Random Effects (in Mixed Model ANOVA)

The term random effects in the context of analysis of variance is used to denote factors in an ANOVA design with levels that were not deliberately arranged by the experimenter (those factors are called fixed effects), but which were sampled from a population of possible samples instead. For example, if one were interested in the effect that the quality of different schools has on academic proficiency, one could select a sample of schools to estimate the amount of variance in academic proficiency (component of variance ) that is attributable to differences between schools.
A simple criterion for deciding whether or not an effect in an experiment is random or fixed is to ask how one would select (or arrange) the levels for the respective factor in a replication of the study. For example, if one wanted to replicate the study described in this example, one would choose (take a sample of) different schools from the population of schools. Thus, the factor "school" in this study would be a random factor. In contrast, if one wanted to compare the academic performance of boys to girls in an experiment with a fixed factor Gender, one would always arrange two groups: boys and girls. Hence, in this case the same (and in this case only) levels of the factor Gender would be chosen when one wanted to replicate the study.
See also, Analysis of Variance and Variance Components and Mixed Model ANOVA/ANCOVA .