Education 231C

Applied Categorical & Nonnormal Data Analysis

A Rasch Model Example

Item response models may be used to model the responses of subjects to a number of questions or test items. An item response model with one parameter for item difficulty is known as a Rasch model. Georg Rasch (1901-1980), a Danish statistician, gave an axiomatic derivation of the model in the 1960s. We will be using a conditional (fixed-effects) logit model to illustrate the model, however, Rasch's derivation used a different approach, but one that turns out to be equivalent to the fixed-effects logit. Rasch models are one of the dominant models for binary items (e.g., success/failure on test items) in psychometrics.

In the Rasch model the log odds of subject i giving a correct response to item j may be modeled using a one-parameter logistic response model

where the βj represent the difficulty of the item and the ui represent the ability of the subject.

In terms of probability the model looks like this,

We will use an example that consists of five items from the lsat exam for 1,000 subjects.

Example

Item 1 is the easiest item, responded correctly by the most subjects, so we will use it as the reference item. We will run an fixed-effects xtlogit using the negative indicators for each of the remaining items. The fixed effects xtlogit is equivalent to running the conditional logistic command clogit. Item 1, the easiest item, has a difficulty value fixed at zero. Item 3 is the most difficult item with a coefficient of 2.49. Note that the fixed-effects xtlogit has dropped 301 subjects from the analysis. These subjects either responded to all items correctly or to all items incorrectly; in a conditional likelihood these subjects carry no information about the difficulty of the items.

We can also look at the item difficulty in terms of the probability of getting an items correct.

Looking at the item probabilities, we see that Item 1 has the highest probability (P = .49) of a correct response and Item 3 has the lowest probability (P = .04) of a correct response. The probabilities follow the same difficulty patterns and the coefficients.

We can check our model specification that the difficulty parameters are the same for the "poor" (low scoring) subjects and the "good" (high scoring) subjects, distinguished by their total score. We will do this using Hausman tests versus low scoring subjects (total = 0,1 or 2) and versus high scoring subjects (total = 3,4 or 5). The hausman command with the less option compares the fully efficient model with the less efficient, but consistent model.

Rasch models, along with other item response models, have an assumption of local independence, that is, the responses to a given item are independent of the responses to other items in the test. In practical terms, this implies Pr(yij=1 & yik=1) = Pr(yij=1)*Pr(yik=1). Further, Rasch models, because they are one-parameter models, assume that all of the items have equal discrimination, that is, the items discriminate equally well for "good" subjects as they do for "poor" subjects. Two-parameter and three-parameter item response models include measures of item discrimination along with item difficulty.

It is also possible to estimate the model using the gllamm command. gllamm is short for generalized linear latent and mixed models. This is, in fact, a random effects model. Note that the model includes a constant.

One advantage to using gllamm is that we can include all of the items by specifying the nocons option. Now we can compute the conditional probabilities using gllapred.

Creating the Rasch Dataset

A Stata Program: raschtest

Jean-Benoit Hardouin of the Regional Health Observatory in France has written several ado programs that will perform a maximum likelihood Rasch analysis. You will need the ado files raschtest.ado and gammasym.ado. The example below uses version 7.3 of raschtest dated 2july2005.

The data are organized differently from the analysis above. The data are organized by individual with each item scored right or wrong (0 or 1). We will use the dataset lsat3 reshaping it to the proper form.

The raschtest program computes both the item difficulties and the ability parameters. By default, raschtest sets the difficulty of the last item in the list to zero, so the values of the difficulty parameters are different in this analysis. But note that item1 is the easiest item. We can make item1 have a difficulty of zero by changing the order of the variables in the command so that item1 comes last. Now the items difficulties are the same as in our original xtlogit (conditional logistic) example. The genlt option generates latent trait (ability) scores for each subject and the genscore option gives a total correct for each observation. We can graph both item difficulty and ability on the same latent dimension. Here is a partial listing with the total score and latent trait score.

Categorical Data Analysis Course

Phil Ender revised 21feb06