Factorial Experiments and Interaction Effects Cheat Sheet

Key Facts

• A factorial design with factor $A$ having $a$ levels and factor $B$ having $b$ levels has $ab$ treatment combinations.
• With $n$ replicates per treatment combination, the total sample size in a balanced two-factor design is $N=abn$.
• A common two-factor fixed-effects model is $Y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}$, where $\varepsilon_{ijk}$ represents random error.
• The main effect of factor $A$ compares marginal means, such as $\bar{Y}_{i..}-\bar{Y}_{...}$.
• An interaction effect exists when the difference between levels of one factor changes across levels of another factor, meaning $(\alpha\beta)_{ij}\neq 0$ for at least one combination.
• The ANOVA test statistic for any effect is $F=\frac{MS_{\text{effect}}}{MS_E}$, where $MS_E$ estimates within-cell error variance.
• For a balanced two-factor ANOVA, the degrees of freedom are $df_A=a-1$, $df_B=b-1$, $df_{AB}=(a-1)(b-1)$, and $df_E=ab(n-1)$.
• In a $2^k$ factorial design, $k$ factors each have $2$ levels, producing $2^k$ treatment combinations before replication.

Vocabulary

Factor

A factor is an explanatory variable that the experimenter studies, such as temperature, fertilizer type, or teaching method.

Level

A level is a specific value or category of a factor, such as low and high temperature.

Treatment Combination

A treatment combination is one specific pairing or grouping of factor levels used in the experiment.

Main Effect

A main effect is the average effect of one factor on the response after averaging over the levels of other factors.

Interaction Effect

An interaction effect occurs when the effect of one factor on the response depends on the level of another factor.

Balanced Design

A balanced design has the same number of observations in every treatment combination.