set.seed(123)
n_kisi <- 1000
n_madde <- 10
# Rastgele 0/1 matris oluştur
test_veri <- as.data.frame(
matrix(rbinom(n_kisi * n_madde, size = 1, prob = 0.5),
nrow = n_kisi, ncol = n_madde)
)
colnames(test_veri) <- paste0("Madde", 1:n_madde)
# Kontrol et
cat("Benzersiz degerler:", unique(unlist(test_veri)), "\n") # sadece 0 ve 1## Benzersiz degerler: 0 1## Boyut: 1000 10Fonksiyon
irt_simulate <- function(gercek_veri, n_kisi = 500, model = "2PL", seed = 42) {
set.seed(seed)
# ÖN KONTROL
cat("==========================================================\n")
cat(" ÖN KONTROL: Maddeler kontrol ediliyor...\n")
cat("==========================================================\n")
unique_kategori <- sapply(gercek_veri, function(x) length(unique(na.omit(x))))
problemli <- which(unique_kategori < 2)
if (length(problemli) > 0) {
cat(" UYARI: Asagidaki maddeler tek kategorili, cikarildi:\n")
cat(" ", names(problemli), "\n")
gercek_veri <- gercek_veri[, -problemli]
} else {
cat(" Tum maddeler OK. Devam ediliyor...\n")
}
# ADIM 1: Madde Parametrelerini Kestirim (a ve b)
cat("\n==========================================================\n")
cat(" ADIM 1: Madde parametreleri kestiriliyor...\n")
cat("==========================================================\n")
mirt_model <- mirt(data = gercek_veri,
model = 1,
itemtype = model,
verbose = FALSE)
madde_par <- coef(mirt_model, IRTpars = TRUE, simplify = TRUE)$items
madde_par <- as.data.frame(madde_par[, c("a", "b")])
cat(" Kestirim tamamlandi. Madde parametreleri (a = ayirt edicilik, b = gucluk):\n\n")
print(round(madde_par, 3))
# ADIM 2: Yetenek Parametrelerini Kestirim (theta)
cat("\n==========================================================\n")
cat(" ADIM 2: Yetenek parametreleri (theta) kestiriliyor...\n")
cat("==========================================================\n")
yetenek_par <- as.vector(fscores(mirt_model, method = "EAP"))
cat(" Kestirim tamamlandi. Theta dagilimi ozeti:\n\n")
print(summary(yetenek_par))
# ADIM 3: Madde Yanitlarini Uret (0/1 Veri)
cat("\n==========================================================\n")
cat(" ADIM 3: Yeni madde yanitlari uretiliyor...\n")
cat("==========================================================\n")
n_madde <- nrow(madde_par)
yeni_theta <- rnorm(n_kisi, mean = mean(yetenek_par), sd = sd(yetenek_par))
# 2PL Olasilik Fonksiyonu
p_dogru <- function(theta, a, b) {
exp(a * (theta - b)) / (1 + exp(a * (theta - b)))
}
# Her kisi x madde icin 0/1 yanit uret
uretilen_veri <- matrix(NA, nrow = n_kisi, ncol = n_madde)
for (j in 1:n_madde) {
p_j <- p_dogru(yeni_theta, madde_par$a[j], madde_par$b[j])
uretilen_veri[, j] <- rbinom(n_kisi, size = 1, prob = p_j)
}
colnames(uretilen_veri) <- colnames(gercek_veri)
uretilen_veri <- as.data.frame(uretilen_veri)
cat(" Uretim tamamlandi!\n")
cat(" Uretilen veri boyutu:", nrow(uretilen_veri), "kisi x", ncol(uretilen_veri), "madde\n")
# SONUÇ: Tum Ciktilari Listele
cat("\n==========================================================\n")
cat(" SONUC: Tum ciktilar hazir.\n")
cat(" - sonuc$madde_parametreleri --> a ve b degerleri\n")
cat(" - sonuc$yetenek_parametreleri --> theta degerleri\n")
cat(" - sonuc$uretilen_veri --> 0/1 yanit matrisi\n")
cat("==========================================================\n")
return(list(
madde_parametreleri = madde_par,
yetenek_parametreleri = yetenek_par,
uretilen_veri = uretilen_veri,
yeni_theta = yeni_theta,
mirt_model = mirt_model
))
