022   Median :-0.02073  
##  Mean   : 0.005719   Mean   : 0.007291   Mean   : 0.01970  
##  3rd Qu.:-0.016037   3rd Qu.: 0.001783   3rd Qu.: 0.01336  
##  Max.   : 1.000000   Max.   : 1.000000   Max.   : 1.00000  
##     madde_10             madde_11           madde_12        
##  Min.   :-0.1143308   Min.   :-0.06161   Min.   :-0.110695  
##  1st Qu.:-0.0608230   1st Qu.:-0.05183   1st Qu.:-0.053587  
##  Median :-0.0360150   Median :-0.04392   Median :-0.032432  
##  Mean   : 0.0008512   Mean   : 0.01261   Mean   : 0.004967  
##  3rd Qu.:-0.0083731   3rd Qu.:-0.01730   3rd Qu.:-0.011230  
##  Max.   : 1.0000000   Max.   : 1.00000   Max.   : 1.000000  
##     madde_13            madde_14            madde_15        
##  Min.   :-0.075675   Min.   :-0.068274   Min.   :-0.087040  
##  1st Qu.:-0.044654   1st Qu.:-0.036015   1st Qu.:-0.059863  
##  Median :-0.018638   Median :-0.021621   Median :-0.028686  
##  Mean   : 0.024757   Mean   : 0.025939   Mean   : 0.007759  
##  3rd Qu.: 0.003354   3rd Qu.: 0.005306   3rd Qu.:-0.007446  
##  Max.   : 1.000000   Max.   : 1.000000   Max.   : 1.000000  
##     madde_16            madde_17            madde_18        
##  Min.   :-0.122419   Min.   :-0.115529   Min.   :-0.086373  
##  1st Qu.:-0.071059   1st Qu.:-0.051246   1st Qu.:-0.059708  
##  Median :-0.040583   Median :-0.032432   Median :-0.040981  
##  Mean   :-0.005227   Mean   : 0.005197   Mean   : 0.002971  
##  3rd Qu.:-0.022271   3rd Qu.:-0.020542   3rd Qu.:-0.020056  
##  Max.   : 1.000000   Max.   : 1.000000   Max.   : 1.000000  
##     madde_19            madde_20             madde_21        
##  Min.   :-0.083002   Min.   :-0.1099056   Min.   :-0.114401  
##  1st Qu.:-0.056825   1st Qu.:-0.0662455   1st Qu.:-0.068274  
##  Median :-0.035888   Median :-0.0390570   Median :-0.047020  
##  Mean   : 0.006562   Mean   :-0.0001591   Mean   :-0.003059  
##  3rd Qu.:-0.004650   3rd Qu.:-0.0120648   3rd Qu.:-0.015601  
##  Max.   : 1.000000   Max.   : 1.0000000   Max.   : 1.000000  
##     madde_22            madde_23            madde_24        
##  Min.   :-0.114401   Min.   :-0.122419   Min.   :-0.114234  
##  1st Qu.:-0.052130   1st Qu.:-0.048057   1st Qu.:-0.048206  
##  Median :-0.035520   Median :-0.029672   Median :-0.034989  
##  Mean   : 0.006547   Mean   : 0.008037   Mean   : 0.005551  
##  3rd Qu.:-0.012944   3rd Qu.:-0.011230   3rd Qu.:-0.007898  
##  Max.   : 1.000000   Max.   : 1.000000   Max.   : 1.000000  
##     madde_25        
##  Min.   :-0.114234  
##  1st Qu.:-0.055340  
##  Median :-0.038739  
##  Mean   : 0.004271  
##  3rd Qu.:-0.011323  
##  Max.   : 1.000000

Ortalama Q3 (-0.034) ≈ 0 → Yerel bağımsızlık genelde sağlanıyor diyebiliriz. Artık korelasyonlar küçük ve beklenen sınırlar içindedir.

mod_1pl <- mirt(binary_data, 1, itemtype = "Rasch")

