Factorial Laboratory Experiment — Multi-Factor Controlled Experimental Design

A main effect is the average influence of one factor across all levels of the other factors. An interaction effect occurs when the effect of one factor differs depending on the level of another factor. For example, caffeine might improve performance under high workload but have no effect under low workload — that pattern is an interaction. Always examine the interaction term first; a significant interaction means the main effects must be interpreted cautiously and separately within each level of the other factor.

Sample size must be determined by an a priori power analysis that specifies the expected effect size for the interaction (typically smaller than main effects), desired power (0.80 minimum), alpha level, and number of conditions. As a rough guideline, a 2×2 between-subjects factorial design detecting a medium interaction effect (f = 0.25) at 80% power requires approximately 20 participants per cell (80 total). Add more cells or factors and total N rises accordingly. Never use rule-of-thumb minimums without a formal power calculation.

Within-subjects designs require fewer participants and have higher power because each participant serves as their own control, but they risk carryover and order effects. Between-subjects designs avoid these contamination risks but need larger samples. Choose within-subjects when the treatment effects are unlikely to persist across conditions and counterbalancing is practical. Choose between-subjects when carryover is likely, when demand characteristics are a concern, or when conditions involve irreversible changes.

A full factorial design tests every possible combination of factor levels. A fractional factorial design tests a carefully chosen subset of combinations — typically a half or quarter fraction — to reduce participant burden when many factors are involved. Fractional designs can estimate main effects and low-order interactions efficiently but cannot estimate all higher-order interactions. They are common in engineering and screening experiments; in behavioral laboratory research the full factorial is standard unless the number of factors is large.

Yes. A factorial ANCOVA (analysis of covariance) allows you to statistically control a continuous covariate — such as baseline anxiety, IQ, or prior task experience — that may account for some outcome variance. This increases precision and removes a potential confound. The covariate must be measured before the manipulation and must meet the assumption of homogeneity of regression slopes across conditions.