}## ==========================================================
## ÖN KONTROL: Maddeler kontrol ediliyor...
## ==========================================================
## Tum maddeler OK. Devam ediliyor...
##
## ==========================================================
## ADIM 1: Madde parametreleri kestiriliyor...
## ==========================================================## Kestirim tamamlandi. Madde parametreleri (a = ayirt edicilik, b = gucluk):
##
## a b
## Madde1 0.092 0.304
## Madde2 0.010 1.174
## Madde3 2.730 0.018
## Madde4 -0.023 1.048
## Madde5 -0.057 -0.771
## Madde6 0.173 0.163
## Madde7 0.105 0.536
## Madde8 -0.149 -0.027
## Madde9 0.024 1.173
## Madde10 0.204 0.139
##
## ==========================================================
## ADIM 2: Yetenek parametreleri (theta) kestiriliyor...
## ==========================================================
## Kestirim tamamlandi. Theta dagilimi ozeti:
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.8799776 -0.6660619 -0.4658709 -0.0000037 0.6786861 0.8990198
##
## ==========================================================
## ADIM 3: Yeni madde yanitlari uretiliyor...
## ==========================================================
## Uretim tamamlandi!
## Uretilen veri boyutu: 300 kisi x 10 madde
##
## ==========================================================
## SONUC: Tum ciktilar hazir.
## - sonuc$madde_parametreleri --> a ve b degerleri
## - sonuc$yetenek_parametreleri --> theta degerleri
## - sonuc$uretilen_veri --> 0/1 yanit matrisi
## ==========================================================## a b
## Madde1 0.09240790 0.30408039
## Madde2 0.01020969 1.17382969
## Madde3 2.73029685 0.01774513
## Madde4 -0.02287794 1.04782305
## Madde5 -0.05738558 -0.77094769
## Madde6 0.17326537 0.16286391
## Madde7 0.10499395 0.53581089
## Madde8 -0.14907638 -0.02719956
## Madde9 0.02384501 1.17261692
## Madde10 0.20360251 0.13897281## [1] -0.7103485 -0.6175164 -0.6767921 0.7605313 -0.5754601 0.5490654
## [7] -0.7413926 -0.5543508 -0.5677122 -0.8719365 0.7285212 -0.7134379
## [13] 0.7045343 0.8054067 0.6745075 -0.4745256 0.6967744 -0.7364665
## [19] -0.7398303 -0.6327117 0.8990198 -0.6048964 0.6988265 -0.6895757
## [25] -0.5396047 -0.7718638 0.6663684 0.6448349 -0.5388370 0.6662130
## [31] 0.5669884 0.8440461 0.7996521 -0.7456178 -0.7859187 0.6261422
## [37] 0.8990198 0.5872363 0.5541295 0.7172773 -0.5209073 -0.7085363
## [43] -0.7725748 0.6672224 0.7165638 0.5718211 0.6083238 -0.7667522
## [49] -0.6368670 -0.7147485 0.6839791 -0.8056022 0.7474023 -0.6403402
## [55] 0.7335802 -0.6857094 0.6927510 0.7119888 0.7045343 0.8055477
## [61] -0.7930793 0.8317670 0.5716442 0.6544742 -0.8186939 -0.6836834
## [67] -0.7286661 -0.6853185 0.6808356 0.7782886 -0.5908791 0.7850750
## [73] 0.8440461 -0.5563333 0.5631738 0.4606629 0.7864052 -0.6229391
## [79] 0.6914983 -0.7993132 -0.6967675 0.8850713 0.7043323 -0.6471902
## [85] -0.5688984 0.7165638 -0.5381891 