The Rasch model for speed tests and some extensions with applications to incomplete designs

In psychological measurement a distinction can be made between speed and power tests. Although most tests are partially speeded, the speed element is usually neglected. Here, the focus will be on latent trait models for pure speed tests. A particularly simple model has been developed by Rasch for the total response time on a (set of) pure speed test(s), based on the assumption that the test response times are approximately gamma distributed, with known index parameters and scale parameters depending on subject ability and test difficulty parameters, In the approach presented here the subject parameters are treated as random variables having a common gamma distribution. From this, maximum marginal likelihood estimators are derived for the test difficulties and the parameters of the latent subject distribution. This basic model can be extended in a number of ways, In a numerical example, art application of the Rasch model to reading data, which were incomplete by design, will be discussed.