📊 Bilgisayar Ortamında Bireye Uyarlanmış Test Nasıl Geliştirilir

class: center, middle, inverse, title-slide

.title[
# 📊 Bilgisayar Ortamında Bireye Uyarlanmış Test Nasıl Geliştirilir
]
.subtitle[
## 📊 Madde Tepki Kuramı ile Madde Analizi Uygulaması
]
.author[
### Dr. Kubra Atalay Kabasakal
]
.date[
### 13 Eylül 2022 - Boğaziçi Üniversitesi
]

---

.center[

## <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M0 383.9l64 .0404c17.75 0 32-14.29 32-32.03V128.3L0 128.3V383.9zM48 320.1c8.75 0 16 7.118 16 15.99c0 8.742-7.25 15.99-16 15.99S32 344.8 32 336.1C32 327.2 39.25 320.1 48 320.1zM348.8 64c-7.941 0-15.66 2.969-21.52 8.328L228.9 162.3C228.8 162.5 228.8 162.7 228.6 162.7C212 178.3 212.3 203.2 226.5 218.7c12.75 13.1 39.38 17.62 56.13 2.75C282.8 221.3 282.9 221.3 283 221.2l79.88-73.1c6.5-5.871 16.75-5.496 22.62 1c6 6.496 5.5 16.62-1 22.62l-26.12 23.87L504 313.7c2.875 2.496 5.5 4.996 7.875 7.742V127.1c-40.98-40.96-96.48-63.88-154.4-63.88L348.8 64zM334.6 217.4l-30 27.49c-29.75 27.11-75.25 24.49-101.8-4.371C176 211.2 178.1 165.7 207.3 138.9L289.1 64H282.5C224.7 64 169.1 87.08 128.2 127.9L128 351.8l18.25 .0369l90.5 81.82c27.5 22.37 67.75 18.12 90-9.246l18.12 15.24c15.88 12.1 39.38 10.5 52.38-5.371l31.38-38.6l5.374 4.498c13.75 11 33.88 9.002 45-4.748l9.538-11.78c11.12-13.75 9.036-33.78-4.694-44.93L334.6 217.4zM544 128.4v223.6c0 17.62 14.25 32.05 31.1 32.05L640 384V128.1L544 128.4zM592 352c-8.75 0-16-7.246-16-15.99c0-8.875 7.25-15.99 16-15.99S608 327.2 608 336.1C608 344.8 600.8 352 592 352z"/></svg>: Kübra Atalay Kabasakal
 
--

## <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Hacettepe Üniversitesi](https://avesis.hacettepe.edu.tr/katalay)
 
--
 
]

---

## Madde Tepki Kuramı (MTK) ile Madde Analizi Uygulaması

- iki kategorili (doğru-yanlış) MTK modelleri

- madde ve yetenek parametresi kestirimi

- madde karakteristik eğrisi çizimi ve yorumlanması

- model-veri uyumunun incelenmesi

- madde ve test bilgi fonksiyonu

---
## Bugun size ne lazım?
- paketler

```r
library(mirt)
library(ggplot2)
library(psych)
library(readr)
```

- kullanılacak veriler
 - <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Sunu veri](https://raw.githubusercontent.com/atalay-k/mirt_k/main/dichotomous.csv)
 
 - <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Ödev Veri](https://raw.githubusercontent.com/atalay-k/mirt_k/main/dat1.csv)

- kodlar
 - <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Kodlar](https://raw.githubusercontent.com/atalay-k/mirt_k/main/kodlar.R) 
 
 - <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Sunum](https://atalay-k.github.io/mirt_k/#1)

---
## İki kategorili (doğru-yanlış) MTK modelleri

- MTK **gizil özelliğin farklı düzeylerindeki bireylerin maddeyi nasıl yanıtlayacağını matematiksel olarak gösterir.**

- **Sonsuz sayıda** MTK modeli tasarlamak mümkün olmakla birlikte, **az sayıda** model uygulamada kullanılmaktadır.

- En popüler tek boyutlu MTK modelleri arasındaki temel ayrım, maddeleri tanımlamak için kullanılan **parametrelerin sayısındadır. **

- En popüler dört tek boyutlu *iki kategorili madde yanıt verisi* MTK modelleri
  - bir-parametreli lojistik  (1-PL)
  - iki-parametreli lojistik  (2-PL)
  - üç-parametreli lojistik   (3-PL) 
  - dört-parametreli lojistik (4-PL) modellerdir.

---
## Bir-Parametreli Lojistik (1-PL) Model

- **1PL model** yaygın olarak kullanılan MTK modellerindendir.

- **1PL model** için madde karakteristik eğrileri aşağıdaki eşitlikle elde edilir:

`$$P_i(\theta) = \frac{exp(\theta-b_i)}{1+exp(\theta-b_i)} = \frac{1}{1+exp[-(\theta-b_i)]}$$`
`$$ln(\frac{P_i(\theta)}{1-P_i(\theta)})=\theta - b_{i}$$`

Burada,
- `$P_i(θ)$` :  θ yetenek düzeyindeki bir bireyin i maddesini doğru 	yanıtlama olasılığı
- `$b_i$`:       i maddesinin güçlük parametresi

---
## 1-PL Model

- Bir madde için `$b_i$` parametresi **yetenek ölçeğinde maddeyi doğru yanıtlama olasılığının 0.5** olduğu noktadır.

- Bu parametre yer parametresi olup **yetenek ölçeğiyle ilişkili olarak madde karakteristik eğrisinin pozisyonunu** belirtir.

-  `$b_i$` parametresinin **daha büyük değerleri**, bir bireyin maddeyi doğru yanıtlamak için **%50 şansa sahip olması için daha büyük yetenek düzeyine sahip olmasını** gerektirir. Diğer bir ifadeyle  `$b_i$` parametresinin daha büyük değerleri, **daha zor** maddeyi ifade eder.

- **Zor maddeler yetenek ölçeğinin sağında** veya daha yüksek ucundadır.
  - **Kolay maddeler yetenek ölçeğinin solunda** veya daha düşük ucundadır.

---
## 1-PL Model

- Bir grubun **yetenek düzeyleri ortalaması 0 ve standart sapması 1 olacak** şekilde ölçeklendiğinde, `$b_i$` değerleri genel olarak **-2.0 ile +2.0** arasında değişir.

- `$b_i$` değerleri **-2.0’ye yakın** olan maddeler bireyler için oldukça **kolay,***

- `$b_i$` değerleri **+2.0’ye** yakın olan maddeler bireyler için oldukça **zor** maddelerdir.

--
  
-   `$b_i$` yetenek düzeyiyle **aynı ölçektedir.**

---
## 1-PL Model

.pull-left[
- Elimizde 1PL modelde uygun dört maddelik bir testte yer alan madde parametreleri bulunsun.

- Madde 1 için `$b_1 =  1.0$`

- Madde 2 için `$b_2 =  2.0$`

- Madde 3 için `$b_3 = -1.0$`

- Madde 4 için `$b_4 =  0.0$`

`$$P_i(\theta) = \frac{1}{1+exp[-(\theta-b_i)]}$$`
]

.pull-right[
![](index_files/figure-html/unnamed-chunk-2-1.png)
]

---
## 1-PL Model

.pull-left[

- **Eğriler yetenek ölçeğinde** sadece **yerleri bakımından farklılık gösterir.**

- 1-PL modelinde birey performansını etkileyen tek madde özelliği **madde güçlüğüdür. **

]
.pull-right[

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---
## mirt paketi yüklenmesi

- Analizler **mirt** paketinde yapılacaktır.
- Paketin yüklenmesi ve aktivite edilmesi aşağıdaki kodlarla sağlanır.

```r
# install.packages("mirt")
library("mirt")
```

- MTK analizlerinin yapılacağı paketlere **ltm** Rizopoulos (2006) ve **irtoys** Partchev vd. (2017) örnek verilebilir. Choi ve Asilkalkan (2019) <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg> [makalesinde](https://doi.org/10.1080/15366367.2019.1586404) 45 farklı MTK paketine ilişkin açıklamalar bulunmaktadır.

---
## 1PL modelin hazırlanması

- İlk olarak test edilecek model hazırlanmalıdır.

```r
birpl_model <- "F = 1-15
 CONSTRAIN = (1-15, a1)"
```

- Kodun ilk satırı,  tek bir gizil özelliğin (F'nin) veri setindeki 1 ile 15 arasındaki sütunlardaki maddeler tarafından ölçüldüğünü göstermektedir

- CONSTRAIN ile başlayan ikinci satır ise 1'den 15'e kadar olan sütunlardaki maddeleri aynı madde ayırt ediciliğine (a1) sahip olacak şekilde sınırlar.