0.7097246 0.5599964 -0.7141609
## [91] -0.7305388 -0.6579596 -0.7271119 -0.6200165 -0.6894328 0.5631738
## [97] 0.7468695 -0.6546775 0.6779835 -0.7211125 0.7468695 -0.6234536
## [103] -0.5740405 0.6109756 0.6927510 0.7789911 -0.7281296 -0.5664600
## [109] 0.8245505 0.6261422 0.5892168 -0.6771594 -0.6950138 -0.6595017
## [115] 0.6441417 -0.7667522 0.6933166 -0.6338469 0.7104362 0.6533457
## [121] 0.6015901 0.7992484 -0.5577021 -0.7667522 0.5546238 -0.5678913
## [127] 0.6247562 0.5715882 -0.5915978 0.7175403 -0.6838196 -0.5999640
## [133] -0.4453145 -0.7991623 0.7605313 0.8054067 -0.7137654 0.7691484
## [139] 0.6808356 0.6912690 0.7470072 -0.8114569 -0.4973731 -0.7595206
## [145] -0.7723640 -0.7421560 -0.6640519 -0.8463154 0.6367312 0.6751810
## [151] -0.6267532 0.5011272 0.6407246 -0.7085363 -0.6654794 -0.7141609
## [157] -0.5262624 0.6655139 -0.6477097 -0.6892514 0.8850713 -0.5866907
## [163] 0.7732988 0.6544742 0.6207874 0.6624513 -0.6267532 0.6622961
## [169] 0.6909774 -0.7804854 -0.6579596 -0.6169513 -0.6970325 -0.5329390
## [175] 0.7169541 -0.7399203 -0.7305388 0.7279925 -0.4504046 0.6806511
## [181] 0.6015901 0.7413890 -0.8122662 -0.6787135 0.6497708 -0.6021337
## [187] 0.6617812 0.7335802 -0.5747758 0.7605313 0.6389307 -0.7799374
## [193] 0.5421606 0.6502400 0.6357174 0.6496308 0.6295529 0.7104362
## [199] -0.6108475 -0.5717766 0.8512740 0.7923377 0.6181026 -0.7399203
## [205] -0.8114569 0.5182582 -0.5388370 0.5964210 0.4720044 -0.6093348
## [211] -0.6825254 0.6295529 -0.5506416 -0.5783884 0.6657554 -0.6417154
## [217] -0.6162559 0.6438120 0.8387053 -0.6146000 -0.7082871 0.6244446
## [223] 0.5426527 -0.6267532 0.5769004 -0.6950138 -0.6169513 -0.5662244
## [229] 0.7562123 -0.7095950 0.8245973 0.7167510 -0.5806150 0.6015473
## [235] -0.8719365 -0.7917257 0.6786861 -0.5872492 -0.7934879 0.6195997
## [241] -0.5037635 0.5135988 -0.4917104 0.7155174 0.6932723 0.7696861
## [247] -0.7626131 0.8116191 -0.5558319 -0.6176989 0.6927510 0.5659952
## [253] -0.5872492 0.5696768 -0.6053495 -0.7152821 0.5663423 -0.5804842
## [259] -0.7799374 0.7342986 0.6604579 0.5473604 -0.6327117 -0.6231220
## [265] 0.7466800 -0.7470936 -0.8659330 0.5764021 -0.6586382 -0.6582038
## [271] -0.5967116 -0.6787135 -0.6596590 -0.7729568 0.6411032 0.6019203
## [277] 0.8113628 -0.6368670 -0.6288889 -0.5264387 -0.7343373 0.7221256
## [283] -0.5687193 0.5476632 0.6190502 -0.7421560 0.5011272 -0.7804854
## [289] 0.6481162 0.7023365 -0.6901487 0.6964822 -0.7652644 -0.6274514
## [295] 0.7541574 -0.5967116 -0.6963146 0.7056357 0.8512740 0.7305101
## [301] -0.6635290 0.6439876 -0.7991623 0.5187462 0.5951877 -0.7934879
## [307] 0.7283172 0.6657554 0.7596437 -0.5630010 0.6617812 -0.7549535
## [313] -0.4339398 0.5369287 0.7543471 0.5787604 -0.6159998 0.6083238
## [319] -0.5862408 0.6071417 0.7753699 0.6510302 -0.7934879 0.7673682
## [325] -0.5806580 -0.7082871 -0.6599794 -0.5383812 -0.7554959 0.7210769
## [331] -0.7800790 -0.6041525 0.7030459 