Item Response Theory for beginners

.title[
# Item Response Theory for beginners
]
.subtitle[
## Stimare, scegliere i modelli e verificare le assunzioni in R
]
.author[
### Dr. Ottavia M. Epifania
]
.institute[
### Bressanone
]
.date[
### Corso IRT @ Università Libera di Bolzano,  17-18 Gennaio 2023
]

---

.inverse {
  background-color: #272822;
  color: #d6d6d6;
  text-shadow: 0 0 20px #333;
}

.scrollable {
  height: 500px;
  overflow-y: auto;
}

.scrollable-auto {
  height: 80%;
  overflow-y: auto;
}

.remark-slide-number {
  display: none;
}
</style>

---
class: section, center, middle

# Getting started

---

## Create a new project I

New file `$\rightarrow$` New project

<img src="img/project1.png" width="90%" style="display: block; margin: auto;" />
---

## Create a new project II

]

]

---

## Pacchetti

```r
install.packages("lavaan")

install.packages("TAM")

install.packages("mokken")

install.packages("difR")

install.packages("ggplot2")
```

]

```r
library("lavaan")

library("TAM")

library("mokken")

library("difR")

library("ggplot2")
```

]

---

## Codici utili

```r
# questa funzione calcola la probabilità di risposta corretta dato un certo theta e determinati valori dell'item
IRT <- function(theta, a = 1, b = 0, c = 0,e = 1) {
  y <- c + (e - c) * exp(a * (theta - b)) / (1 + exp(a * (theta - b)))
  y[is.na(y)] = 1
  return(y)
}
```

---

## Codici utili II

```r
# questa funzione estrae tutti gli item e i loro parametri e calcola la probabilità di risposta corretta per ogni item per ogni livello di theta
# restituisce una lista con dentro il dataset usatoe un grafico in ggplot
irt.icc = function(model) {
  item_par = model$item
  est_theta = seq(-4,4, length.out=1000)
  item_prob = list()
  if (any(grep("guess", colnames(item_par))) == F) {
    for (i in 1:nrow(item_par)) {
      item_prob[[i]] = data.frame(theta = est_theta)
      item_prob[[i]]$it_p = IRT(item_prob[[i]]$theta, 
                          b = item_par[i, "xsi.item"], 
                          a = item_par[i, "B.Cat1.Dim1"])
      item_prob[[i]]$item = item_par[i, "item"]

}
  } else {
     for (i in 1:nrow(item_par)) {
      item_prob[[i]] = data.frame(theta = est_theta)
      item_prob[[i]]$it_p = IRT(item_prob[[i]]$theta, 
                          b = item_par[i, "AXsi_.Cat1"], 
                          a = item_par[i, "B.Cat1.Dim1"], 
                          c = item_par[i, "guess"])
      item_prob[[i]]$item = item_par[i, "item"]

}
  }
  p = do.call("rbind", item_prob)
  gp = ggplot(p, 
       aes(x = theta, y = it_p, group = item, col =
             item)) + geom_line(lwd = 1)
  object = list(prob.data = p, 
              icc.graph = gp)

return(object)
}
```
]

---

## Codici utili III

```r
irt.iif = function(model) {
  est_theta = IRT.factor.scores(model, 
                              type = "EAP")$EAP
ii = IRT.informationCurves(model, theta = est_theta)

test_info = data.frame(theta = est_theta, 
               info = ii$test_info_curve, 
               se = ii$se_curve)

iif_info = list()
for(i in 1:nrow(ii$info_curves_item)) {
    iif_info[[i]] = data.frame(theta = est_theta)
    iif_info[[i]]$ii_item = ii$info_curves_item[i, ]
    iif_info[[i]]$item = dimnames(ii$info_curves_item)[[1]][i]
}

dat_info = do.call("rbind", iif_info)

info_tot = list(test_info = test_info, 
                item_info = dat_info)

return(info_tot)
}
```

]

---

---
class: section, center, middle

# Importare i dati

---

## Caricare i dati in R

.scrollable[
Scaricare il file  `dataClass.csv` dalla cartella [Dati](https://drive.google.com/drive/folders/1PXDG7HhjRDMdFEQjk5WQQWdiorDpSgtw?usp=sharing)