- Sadece ilk 10 maddede madde ayırt ediciliğini aynı olacak şekilde sınırlamak isterseniz modeli aşağıdaki gibi düzenleyebilirsiniz.

```r
birpl_model_v1 <- "F = 1-15
 CONSTRAIN = (1-10, a1)"
```

---
## Veri aktarımı

```r
library(readr)
ikikategorili <- read_csv("dichotomous.csv")[,-1]
head(ikikategorili[,1:5])
```

```
## # A tibble: 6 × 5
## V1 V2 V3 V4 V5
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0 0 0 0 0
## 2 0 0 0 0 0
## 3 0 0 0 1 0
## 4 0 0 0 0 0
## 5 1 0 0 1 1
## 6 0 0 0 0 0
```
]
--

.pull-right[

```r
summary(ikikategorili)
```

```
##        V1              V2              V3              V4              V5      
##  Min.   :0.000   Min.   :0.000   Min.   :0.000   Min.   :0.000   Min.   :0.00  
##  1st Qu.:0.000   1st Qu.:0.000   1st Qu.:0.000   1st Qu.:0.000   1st Qu.:0.00  
##  Median :0.000   Median :0.000   Median :0.000   Median :0.000   Median :0.00  
##  Mean   :0.182   Mean   :0.074   Mean   :0.175   Mean   :0.164   Mean   :0.28  
##  3rd Qu.:0.000   3rd Qu.:0.000   3rd Qu.:0.000   3rd Qu.:0.000   3rd Qu.:1.00  
##  Max.   :1.000   Max.   :1.000   Max.   :1.000   Max.   :1.000   Max.   :1.00  
##        V6              V7             V8              V9             V10       
##  Min.   :0.000   Min.   :0.00   Min.   :0.000   Min.   :0.000   Min.   :0.000  
##  1st Qu.:0.000   1st Qu.:0.00   1st Qu.:0.000   1st Qu.:0.000   1st Qu.:1.000  
##  Median :1.000   Median :0.00   Median :0.000   Median :0.000   Median :1.000  
##  Mean   :0.566   Mean   :0.44   Mean   :0.479   Mean   :0.435   Mean   :0.915  
##  3rd Qu.:1.000   3rd Qu.:1.00   3rd Qu.:1.000   3rd Qu.:1.000   3rd Qu.:1.000  
##  Max.   :1.000   Max.   :1.00   Max.   :1.000   Max.   :1.000   Max.   :1.000  
##       V11             V12            V13             V14             V15       
##  Min.   :0.000   Min.   :0.00   Min.   :0.000   Min.   :0.000   Min.   :0.000  
##  1st Qu.:0.000   1st Qu.:1.00   1st Qu.:1.000   1st Qu.:0.000   1st Qu.:0.000  
##  Median :0.000   Median :1.00   Median :1.000   Median :1.000   Median :1.000  
##  Mean   :0.123   Mean   :0.76   Mean   :0.936   Mean   :0.612   Mean   :0.541  
##  3rd Qu.:0.000   3rd Qu.:1.00   3rd Qu.:1.000   3rd Qu.:1.000   3rd Qu.:1.000  
##  Max.   :1.000   Max.   :1.00   Max.   :1.000   Max.   :1.000   Max.   :1.000
```
]

---
## Veri aktarımı

- Veriyi 1-0 olarak puanlamak için **key2binary()**
fonksiyonunu kullanabilirsiniz.

```r
veri <- read_csv("veri.csv")
library(mirt)
dat1 <- key2binary(veri[,-1],
 key = c(2,3,4,5,2,3,4,5,2,3,4,5))
```
.pull-left[

```r
head(veri[,1:5])
```

```
## # A tibble: 6 × 5
## Subject Rot1_2 Rot1_3 Rot1_4 Rot1_5
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 2 1 0 0
## 2 2 2 2 1 1
## 3 3 2 1 4 1
## 4 4 2 2 3 1
## 5 5 2 3 4 5
## 6 7 2 3 0 2
```
]
.pull-right[

```r
head(dat1)
```

```
##      Rot1_2 Rot1_3 Rot1_4 Rot1_5 Rot2_2 Rot2_3 Rot2_4 Rot2_5 Rot3_2 Rot3_3
## [1,]      0      0      0      0      0      0      0      0      0      0
## [2,]      0      0      0      0      0      0      0      0      0      0
## [3,]      0      0      0      0      0      0      0      0      0      0
## [4,]      0      0      0      0      0      0      0      0      0      0
## [5,]      0      0      0      0      0      0      0      0      0      0
## [6,]      0      0      0      0      0      0      0      0      0      0
##      Rot3_4 Rot3_5
## [1,]      0      0
## [2,]      0      0
## [3,]      0      0
## [4,]      0      0
## [5,]      0      0
## [6,]      0      0
```
]

---
## Madde Istatistikleri

```r
itemstats(ikikategorili)
```

```
## $overall
##     N mean_total.score sd_total.score ave.r sd.r alpha
##  1000             6.68            2.7 0.114 0.11 0.688
## 
## $itemstats
##        N  mean    sd total.r total.r_if_rm alpha_if_rm
## V1  1000 0.182 0.386   0.626         0.526       0.645
## V2  1000 0.074 0.262  -0.169        -0.261       0.717
## V3  1000 0.175 0.380   0.382         0.253       0.678
## V4  1000 0.164 0.370   0.438         0.317       0.671
## V5  1000 0.280 0.449   0.427         0.277       0.676
## V6  1000 0.566 0.496   0.501         0.344       0.666
## V7  1000 0.440 0.497   0.536         0.384       0.660
## V8  1000 0.479 0.500   0.532         0.379       0.661
## V9  1000 0.435 0.496   0.525         0.372       0.662
## V10 1000 0.915 0.279   0.291         0.193       0.684
## V11 1000 0.123 0.329   0.375         0.263       0.677
## V12 1000 0.760 0.427   0.414         0.270       0.676
## V13 1000 0.936 0.245   0.277         0.190       0.684
## V14 1000 0.612 0.488   0.517         0.366       0.663
## V15 1000 0.541 0.499   0.522         0.368       0.663
## 
## $proportions
##         0     1
## V1  0.818 0.182
## V2  0.926 0.074
## V3  0.825 0.175
## V4  0.836 0.164
## V5  0.720 0.280
## V6  0.434 0.566
## V7  0.560 0.440
## V8  0.521 0.479
## V9  0.565 0.435
## V10 0.085 0.915
## V11 0.877 0.123
## V12 0.240 0.760
## V13 0.064 0.936
## V14 0.388 0.612
## V15 0.459 0.541
```

---
## Parametre Kestirimleri

- mirt paketinin **mirt()** fonksiyonu temel olarak data ve model olarak iki argümanla çalışır.

- **ikikategorili** veri setinin **birpl_model** modeli için testi aşağıdaki gibi yapılabilir.

```r
birpl_model <- "F = 1-15
 CONSTRAIN = (1-15, a1)"
birpl_uyum <- mirt(data = ikikategorili, model = birpl_model,SE=TRUE)
```

- birpl_uyum nesnesi
 - parametre kestirimlerini
 - gizil özelliğin ortalamasını
 - gizil özelliğin varyans-kovaryans matrisini
 - kestirim sürecine ilişkin ek bilgileri içerir.

