score.irt: Find Item Response Theory (IRT) based scores for dichotomous or polytomous items

Description

irt.fa finds Item Response Theory (IRT) parameters through factor analysis of the tetrachoric or polychoric correlations of dichtomous or polytomous items. score.irt uses these parameter estimates of discrimination and location to find IRT based scores for the responses. As many factors as found for the correlation matrix will be scored.

Usage

score.irt(stats, items, keys=NULL,cut = 0.3,bounds=c(-3.5,3.5))  #the higher order call just calls 
score.irt.2(stats, items,keys=NULL,cut = 0.3,bounds=c(-3.5,3.5))  #for dichotomous items 
score.irt.poly(stats, items, keys=NULL, cut = 0.3,bounds=c(-3.5,3.5)) #for polytomous items
irt.stats.like(items,stats,keys=NULL,cut=.3) #to create irt like statistics for plotting
irt.tau(x)

Arguments

stats

Output from irt.fa is used for parameter estimates of location and discrimination. Stats may also be the output from a normal factor analysis (fa)

items

The raw data, may be either dichotomous or polytomous.

keys

A keys matrix of which items should be scored for each factor

cut

Only items with discrimination values > cut will be used for scoring.

The raw data to be used to find the tau parameter in irt.tau

bounds

The lower and upper estimates for the fitting function

Value

scoresA data frame of theta estimates, total scores based upon raw sums, and estimates of fit.

Details

Although there are more elegant ways of finding subject scores given a set of item locations (difficulties) and discriminations, simply finding that value of theta $\theta$ that best fits the equation $P(x|\theta) = 1/(1+exp(\beta(\delta - \theta) )$ for a score vector X, and location $\delta$ and discrimination $\beta$ provides more information than just total scores. With complete data, total scores and irt estimates are almost perfectly correlated. However, the irt estimates provide much more information in the case of missing data.

The bounds parameter sets the lower and upper limits to the estimate. This is relevant for the case of a subject who gives just the lowest score on every item, or just the top score on every item.

There are several more elegant packages in R that provide Full Information Maximum Likeliood IRT based estimates. The estimates from score.irt do not do so. However, the score.irt seems to do a good job of recovering the basic structure.

The keys matrix is a matrix of 1s, 0s, and -1s reflecting whether an item should be scored or not scored for a particular factor. See score.items or make.keys for details. The default case is to score all items with absolute discriminations > cut.

If one wants to score scales taking advantage of differences in item location but not do a full irt analysis, then find the item difficulties from the raw data using irt.tau or combine this information with a scoring keys matrix (see score.items and code{make.keys} and create quasi-irt statistics using irt.stats.like.

References

Kamata, Akihito and Bauer, Daniel J. (2008) A Note on the Relation Between Factor Analytic and Item Response Theory Models Structural Equation Modeling, 15 (1) 136-153. McDonald, Roderick P. (1999) Test theory: A unified treatment. L. Erlbaum Associates.

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at http://personality-project.org/r/book/

Examples

Run this code

d9 <- sim.irt(9,1000,-2.5,2.5,mod="normal") #dichotomous items
test <- irt.fa(d9$items)
scores <- score.irt(test,d9$items)
scores.df <- data.frame(scores,d9$theta) #combine the estimates with the true thetas.
pairs.panels(scores.df) #with all the data, why bother 
#now delete some of the data
d9$items[1:333,1:3] <- NA
d9$items[334:666,4:6] <- NA
d9$items[667:1000,7:9] <- NA
scores <- score.irt(test,d9$items)
scores.df <- data.frame(scores,d9$theta) #combine the estimates with the true thetas.
pairs.panels(scores.df) #with missing data, the theta estimates are noticably better.

v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items
items <- v9$items
test <- irt.fa(items)
total <- rowSums(items)
ord <- order(total)
items <- items[ord,]
#now delete some of the data - note that they are ordered by score
items[1:333,5:9] <- NA
items[334:666,3:7] <- NA
items[667:1000,1:4] <- NA
scores <- score.irt(test,items)
unitweighted <- score.irt(items=items,keys=rep(1,9))
scores.df <- data.frame(scores,v9$theta[ord],unitweighted) #combine the estimates with the true thetas.
pairs.panels(scores.df) #with missing data, the theta estimates are noticably better estimates of the generating theta

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