Salvarlo all'interno della sottocartella `Dati` del progetto `R` che avete creato per il corso

Usate questo codice:

```r
data = read.csv("Dati/dataClass.csv", header = T, sep = ",")
```

]
---

## Look at the data - Item I

```r
vis_data = data
vis_data$id = paste0("s", 1:nrow(data))

vis_data = vis_data[, c(ncol(vis_data), 1:ncol(data))]

vis_data = reshape(vis_data, idvar = "id", direction = "long", varying = list(2:ncol(vis_data)), v.names = "resp", 
        times = names(vis_data)[-1], timevar = "item")

prop_data = data.frame(table(vis_data$item, vis_data$resp))

colnames(prop_data)[1:2] = c("item", "response")

prop_data$proportion = prop_data$Freq/nrow(data)

ggplot(prop_data, 
       aes(x = item, y=proportion,  
 
           fill = response)) + geom_bar(stat = "identity") + 
  theme(axis.text = element_text(angle = 90)) + geom_hline(aes(yintercept=.50)) + theme_bw() 
```

]

---

## Look at the data - Respondents
.scrollable[

```r
sbj_data = data
sbj_data$id = paste0("s", 1:nrow(data))
sbj_data = sbj_data[, c(ncol(sbj_data), 1:ncol(data))]

sbj_data$sum = rowSums(sbj_data[,-1])

boxplot(sbj_data$sum)
```

<img src="stimareImodelli_files/figure-html/unnamed-chunk-12-1.png" width="90%" style="display: block; margin: auto;" />
]

---

# Stima dei modelli

---

## 1PL - Stima

```r
m1pl = tam.mml(data, verbose = F)

summary(m1pl)
```

```{.scroll-100}
## ------------------------------------------------------------
## TAM 4.0-16 (2022-05-13 13:23:23) 
## R version 4.2.2 (2022-10-31 ucrt) x86_64, mingw32 | nodename=LAPTOP-OTTAVIA | login=huawei 
## 
## Date of Analysis: 2023-01-17 08:09:53 
## Time difference of 0.09267402 secs
## Computation time: 0.09267402 
## 
## Multidimensional Item Response Model in TAM 
## 
## IRT Model: 1PL
## Call:
## tam.mml(resp = data, verbose = F)
## 
## ------------------------------------------------------------
## Number of iterations = 34 
## Numeric integration with 21 integration points
## 
## Deviance = 5413.25 
## Log likelihood = -2706.63 
## Number of persons = 1000 
## Number of persons used = 1000 
## Number of items = 5 
## Number of estimated parameters = 6 
##     Item threshold parameters = 5 
##     Item slope parameters = 0 
##     Regression parameters = 0 
##     Variance/covariance parameters = 1 
## 
## AIC = 5425  | penalty=12    | AIC=-2*LL + 2*p 
## AIC3 = 5431  | penalty=18    | AIC3=-2*LL + 3*p 
## BIC = 5455  | penalty=41.45    | BIC=-2*LL + log(n)*p 
## aBIC = 5436  | penalty=22.37    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC) 
## CAIC = 5461  | penalty=47.45    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC) 
## AICc = 5425  | penalty=12.08    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC) 
## GHP = 0.54253     | GHP=( -LL + p ) / (#Persons * #Items)  (Gilula-Haberman log penalty) 
## 
## ------------------------------------------------------------
## EAP Reliability
## [1] 0.713
## ------------------------------------------------------------
## Covariances and Variances
##       [,1]
## [1,] 3.937
## ------------------------------------------------------------
## Correlations and Standard Deviations (in the diagonal)
##       [,1]
## [1,] 1.984
## ------------------------------------------------------------
## Regression Coefficients
##      [,1]
## [1,]    0
## ------------------------------------------------------------
## Item Parameters -A*Xsi
##   item    N     M xsi.item AXsi_.Cat1 B.Cat1.Dim1
## 1  I01 1000 0.572   -0.498     -0.498           1
## 2  I02 1000 0.442    0.370      0.370           1
## 3  I03 1000 0.729   -1.623     -1.623           1
## 4  I04 1000 0.262    1.685      1.685           1
## 5  I05 1000 0.295    1.421      1.421           1
## 
## Item Parameters in IRT parameterization
##   item alpha   beta
## 1  I01     1 -0.498
## 2  I02     1  0.370
## 3  I03     1 -1.623
## 4  I04     1  1.685
## 5  I05     1  1.421
```