---
## Varsayımlar
- Tek boyutluluk tek boyutlu MTK modelleri, tüm maddelerin tek bir sürekli gizli değişkeni ölçtüğünü varsayar. 
- Tek boyutluluk varsayımını test etmenin farklı yolları vardır. Örneğin, kavramsal olarak genel bir faktör tarafından hesaplanan ölçek puanlarındaki varyans yüzdesini yansıtan McDonald's hiyerarşik Omega'sını  değerlendirebiliriz.

```r
library(psych)
summary(omega(ikikategorili, plot = F))
```

```
## Omega 
## omega(m = ikikategorili, plot = F)
## Alpha: 0.72 
## G.6: 0.71 
## Omega Hierarchical: 0.58 
## Omega H asymptotic: 0.78 
## Omega Total 0.74 
## 
## With eigenvalues of:
## g F1* F2* F3* 
## 1.86 0.69 0.20 0.28 
## The degrees of freedom for the model is 63 and the fit was 0.05 
## The number of observations was 1000 with Chi Square = 52.2 with prob < 0.83 
## 
## The root mean square of the residuals is 0.02 
## The df corrected root mean square of the residuals is 0.03 
## 
## RMSEA and the 0.9 confidence intervals are 0 0 0.012
## BIC = -383Explained Common Variance of the general factor = 0.61 
## 
## Total, General and Subset omega for each subset
## g F1* F2* F3*
## Omega total for total scores and subscales 0.74 0.60 0.53 0.39
## Omega general for total scores and subscales 0.58 0.36 0.43 0.26
## Omega group for total scores and subscales 0.09 0.25 0.09 0.14
```

---
## Varsayımlar

- Madde çiftlerinin yerel bağımsızlığını kontrol etmek için ise Yen'in Q3 istatistiği kullanılabilir.

- i. ve j. maddelerinden elde edilen artıklar arasındaki korelasyon matrisi

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

```
## Q3 summary statistics:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -0.183  -0.099  -0.064  -0.069  -0.040   0.027 
## 
##         V1     V2     V3     V4     V5     V6     V7     V8     V9    V10
## V1   1.000 -0.183 -0.127  0.016 -0.142 -0.089 -0.049 -0.054  0.026 -0.053
## V2  -0.183  1.000  0.027 -0.040 -0.034 -0.083 -0.062 -0.056 -0.057 -0.014
## V3  -0.127  0.027  1.000 -0.134 -0.049 -0.031 -0.100 -0.068 -0.091 -0.009
## V4   0.016 -0.040 -0.134  1.000 -0.099 -0.056 -0.019 -0.102 -0.098 -0.016
## V5  -0.142 -0.034 -0.049 -0.099  1.000 -0.117 -0.064 -0.062 -0.082  0.003
## V6  -0.089 -0.083 -0.031 -0.056 -0.117  1.000 -0.106 -0.143 -0.114 -0.109
## V7  -0.049 -0.062 -0.100 -0.019 -0.064 -0.106  1.000 -0.155 -0.128  0.007
## V8  -0.054 -0.056 -0.068 -0.102 -0.062 -0.143 -0.155  1.000 -0.089 -0.046
## V9   0.026 -0.057 -0.091 -0.098 -0.082 -0.114 -0.128 -0.089  1.000 -0.094
## V10 -0.053 -0.014 -0.009 -0.016  0.003 -0.109  0.007 -0.046 -0.094  1.000
## V11 -0.005 -0.020 -0.167 -0.120 -0.076 -0.036 -0.056 -0.072 -0.009 -0.050
## V12 -0.081 -0.096 -0.041 -0.061 -0.068 -0.090 -0.111 -0.055 -0.104 -0.117
## V13  0.006 -0.016 -0.039 -0.019 -0.040 -0.062 -0.052 -0.036 -0.053  0.001
## V14 -0.048 -0.059 -0.067 -0.042 -0.095 -0.083 -0.078 -0.082 -0.130 -0.106
## V15 -0.018 -0.086 -0.074 -0.070 -0.169 -0.060 -0.127 -0.114 -0.102 -0.063
##        V11    V12    V13    V14    V15
## V1  -0.005 -0.081  0.006 -0.048 -0.018
## V2  -0.020 -0.096 -0.016 -0.059 -0.086
## V3  -0.167 -0.041 -0.039 -0.067 -0.074
## V4  -0.120 -0.061 -0.019 -0.042 -0.070
## V5  -0.076 -0.068 -0.040 -0.095 -0.169
## V6  -0.036 -0.090 -0.062 -0.083 -0.060
## V7  -0.056 -0.111 -0.052 -0.078 -0.127
## V8  -0.072 -0.055 -0.036 -0.082 -0.114
## V9  -0.009 -0.104 -0.053 -0.130 -0.102
## V10 -0.050 -0.117  0.001 -0.106 -0.063
## V11  1.000 -0.017 -0.035 -0.054 -0.093
## V12 -0.017  1.000 -0.076 -0.116 -0.032
## V13 -0.035 -0.076  1.000 -0.106 -0.055
## V14 -0.054 -0.116 -0.106  1.000 -0.087
## V15 -0.093 -0.032 -0.055 -0.087  1.000
```

???

---
## Varsayımlar

- Yen .20'den yüksek korelasyonlara problemli olarak yaklaşmayı tavsiye etmiştir.

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

```
## [1] 0
```

---
## Model Uyumu

```r
 M2(birpl_uyum)
```

```
##        M2  df p  RMSEA RMSEA_5 RMSEA_95  SRMSR   TLI   CFI
## stats 540 104 0 0.0648  0.0594   0.0702 0.0929 0.825 0.827
```

- Elde edilen 
- **RMSEA** değeri = 0.0648 (%95 CI[0.0594, 0.0702]) ve - **SRMSR** değeri = 0.0929, önerilen eşik değerleri olan **RMSEA <= .06** 
- **SRMSR <= .08** kullanılarak verilerin modelin iyi uyum **sağlamadığını** göstermektedir.

---
## Model Uyumu

Madde uyumlarına baktığımızda özellikle V1, V2 maddelerinde yüksek ki-kare ve ve RMSEA değerleri gözlenmektedir.

```r
itemfit(birpl_uyum)
```

```
##    item   S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
## 1    V1  85.28       9      0.092  0.000
## 2    V2 548.10       8      0.260  0.000
## 3    V3  15.16       9      0.026  0.087
## 4    V4  13.07       9      0.021  0.160
## 5    V5  17.62       9      0.031  0.040
## 6    V6  16.94       9      0.030  0.050
## 7    V7  10.08       9      0.011  0.344
## 8    V8   9.69       9      0.009  0.376
## 9    V9   1.95       9      0.000  0.992
## 10  V10   1.78       8      0.000  0.987
## 11  V11  10.54       9      0.013  0.308
## 12  V12   5.51       8      0.000  0.702
## 13  V13  10.47       8      0.018  0.234
## 14  V14  12.53       9      0.020  0.185
## 15  V15   5.35       9      0.000  0.803
```

---
## Parametre Kestirimleri

.pull-left[
- Kestirim süreci **birpl_uyum** nesnesine atandıktan sonra, parametreleri inceleme için **coef()**  fonksiyonunun kullanabilir.

- Çok boyutlu MTK'da yer alan, eğim ve kesişim parametrelerini geleneksel MTK parametrelerine dönüştürmek için **IRTpars**  argümanı **TRUE** değeri ile kullanılır.

- **simplify** argümanı **TRUE** değeri ile kullanıldığında parametreler liste yapısı yerine veri seti olarak elde edilir.

```
## 
Iteration: 1, Log-Lik: -7355.589, Max-Change: 0.07977
Iteration: 2, Log-Lik: -7346.864, Max-Change: 0.03952
Iteration: 3, Log-Lik: -7344.574, Max-Change: 0.02206
Iteration: 4, Log-Lik: -7343.777, Max-Change: 0.00919
Iteration: 5, Log-Lik: -7343.636, Max-Change: 0.00536
Iteration: 6, Log-Lik: -7343.590, Max-Change: 0.00345
Iteration: 7, Log-Lik: -7343.564, Max-Change: 0.00151
Iteration: 8, Log-Lik: -7343.563, Max-Change: 0.00054
Iteration: 9, Log-Lik: -7343.563, Max-Change: 0.00033
Iteration: 10, Log-Lik: -7343.562, Max-Change: 0.00012
Iteration: 11, Log-Lik: -7343.562, Max-Change: 0.00009
```
]
--

.pull-right[
- parametre kestirimlerini olusturulan **birpl_par** nesnesinin **items** bileşeninden alabiliriz.

```r
birpl_par <- coef(birpl_uyum, 
 IRTpars = TRUE, 
 simplify = TRUE)
birpl_par$items
```

```
##        a       b g u
## V1  1.01  1.7695 0 1
## V2  1.01  2.8974 0 1
## V3  1.01  1.8237 0 1
## V4  1.01  1.9120 0 1
## V5  1.01  1.1243 0 1
## V6  1.01 -0.3201 0 1
## V7  1.01  0.2882 0 1
## V8  1.01  0.0996 0 1
## V9  1.01  0.3126 0 1
## V10 1.01 -2.7311 0 1
## V11 1.01  2.2865 0 1
## V12 1.01 -1.3673 0 1
## V13 1.01 -3.0578 0 1
## V14 1.01 -0.5475 0 1
## V15 1.01 -0.1988 0 1
```
]

---
## Parametre Kestirimleri
.pull-left[

```r
birpl_par <- coef(birpl_uyum, 
IRTpars = TRUE, simplify = TRUE)
birpl_par$items
```

- Her satır, madde adıyla başlar.

