Constrained statistical inference: sample-size tables for ANOVA and regression

Researchers in the social and behavioral sciences often have clear expectations about the order/direction of the parameters in their statistical model. For example, a researcher might expect that regression coefficient β(1) is larger than β(2) and β(3). The corresponding hypothesis is H: β(1) > {β(2), β(3)} and this is known as an (order) constrained hypothesis. A major advantage of testing such a hypothesis is that power can be gained and inherently a smaller sample size is needed. This article discusses this gain in sample size reduction, when an increasing number of constraints is included into the hypothesis. The main goal is to present sample-size tables for constrained hypotheses. A sample-size table contains the necessary sample-size at a pre-specified power (say, 0.80) for an increasing number of constraints. To obtain sample-size tables, two Monte Carlo simulations were performed, one for ANOVA and one for multiple regression. Three results are salient. First, in an ANOVA the needed sample-size decreases with 30–50% when complete ordering of the parameters is taken into account. Second, small deviations from the imposed order have only a minor impact on the power. Third, at the maximum number of constraints, the linear regression results are comparable with the ANOVA results. However, in the case of fewer constraints, ordering the parameters (e.g., β(1) > β(2)) results in a higher power than assigning a positive or a negative sign to the parameters (e.g., β(1) > 0).