]

---

## 1PL - FIT

```r
f.m1pl = tam.modelfit(m1pl, progress = F)

f.m1pl$statlist
```

```{.scroll-100}
##   X100.MADCOV       SRMR      SRMSR     MADaQ3      pmaxX2
## 1    1.415162 0.06452616 0.08729947 0.06260904 3.64464e-11
```

```r
f.m1pl$modelfit.test
```

```{.scroll-100}
##     maxX2 Npairs      p.holm
## 1 48.3069     10 3.64464e-11
```

---

## 1PL - Item fit

```r
item.fit.1pl = IRT.itemfit(m1pl)

item.fit.1pl$chisquare_stat
```

```{.scroll-100}
##   item     Group1
## 1  I01 11.2578889
## 2  I02 14.6178740
## 3  I03  6.8620373
## 4  I04  0.5385041
## 5  I05 10.7291513
```

```r
item.fit.1pl$RMSD
```

```{.scroll-100}
##   item      Group1
## 1  I01 0.033633418
## 2  I02 0.045644930
## 3  I03 0.020356685
## 4  I04 0.008875834
## 5  I05 0.035963344
```

```r
item.fit.1pl$RMSD_summary
```

```{.scroll-100}
##     Parm          M         SD         Min        Max
## 1 Group1 0.02889484 0.01437571 0.008875834 0.04564493
```

]

---

## 1PL - ICC

```r
irt.icc(m1pl)$icc.graph + theme_classic() 
```

]

---
## 2PL

```r
m2pl = tam.mml.2pl(data, irtmodel = "2PL", verbose = F)

summary(m2pl)
```

```{.scroll-100}
## ------------------------------------------------------------
## TAM 4.0-16 (2022-05-13 13:23:23) 
## R version 4.2.2 (2022-10-31 ucrt) x86_64, mingw32 | nodename=LAPTOP-OTTAVIA | login=huawei 
## 
## Date of Analysis: 2023-01-17 08:09:54 
## Time difference of 0.247611 secs
## Computation time: 0.247611 
## 
## Multidimensional Item Response Model in TAM 
## 
## IRT Model: 2PL
## Call:
## tam.mml.2pl(resp = data, irtmodel = "2PL", verbose = F)
## 
## ------------------------------------------------------------
## Number of iterations = 66 
## Numeric integration with 21 integration points
## 
## Deviance = 5308.52 
## Log likelihood = -2654.26 
## Number of persons = 1000 
## Number of persons used = 1000 
## Number of items = 5 
## Number of estimated parameters = 10 
##     Item threshold parameters = 5 
##     Item slope parameters = 5 
##     Regression parameters = 0 
##     Variance/covariance parameters = 0 
## 
## AIC = 5329  | penalty=20    | AIC=-2*LL + 2*p 
## AIC3 = 5339  | penalty=30    | AIC3=-2*LL + 3*p 
## BIC = 5378  | penalty=69.08    | BIC=-2*LL + log(n)*p 
## aBIC = 5346  | penalty=37.28    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC) 
## CAIC = 5388  | penalty=79.08    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC) 
## AICc = 5329  | penalty=20.22    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC) 
## GHP = 0.53285     | GHP=( -LL + p ) / (#Persons * #Items)  (Gilula-Haberman log penalty) 
## 
## ------------------------------------------------------------
## EAP Reliability
## [1] 0.729
## ------------------------------------------------------------
## Covariances and Variances
##      [,1]
## [1,]    1
## ------------------------------------------------------------
## Correlations and Standard Deviations (in the diagonal)
##      [,1]
## [1,]    1
## ------------------------------------------------------------
## Regression Coefficients
##      [,1]
## [1,]    0
## ------------------------------------------------------------
## Item Parameters -A*Xsi
##   item    N     M xsi.item AXsi_.Cat1 B.Cat1.Dim1
## 1  I01 1000 0.572   -0.382     -0.382       1.299
## 2  I02 1000 0.442    0.695      0.695       4.178
## 3  I03 1000 0.729   -1.365     -1.365       1.460
## 4  I04 1000 0.262    1.620      1.620       1.855
## 5  I05 1000 0.295    2.206      2.206       3.616
## 
## Item Parameters in IRT parameterization
##   item alpha   beta
## 1  I01 1.299 -0.294
## 2  I02 4.178  0.166
## 3  I03 1.460 -0.935
## 4  I04 1.855  0.874
## 5  I05 3.616  0.610
```