- Sütunlar ise sırasıyla

- ilk sütun **a** madde ayırtedicliği
  
  - ikinci sütun **b** madde güçlüğü
  - üçüncü sütun **g** alt asimptot (yani tahmin)
  - son sütun **u** üst asimptottur. 
  
  - 1PL modeli alt ve üst asimptot parametrelerini içermediğinden, sırasıyla her zaman 0 ve 1 dir.
]

---
## Parametre Kestirimleri
.pull-left[

```r
birpl_par <- coef(birpl_uyum, 
 IRTpars = TRUE, 
 simplify = TRUE)
birpl_par$items
```

- İlk sütun, 1.01 tahmini ile madde ayırt ayırtedicliği parametresini göstermektedir.

- ikinci sütun, madde güçlük parametrelerini göstermektedir.

- Peki en kolay madde hangisidir?

- En zor madde hangisidir?
]

---
## Madde Karakteristik Eğrisi (MKE)

.pull-left-narrow[
- **plot()** fonksiyonu ile oluşturulan nesne içindeki maddeler için tek tek ya da istenilen maddeler için MKE çizdirilebilir.
]
.pull-right-wide[

```r
plot(birpl_uyum,type = "trace", which.items = 1:15)
```

![](index_files/figure-html/unnamed-chunk-24-1.png)
]

---
## MKE

.pull-left-narrow[

- mirt paketi grafik çiziminde **lattice** paketini kullanmaktadır. lattice paketi özellikleri ile grafiklerinizi özelleştirebilirsiniz.

- Panelin oluşum şekli **layout** argümanı ile x ekseni limitlerini ise **theta_lim** argümanı ile değiştirilebilir.
]
.pull-right-wide[

```r
plot(birpl_uyum, type = "trace", which.items = 1:15,
     layout=c(5, 3),theta_lim = c(-4, 4))
```

![](index_files/figure-html/unnamed-chunk-25-1.png)
]

---
## MKE

- 1-PL modelinde KTK  **madde ayırt edicilik indeksine** karşılık gelen bir madde parametresi yoktur.

- Bu bütün maddelerin **eşit ayırt ediciliğe** sahip olduğunu varsaymaya eşdeğerdir.

- 1-PL modelinde madde karakteristik eğrilerinin **alt asimptotu sıfırdır.**

- Bu **çok düşük yetenek düzeyine** sahip bireylerin maddeyi **doğru yanıtlama olasılığının sıfır** olduğunu belirtir.

---
## MKE

.pull-left-narrow[

```r
plot(birpl_uyum,
 type = "trace", 
which.items = 1:15, 
layout=c(5, 3),
panel=function(x, y){
panel.grid(h=-1, v=-1)
panel.xyplot(x, y)
panel.abline(h=0.5, lwd=1,
 lty=1)})
```
]
.pull-right-wide[
![](index_files/figure-html/unnamed-chunk-27-1.png)

]
???

- Böylece çoktan seçmeli maddelerde **düşük yetenek düzeyine** sahip bireylerin **tahmin olasılığına izin verilmez.** Tahmin olmaması sayıltısı çoktan-seçmeli maddeleri içeren bir testin çok kolay olduğu durumlarda karşılanabilir.

---
## MKE

.pull-left-narrow[

- **facet_items** argümanının FALSE değeri ile tüm maddelerin MKE tek bir grafikte elde edilebilir.
]
.pull-right-wide[

```r
plot(birpl_uyum, type = "trace", which.items = 1:15,
facet_items = FALSE)
```

![](index_files/figure-html/unnamed-chunk-28-1.png)
]

---
## İki-Parametreli Lojistik (2-PL) Model

- **2-PL** model yaygın olarak kullanılan MTK modellerindendir.

- **2-PL** model için madde karakteristik eğrileri aşağıdaki eşitlikle elde edilir:

`$$P_i(\theta) = \frac{exp[a_i(\theta-b_i)]}{1+exp[a_i(\theta-b_i)]}=\frac{1}{1+exp(-[a_i(\theta-b_i)])}$$`

`$$ln(\frac{P_i(\theta)}{1-P_i(\theta)})=a_i(\theta - b_{i})$$`

Burada,
- `$P_i(\theta)$`:  θ yetenek düzeyindeki bir bireyin i maddesini doğru 	yanıtlama olasılığı
- `$b_i$`:       i maddesinin güçlük parametresi
- `$a_i$`:       i maddesinin ayırt edicilik parametresi
- .xsmall[Çoğu durumda ai(θ - bi), D = 1.7 normalleştirme sabitiyle çarpılır.]

---
##  2-PL Model

- Tarihsel olarak MTK modeli **kümülatif normal model (normal ogive model)** olarak geliştirilmiştir.

- Ancak zamanla kümülatif normal model yerine, matematiksel olarak daha kolay ele alındığından, **kümülatif lojistik model** kullanılmaya başlamıştır.

- Eğer `$a_i(θ - b_i)$` **1.7** normalleştirme sabitiyle çarpılırsa, iki model arasındaki **fark neredeyse ihmal edilir düzeyde olacaktır.** Yetenek düzeyinin bütün değerleri için iki modelle elde edilen olasılık değerleri arasındaki fark 0.01’den küçük olacaktır.

???

- MTK modeli başlangıçta **normal ogive modeli olarak** geliştirildiğinden, çoğu psikometrisyen geleneksel olarak lojistik modeli normal ogive model gibi yapma eğilimindedir.

- BILOG ve MULTILOG gibi özelleşmiş çoğu MTK yazılımı sadece lojistik modeli kullanır.

- D sabitinin kullanılıp kullanılmaması tercihe kalmıştır.

---
##  2-PL Model

- Bir madde için `$a_i$` parametresi yetenek ölçeğinde `$b_i$`  noktasında **madde karakteristik eğrisinin eğimiyle orantılıdır.**

- Daha **dik eğimli** maddeler farklı yetenek düzeylerindeki bireyleri ayırmada **daha az eğimli** maddelere göre daha kullanışlıdır. 
  
--

- Bir maddenin bir θ yetenek düzeyinin yakınındaki bireyler arasındaki ayırt ediciliği 
  - (θ düzeyine eşit veya daha düşük yetenek düzeyine sahip bireyleri θ düzeyinden yüksek yetenek düzeyine sahip bireylerden ayırma gücü) 
  θ değerindeki madde karakteristik **eğrisinin eğimiyle** belirlenir.

-   `$a_i$` değerleri kuramsal olarak (-∞, +∞) ölçeğindedir.

???

- Başarı testlerinde **eksi yönde ayırt ediciliğe sahip maddeler**, testten çıkarılır.

- Çünkü yetenek düzeyi arttıkça maddenin doğru yanıtlanma olasılığının düşmesi maddeyle ilgili bir probleme (yanlış anahtarlama gibi) işaret eder.

- Ayrıca uygulamada genellikle 2.0’den büyük ayırt edicilik değerlerine rastlanmaz. Bu nedenle `$a_i$` parametresi için olağan aralık (0, 2)’dir.

---
##  2-PL Model MKE

.pull-left[
- Elimizde 2PL modelde uygun dört maddelik bir testte yer alan madde parametreleri bulunsun.

- Madde 1 için `$b_1$` = 1.0 ve `$a_1$`  = 1.0

- Madde 2 için `$b_2$` = 2.0 ve `$a_2$`  = 0.5

- Madde 3 için `$b_3$` = -1.0 ve `$a_3$`  = 1.5

- Madde 4 için `$b_4$` = 0.0 ve `$a_4$`  = 1.2

]

.pull-right[
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---
##  2-PL Model MKE

.pull-left[

- Eğriler 1-PL modelinde olduğu gibi **paralel değildir.** Her eğrinin **eğimi farklılık gösterir.** Bu da madde ayırt edicilik parametrelerinin farklı olduğunu yansıtır.

- 2-PL modelinde birey performansını etkileyen madde özellikleri **madde güçlüğü ve madde ayırt ediciliğidir. **
]

.pull-right[
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]

???