## Iteration: 1, Log-Lik: -23303.739, Max-Change: 0.09815Iteration: 2, Log-Lik: -23297.714, Max-Change: 0.03942Iteration: 3, Log-Lik: -23296.554, Max-Change: 0.01672Iteration: 4, Log-Lik: -23296.277, Max-Change: 0.00737Iteration: 5, Log-Lik: -23296.188, Max-Change: 0.00328Iteration: 6, Log-Lik: -23296.148, Max-Change: 0.00197Iteration: 7, Log-Lik: -23296.093, Max-Change: 0.00090Iteration: 8, Log-Lik: -23296.089, Max-Change: 0.00060Iteration: 9, Log-Lik: -23296.087, Max-Change: 0.00047Iteration: 10, Log-Lik: -23296.083, Max-Change: 0.00014Iteration: 11, Log-Lik: -23296.083, Max-Change: 0.00009

mod_2pl <- mirt(binary_data, 1, itemtype = "2PL")

## Iteration: 1, Log-Lik: -23301.605, Max-Change: 0.36415Iteration: 2, Log-Lik: -23195.704, Max-Change: 0.12903Iteration: 3, Log-Lik: -23187.443, Max-Change: 0.05170Iteration: 4, Log-Lik: -23186.110, Max-Change: 0.02777Iteration: 5, Log-Lik: -23185.667, Max-Change: 0.01436Iteration: 6, Log-Lik: -23185.488, Max-Change: 0.00798Iteration: 7, Log-Lik: -23185.364, Max-Change: 0.00208Iteration: 8, Log-Lik: -23185.346, Max-Change: 0.00136Iteration: 9, Log-Lik: -23185.337, Max-Change: 0.00109Iteration: 10, Log-Lik: -23185.327, Max-Change: 0.00053Iteration: 11, Log-Lik: -23185.326, Max-Change: 0.00042Iteration: 12, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 13, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 14, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 15, Log-Lik: -23185.324, Max-Change: 0.00016Iteration: 16, Log-Lik: -23185.323, Max-Change: 0.00010

mod_3pl <- mirt(binary_data, 1, itemtype = "3PL")

## Iteration: 1, Log-Lik: -23475.118, Max-Change: 0.87773Iteration: 2, Log-Lik: -23196.171, Max-Change: 0.70725Iteration: 3, Log-Lik: -23131.871, Max-Change: 0.52430Iteration: 4, Log-Lik: -23108.030, Max-Change: 0.27898Iteration: 5, Log-Lik: -23096.875, Max-Change: 0.19118Iteration: 6, Log-Lik: -23090.447, Max-Change: 0.13125Iteration: 7, Log-Lik: -23086.142, Max-Change: 0.12323Iteration: 8, Log-Lik: -23083.646, Max-Change: 0.09615Iteration: 9, Log-Lik: -23082.131, Max-Change: 0.06493Iteration: 10, Log-Lik: -23080.478, Max-Change: 0.08297Iteration: 11, Log-Lik: -23079.866, Max-Change: 0.07340Iteration: 12, Log-Lik: -23079.373, Max-Change: 0.02644Iteration: 13, Log-Lik: -23078.931, Max-Change: 0.06015Iteration: 14, Log-Lik: -23078.693, Max-Change: 0.04940Iteration: 15, Log-Lik: -23078.516, Max-Change: 0.04514Iteration: 16, Log-Lik: -23078.019, Max-Change: 0.01058Iteration: 17, Log-Lik: -23077.994, Max-Change: 0.00197Iteration: 18, Log-Lik: -23077.990, Max-Change: 0.00148Iteration: 19, Log-Lik: -23077.987, Max-Change: 0.00154Iteration: 20, Log-Lik: -23077.982, Max-Change: 0.00122Iteration: 21, Log-Lik: -23077.978, Max-Change: 0.00128Iteration: 22, Log-Lik: -23077.971, Max-Change: 0.00260Iteration: 23, Log-Lik: -23077.971, Max-Change: 0.00101Iteration: 24, Log-Lik: -23077.970, Max-Change: 0.00030Iteration: 25, Log-Lik: -23077.970, Max-Change: 0.00027Iteration: 26, Log-Lik: -23077.970, Max-Change: 0.00076Iteration: 27, Log-Lik: -23077.970, Max-Change: 0.00086Iteration: 28, Log-Lik: -23077.970, Max-Change: 0.00033Iteration: 29, Log-Lik: -23077.970, Max-Change: 0.00039Iteration: 30, Log-Lik: -23077.970, Max-Change: 0.00052Iteration: 31, Log-Lik: -23077.970, Max-Change: 0.00021