]

---

## 2PL - Fit

```r
f.m2pl = tam.modelfit(m2pl, progress = F)

f.m2pl$statlist
```

```{.scroll-100}
##   X100.MADCOV        SRMR      SRMSR     MADaQ3 pmaxX2
## 1   0.2043301 0.009127058 0.01265885 0.07103058      1
```

```r
f.m2pl$modelfit.test
```

```{.scroll-100}
##       maxX2 Npairs p.holm
## 1 0.8595675     10      1
```

---

## 2PL - Item fit

```r
item.fit.2pl = IRT.itemfit(m2pl)

item.fit.2pl$chisquare_stat
```

```{.scroll-100}
##   item    Group1
## 1  I01 0.4296817
## 2  I02 0.2003800
## 3  I03 1.4600113
## 4  I04 1.3387959
## 5  I05 0.3435749
```

```r
item.fit.1pl$RMSD
```

```{.scroll-100}
##   item      Group1
## 1  I01 0.033633418
## 2  I02 0.045644930
## 3  I03 0.020356685
## 4  I04 0.008875834
## 5  I05 0.035963344
```

```r
item.fit.1pl$RMSD_summary
```

```{.scroll-100}
##     Parm          M         SD         Min        Max
## 1 Group1 0.02889484 0.01437571 0.008875834 0.04564493
```

]

---

## 2PL - ICC

```r
irt.icc(m2pl)$icc.graph + theme_classic()
```

]

---

## 3PL

```r
m3pl = tam.mml.3pl(data, est.guess = colnames(data),
                     verbose = F)

summary(m3pl)
```

```{.scroll-100}
## ------------------------------------------------------------
## TAM 4.0-16 (2022-05-13 13:23:23) 
## R version 4.2.2 (2022-10-31 ucrt) x86_64, mingw32 | nodename=LAPTOP-OTTAVIA | login=huawei 
## 
## Date of Analysis: 2023-01-17 08:10:05 
## Time difference of 10.2115 secs
## Computation time: 10.2115 
## 
## Multidimensional Item Response Model in TAM 
## 
## IRT Model 2PL  (Function 'tam.mml.3pl')
## Call:
## tam.mml.3pl(resp = data, est.guess = colnames(data), verbose = F)
## 
## ------------------------------------------------------------
## Number of iterations = 576 
## 
## Skill space: Normal Distribution 
## Numeric integration with 21 integration points
## 
## Deviance = 5309.64  | Log Likelihood = -2654.82 
## Number of persons = 1000 
## Number of persons used = 1000 
## Number of items = 5 
## Number of estimated parameters = 16 
##     Item threshold parameters = 5 
##     Item slope parameters = 5 
##       Non-active item slopes = 0 
##     Item guessing parameters = 5 
##     Regression parameters = 0 
##     Variance/covariance parameters = 1 
##     Delta parameters     = 0 
## 
## AIC = 5342  | penalty=32    | AIC=-2*LL + 2*p 
## AIC3 = 5358  | penalty=48    | AIC3=-2*LL + 3*p 
## BIC = 5420  | penalty=110.52    | BIC=-2*LL + log(n)*p 
## aBIC = 5369  | penalty=59.64    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC) 
## CAIC = 5436  | penalty=126.52    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC) 
## AICc = 5342  | penalty=32.55    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC) 
## GHP = 0.53416     | GHP=( -LL + p ) / (#Persons * #Items)  (Gilula-Haberman log penalty) 
## 
## ------------------------------------------------------------
## EAP Reliability
## [1] 0.728
## ------------------------------------------------------------
## Covariances and Variances
##       [,1]
## [1,] 1.481
## ------------------------------------------------------------
## Correlations and Standard Deviations (in the diagonal)
##       [,1]
## [1,] 1.217
## ------------------------------------------------------------
## Regression Coefficients
##      [,1]
## [1,]    0
## ------------------------------------------------------------
## Item Parameters -A*Xsi
##   item    N     M est.guess guess AXsi_.Cat1 B.Cat1.Dim1
## 1  I01 1000 0.572         1 0.072     -0.332       1.095
## 2  I02 1000 0.442         2 0.030      0.802       3.544
## 3  I03 1000 0.729         3 0.068     -1.324       1.195
## 4  I04 1000 0.262         4 0.053      1.809       1.669
## 5  I05 1000 0.295         5 0.062      2.278       3.016
## 
## Gammaslope Parameters
## [1] 1.095 3.544 1.195 1.669 3.016
```

