Test Theory With Minimal Assumptions

Using the concepts of conditional probability, conditional expectation, and conditional independence, the main results of the classical test theory model can be derived in a very few steps with minimal assumptions. It is well known that the variance of a random variable is the sum of the variance of its conditional expectation and the expectation of its conditional variance. These concepts lead to another fundamental relation: the reliability coefficient, regarded as a correlation between two conditionally independent, conditionally identically distributed random variables (exchangeable random variables, parallel measurements) is the ratio of the variance of their common conditional expectation to their common unconditional variance. Formulas describing the reliability of lengthened tests follow easily, and the same result also applies to stochastic processes involving joint distributions of many dependent random variables.