anova(mod_1pl, mod_2pl, mod_3pl)

##              AIC    SABIC       HQ      BIC    logLik      X2 df p
## mod_1pl 46644.17 46699.21 46695.49 46781.80 -23296.08             
## mod_2pl 46470.65 46576.50 46569.35 46735.33 -23185.32 221.519 24 0
## mod_3pl 46305.94 46464.71 46453.99 46702.97 -23077.97 214.707 25 0

3PL modelinin AIC, SABIC, HQ, BIC değerleri daha düşük olduğundan ve logLik değeri daha yüksek olduğundan, diğer modellere göre model veri uyumu daha iyi durumdadır diyebiliriz. Ayrıca, 3PL modelinin X² değeri 2PL modeline göre daha düşüktür, yani bu kriter açısından da veriye daha uygun olan modelin 3PL olduğunu söyleyebiliriz. Son olarak, p-değerleri 0 çıktığı için, her iki karmaşık modelin daha basit modellere göre anlamlı derecede daha iyi uyum sağladığı anlaşılır (yani 2PL > 1PL, 3PL > 2PL).

CEVAP 3.b Madde ve birey parametrelerini raporlayınız.

mod_2pl <- mirt(binary_data, 1, itemtype = "2PL")

## Iteration: 1, Log-Lik: -23301.605, Max-Change: 0.36415Iteration: 2, Log-Lik: -23195.704, Max-Change: 0.12903Iteration: 3, Log-Lik: -23187.443, Max-Change: 0.05170Iteration: 4, Log-Lik: -23186.110, Max-Change: 0.02777Iteration: 5, Log-Lik: -23185.667, Max-Change: 0.01436Iteration: 6, Log-Lik: -23185.488, Max-Change: 0.00798Iteration: 7, Log-Lik: -23185.364, Max-Change: 0.00208Iteration: 8, Log-Lik: -23185.346, Max-Change: 0.00136Iteration: 9, Log-Lik: -23185.337, Max-Change: 0.00109Iteration: 10, Log-Lik: -23185.327, Max-Change: 0.00053Iteration: 11, Log-Lik: -23185.326, Max-Change: 0.00042Iteration: 12, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 13, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 14, Log-Lik: -23185.325, Max-Change: 0.00017Iteration: 15, Log-Lik: -23185.324, Max-Change: 0.00016Iteration: 16, Log-Lik: -23185.323, Max-Change: 0.00010

item_params <- coef(mod_2pl, IRTpars = TRUE, simplify = TRUE)$items
print(round(item_params, 3))

##              a      b g u
## madde_1  1.364 -0.382 0 1
## madde_2  0.780  0.290 0 1
## madde_3  0.414  0.306 0 1
## madde_4  1.008  0.215 0 1
## madde_5  1.157 -0.380 0 1
## madde_6  0.931  0.148 0 1
## madde_7  0.938  0.189 0 1
## madde_8  0.845  0.038 0 1
## madde_9  0.605  0.662 0 1
## madde_10 1.096 -0.316 0 1
## madde_11 0.759  0.054 0 1
## madde_12 1.016  0.132 0 1
## madde_13 0.555  0.996 0 1
## madde_14 0.518  1.906 0 1
## madde_15 0.919  0.193 0 1
## madde_16 1.277  0.112 0 1
## madde_17 1.010 -0.396 0 1
## madde_18 1.036  0.113 0 1
## madde_19 0.930  0.338 0 1
## madde_20 1.148 -0.255 0 1
## madde_21 1.221  0.362 0 1
## madde_22 1.058  0.882 0 1
## madde_23 0.921  0.035 0 1
## madde_24 0.914  0.050 0 1
## madde_25 1.038  0.346 0 1

theta_scores <- fscores(mod_2pl)
head(round(theta_scores, 3))

