IRTest can be a useful tool for IRT (item response
theory) parameter estimation, especially when the violation of normality
assumption on latent distribution is suspected.
IRTest deals with uni-dimensional latent
variable.
In IRTest, along with the conventional approach that
assumes normality on latent distribution, several methods can be applied
for estimation of latent distribution:
+ empirical histogram method,
+ two-component Gaussian mixture distribution,
+ Davidian curve,
+ kernel density estimation.
You can install IRTest on R-console with:
install.packages("IRTest")Followings are functions of IRTest available for users.
IRTest_Dich is the estimation function when all
items are dichotomously scored.
IRTest_Poly is the estimation function when all
items are polytomously scored.
IRTest_Mix is the estimation function for a
mixed-format test, a combination of dichotomous item(s) and polytomous
item(s).
DataGeneration generates several objects that are
useful for computer simulation studies. Among these are starting values
for an algorithm and artificial item-response data that can be passed to
IRTest_Dich, IRTest_Poly, or
IRTest_Mix
plot_LD draws a plot of the estimated latent
distribution.
dist2 is a probability density function of
two-component Gaussian mixture distribution.
original_par_2GM converts re-parameterized
parameters of two-component Gaussian mixture distribution into original
parameters.
A simulation study for a Rasch model can be done in following manners:
library(IRTest)Alldata <- DataGeneration(seed = 123456789,
model_D = rep(1, 20),
N=1000,
nitem_D = 20,
nitem_P = 0,
d = 1.664,
sd_ratio = 2,
prob = 0.3)
data <- Alldata$data_D
item <- Alldata$item_D
initialitem <- Alldata$initialitem_D
theta <- Alldata$thetaFor an illustrative purpose, empirical histogram method is used for the estimation of latent distribution.
Mod1 <- IRTest_Dich(initialitem = initialitem,
data = data,
model = rep("1PL", 20),
latent_dist = "EHM",
threshold = .001
)### True item parameters
colnames(item) <- c("a", "b", "c")
knitr::kable(item, format='simple', caption = "True item parameters")| a | b | c |
|---|---|---|
| 1 | -0.96 | 0 |
| 1 | 1.04 | 0 |
| 1 | 0.47 | 0 |
| 1 | -0.16 | 0 |
| 1 | -0.81 | 0 |
| 1 | -0.40 | 0 |
| 1 | 0.82 | 0 |
| 1 | -0.37 | 0 |
| 1 | -1.11 | 0 |
| 1 | 0.50 | 0 |
| 1 | -0.97 | 0 |
| 1 | -1.05 | 0 |
| 1 | 0.02 | 0 |
| 1 | 1.32 | 0 |
| 1 | -0.50 | 0 |
| 1 | 0.18 | 0 |
| 1 | -1.39 | 0 |
| 1 | 0.59 | 0 |
| 1 | -0.58 | 0 |
| 1 | -1.59 | 0 |
True item parameters
### Estimated item parameters
knitr::kable(Mod1$par_est, format='simple', caption = "Estimated item parameters")| a | b | c |
|---|---|---|
| 1 | -0.8177894 | 0 |
| 1 | 0.9514716 | 0 |
| 1 | 0.4703169 | 0 |
| 1 | -0.0574434 | 0 |
| 1 | -0.8503595 | 0 |
| 1 | -0.4316589 | 0 |
| 1 | 0.8852200 | 0 |
| 1 | -0.3157931 | 0 |
| 1 | -1.1680628 | 0 |
| 1 | 0.5366363 | 0 |
| 1 | -1.0744048 | 0 |
| 1 | -1.1621301 | 0 |
| 1 | 0.0709861 | 0 |
| 1 | 1.2536640 | 0 |
| 1 | -0.4265914 | 0 |
| 1 | 0.2046360 | 0 |
| 1 | -1.3770776 | 0 |
| 1 | 0.5984116 | 0 |
| 1 | -0.7533302 | 0 |
| 1 | -1.6965297 | 0 |
Estimated item parameters
### Plotting
par(mfrow=c(1,2))
plot(item[,2], Mod1$par_est[,2], xlab = "true", ylab = "estimated", main = "item parameters")
abline(a=0,b=1)
plot(theta, Mod1$theta, xlab = "true", ylab = "estimated", main = "ability parameters")
abline(a=0,b=1)
plot_LD(Mod1)+
geom_line(mapping = aes(colour="Estimated"))+
geom_line(mapping=aes(x=seq(-6,6,length=121),
y=dist2(seq(-6,6,length=121),prob = .3, d=1.664, sd_ratio = 2),
colour="True"))+
labs(title="The estimated latent density using 'EHM'", colour= "Type")+
theme_bw()
Each examinee’s posterior distribution is identified in the E-step of
the estimation algorithm (i.e., EM algorithm). Posterior distributions
can be found in Mod1$Pk.
set.seed(1)
selected_examinees <- sample(1:1000,6)
post_sample <- data.frame(X=rep(seq(-6,6, length.out=121),6),
posterior = 10*c(t(Mod1$Pk[selected_examinees,])),
ID=rep(paste("examinee", selected_examinees), each=121))
ggplot(data=post_sample, mapping=aes(x=X, y=posterior, group=ID))+
geom_line()+
labs(title="Posterior densities for selected examinees", x=expression(theta))+
facet_wrap(~ID, ncol=2)+
annotate(geom="line", x=seq(-6,6,length=121),
y=dist2(seq(-6,6,length=121),prob = .3, d=1.664, sd_ratio = 2), colour="grey")+
theme_bw()