Chapter 9 Two-way between-groups ANOVA Psyc301- Spring 2013 SPSS Session TA: Ezgi Aytürk

One-way ANOVA: One IV Two-way ANOVA: Two IVs (factors)

Why do we use two-way ANOVA instead of running two one-way ANOVAs Two-way ANOVA gives us both the MAIN EFFECTS and INTERACTION EFFECT if any. Main effect: The effect of one independent variable averaged across the levels of the other independent variable (The effect of one independent variable ignoring the other variable). Interaction effect: If the effect of one factor depends on the level of the other factor.

Example… Reward magnitude (IV1) Level of mastery on task (IV2) Performance (DV)

Assumptions The populations from which the samples were obtained must be normal. The samples must be independent. The scores in each group should have homogenous variance. The groups must have the same sample size DV must be an interval/ratio variable. Factors must be nominal/ordinal variable with at least two levels.

Hypotheses Three hypotheses are tested simultaneously: 1) The effect of independent variable #1 on the dependent variable (main effect). 2) The effect of independent variable #2 on the dependent variable (main effect). 3) The combined (joint) effect of independent variables #1 and #2 on the dependent variable (interaction effect).

Working example Data: Children_autonomy_data.sav Questions: 1)Does the sex of children influence their autonomy? (main effect) 2)Does the maternal education influence children’s autonomy? (main effect) 3) Does the effect of maternal education on children’s autonomy depends on children’s sex? (Interaction effect)

Analyze-General Linear Model- Univariate… Move autonomy (DV) into dependent variable box. Move both sex (factor1) and educ (factor2) into Fixed factors box. Click on Plots Move educ to horizontal axis, click on Add button (main effect plot for education) Move sex to horizontal axis, click on Add button (main effect plot for sex) Move educ to horizontal axis AND sex to separate lines, click on Add button. ( This is the interaction plot!). Continue.. Click on Post-hoc and select only educ (since sex has only two levels, we do not need a post-hoc analysis for it) for Post-hoc analysis. Select Tukey. Finally, click on Options… Display Means For: sex, educ, and sex*educ. Display Descriptive statistics and homogeneity tests OK.

It is not significant. We are not violating the homogeneity of variance assumption.

Main effect of sex is significant. F(1, 354)= P=.000

Main effect of education is significant. F(2,354)=19.08 p=.000

Interaction effect of sex and education is significant F(2, 354)=3,942, p= 0.2.

Lines are not parallel because there is a significant interaction effect. As educ increases, the difference between sexes on autonomy scores increases.

Do they overlap??

We have a significant interaction effect. Next, we need to conduct a simple effect analysis. Simple effect: The effect of one variable at one level of another variable. The effects of Sex at each level of Maternal education ▫The effect of sex when maternal education is low (=1) ▫The effect of sex when maternal education is moderate (=2) ▫The effect of sex when maternal education is high (=3) The effects of Maternal education at each level of Sex ▫The effect of maternal education for females ▫The effect of maternal education for males

Simple effects… 1)Data  Select cases  if educ = 1 (Run one-way ANOVA; DV: autonomy, IV:sex) 2)Data  Select cases  if educ = 2 (Run one-way ANOVA; DV: autonomy, IV:sex) 3)Data  Select cases  if educ = 3 (Run one-way ANOVA; DV: autonomy, IV:sex) 4)Data  Select cases  if sex= 1 (Run one-way ANOVA; DV: autonomy, IV:educ) (You also need a post-hoc here) 5)Data  Select cases  if sex= 2 (Run one-way ANOVA; DV: autonomy, IV:educ) (You also need a post-hoc here)

How to report main effects? The main effect of sex on children’s autonomy is significant (F(1, 354)= , p=.000). Males autonomy scores (M=106.06, SD=16.05) are significantly greater than those of females (M=89.62, SD=16.89). The main effect of maternal education on children’s autonomy is also significant (F(2,354)=19.08, p=.000). Multiple comparison analysis shows that the difference between low (M=94.61, SD= 15.61) and moderate (M=93.90, SD=18.14) level of education is not significant, whereas the difference between low and high levels of education(M=105.01, SD= 19.25) and the difference between moderate and high levels of education is significant.

How to report interaction effect? “ The interaction effect of mothers’ education and children’s sex was significant (F(2,354) = 3,94, p=.02). The effect of maternal education on children’s autonomy depends on children’s sex (or the effect of gender on autonomy depends on mother’s education). As mothers’ education level increased, the difference between girls and boys in their automomy scores increased. For highly educated mothers, the difference between boy’s autonomy scores (M=115.82, SD=16,40) and girl’s autonomy scores (M=94.20, SD=15,54) was greater compared to those with middle (M= , SD= 13.4 for males; M=85.25, SD= for females) and low educated mothers (M= 99.8 SD=13.5 for males; M= 89.4, SD=15.9 for females).”