Chapter 10 Factorial Analysis of Variance Part 2 – Apr 3, 2008

F tests in factorial ANOVA REVIEW: Conduct 3 F tests –2 possible main effects (variable 1, variable 2) –1 possible interaction (variable 1 X variable 2) Look at cell means and from them, find marginal means –Marginal means can show main effects –Cell means can show an interaction When subtracting cell means in row 1 versus row 2 (etc. if more rows) and notice a difference in direction or magnitude of the difference  evidence of an interaction Note that any combination can be signif –With signif interaction and main effects, always interpret the interaction 1 st  meaning of main effects may change if there is an interaction!

Main Effects and/or Interaction If no interaction, you can interpret the main effect (or effects, if both…) as you would the one-way ANOVAs –See table 10-4, Result D (p. 390) –There is a main effect of age on income, such that older workers make significantly more than younger workers. –There is no main effect of education and no interaction –The impact of age on income did not depend on level of education If significant interaction and main effects – report F statistics for interaction & main effects, but… – only interpret the interaction –See Results B and F (Table 10-4 p. 390)

Result B: –No main effects (50 v. 50; 50 v. 50) but an interaction (row 1 diff = 20, row 2 diff = -20) Younger workers with high school education make more than younger workers with college education; But…older workers with high school education make less than older workers with college education. Result F: –2 main effects (37.5 v and 32.5 v. 52.5) and an interaction (row 1 diff = -15, row 2 diff = -25) Workers with college education make more than those with high school BUT the difference is more extreme for older than younger workers.

Recognizing and Interpreting Interaction Effects Graphically, will see an interaction if pattern of bars for 1 st section of graph differs from 2 nd section… For inappropriate group, hiring chances decreased from neutral  sad… For appropriate group, hiring chances increased from neutral  sad

Basic Logic of 2-way ANOVA Calculate 3 F ratio’s comparing between group variance to within-group variance –1 st F ratio for column main effect –2 nd F ratio for row main effect –3 rd F ratio for interaction effect –Only the numerator will change (betw-gp variance); denominator (within-gp variance) is same for all 3, just an average of pop variance based on all scores in sample

Two-Way ANOVA table Reports SS, df, MS, and F ratios for all 3 sources (column main effect, row main effect, interaction)…SPSS output looks like this, too!

Factorial ANOVA in research articles (see book for example!) A two-factor ANOVA yielded a significant main effect of voice, F(2, 245) = 26.30, p <.001. As expected, participants responded less favorably in the low voice condition (M = 2.93) than in the high voice condition (M = 3.58). The mean rating in the control condition (M = 3.34) fell between these two extremes. Of greater importance, the interaction between culture and voice was also significant, F(2, 245) = 4.11, p <.02. See Figure 1 for a graph of the interaction effect. (note: Fig 1 isn’t actually included in these notes, just written as an example of how you would cite this).

SPSS example Analyze  General Linear Model  Univariate Pop-up box, indicate ‘Dependent Variable” (here, ‘harass’), and “Fixed factor” (here, ‘year’ & ‘working’) –Don’t need to specify anything in the other boxes… Choose ‘options’  ‘display means for’ choose year, working, & year*working Choose ‘plots’  ‘horizontal axis’ choose year; ‘separate lines’ choose working, hit ‘Add’, then ‘Continue’, then ‘OK’ See Output handout for interpretation