0.5951877 0.6786861 -0.7271119
## [337] -0.5783884 0.7100692 0.7341101 0.7341101 0.6745075 -0.7266210
## [343] 0.6357174 0.4955269 -0.7178687 -0.8799776 0.7169541 -0.8050484
## [349] -0.7173346 -0.7800790 0.5841139 0.6874551 -0.5927890 0.6241911
## [355] -0.6637149 -0.5801092 -0.6506777 -0.5933404 -0.6453370 -0.6229391
## [361] 0.7096800 0.8317670 -0.6524613 0.7997928 -0.6827127 -0.6270913
## [367] -0.6727433 0.5593520 0.5663423 0.6194043 0.6423584 -0.6546189
## [373] -0.7026308 -0.7042462 -0.7286661 -0.6338469 0.5961127 -0.6093348
## [379] -0.7023048 0.7850750 -0.6393047 -0.6327117 0.7782886 -0.7858720
## [385] 0.7225167 -0.7077698 0.7041461 0.7670217 -0.6732678 0.5546238
## [391] 0.6629083 -0.7364665 -0.7137654 -0.8186939 0.6909774 0.4891921
## [397] -0.6672123 0.7172773 0.5579441 0.5773991 0.7097246 0.6314123
## [403] -0.5386869 0.7691484 0.5074880 -0.5632364 -0.6835247 0.6528326
## [409] 0.5301805 -0.7021598 0.6438120 0.8915993 -0.6546775 -0.5506416
## [415] 0.7354128 -0.6911825 -0.7095950 -0.6295877 0.6300614 -0.5266299
## [421] 0.5840305 0.8582688 -0.6466711 -0.7626131 0.7994585 -0.7209223
## [427] -0.8109017 0.7410173 -0.7211580 0.6357174 0.7748310 -0.6092913
## [433] -0.5214020 0.7925308 -0.6825254 0.8518306 -0.6894328 0.5893946
## [439] -0.7211580 0.5469949 0.6263800 -0.7930793 -0.5837292 -0.6036421
## [445] -0.7666033 0.7881547 -0.5420313 -0.6660619 -0.7859187 0.5664919
## [451] 0.7979112 -0.8192512 -0.6115414 0.7297613 0.5887159 0.8715014
## [457] 0.7097246 -0.6232986 0.7279476 -0.5740405 -0.6159998 0.5754272
## [463] 0.7369214 -0.5866907 -0.5325916 -0.7042462 -0.6967073 -0.6471902
## [469] -0.7591407 -0.5330884 -0.7668136 -0.7042010 -0.7341463 0.7732988
## [475] -0.5208653 -0.6361661 0.6441417 -0.5687193 0.6712051 -0.5205405
## [481] -0.8330253 -0.5801092 0.6655139 -0.5278481 0.6622102 -0.6660619
## [487] -0.6654794 0.7680966 -0.8324652 0.8445539 0.7994585 -0.6695759
## [493] 0.6295529 0.8309631 0.6671637 0.6678939 0.5598472 -0.5383812
## [499] 0.6971616 -0.5681731 0.6436302 0.6598997 0.5998357 -0.8585046
## [505] 0.6564484 -0.7474422 0.7881547 0.6874551 0.6973409 0.5891742
## [511] -0.7134379 -0.7595206 -0.6232986 -0.6767921 0.8856355 0.4955269
## [517] -0.6294043 0.7996521 0.6479341 -0.6270913 0.5764021 0.5079331
## [523] 0.6726165 -0.6767921 -0.5334358 0.7114636 -0.5386869 -0.7266210
## [529] 0.5950093 0.6389307 0.6859768 0.6502400 0.6932723 0.6574764
## [535] 0.6914983 -0.5140696 -0.6288889 0.7169541 0.7979112 0.7539963
## [541] 0.6190933 -0.7859187 0.6316086 0.6604142 0.7850750 0.5897175
## [547] -0.6710065 0.6844989 -0.7991623 -0.7724104 0.7655555 -0.6203973
## [553] 0.6881603 -0.6160581 0.7244460 0.7866141 -0.5278481 -0.5982169
## [559] -0.6728789 -0.7348754 -0.7476346 0.7114636 -0.6038975 -0.8522700
## [565] -0.6033872 0.5897175 -0.7103485 -0.6835247 -0.7470936 0.7605313
## [571] -0.5538501 0.8387053 -0.6703836 -0.5089999 -0.6890923 -0.6048964
## [577] -0.7008559 -0.6330945 -0.6102923 