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Multi-Factor ANOVA

NIST/SEMATECH Section 1.3.5.5 Multi-Factor ANOVA

What It Is

Multi-factor analysis of variance extends one-factor ANOVA to simultaneously test for the effects of two or more factors and their interactions. It partitions the total variability into components attributable to each factor, each interaction, and the residual error.

When to Use It

Use multi-factor ANOVA in designed experiments where multiple factors may influence the response variable. It is more efficient than running separate one-factor analyses because it can detect interaction effects -- situations where the effect of one factor depends on the level of another. This technique is central to factorial experimental design and industrial process optimization.

How to Interpret

Examine the F-statistics for each main effect and interaction term, comparing them against the F-distribution critical values at the appropriate degrees of freedom. If the interaction term is significant, interpret main effects cautiously because the effect of each factor depends on the other factor levels. Interaction plots (plotting mean response versus one factor at each level of the other) are essential for understanding significant interactions. When no significant interaction exists, interpret main effects independently. The residual mean square provides an estimate of experimental error.

Assumptions and Limitations

Multi-factor ANOVA assumes normality, independence, and homogeneity of variance across all factor-level combinations. Balanced designs (equal sample sizes per cell) are preferred because they make the analysis robust to moderate violations of assumptions and simplify the sum-of-squares decomposition. Unbalanced designs require Type III (or similar) sum-of-squares adjustments.

Reference: NIST/SEMATECH e-Handbook, Section 1.3.5.5

Formulas

Two-Factor F-Statistic (Factor A)

FA=MSAMSE=SSA/(a−1)SSE/(N−ab)F_A = \frac{MS_A}{MS_E} = \frac{SS_A / (a-1)}{SS_E / (N - ab)}

Tests whether the main effect of factor A is significant. MS_A is the mean square for factor A and MS_E is the mean square error.

Interaction F-Statistic

FAB=MSABMSE=SSAB / [(a−1)(b−1)]SSE / (N−ab)F_{AB} = \frac{MS_{AB}}{MS_E} = \frac{SS_{AB} \,/\, [(a-1)(b-1)]}{SS_E \,/\, (N - ab)}

Tests whether the interaction between factors A and B is significant. A significant interaction means the effect of one factor depends on the level of the other.

Total Decomposition

SST=SSA+SSB+SSAB+SSESS_T = SS_A + SS_B + SS_{AB} + SS_E

The total sum of squares is partitioned into main effects for each factor, the interaction term, and the residual error.