]

---

## 3PL - Fit

```r
f.m3pl = tam.modelfit(m3pl)
```

```{.scroll-100}
## **** Calculate Residuals 
## **** Calculate Counts 
## **** Calculate Covariances
```

```r
f.m3pl$statlist
```

```{.scroll-100}
##   X100.MADCOV     SRMR    SRMSR     MADaQ3 pmaxX2
## 1    3.481883 0.135382 0.137495 0.06034752      0
```

```r
f.m3pl$modelfit.test
```

```{.scroll-100}
##      maxX2 Npairs p.holm
## 1 116.1181     10      0
```

]

---

## 3PL - Item fit

```r
item.fit.3pl = IRT.itemfit(m3pl)

item.fit.3pl$chisquare_stat
```

```{.scroll-100}
##   item   Group1
## 1  I01 21.67682
## 2  I02      Inf
## 3  I03 27.58752
## 4  I04 86.01963
## 5  I05      Inf
```

```r
item.fit.3pl$RMSD
```

```{.scroll-100}
##   item     Group1
## 1  I01 0.06361902
## 2  I02 0.13848673
## 3  I03 0.07387837
## 4  I04 0.12824780
## 5  I05 0.22998383
```

```r
item.fit.3pl$RMSD_summary
```

```{.scroll-100}
##     Parm         M         SD        Min       Max
## 1 Group1 0.1268432 0.06629122 0.06361902 0.2299838
```

]

---

## 3PL - ICC
.scrollable[

```r
irt.icc(m3pl)$icc.graph + theme_bw()
```

]

---

# Scelta del modello

---

## Confronto tra modelli

```r
IRT.compareModels(m1pl, m2pl, m3pl)
```

```{.scroll-100}
## $IC
##   Model   loglike Deviance Npars Nobs      AIC      BIC     AIC3     AICc
## 1  m1pl -2706.625 5413.250     6 1000 5425.250 5454.697 5431.250 5425.335
## 2  m2pl -2654.262 5308.525    10 1000 5328.525 5377.602 5338.525 5328.747
## 3  m3pl -2654.822 5309.644    16 1000 5341.644 5420.168 5357.644 5342.197
##       CAIC       GHP
## 1 5460.697 0.5425250
## 2 5387.602 0.5328525
## 3 5436.168 0.5341644
## 
## $LRtest
##   Model1 Model2       Chi2 df p
## 1   m1pl   m2pl 104.725382  4 0
## 2   m1pl   m3pl 103.606273 10 0
## 3   m2pl   m3pl  -1.119109  6 1
## 
## attr(,"class")
## [1] "IRT.compareModels"
```