##          F1
## [1,] -0.800
## [2,] -0.721
## [3,]  0.510
## [4,] -0.812
## [5,] -1.057
## [6,] -0.985

CEVAP 3.c Madde karakteristik ve test karakteristik eğrilerini oluşturunuz. Oluşturduğunuz grafikleri yorumlayınız.

# Tüm maddelerin ICC'lerini çizdir
plot(mod_2pl, type = "trace")

# Test karakteristik eğrisi (toplam test için)
plot(mod_2pl, type = "info")

Madde karakteristik eğrilerine baktığımızda 1.,16. ve 21. maddelerin grafiklerinin daha dikey olduğunu gözlemliyoruz. Bu maddelerin ayırıcılık değerlerinin en yükseklerden olduğunu söyleyebiliriz. Bunun aksine, 3., 9. ve 2. maddelerin ise gragikleri daha yatay bir görünümde. Bu da bu maddelerin ayırıcılık güçlerinin az olduğunu gösteriyor. 14., 13. ve 9. maddelerin grafikleri diğerlerine göre daha sağa kaymış gibi gözükmektedir. Bu maddelerin güçlük düzeyleri yüksek olarak yorumlanabilir. 1.,5. ve 17. maddelerin grafikleri ise sola kaymış gözüküyor. Bu maddelerin diğerlerine göre daha kolay maddeler olduğunu söyleyebiliriz.

CEVAP 3.d KTK’ya dayalı madde parametrelerini kestirip, KTK’ya ve MTK’ya dayalı parametrelerin ilişkisini inceleyiniz.

# Madde güçlükleri (p-değeri): Her maddenin ortalaması
p_values <- colMeans(binary_data)

# Madde toplam korelasyonu (ayırt edicilik)
# Toplam puanı hesapla
total_score <- rowSums(binary_data)

# Her madde ile toplam puan arasındaki korelasyon
r_it <- sapply(binary_data, function(x) cor(x, total_score))

print(p_values)

##   madde_1   madde_2   madde_3   madde_4   madde_5   madde_6   madde_7   madde_8 
## 0.5873555 0.4473148 0.4690687 0.4500340 0.5798776 0.4663494 0.4581917 0.4894630 
##   madde_9  madde_10  madde_11  madde_12  madde_13  madde_14  madde_15  madde_16 
## 0.4072060 0.5635622 0.4881033 0.4670292 0.3725357 0.2821210 0.4581917 0.4643100 
##  madde_17  madde_18  madde_19  madde_20  madde_21  madde_22  madde_23  madde_24 
## 0.5771584 0.4704283 0.4296397 0.5513256 0.4078858 0.3140721 0.4887831 0.4860639 
##  madde_25 
## 0.4214820

print(r_it)

##   madde_1   madde_2   madde_3   madde_4   madde_5   madde_6   madde_7   madde_8 
## 0.4953716 0.3972788 0.2713392 0.4556106 0.4612861 0.4384830 0.4364790 0.4062495 
##   madde_9  madde_10  madde_11  madde_12  madde_13  madde_14  madde_15  madde_16 
## 0.3406210 0.4573527 0.3864341 0.4579579 0.3243673 0.2943838 0.4333691 0.5113066 
##  madde_17  madde_18  madde_19  madde_20  madde_21  madde_22  madde_23  madde_24 
## 0.4382446 0.4590592 0.4364781 0.4749399 0.5014369 0.4609499 0.4322469 0.4269816 
##  madde_25 
## 0.4668952

