Sequential probability ratio test

Overview
The sequential probability ratio test (SPRT) or likelihood-ratio test was developed by Abraham Wald as a hypothesis test for sequential analysis. While originally developed for use in quality control studies in the realm of manufacturing, it has been formulated for use in the computerized testing of human examinees as a termination criterion (Ferguson, 1967; Reckase, 1983; Eggen, 1999). This article describes that application; a statistical treatment is found under likelihood-ratio test.

The test is done on the proportion metric, and tests that a variable p is equal to one of two desired points, p1 or p2. The region between these two points is known as the indifference region (IR). For example, suppose you are performing a quality control study on a factory lot of widgets. Management would like the lot to have 3% or less defective widgets, but 1% or less is the ideal lot that would pass with flying colors. In this example, p1 = 0.01 and p2 = 0.03 and the region between them is the IR because management considers these lots to be marginal and is OK with them being classified either way. Widgets would be sampled one at a time from the lot (sequential analysis) until the test determines, within an acceptable error level, that the lot is ideal or should be rejected.

Application to testing of human examinees
The SPRT is currently the predominant method of classifying examinees in a variable-length computerized classification test (CCT). The two parameters are p1 and p2 are specified by determining a cutscore for examinees on the proportion correct metric, and selecting a point above and below that cutscore. For instance, suppose the cutscore is set at 70% for a test. We could select p1 = 0.65 and p2 = 0.75. The test then evaluates the likelihood that an examinee's true score on that metric is equal to one of those two points. If the examinee is determined to be at 75%, they pass, and they fail if they are determined to be at 65%.

These points are not specified completely arbitrarily. A cutscore should always be set with a legally defensible method, such as a modified Angoff procedure. Again, the indifference region represents the region of scores that the test designer is OK with going either way (pass or fail). The upper parameter p2 is conceptually the highest level that the test designer is willing to accept for a Fail (because everyone below it has a good chance of failing), and the lower parameter p1 is the lowest level that the test designer is willing to accept for a pass (because everyone above it has a decent chance of passing). While this definition may seem to be a relatively small burden, consider the high-stakes case of a licensing test for medical doctors: at just what point should we consider somebody to be at one of these two levels?

Application of Item Response Theory
While the SPRT was first applied to testing in the days of classical test theory, Reckase (1983) suggested that item response theory be used to determine the p1 and p2 parameters. The cutscore and indifference region are defined on the latent ability (theta) metric, and translated onto the proportion metric for computation. Research on CCT since then has applied this methodology for several reasons:

1. Large item banks tend to be calibrated with IRT

2. This allows more accurate specification of the parameters

3. By using the item response function for each item, the parameters are easiliy allowed to vary between items.