Analysis of Variance (Two Factors)

Two Factor Analysis of Variance Main effect The effect of a single factor when any other factor is ignored. Example page 386 – Responsibility in crowds Main effects Crowd size Gender Table 18.1

Two Factor Analysis of Variance Interaction effect Combination or interaction of crowd size and gender on reaction time Interaction occurs whenever the effects of one factor on the dependent variable are not consistent for all values (or levels) of the second factor.

Two factor hypotheses Three different null hypotheses are tested one at a time, with three different F tests. F column Variability between columns (crowd size) F row Variability between rows (gender) F interaction Any remaining variability between cells not attributed to either of the other types of variability

Simple effect This represents the effect of one factor on the dependent variable at a single level of the second factor. Page 392 – two simple effects of crowd size, one for males, one for females Example C - since both simple effects are in the same direction the main effect can be interpreted without referring to its two simple effects. Interaction can be viewed as the product of inconsistent simple effects.

SS total = SS column + SS row + SS interaction +Ss within SS total = SS between +Ss within SS between = SS column + SS row + Ss interaction SS interaction = SS between - (SS colomn + SS row )

Computation formulas T 2 cell G 2 SS between = Σ n N T 2 cell SS within = Σ X 2 – Σ n T 2 column G 2 SS column = Σ rn N

Computation formulas T 2 row G 2 SS row = Σ cn N

Degrees of freedom df total = N – 1N = all data elements df column = c – 1c = columns or groups df row = r – 1r = rows or categories df interaction = (c – 1)(r – 1) df within = N – (c)(r)

Estimating Effect Size _________SS column ___________ ____SS column _______ η 2 p (column)= SS total – (SS row + SS interaction ) = SS column + SS within ____SS row ___ η 2 p (row) = SS row + SS within ____SS interaction ___ η 2 p (interaction) = SS interaction + SS within Each η 2 is a partial eta squared, accounting for only part of the total variance.

Calculating Simple effects Page 404 T 2 se G 2 se SS se = Σ n N se Where SS se signifies the sum of squares for the simple effect, or a single row (or column)

Tukey’s HSD for Multiple Comparisons Page 405 Use Tukey’s HSD test to find differences between pairs of means. Tukey’s “honestly significant difference” test MS within HSD = q√ n Where q (studentized range statistic) comes from Table G, Appendix C, page 529

Estimating effect Size Use Cohen’s d (defined on page 355, Equation 16.10) X 1 – X 2 D = √ MS within

Interpreting Two Factor ANOVA Page 406 If interaction is significant, Estimate its effect size with η 2 p and conduct F se tests for at least one set of simple effects Further analyze any significant simple effects with HSD tests and any significant HSD test with an estimate of its effect size, d.

Interpreting Two Factor ANOVA Page 406 If interaction is NOT significant, Estimate its effect size with η 2 p and conduct F test Further analyze any significant effects with HSD tests and any significant HSD test with an estimate of its effect size, d. (See flow chart in Word format)