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```
## Here is a sample analysis of the LSAT data using the Rasch model
## First some descriptives for LSAT
dsc <- descript(LSAT)
dsc
plot(dsc, type = "b", lty = 1)
## First we fit the original form of the Rasch model assuming
## fixed discrimination parameter equal to 1; results are reported
## under the usual IRT parameterization; in order to fix the
## discrimination parameter the 'constraint' argument is used
m1 <- rasch(LSAT, constr = cbind(length(LSAT) + 1, 1))
summary(m1)
## In order to check the fit of the model the GoF.rasch() function
## is used; This computes a Bootstrap p-value for the Pearson's
## Chi-squared statistic
GoF.rasch(m1, B = 199) # B specifies the number of Bootstrap samples
## Alternatively, we could also check the fit on the margins
margins(m1)
margins(m1, "three-way")
## The Item Characterstic Curves are produced by the plot() function
plot(m1, lwd = 3, cex = 1.2)
# or
plot(m1, legend = TRUE, lwd = 3, cx = 1, cy = 0.7) # 'cx' and 'cy' define the coordinates of the legend
## The Item Information Curves are produced using type = "IIC"
## increase cex.lab and cex.main
plot(m1, type = "IIC", legend = TRUE, cx = "topright", lwd = 2.3, cex = 1.3, cex.lab = 1.2, cex.main = 1.6)
## The Test Information Function is produced using type = "IIC" and items = 0
plot(m1, type = "IIC", items = 0, lwd = 2.3)
## We repeat the analysis without constaining discrimination parameter
m2 <- rasch(LSAT)
summary(m2)
## The Goodness-of-Fit is checked again
GoF.rasch(m2, B = 199) # B specifies the number of Bootstrap samples
## The fit on the margins
margins(m2)
margins(m2, "three-way")
## The Likelihood Ratio Test of the two models is computed again with
## the anova() function; remember to put first the model under the null
## hypothesis -- in this case the constrained Rasch model m1
anova(m1, m2)
## The Item Characterstic Curves for the unconstrained model
## plot only items 1, 3 and 5
plot(m2, items = c(1, 3, 5), lwd = 3, cex = 1.2)
# or
plot(m2, items = c(1, 3, 5), legend = TRUE, lwd = 3, cx = 1, cy = 0.7)
## The Item Information Curves are produced using type = "IIC";
## plot only items 1, 3 and 5
plot(m2, type = "IIC", items = c(1, 3, 5), legend = TRUE, cx = "topright", lwd = 2.3, cex = 1.3)
## The Test Information Function is produced using type = "IIC" and items = 0
plot(m2, type = "IIC", items = 0, lwd = 2.3)
## Finally, the ability estimates can be obtained using the factor.scores() function
factor.scores(m2)
# or
factor.scores(m2, method = "MI", B = 20) # using Multiple Imputation
```

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