The Basics of Item Response Theory Using R by Frank B. Baker & Seock-Ho Kim

Author:Frank B. Baker & Seock-Ho Kim
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

5.4.4 Procedure for Investigating the Item Invariance of an Examinee’s Ability

In this example a given examinee will be administered a number of different tests. The intent is to illustrate that the estimated abilities should cluster about the value of the examinee’s ability parameter. The function ability will be employed in each replication to obtain the ability estimate and the standard error.

The model, the value of the ability parameter, and the number of items in a test are assumed to be known. The computer will generate the random values of the item parameters. Using these parameters and the specified item characteristic curve model, the item response vector will be randomly generated. The examinee’s ability will be estimated and shown in the R console window accompanied by its standard error. The process will be replicated several times. The maximum number of iterations is specified by the value of R. Theoretically, the ability estimates from replications should be very close to the ability parameter that was used to generate the data. The average of the ability estimates can be obtained and compared to the value of the ability parameter. There should not be a large amount of scatter in the estimates. Again, due to the small number of items and the limited number of ability estimates from the replications used, the item invariance of the ability estimate may not be readily apparent.

The R command lines for an example run are as follows:

mdl <- 2

theta <- 0.5

J <- 5

R <- 10

thr <- rep(0, R)

ser <- ref(0, R)

for (r in 1:R) {

b <- round(runif(J,-3,3), 2)

a <- round(runif(J,0.2,2.8), 2)

c <- round(runif(J,0,.35), 2)

if (mdl == 1 | mdl == 2) { c <- rep(0, J) }

if (mdl == 1) { a <- rep(1, J) }

u <- rep(0, J)

for (j in 1:J) {

P <- c[j] + (1 - c[j]) /

(1 + exp(-a[j] * (theta - b[j])))

u[j] <- rbinom(1, 1, P)

}

thse <- ability(mdl, u, b, a, c)

thr[r] <- thse[1]

ser[r] <- thse[2]

}

theta

thr

mean(thr)

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