KTK ve MTK ile elde ettiğimiz madde parametrelerinin büyük oranda paralellik gösterdiğini söyleyebiriz. MTK da en ayırt edici maddeler 1,16 ve 21. maddeler iken KTK da da bu maddeler en ayırt edici maddeler olarak gözükmekte.MTK da en az ayırt edici maddeler 2,3 ve 9. maddeler iken KTK da 3,14 ve 9. maddeler en az ayırt edici maddeler olmuştur. MTK da en zor maddeler 9,13 ve 14. maddeler iken KTK da 13,14 ve 22. maddeler en zor maddelerdir. Son olarak MTK değerlerine göre en kolay maddeler 1,5 ve 17. maddeler iken KTK da ise 1,10 ve 17. maddeler en kolay maddeler olarak karşımıza çıkmaktadır. Her iki kuramla hesaplanan değerler açısından da 1.soru hem çok ayırt edici hem de orta güçlü bir madde olarak değerlendirilebilir.

CEVAP 4.a Üretilen madde cevaplarını kullanarak madde parametrelerini kestiriniz. Parametre kestiriminin hatasını RMSE değerleri üzerinden değerlendiriniz. RMSE değerlerini a ve b parametreleri için ayrı ayrı hesaplayınız. replikasyon sayısınına bağlı olarak hata değerlerinin nasıl değiştiği tablo ve grafikler ile gösteriniz.

madde_par <- readRDS("D:/OLC_733/final/maddepar.Rds")

# Parametreleri ayır
a_true <- madde_par[, "a"]
b_true <- madde_par[, "b"]

# Simülasyon ve tahmin (örnek)
library(mirt)

simulate_2PL <- function(n_persons, a, b) {
  theta <- rnorm(n_persons, 0, 1)
  n_items <- length(a)
  P <- sapply(1:n_items, function(i) 1 / (1 + exp(-a[i] * (theta - b[i]))))
  responses <- matrix(rbinom(n_persons * n_items, 1, as.vector(P)), nrow = n_persons, ncol = n_items)
  colnames(responses) <- paste0("Item", 1:n_items)
  list(responses = as.data.frame(responses), theta = theta)
}

# Replikasyon sayıları
replikasyon_sayilari <- c(10, 50, 100, 200)
n_persons <- 1000

results <- data.frame(Replikasyon = integer(), RMSE_a = double(), RMSE_b = double())

set.seed(123)
for (rep_count in replikasyon_sayilari) {
  rmse_a_vec <- numeric(rep_count)
  rmse_b_vec <- numeric(rep_count)
  
  for (i in 1:rep_count) {
    sim <- simulate_2PL(n_persons, a_true, b_true)
    mod <- mirt(sim$responses, 1, itemtype = "2PL", verbose = FALSE)
    est_par <- coef(mod, IRTpars = TRUE, simplify = TRUE)$items
    a_est <- est_par[, "a"]
    b_est <- est_par[, "b"]
    rmse_a_vec[i] <- sqrt(mean((a_est - a_true)^2))
    rmse_b_vec[i] <- sqrt(mean((b_est - b_true)^2))
  }
  
  results <- rbind(results, data.frame(
    Replikasyon = rep_count,
    RMSE_a = mean(rmse_a_vec),
    RMSE_b = mean(rmse_b_vec)
  ))
}

print(results)

##   Replikasyon    RMSE_a    RMSE_b
## 1          10 0.1286859 0.1466589
## 2          50 0.1256007 0.1453437
## 3         100 0.1265617 0.1615920
## 4         200 0.1262707 0.1493059

library(tidyr)
library(ggplot2)