---

## Scelta del modello

Il 2PL è il modello migliore

---
class: section, center, middle

# Verifica delle assunzioni

---

## Unidimensionalità

```r
item_lab = paste(colnames(data), collapse = " + ")
form = paste("latent =~", item_lab)

form
```

```{.scroll-100}
## [1] "latent =~ I01 + I02 + I03 + I04 + I05"
```

```r
model = cfa(form, data = data,  ordered = colnames(data))
summary(model, fit.measures = T)
```

```{.scroll-100}
## lavaan 0.6-12 ended normally after 15 iterations
## 
##   Estimator                                       DWLS
##   Optimization method                           NLMINB
##   Number of model parameters                        10
## 
##   Number of observations                          1000
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                 1.643       3.095
##   Degrees of freedom                                 5           5
##   P-value (Chi-square)                           0.896       0.685
##   Scaling correction factor                                  0.551
##   Shift parameter                                            0.115
##     simple second-order correction                                
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2945.814    2439.475
##   Degrees of freedom                                10          10
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.208
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.002       1.002
##                                                                   
##   Robust Comparative Fit Index (CFI)                            NA
##   Robust Tucker-Lewis Index (TLI)                               NA
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.019       0.034
##   P-value RMSEA <= 0.05                          0.999       0.995
##                                                                   
##   Robust RMSEA                                                  NA
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                        NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.014       0.014
## 
## Parameter Estimates:
## 
##   Standard errors                           Robust.sem
##   Information                                 Expected
##   Information saturated (h1) model        Unstructured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   latent =~                                           
##     I01               1.000                           
##     I02               1.516    0.096   15.845    0.000
##     I03               1.035    0.088   11.716    0.000
##     I04               1.180    0.083   14.202    0.000
##     I05               1.436    0.088   16.291    0.000
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .I01               0.000                           
##    .I02               0.000                           
##    .I03               0.000                           
##    .I04               0.000                           
##    .I05               0.000                           
##     latent            0.000                           
## 
## Thresholds:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     I01|t1           -0.181    0.040   -4.549    0.000
##     I02|t1            0.146    0.040    3.665    0.000
##     I03|t1           -0.610    0.042  -14.364    0.000
##     I04|t1            0.637    0.043   14.915    0.000
##     I05|t1            0.539    0.042   12.885    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .I01               0.618                           
##    .I02               0.123                           
##    .I03               0.591                           
##    .I04               0.468                           
##    .I05               0.212                           
##     latent            0.382    0.044    8.740    0.000
## 
## Scales y*:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     I01               1.000                           
##     I02               1.000                           
##     I03               1.000                           
##     I04               1.000                           
##     I05               1.000
```

]

---

## Monotonicità

```r
mono_check = check.monotonicity(data)
summary(mono_check)
```

```{.scroll-100}
##     ItemH #ac #vi #vi/#ac maxvi  sum sum/#ac zmax #zsig crit
## I01  0.47  10   0     0.0  0.00 0.00  0.0000 0.00     0    0
## I02  0.65  10   0     0.0  0.00 0.00  0.0000 0.00     0    0
## I03  0.55  10   1     0.1  0.05 0.05  0.0049 1.56     0   18
## I04  0.55  10   0     0.0  0.00 0.00  0.0000 0.00     0    0
## I05  0.63   6   0     0.0  0.00 0.00  0.0000 0.00     0    0
```

---
## Indipendenza Locale

```r
f.m2pl$Q3_summary
```

```{.scroll-100}
##   type             M         SD        min         max      SGDDM     wSGDDM
## 1   Q3 -1.686040e-01 0.09870174 -0.4230942 -0.06357212 0.16860403 0.16860403
## 2  aQ3 -3.552714e-18 0.09870174 -0.2544902  0.10503191 0.07103058 0.07103058
```

---
class: section, center, middle

#  Item Information Function

---

## IIF per il 2PL

```r
info2pl = irt.iif(m2pl)

ggplot(info2pl$item_info, 
       aes(x = theta, y = ii_item, group = item, color = item)) + geom_line(lwd = 1) + theme_bw()
```

<img src="stimareImodelli_files/figure-html/unnamed-chunk-30-1.png" width="90%" style="display: block; margin: auto;" />
]

---

## TIF per il 2PL

```r
ggplot(info2pl$test_info, 
       aes(x = theta, y = info)) + geom_line(lwd = 2) + theme_bw()
```

]

---

## SE per il 2PL

```r
ggplot(info2pl$test_info, 
       aes(x = theta, y = se, col = "red")) + geom_line(lwd = 2) + theme_bw() + theme(legend.position = "none")
```

]

---

---
class: section, center, middle

#  Esercitazione

---

## Esercitazione stima

Scaricare il dataset `dataES_stima.csv` dalla cartella [Dati](https://drive.google.com/drive/folders/1PXDG7HhjRDMdFEQjk5WQQWdiorDpSgtw?usp=sharing)

Look at the data!

Fittare i modelli IRT

Scegliere il modello più appropriato

Valutarne la fit (sia del modello sia degli item)

Valutare le assunzioni

IIC, IIF, TIF

---