- **2-PL** modelinde 1-PL modelinde olduğu gibi madde karakteristik eğrilerinin **alt asimptotu sıfırdır. **

- Bu çok düşük yetenek düzeyine sahip bireylerin maddeyi doğru yanıtlama olasılığının sıfır olduğunu belirtir. Böylece çoktan seçmeli maddelerde **düşük yetenek düzeyine** sahip bireylerin **tahmin olasılığına izin verilmez.**

- Tahmin olmaması sayıltısı çoktan-seçmeli maddeleri içeren bir testin çok zor olmadığı durumlarda karşılanabilir.

---
##  2-PL Model için Analiz

- Modelin hazırlanması

```r
ikipl_model <- "F = 1 - 15"
```

- Modelin testi

```r
ikipl_uyum <- mirt(data = ikikategorili, model = ikipl_model,
itemtype = "2PL", SE=TRUE)
```

```
## 
Iteration: 1, Log-Lik: -7355.589, Max-Change: 1.36358
Iteration: 2, Log-Lik: -7198.383, Max-Change: 0.65912
Iteration: 3, Log-Lik: -7170.537, Max-Change: 0.46260
Iteration: 4, Log-Lik: -7162.738, Max-Change: 0.38909
Iteration: 5, Log-Lik: -7159.685, Max-Change: 0.29704
Iteration: 6, Log-Lik: -7158.349, Max-Change: 0.22782
Iteration: 7, Log-Lik: -7157.056, Max-Change: 0.10551
Iteration: 8, Log-Lik: -7156.900, Max-Change: 0.09235
Iteration: 9, Log-Lik: -7156.824, Max-Change: 0.07639
Iteration: 10, Log-Lik: -7156.702, Max-Change: 0.03322
Iteration: 11, Log-Lik: -7156.687, Max-Change: 0.02994
Iteration: 12, Log-Lik: -7156.681, Max-Change: 0.03699
Iteration: 13, Log-Lik: -7156.667, Max-Change: 0.00433
Iteration: 14, Log-Lik: -7156.661, Max-Change: 0.00143
Iteration: 15, Log-Lik: -7156.661, Max-Change: 0.00070
Iteration: 16, Log-Lik: -7156.661, Max-Change: 0.00037
Iteration: 17, Log-Lik: -7156.661, Max-Change: 0.00185
Iteration: 18, Log-Lik: -7156.661, Max-Change: 0.00130
Iteration: 19, Log-Lik: -7156.661, Max-Change: 0.00080
Iteration: 20, Log-Lik: -7156.661, Max-Change: 0.00065
Iteration: 21, Log-Lik: -7156.661, Max-Change: 0.00039
Iteration: 22, Log-Lik: -7156.661, Max-Change: 0.00129
Iteration: 23, Log-Lik: -7156.661, Max-Change: 0.00063
Iteration: 24, Log-Lik: -7156.661, Max-Change: 0.00045
Iteration: 25, Log-Lik: -7156.661, Max-Change: 0.00127
Iteration: 26, Log-Lik: -7156.661, Max-Change: 0.00079
Iteration: 27, Log-Lik: -7156.660, Max-Change: 0.00052
Iteration: 28, Log-Lik: -7156.660, Max-Change: 0.00025
Iteration: 29, Log-Lik: -7156.660, Max-Change: 0.00123
Iteration: 30, Log-Lik: -7156.660, Max-Change: 0.00071
Iteration: 31, Log-Lik: -7156.660, Max-Change: 0.00024
Iteration: 32, Log-Lik: -7156.660, Max-Change: 0.00120
Iteration: 33, Log-Lik: -7156.660, Max-Change: 0.00071
Iteration: 34, Log-Lik: -7156.660, Max-Change: 0.00024
Iteration: 35, Log-Lik: -7156.660, Max-Change: 0.00118
Iteration: 36, Log-Lik: -7156.660, Max-Change: 0.00069
Iteration: 37, Log-Lik: -7156.660, Max-Change: 0.00023
Iteration: 38, Log-Lik: -7156.660, Max-Change: 0.00115
Iteration: 39, Log-Lik: -7156.660, Max-Change: 0.00067
Iteration: 40, Log-Lik: -7156.660, Max-Change: 0.00023
Iteration: 41, Log-Lik: -7156.660, Max-Change: 0.00112
Iteration: 42, Log-Lik: -7156.660, Max-Change: 0.00066
Iteration: 43, Log-Lik: -7156.660, Max-Change: 0.00022
Iteration: 44, Log-Lik: -7156.660, Max-Change: 0.00110
Iteration: 45, Log-Lik: -7156.660, Max-Change: 0.00064
Iteration: 46, Log-Lik: -7156.660, Max-Change: 0.00022
Iteration: 47, Log-Lik: -7156.660, Max-Change: 0.00107
Iteration: 48, Log-Lik: -7156.660, Max-Change: 0.00063
Iteration: 49, Log-Lik: -7156.660, Max-Change: 0.00021
Iteration: 50, Log-Lik: -7156.660, Max-Change: 0.00105
Iteration: 51, Log-Lik: -7156.660, Max-Change: 0.00062
Iteration: 52, Log-Lik: -7156.660, Max-Change: 0.00021
Iteration: 53, Log-Lik: -7156.660, Max-Change: 0.00102
Iteration: 54, Log-Lik: -7156.660, Max-Change: 0.00060
Iteration: 55, Log-Lik: -7156.660, Max-Change: 0.00020
Iteration: 56, Log-Lik: -7156.660, Max-Change: 0.00100
Iteration: 57, Log-Lik: -7156.660, Max-Change: 0.00059
Iteration: 58, Log-Lik: -7156.660, Max-Change: 0.00020
Iteration: 59, Log-Lik: -7156.660, Max-Change: 0.00097
Iteration: 60, Log-Lik: -7156.660, Max-Change: 0.00058
Iteration: 61, Log-Lik: -7156.660, Max-Change: 0.00019
Iteration: 62, Log-Lik: -7156.660, Max-Change: 0.00095
Iteration: 63, Log-Lik: -7156.660, Max-Change: 0.00056
Iteration: 64, Log-Lik: -7156.660, Max-Change: 0.00019
Iteration: 65, Log-Lik: -7156.660, Max-Change: 0.00093
Iteration: 66, Log-Lik: -7156.660, Max-Change: 0.00055
Iteration: 67, Log-Lik: -7156.660, Max-Change: 0.00018
Iteration: 68, Log-Lik: -7156.660, Max-Change: 0.00091
Iteration: 69, Log-Lik: -7156.660, Max-Change: 0.00054
Iteration: 70, Log-Lik: -7156.660, Max-Change: 0.00018
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Iteration: 148, Log-Lik: -7156.659, Max-Change: 0.00010
## 
## Calculating information matrix...
```

---
##  2-PL Model için Model Uyum

```r
M2(ikipl_uyum)
```

```
##         M2 df     p RMSEA RMSEA_5 RMSEA_95  SRMSR TLI CFI
## stats 88.1 90 0.538     0       0   0.0163 0.0247   1   1
```

---
##  2-PL Model için Madde Uyum

```r
itemfit(ikipl_uyum)
```

```
##    item  S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
## 1    V1  5.56       5      0.011  0.352
## 2    V2 12.23       9      0.019  0.201
## 3    V3 16.67       9      0.029  0.054
## 4    V4  9.71       8      0.015  0.286
## 5    V5  9.36       9      0.006  0.405
## 6    V6 15.27       9      0.026  0.084
## 7    V7  8.63       9      0.000  0.472
## 8    V8  8.50       9      0.000  0.485
## 9    V9  2.62       9      0.000  0.977
## 10  V10  1.70       8      0.000  0.989
## 11  V11  7.63       8      0.000  0.470
## 12  V12  4.07       8      0.000  0.851
## 13  V13 11.14       8      0.020  0.194
## 14  V14 12.27       9      0.019  0.199
## 15  V15  5.33       9      0.000  0.805
```

---
##  2-PL Model Parametrelerin incelenmesi

```r
ikipl_par <- coef(ikipl_uyum, IRTpars = TRUE, simplify = TRUE)
```

