The expected mean squares (EMS) for the effects in a two-way ANOVA model are derived when sampling from a finite population, for factors A and B. The A-effects and B-effects are represented by τi and βj , respectively, in the model. Model effects are studied for the random and mixed effects models. Thus, in terms of hypothesis testing, we are interested in the expected mean square formulas when the random effects are sampled from finite populations. For balanced data, the EMS for the A, B and AB interaction effects when the effects are sampled from a finite population are the same as the EMS for an infinite population. For unbalanced data, the EMS when the effects are sampled from a finite population in factors A, B and the AB interaction, the EMS are not the same as for an infinite population because the values that multiply the variance components differ.