0.8317670 0.8309631 -0.5806150
## [583] 0.8055941 -0.5383812 -0.8260003 0.5591759 0.8375987 0.6975267
## [589] 0.7379831 -0.6635290 0.7471678 0.5421188 0.6899894 -0.8525345
## [595] -0.8251392 0.6751810 -0.4572162 0.5874706 0.5303143 -0.7857300
## [601] -0.7534113 0.7546918 -0.6531976 0.5537840 -0.7919836 -0.5662244
## [607] 0.6372990 -0.5553309 0.6677384 0.6384205 0.5187462 -0.5453672
## [613] -0.8109017 -0.6710940 0.7865531 0.6933166 -0.7339842 0.6881603
## [619] -0.6200165 0.5964210 0.5773991 -0.6160581 -0.6109860 0.5469949
## [625] 0.6481162 0.5598472 -0.4921999 -0.8050013 -0.7346841 -0.8525345
## [631] 0.7730466 0.6623933 -0.7554959 -0.6797594 -0.6801494 -0.8056022
## [637] 0.7415492 0.6876401 0.7696861 -0.6236799 -0.7723640 0.6929521
## [643] -0.5999640 -0.7534113 0.5701739 0.7474023 0.5717381 -0.7216929
## [649] 0.7283172 -0.6466711 0.6881603 0.8054067 0.5315187 -0.5677122
## [655] -0.6853185 0.7789911 -0.6635290 0.5421188 -0.5325916 0.5891742
## [661] -0.6411974 -0.7337995 0.4669379 -0.5329390 -0.5813438 0.6117539
## [667] -0.7081417 0.6474221 -0.6039556 -0.6892514 -0.6491278 -0.7154719
## [673] 0.6207874 -0.7476346 -0.6710065 -0.4572162 -0.6394824 0.6804510
## [679] -0.6712520 -0.7405612 -0.7989036 -0.5999640 0.8052126 0.6446530
## [685] -0.6327117 0.5787604 -0.7527060 0.7096800 -0.5331307 0.7916314
## [691] -0.6026437 -0.5967116 -0.5262624 0.6438120 0.5537840 0.5717381
## [697] -0.6600238 0.6839791 0.6382390 0.6241911 -0.5367737 0.7297613
## [703] -0.7266210 0.7170139 -0.6911825 0.7435935 0.7543926 0.7091553
## [709] 0.5476632 -0.6115414 0.6142014 -0.7137654 -0.8181371 -0.6394824
## [715] -0.7531575 0.8512740 -0.7305388 -0.5038053 0.7468695 0.5962425
## [721] 0.6407246 -0.4916688 0.7979112 0.7866141 0.6441417 0.8582688
## [727] 0.8850713 -0.5212111 -0.7723640 -0.6767921 -0.6236799 0.7793685
## [733] 0.6015901 -0.6343634 0.7238733 -0.7078150 0.6574764 0.6859768
## [739] 0.6051478 0.5591759 -0.5380121 -0.6368670 -0.7610048 -0.7042010
## [745] 0.6914983 0.6844989 -0.6289749 -0.6241509 -0.6980692 0.6301046
## [751] 0.5598472 -0.6712520 -0.5331307 0.5246154 -0.5664600 -0.6725567
## [757] -0.7042462 0.6175965 0.7223727 -0.6241509 0.6014173 -0.6725567
## [763] 0.6190933 -0.6892514 0.6881603 -0.5808379 0.6493908 -0.6967675
## [769] 0.6537262 0.6070849 -0.5625825 -0.5500472 -0.6595017 -0.6343634
## [775] 0.8317670 0.8323188 0.5718211 -0.5740405 -0.6710940 -0.5678913
## [781] -0.7554959 0.6085184 -0.5577021 0.7748310 0.7546918 0.6617812
## [787] 0.6421191 -0.7273479 0.5893946 0.7341101 -0.5806150 -0.4916688
## [793] 0.5715882 -0.6236799 0.7541574 -0.7917257 0.6136531 0.4724848
## [799] -0.8799776 0.7238733 -0.6546189 0.7788284 -0.6957313 -0.5563333
## [805] 0.7093862 0.6751810 -0.6361661 0.7782886 0.7730466 0.7748310
## [811] 0.4669379 -0.6668242 0.7655555 0.6658962 0.6317896 -0.7857300
## [817] -0.8330253 0.6367312 -0.7527060 0.6316086 0.7621158 -0.6418131
## [823] 0.7004984 0.7737767 -0.6540000 -0.6836834 0.7539963 -0.7724104
## [829] 0.6839791 -0.5630849 0.5927579 0.7881547 0.7788284 0.6190933
## [835] -0.5683947 -0.7264142 0.6622102 0.6968188 0.7412064 0.5631738
## [841] -0.5420313 -0.6108040 0.6195997 0.6672224 0.7169541 0.5424628
## [847] 0.8059524 -0.7591407 -0.7534113 -0.6162559 0.6604142 -0.7991623
## [853] -0.8181371 -0.4390107 0.6936589 -0.6239679 -0.7795310 0.5417958
## [859] -0.7085363 0.7925308 0.7670217 -0.6670261 -0.7584337 -0.8324652
## [865] 0.5187462 0.7413890 0.6367312 0.8245505 -0.7152220 -0.7095950
## [871] 0.5819598 -0.7042462 -0.5982169 -0.6764025 0.7732988 0.6497708
## [877] -0.6295877 -0.5380121 -0.7212963 0.6357174 -0.6526181 0.6015901
## [883] 0.6175965 0.7041461 -0.7396678 0.8440461 0.7613573 -0.5740405
## [889] 0.6749813 0.6656973 -0.4860587 0.7300256 -0.6764025 -0.6673475
## [895] -0.6272682 -0.6705700 0.6602594 0.6438120 -0.8522700 -0.6544766
## [901] 0.7548819 0.6804951 0.7379831 -0.6582038 0.6014173 0.8304116
## [907] 0.7748310 -0.6048964 -0.5749979 0.6712051 0.5959614 -0.7209677
## [913] 0.7471678 -0.5754601 0.5421188 0.8856355 0.7675593 0.6495453
## [919] 0.7412064 0.7354128 -0.6451953 0.6263800 0.8381518 -0.4916688
## [925] 0.7468695 -0.5208653 0.7412064 0.8717641 0.4720044 0.7737767
## [931] -0.6705700 -0.8109017 -0.7337995 -0.8186939 0.7228856 -0.6725567
## [937] 0.6859177 -0.7666033 -0.7021000 0.5011272 -0.6832928 0.7038529
## [943] -0.6957313 -0.6241509 -0.6601808 0.6662130 -0.5987121 0.6015473
## [949] -0.6531384 -0.6041525 -0.7336083 0.5542782 0.7546918 -0.7727691
## [955] -0.7178687 0.6804951 -0.6108040 0.7782886 -0.5209073 -0.7405612
## [961] 0.5950093 -0.7857300 0.7417831 -0.7667522 0.6142014 -0.6270913
## [967] 0.7860727 -0.7403694 -0.7095950 -0.5458664 0.5664919 -0.6654794
## [973] -0.6358328 -0.6672123 0.6550460 -0.6411974 0.8131134 -0.6453370
## [979] 0.6443233 -0.4968827 0.6367891 -0.6160581 -0.7036698 -0.7042010
## [985] -0.6396665 0.8251474 0.6872544 -0.5866907 -0.8330253 0.5711469
## [991] 0.6194043 -0.6295877 0.6109756 -0.6453370 0.5426527 -0.6038975
## [997] 0.7927173 -0.7341463 -0.5558319 0.8454091## Madde1 Madde2 Madde3 Madde4 Madde5 Madde6 Madde7 Madde8 Madde9 Madde10
## 1 1 0 1 1 1 0 1 0 0 1
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## 298 1 0 1 1 0 0 1 1 0 0
## 299 1 1 1 0 1 1 0 0 0 0
## 300 1 0 0 1 1 1 1 0 0 1Bu ödev kapsamında, hazır bir veri setinden IRT parametrelerini kestirip yeni veri üreten bir R fonksiyonu yazmayı öğrendim. Teoride bildiğim 2PL modelinin (ayırt edicilik (a) ve güçlük (b) parametrelerinin) pratikte nasıl işlediğini, mirt paketiyle nasıl kestirildiğini ve bu parametreler kullanılarak gerçekçi 0-1 yanıt verilerinin nasıl simüle edilebileceğini bizzat deneyimledim. Süreç boyunca karşılaştığım hatalar (varyansı sıfır maddeler, politomik veri sorunları, sözdizimi hataları) başlangıçta zorlayıcı olsa da her birini çözerken hem R’a hem de IRT’nin arka planındaki mantığa daha iyi hâkim olduğumu fark ettim.