---
##  2-PL Model Madde Parametreleri

- Madde parametreleri oluşturulan nesnenin **items** bileşeninde yer almaktadır.

```r
ikipl_par$items
```

```
##          a       b g u
## V1   4.977  0.9739 0 1
## V2  -1.506 -2.2354 0 1
## V3   0.827  2.1212 0 1
## V4   1.247  1.6606 0 1
## V5   0.817  1.3180 0 1
## V6   1.017 -0.3152 0 1
## V7   1.141  0.2702 0 1
## V8   1.104  0.0982 0 1
## V9   1.160  0.2897 0 1
## V10  0.896 -3.0037 0 1
## V11  1.141  2.1006 0 1
## V12  0.922 -1.4612 0 1
## V13  1.048 -2.9899 0 1
## V14  1.118 -0.5082 0 1
## V15  1.160 -0.1772 0 1
```

---
##  2-PL Model MKE

```r
plot(ikipl_uyum, type = "trace", which.items = 1:15)
```

![](index_files/figure-html/unnamed-chunk-37-1.png)

---
##  2-PL Model MKE

```r
plot(ikipl_uyum, type = "trace", which.items = 1:15,facet_items = FALSE,
     abline=c(h=0.5))
```

![](index_files/figure-html/unnamed-chunk-38-1.png)

---
##  3-PL Model

- 3-PL model için madde karakteristik eğrileri aşağıdaki eşitlikle elde edilir:

`$$P_i(\theta) = c_i + (1- ci)* \frac{exp[a_i(\theta-b_i)]}{1+exp[a_i(\theta-b_i)]}=c_i +\frac{1-c_i}{1+exp(-[a_i(\theta-b_i)])}$$`
- `$P_i(θ):$`  θ yetenek düzeyindeki bir bireyin i maddesini doğru 	yanıtlama olasılığı
  - `$b_i$` :       i maddesinin güçlük parametresi
  - `$a_i$` :       i maddesinin ayırt edicilik parametresi
  - `$c_i$` :      i maddesinin sahte-tahmin parametresi

- Tahmin yerine sahte-tahmin denmesinin nedeni, parametrenin tahminden fazlasını içermesidir.  Örneğin, madde yazarları çekici ancak yanlış seçenekler geliştirebilir.

???

- Seçmeli-yanıtlı (çoktan-seçmeli gibi) maddeler gibi tahmin yoluyla doğru yanıtlara izin veren madde formatlarından elde edilen verilere **1-PL** ve **2-PL** modellerin uygulanmasında problemle karşılaşılabilir.

- 1-PL ve 2-PL modellerinde **maddeyi doğru yanıtlama olasılığı yetenek düzeyi düştükçe sıfıra** yaklaşır. Ancak **çok düşük yetenek** düzeyindeki bireyler için bile maddeyi doğru yanıtlama olasılığı, bireyler **doğru yanıtı tahmin edebileceklerinden sıfırdan büyüktür. **

- **3-PL** modelinde yer alan `$c_i$` parametresi, seçmeli-yanıtlı test maddelerindeki performansta tahminin bir etken olduğu durumlarda, yetenek ölçeğinin düşük ucundaki performansı hesaba katar.

- Sıfırdan farklı `$c_i$` parametresi, testi alan herhangi bir bireyin maddeyi doğru yanıtlama olasılığının sıfırdan farklı olduğunu yansıtır.

- Yetenek düzeyinin çok düşük değerleri için bile bireylerin en az %20’si maddeyi doğru yanıtlayacaktır.

---
##  3-PL Model için Analiz

- Modelin hazırlanması

```r
ucpl_model <- "F = 1 - 15"
```

- Modelin testi

```r
ucpl_uyum <- mirt(data = ikikategorili, model = ucpl_model,
itemtype = "3PL")
```

```
## 
Iteration: 1, Log-Lik: -7658.778, Max-Change: 2.58886
Iteration: 2, Log-Lik: -7257.494, Max-Change: 1.80258
Iteration: 3, Log-Lik: -7196.191, Max-Change: 0.86544
Iteration: 4, Log-Lik: -7172.568, Max-Change: 0.36251
Iteration: 5, Log-Lik: -7163.490, Max-Change: 0.57814
Iteration: 6, Log-Lik: -7159.566, Max-Change: 0.74891
Iteration: 7, Log-Lik: -7164.697, Max-Change: 0.66705
Iteration: 8, Log-Lik: -7153.433, Max-Change: 0.75366
Iteration: 9, Log-Lik: -7152.258, Max-Change: 0.38706
Iteration: 10, Log-Lik: -7151.965, Max-Change: 1.18505
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Iteration: 475, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 476, Log-Lik: -7151.078, Max-Change: 0.00094
Iteration: 477, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 478, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 479, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 480, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 481, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 482, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 483, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 484, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 485, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 486, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 487, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 488, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 489, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 490, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 491, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 492, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 493, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 494, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 495, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 496, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 497, Log-Lik: -7151.078, Max-Change: 0.00093
Iteration: 498, Log-Lik: -7151.078, Max-Change: 0.00061
Iteration: 499, Log-Lik: -7151.078, Max-Change: 0.00019
Iteration: 500, Log-Lik: -7151.078, Max-Change: 0.00093
```

- Parametrelerin incelenmesi

```r
ucpl_par <- coef(ucpl_uyum, IRTpars = TRUE, simplify = TRUE)
```

---

##  3-PL Model için Model Uyum

```r
M2(ucpl_uyum)
```

```
##         M2 df    p RMSEA RMSEA_5 RMSEA_95 SRMSR TLI CFI
## stats 67.5 75 0.72     0       0   0.0141 0.025   1   1
```

---
##  3-PL Model için Madde Uyum

```r
itemfit(ucpl_uyum)
```

```
##    item  S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
## 1    V1  5.57       4      0.020  0.233
## 2    V2 12.21       7      0.027  0.094
## 3    V3 15.87       8      0.031  0.044
## 4    V4 11.47       8      0.021  0.177
## 5    V5  9.38       8      0.013  0.312
## 6    V6 15.05       8      0.030  0.058
## 7    V7  8.68       8      0.009  0.370
## 8    V8  8.47       8      0.008  0.389
## 9    V9  2.18       8      0.000  0.975
## 10  V10  1.76       7      0.000  0.972
## 11  V11  7.72       7      0.010  0.358
## 12  V12  3.95       7      0.000  0.786
## 13  V13  5.32       5      0.008  0.379
## 14  V14 12.28       8      0.023  0.139
## 15  V15  5.04       8      0.000  0.753
```

---
##  3-PL Model Madde Parametreleri

- Madde parametreleri oluşturulan nesnenin **items** bileşeninde yer almaktadır.

```r
ucpl_par$items
```

```
##          a      b        g u
## V1   5.010  0.973 6.86e-05 1
## V2  -1.786 -2.125 7.45e-03 1
## V3   0.826  2.124 2.19e-04 1
## V4   1.507  1.613 2.28e-02 1
## V5   0.832  1.303 4.39e-04 1
## V6   1.013 -0.312 9.97e-04 1
## V7   1.143  0.275 1.17e-03 1
## V8   1.114  0.104 1.63e-03 1
## V9   1.400  0.468 8.29e-02 1
## V10  0.886 -2.964 5.06e-02 1
## V11  1.210  2.056 4.70e-03 1
## V12  0.925 -1.450 4.69e-03 1
## V13  5.925 -0.127 8.57e-01 1
## V14  1.134 -0.484 9.60e-03 1
## V15  1.163 -0.173 8.85e-04 1
```

---
##  3-PL Model MKE

```r
plot(ucpl_uyum, type = "trace", which.items = 1:15)
```

![](index_files/figure-html/unnamed-chunk-45-1.png)

---
##  3-PL Model MKE

```r
plot(ucpl_uyum, type = "trace", which.items = 1:15,facet_items = FALSE,
     abline=c(h=0.5))
```

![](index_files/figure-html/unnamed-chunk-46-1.png)

---
##  3-PL Model için MKE

![](index_files/figure-html/unnamed-chunk-47-1.png)

???
- Madde 1 için `$b_1$` = 1.0 ve `$a_1$`  = 1.8 ve `$c_1$`  = 0
- Madde 2 için `$b_2$` = 1.0 ve `$a_2$`  = 0.8 ve `$c_2$`  = 0
- Madde 3 için `$b_3$` = 1.0 ve `$a_3$`  = 1.8 ve `$c_3$`  = 0.25
- Madde 4 için `$b_4$` = -1.5 ve `$a_4$`  = 1.8 ve `$c_4$`  = 0
- Madde 5 için `$b_5$`  = -0.5 ve `$a_5$` = 1.2 ve `$c_5$` = 0.10
- Madde 6 için `$b_6$` = 0.5 ve `$a_6$` = 0.4 ve `$c_6$` = 0.15

## 3-PL Model MKE
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##  3-PL Model MKE

- Madde 1 ve Madde 4 ait MKE arasındaki karşılaştırma hangi parametrenin rolunu vurgulamaktadır?

