Factorial ANOVA and Multiple Comparisons

Vocabulary practice flashcards covering factorial ANOVA designs, main effects, ordinal and disordinal interactions, degrees of freedom, effect sizes, power calculations, planned orthogonal contrasts, and post-hoc multiple comparison procedures.

Factorial ANOVA

An analysis of variance that evaluates the simultaneous effects of two or more independent variables on a single dependent variable to assess main effects and examine whether adding an independent variable impacts the effect of the first.

One-Way ANOVA

An analysis of variance that evaluates the effect of a single independent variable on a single dependent variable.

Main Effect

The individual effect of a single independent variable on the dependent variable, averaged across all levels of the other independent variable(s) and evaluated using row means or column means.

Interaction Effect

The joint effect occurring when the impact of one independent variable on the dependent variable varies depending on the specific level of another independent variable.

Cell

A specific treatment group or condition in a factorial design defined by the intersection of one level of a row factor and one level of a column factor.

Ordinal Interaction

A type of interaction where the degree of superiority changes across conditions, but one subgroup consistently scores higher or better than the other, resulting in lines that do not cross when plotted.

Disordinal Interaction

A type of interaction characterized by a reversal in the degree of superiority across conditions, where one group performs better under one treatment level while the other group performs better under a different level, resulting in intersecting lines when plotted.

Sources of Variation (Two-Way ANOVA)

The four distinct components into which total variance is partitioned: the row main effect ($SS_R$), the column main effect ($SS_C$), the interaction effect ($SS_{RC}$), and the within-group error variance ($SS_W$).

Degrees of Freedom (Two-Way ANOVA)

The values assigned to each source of variance: row effect ($df_R = R - 1$), column effect ($df_C = C - 1$), interaction ($df_{RC} = (R - 1)(C - 1)$), and within-cell error ($df_W = N - RC$).

Eta Squared ($\eta^2$)

An effect size metric representing the proportion of total variability in the dependent variable explained by an independent variable.

Partial Eta Squared ($\eta^2_{\text{partial}}$)

The proportion of variance in the dependent variable explained by a specific factor while controlling for and excluding the variance of other effects, computed as $\eta^2_{\text{partial}} = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_W}$.

Effective Sample Size ($n_{RC}$)

The adjusted sample size value used to enter a factorial power table, calculated as $n_{RC} = \left[\frac{N - RC}{(R - 1)(C - 1) + 1}\right] + 1 = \left[\frac{df_W}{df_{RC} + 1}\right] + 1$.

Planned Comparisons

Hypothesis-driven comparisons between group means specified prior to conducting an ANOVA, which must be independent (orthogonal) of one another.

Post-Hoc Comparisons

Pairwise comparisons conducted after obtaining a significant omnibus ANOVA result to identify exactly which group means differ while controlling the familywise Type I error rate.

Post-Hoc Decision Roadmap (Factorial ANOVA)

The hierarchical testing procedure where the interaction ($R \times C$) is evaluated first; if significant, simple comparisons are tested using cell means, but if not significant, main effects are evaluated using marginal means.

Simple Comparisons

Post-hoc comparisons conducted between individual cell means to pinpoint the specific locus of a statistically significant interaction effect.

Main Comparisons

Post-hoc tests conducted between marginal row or column means following a statistically significant main effect with more than two levels, evaluated only when the interaction is not significant.

Bonferroni Correction

A conservative procedure to control Type I error across multiple tests by establishing an adjusted significance threshold equal to $\alpha_{\text{adjusted}} = \frac{\alpha}{\text{\# of comparisons}}$, which makes critical cutoffs stricter but increases the risk of Type II error.

Tukey's HSD (Honestly Significant Difference)

A post-hoc procedure based on the studentized range statistic ($q$) that tests all pairwise mean comparisons while controlling both Type I and Type II error rates, applicable when the homogeneity of variance assumption is satisfied.

Games-Howell Procedure

A post-hoc testing procedure used when the assumption of homogeneity of variance is violated or sample sizes are unequal, which decreases degrees of freedom to inflate the critical value and prevent Type I errors.

Harmonic Mean ($\tilde{n}$)

A weighted sample size average applied in Tukey's HSD when group sizes are unequal, calculated for any two groups as $\tilde{n} = \frac{2 \times n_1 \times n_2}{n_1 + n_2}$.

Orthogonal Contrasts

A set of independent planned comparisons where the sum of contrast weights equals zero ($\sum w_i = 0$) and the sum of cross-products between any two contrasts equals zero ($\sum w_{1i} w_{2i} = 0$), with a maximum of $k - 1$ possible comparisons.

Scheffé's Method

A highly conservative post-hoc test used to evaluate any and all possible linear combinations of group means while strictly maintaining the experimentwise alpha level.