# Örnek olarak mevcut tabloyu kullanalım
results <- data.frame(
  Replikasyon = c(10, 50),
  RMSE_a = c(0.1286859, 0.1256007),
  RMSE_b = c(0.1466589, 0.1453437)
)

results_long <- results %>%
  pivot_longer(cols = c("RMSE_a", "RMSE_b"), names_to = "Parametre", values_to = "RMSE") %>%
  mutate(log_RMSE = log(RMSE))

ggplot(results_long, aes(x = Replikasyon, y = log_RMSE, color = Parametre)) +
  geom_line(size = 1.2) +
  geom_point(size = 3) +
  scale_x_continuous(breaks = results$Replikasyon) +
  labs(
    title = "Replikasyon Sayısına Göre Madde Parametre Tahmin Hatası (Log(RMSE))",
    x = "Replikasyon Sayısı",
    y = "Log(RMSE)",
    color = "Parametre"
  ) +
  theme_minimal()

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Tablodaki sonuçlara göre, replikasyon sayısı arttıkça madde parametrelerinin tahmin hatası (RMSE) biraz azalıyor, yani tahminler biraz daha doğru oluyor diyebiliriz. a parametresi için hata b parametresine göre biraz daha düşük. Genel olarak, daha çok replikasyon yapıldığında parametre tahminlerinin daha güvenilir olduğu söylenebilir.Bununla birlikte, a parametresi modelin ayırt ediciliğini gösterdiği için, genellikle daha net ve stabil sonuçlar verir.Buna karşın, b parametresi ise madde zorluğu olarak daha değişken ve hassas olduğu için tahmin hatasının biraz daha yüksek olduğunu söyleyebiliriz.

CEVAP 4.b Üretilen madde cevaplarını kullanarak yetenek parametresini üç farklı kestirim yöntemi ile elde ediniz. Yenetenk parametresi kesitrimine ilişkin hata değerlerini RMSE değeri olarak rapolayınız. Replikasyon sayısınına bağlı olarak hata değerlerinin nasıl değiştiği tablo ve grafikler ile gösteriniz.

set.seed(123)
n <- 1000  # birey sayısı
theta_true <- rnorm(n, mean = 0, sd = 1)


# madde_par bir 20x2 matris: [,1] = a, [,2] = b
a <- madde_par[,1]
b <- madde_par[,2]

# Her birey için 20 maddeye yanıt üret
sim_data <- sapply(1:length(a), function(i) {
  p <- 1 / (1 + exp(-1.7 * a[i] * (theta_true - b[i])))
  rbinom(n, size = 1, prob = p)
})
sim_data <- as.data.frame(sim_data)





model <- mirt(sim_data, 1, itemtype = "2PL", verbose = FALSE)

## Warning: EM cycles terminated after 500 iterations.

theta_EAP <- fscores(model, method = "EAP")[,1]
theta_MAP <- fscores(model, method = "MAP")[,1]
theta_MLE <- fscores(model, method = "ML")[,1]




rmse <- function(true, est) sqrt(mean((true - est)^2))

rmse_EAP <- rmse(theta_true, theta_EAP)
rmse_MAP <- rmse(theta_true, theta_MAP)
rmse_MLE <- rmse(theta_true, theta_MLE)

rmse_table <- data.frame(
  Yontem = c("EAP", "MAP", "MLE"),
  RMSE = c(rmse_EAP, rmse_MAP, rmse_MLE)
)
print(rmse_table)

##   Yontem      RMSE
## 1    EAP 0.3607144
## 2    MAP 0.3611647
## 3    MLE       Inf

RMSE (Root Mean Square Error) değerleri gerçek yetenek değerleri ile kestirilen yetenek değerleri arasındaki ortalama hata büyüklüğü olduğundan küçük değerlerin daha iyi kestirim anlamına geldiğini söyleyebiliriz. EAP ve MAP sonuçlarına göre (yaklaşık 0.36 civarında) gerçek yetenek değerlerine ortalama 0.36 hata ile yaklaşıyorlar. Bunun normal bir hata büyüklüğü olduğunu yani kestirimlerin başarılı olduğunu söyleyebiliriz.