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---
##  3-PL Model MKE

- Madde 1 için `$b_1$` = 1.0 ve `$a_1$`  = 1.8 ve `$c_1$`  = 0
- Madde 4 için `$b_4$` = -1.5 ve `$a_4$`  = 1.8 ve `$c_4$`  = 0

---
##  3-PL Model MKE

- Madde 1 ve Madde 2 ait MKE arasındaki karşılaştırma hangi parametrenin rolunu vurgulamaktadır?

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---
##  3-PL Model MKE

- Madde 1 için `$b_1$` = 1.0 ve `$a_1$`  = 1.8 ve `$c_1$`  = 0
- Madde 2 için `$b_2$` = 1.0 ve `$a_2$`  = 0.8 ve `$c_2$`  = 0

---
##  3-PL Model MKE

- Madde 1 ve Madde 3  ait MKE arasındaki karşılaştırma hangi parametrenin rolunu vurgulamaktadır?
--

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---
##  3-PL Model MKE

- Madde 1 için `$b_1$` = 1.0 ve `$a_1$`  = 1.8 ve `$c_1$`  = 0
- Madde 3 için `$b_3$` = 1.0 ve `$a_3$`  = 1.8 ve `$c_3$`  = 0.25

---
##  3-PL Model MKE

- Yandaki 6 maddeden hangi madde θ = 0.0 değerinde en zor maddedir?

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##  3-PL Model MKE

- Yandaki 6 maddeden hangi madde θ = 0.0 değerinde en zor maddedir?

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##  3-PL Model MKE

Yandaki 6 maddeden hangi iki madde  θ = -1.0 değerinde eşit güçlükteki maddelerdir?

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##  3-PL Model MKE

Yandaki 6 maddeden hangi iki madde  θ = -1.0 değerinde eşit güçlükteki maddelerdir?

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##  3-PL Model için Madde Karakteristik Eğrisi

Yandaki 6 maddeden hangi madde θ = 3.0 değerinde en ayırt edici maddedir?

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???
 Madde 2 
 
---
##  Yetenek Parametresi Kestirimi

MTK modellerinde başlıca üç yolla puanlama yapılır:

- Maksimum Likelihood (ML)
- Maksimum a Posteriori (MAP)
- Expected/estimated a Posteriori (EAP)

---
##  Yetenek Parametresi Kestirimi

- Bireylerin faktör puanları veya gizil özellik düzeyi kestirimleri **fscores()** fonksiyonuyla hesaplanabilir.

- **fscores()** fonksiyonunun birinci argümanı object olup bu argümanın değeri **mirt()** fonksiyonunun çıktısı olarak kaydedilen nesnelerdir. Kestirim yönteminin türü method argümanıyla maksimum olabilirlik (ML) olarak belirlenmiştir.

- full.scores.SE argümanı için de TRUE değeri seçilerek kestirimlerin standart hataları istenebilir.

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

---
##  Yetenek Parametresi Kestirimi

.three-column[

```r
head(ML)
```

```
##            F  SE_F
## [1,]  0.7360 0.381
## [2,]  0.5359 0.457
## [3,]  0.8977 0.346
## [4,] -0.5036 0.632
## [5,]  2.3398 0.819
## [6,] -0.0337 0.603
```

]
.three-column[

```r
head(MAP)
```

```
##            F  SE_F
## [1,]  0.6356 0.384
## [2,]  0.4366 0.445
## [3,]  0.7981 0.342
## [4,] -0.3610 0.530
## [5,]  1.5725 0.491
## [6,] -0.0247 0.516
```
]
.three-column[

```r
head(EAP)
```

```
##            F  SE_F
## [1,]  0.5552 0.407
## [2,]  0.3483 0.438
## [3,]  0.7493 0.383
## [4,] -0.3892 0.520
## [5,]  1.7016 0.491
## [6,] -0.0737 0.493
```
]

---
##  Yetenek Parametresi Kestirimi

```r
yetenek <- data.frame(ML= ML[,1],MAP=MAP[,1],EAP=EAP[,1])

apply(yetenek,2,summary)
```

```
##              ML     MAP       EAP
## Min.       -Inf -2.5228 -2.57e+00
## 1st Qu. -0.9165 -0.6457 -6.67e-01
## Median  -0.0645 -0.0472 -9.43e-02
## Mean       -Inf  0.0228 -5.42e-05
## 3rd Qu.  0.7064  0.6054  5.22e-01
## Max.    19.9999  2.2015  2.30e+00
```

---
##  Yetenek Parametresi Kestirimi

```r
yetenek_v1 <- yetenek[!is.infinite(yetenek$ML),]

apply(yetenek_v1,2,summary)
```

```
##              ML     MAP      EAP
## Min.    -3.8881 -2.1960 -2.22860
## 1st Qu. -0.9153 -0.6449 -0.66644
## Median  -0.0637 -0.0466 -0.09374
## Mean     0.0137  0.0254  0.00252
## 3rd Qu.  0.7064  0.6054  0.52177
## Max.    19.9999  2.2015  2.29584
```

---
##  Yetenek Parametresi Kestirimi

.pull-left[

```r
cor(yetenek_v1)
```

```
##        ML   MAP   EAP
## ML  1.000 0.817 0.817
## MAP 0.817 1.000 0.998
## EAP 0.817 0.998 1.000
```
]
.pull-right[

```r
pairs(yetenek_v1)
```

![](index_files/figure-html/unnamed-chunk-64-1.png)
]

---
##  Model Seçimi

```r
anova(birpl_uyum,ikipl_uyum)
```

```
##              AIC SABIC    HQ   BIC logLik      X2 df p
## birpl_uyum 14719 14747 14749 14798  -7344             
## ikipl_uyum 14373 14425 14429 14521  -7157 373.807 14 0
```

```r
anova(ikipl_uyum,ucpl_uyum)
```

```
##              AIC SABIC    HQ   BIC logLik     X2 df     p
## ikipl_uyum 14373 14425 14429 14521  -7157                
## ucpl_uyum  14392 14470 14476 14613  -7151 11.162 15 0.741
```

---
##  Madde Bilgi Fonksiyonu

- Teknik olarak, **bilgi** bir parametre **kestiriminin standart hatasının tersiyle** ilişkili bir değerdir.

- Yüksek bilgi değeri parametre kestirimi hakkında daha fazla bilgiye sahip olunduğunu belirtir.

- MTK'da  **bilgi** birey yeteneğini kestirmek için kullanılan maddelerin toplamından elde edilen bilgiyi ifade eder.

---
##  Test Bilgi Fonksiyonu

- Bilginin miktarı yetenek değerine bağlıdır, bu nedenle **test bilgi fonksiyonu** olarak adlandırılır.

- Bilgi miktarı uygulamada test düzeyinde değerlendirilir.

- Ancak bilgi madde düzeyinde elde edilir ve **test bilgi fonksiyonu** `$I_T(θ)$` madde bilgi fonksiyonlarının `$I_i(θ)$` toplamıdır. `$I_T(θ)= \sum{I_i(θ)}$`

---
##  Madde Bilgi Fonksiyonu

i maddesi için belli bir yetenek düzeyinde (θ değerinde) bilgi miktarı için farklı MTK modellerinde kullanılan eşitlikler aşağıdaki gibidir:

- 1PL
 
 - `$I_i(θ)=P_i(θ)*Q_i(θ)$`    
    - `$Q_i(θ)=1-P_i(θ)$`
 
- 2PL
  - `$I_i(θ)=a_i^2P_i(θ)*Q_i(θ)$`     
 
- 3PL

- `$I_i(θ)=a_i^2 \frac{Q_i(θ)}{P_i(θ)}[\frac{P_i(θ) - c_i}{1- c_i}]^2$`     
 
---
## 1-PL Modeli için Madde Bilgi Fonksiyonu

- 1-PL (a = 1.0, c = 0.0, D = 1.7 sabiti yok)

- b = 1.2 madde güçlüğü ile  θ = 1.0 yetenek düzeyindeki bir birey için

`$$P_i(\theta) = \frac{1}{1+exp[-(\theta-b_i)]}$$`
`$$P_i(1) = \frac{1}{1+exp[-(1-1.2)]} = 0.45$$`
 `$I_i(θ)= 0.45 * (1-0.45) =2.48$`

```r
p <- 1/(1+exp(-(1-1.2)))
p * (1-p)
```

```
## [1] 0.248
```

