Those involved in educational and psychological measurement must deal with a number of issues that arise when one assumes a probabilistic relationship between the observations made of a learner and the learner’s underlying cognitive constructs. The essential idea is that statistical models can be developed to predict the probability that people will behave in certain ways in assessment situations, and that evidence derived from observing these behaviors can be used to draw inferences about students’ knowledge, skills, and strategies (which are not directly observable).^{2} In assessment, aspects of students’ knowledge, skills, and strategies that cannot be directly observed play the role of “that which is to be explained” —generally referred to as “cognition” in Chapter 2 and more specifically as the “construct” in this chapter. The constructs are called “latent” because they are not directly observable. The things students say and do constitute the evidence used in this explanation—the observation element of the assessment triangle.

In broad terms, the construct is seen as “causing” the observations, although generally this causation is probabilistic in nature (that is, the constructs determine the probability of a certain response, not the response itself). More technically there are two elements of probability-based measurement models: (1) unobservable latent constructs and (2) observations or observable variables, which are, for instance, students’ scores on a test intended to measure the given construct. The nature of the construct variables depends partly on the structure and psychology of the subject domain and partly on the purpose of assessment. The nature of the observations is determined by the kinds of things students might say or do in various situations to provide evidence about their values with respect to the construct. Figure 4– 1 shows how the construct is related to the observations. (In the figure, the latent construct is denoted θ [theta] and the observables *x*.) Note that although the latent construct causes the observations, one needs to go the other way when one draws inferences—back from the observations to their antecedents.

Other variables are also needed to specify the formal model of the observations; these are generally called *item parameters.* The central idea of probability models is that these unknown constructs and item parameters do not determine the specifics of what occurs, but they do determine the probability associated with various possible results. For example, a coin might be expected to land as heads and as tails an approximately equal number of