---
##  Madde Bilgi Fonksiyonu

.pull-left[
![](index_files/figure-html/unnamed-chunk-67-1.png)
]
.pull-right[

```r
b <- c(1.2)
theta <- seq(-4,4,0.01)

prob <- c()
 for(j in 1:length(theta)){
 dir <- 1/(1 + exp(-(theta[j] - b)))
 prob[j] <- dir
 j=j+1
 }
bilgi = prob * (1- prob)

p <- data.frame(prob,bilgi)
MBF <- ggplot(p, aes(theta, bilgi)) + 
 geom_line()
```

]

---
## 2-PL Model  Madde Bilgi Fonksiyonu

- 2-PL (a = 0.8, c = 0.0, D = 1.7 sabiti yok)

- b = 1.2 madde güçlüğü ile  θ = 1.0 yetenek düzeyindeki bir birey için

```r
b <- 1.2
a <- 0.8
theta <- seq(-4,4,0.01)
p <- 1/(1+exp(-(0.8*(1-1.2))))
a^2 * p * (1-p)
```

```
## [1] 0.159
```

---
## 2-PL Modeli Madde Bilgi Fonksiyonu
.pull-left[
![](index_files/figure-html/unnamed-chunk-70-1.png)
]
.pull-right[

```r
prob <- c()
 for(j in 1:length(theta)){
 dir <- 1/(1 + exp(-(a*(theta[j] - b))))
 prob[j] <- dir
 j=j+1
 }
bilgi = a*a * prob * (1- prob)

p <- data.frame(prob,bilgi)
MBF2 <- ggplot(p, aes(theta, bilgi)) + 
 geom_line()
```
]

---
##  Madde Bilgi Fonksiyonu

.three-column[

```r
plot(birpl_uyum, 
type = "infotrace", 
which.items = 5)
```

![](index_files/figure-html/unnamed-chunk-72-1.png)

]
.three-column[

```r
plot(ikipl_uyum, 
type = "infotrace", 
which.items = 5)
```

![](index_files/figure-html/unnamed-chunk-73-1.png)
]
.three-column[

```r
plot(ucpl_uyum, 
type = "infotrace", 
which.items = 5)
```

![](index_files/figure-html/unnamed-chunk-74-1.png)
]

---
##  Madde Bilgi Fonksiyonu

![](index_files/figure-html/unnamed-chunk-75-1.png)

---
##  Madde Bilgi Fonksiyonu

![](index_files/figure-html/unnamed-chunk-76-1.png)

---
##  Madde Bilgi Fonksiyonu

```r
madde1 <- extract.item(ikipl_uyum, 1)
Theta <- matrix(seq(-6,6, by = .1))
info.1 <- iteminfo(madde1, Theta)
plot(Theta, info.1, type = 'l', main = 'Item information')
```

![](index_files/figure-html/unnamed-chunk-77-1.png)

---
##  Madde Bilgi Fonksiyonu

- i maddesi için maksimum bilgi farklı MTK modellerinde aşağıdaki yetenek düzeylerinde (θ değerinde) elde edilir:

- 1-PL 
  - `$\theta=b_i$`

- 2-PL 
  - `$\theta=b_i$`

- 3-PL 
  - `$\theta=b_i + \frac{1}{Da_i}[ln\frac{1 + \sqrt{1+8c_i}}{2}]^2$`

---
## Test Bilgi Fonksiyonu

- Bireysel maddelerin teste katkısının miktarı testteki diğer maddelerin bilgisi olmadan belirlenebilir.

- Bu klasik test kuramında mümkün değildir.

- Örneğin, güvenirlik veya nokta-çift serili korelasyon testteki maddelerin geri kalanından bağımsız olarak belirlenemez. 
  
  - Testteki madde sayısı daha fazlaysa, daha yüksek test bilgi fonksiyonu elde edilir.

---
## Test Bilgi Fonksiyonu

- Lord (1977) tarafından önerilen test geliştirme yöntemi:

- Beklenen test bilgi fonksiyonunun şekli belirlenir:
Hedef bilgi fonksiyonu
Örneğin,

- Maddeler seçilir ve test bilgi fonksiyonu hesaplanır ve hedef bilgi fonksiyonuyla karşılaştırılır.

- Bir önceki basamak beklenen sonuçlar elde edilene kadar tekrar edilir.

---
## Test Bilgi Fonksiyonu

.three-column[

```r
tinfo <- testinfo(ikipl_uyum, 
Theta,which.items = 1:5)
plot(Theta, tinfo, type = 'l')
```

![](index_files/figure-html/unnamed-chunk-79-1.png)
]
.three-column[

```r
tinfo <- testinfo(ikipl_uyum, 
Theta,which.items = 1:10)
plot(Theta, tinfo, type = 'l')
```

![](index_files/figure-html/unnamed-chunk-80-1.png)
]
.three-column[

```r
tinfo <- testinfo(ikipl_uyum, 
Theta,which.items = 1:15)
plot(Theta, tinfo, type = 'l')
```

![](index_files/figure-html/unnamed-chunk-81-1.png)
]

---

## Test Bilgi Fonksiyonu

```r
tinfo <- testinfo(ikipl_uyum, Theta,which.items = c(1,3:5,7:10,11))
plot(Theta, tinfo, type = 'l')
```

![](index_files/figure-html/unnamed-chunk-82-1.png)

---

## Test Bilgi Fonksiyonu

```r
plot(ikipl_uyum, type='infoSE')
```

![](index_files/figure-html/unnamed-chunk-83-1.png)

---

## Sıra Sizde

1.<svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M172.5 131.1C228.1 75.51 320.5 75.51 376.1 131.1C426.1 181.1 433.5 260.8 392.4 318.3L391.3 319.9C381 334.2 361 337.6 346.7 327.3C332.3 317 328.9 297 339.2 282.7L340.3 281.1C363.2 249 359.6 205.1 331.7 177.2C300.3 145.8 249.2 145.8 217.7 177.2L105.5 289.5C73.99 320.1 73.99 372 105.5 403.5C133.3 431.4 177.3 435 209.3 412.1L210.9 410.1C225.3 400.7 245.3 404 255.5 418.4C265.8 432.8 262.5 452.8 248.1 463.1L246.5 464.2C188.1 505.3 110.2 498.7 60.21 448.8C3.741 392.3 3.741 300.7 60.21 244.3L172.5 131.1zM467.5 380C411 436.5 319.5 436.5 263 380C213 330 206.5 251.2 247.6 193.7L248.7 192.1C258.1 177.8 278.1 174.4 293.3 184.7C307.7 194.1 311.1 214.1 300.8 229.3L299.7 230.9C276.8 262.1 280.4 306.9 308.3 334.8C339.7 366.2 390.8 366.2 422.3 334.8L534.5 222.5C566 191 566 139.1 534.5 108.5C506.7 80.63 462.7 76.99 430.7 99.9L429.1 101C414.7 111.3 394.7 107.1 384.5 93.58C374.2 79.2 377.5 59.21 391.9 48.94L393.5 47.82C451 6.731 529.8 13.25 579.8 63.24C636.3 119.7 636.3 211.3 579.8 267.7L467.5 380z"/></svg>: [Veriyi açılan linkten farklı kaydet ile alabilirsiniz.](https://raw.githubusercontent.com/atalay-k/mirt_k/main/dat1.csv)
2. MTK varsayamlarını test ediniz.
3. Verinin hangi MTK modeline daha iyi uyum sağladığını inceleyiniz.
4. Madde parametrelerini ve madde karakteristik eğrileri ile birlikte raporlayınız.
5. Yetenek paramterelerini kestiriniz
6. Test bilgi fonskiyonun grafiğini çiziniz.

---

## Kaynaklar

-  Atar, B., Atalay Kabasakal, K, Unsal Ozberk, E. B., Ozberk, E. H. & Kibrislioglu Uysal, N. (2020). R ile Veri Analizi ve Psikometri Uygulamaları, Pegem Akademi, Ankara.

- Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1-29.

- Desjardins, C.D., & Bulut, O. (2017). Handbook of Educational Measurement and Psychometrics Using R (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b20498

---
.center[

.hand[Teşekkürler]

🙏
 
]