Intro to Statistics for the Behavioral Sciences PSYC 1900 Lecture 14: Factorial ANOVA

Factorial ANOVA Generalization of one-way ANOVA to case where there are 2 or more independent variables. Generalization of one-way ANOVA to case where there are 2 or more independent variables. Each iv is referred to as a “factor.” Each iv is referred to as a “factor.” Nature of design is defined by number of conditions, or “groups,” per factor. Nature of design is defined by number of conditions, or “groups,” per factor. Gender and Religion (Christian, Jewish, Muslim) would be a 2X3 design. Gender and Religion (Christian, Jewish, Muslim) would be a 2X3 design. A cell represents a specific combination of the iv’s (e.g., female Christians). A cell represents a specific combination of the iv’s (e.g., female Christians).

Factorial ANOVA Factorials have 2 primary advantages. Factorials have 2 primary advantages. Allow greater tests of genearlizability. Allow greater tests of genearlizability. We can actually see if the effect of one iv holds across different groups defined by other iv’s. We can actually see if the effect of one iv holds across different groups defined by other iv’s. E.g, does increased temperature increase aggression for both men and women? E.g, does increased temperature increase aggression for both men and women? Allows tests of interaction. Allows tests of interaction. Do the iv’s interact to produce effects on the dv, whereby the nature of the effect of one variable depends on the level of the other variable? Do the iv’s interact to produce effects on the dv, whereby the nature of the effect of one variable depends on the level of the other variable?

Factorial ANOVA An example: Effects of Temperature and Gender on Aggression An example: Effects of Temperature and Gender on Aggression Each Factor has marginal means (i.e., means averaged across the other iv) Each Factor has marginal means (i.e., means averaged across the other iv) Main Effect Main Effect The effect of one iv averaged across the levels of the other The effect of one iv averaged across the levels of the other Simply a one-way ANOVA on the marginal means Simply a one-way ANOVA on the marginal means Here, there are two main effects. Here, there are two main effects. They tell us if gender and/or temperature affects aggression. They tell us if gender and/or temperature affects aggression.

Factorial ANOVA Calculations As with one-ways, we calculate SS for each effect. SS total again captures all variance of scores around grand mean. Each main effect captures variance of cell means around grand mean Differences among all cell means is captured by SS cells.

Main Effects

Testing Significance of Main Effects We convert each relevant SS into Mean Squares (MS). We convert each relevant SS into Mean Squares (MS). We divide SS by associated df’s. We divide SS by associated df’s. df total =N-1 df total =N-1 df factor = (#conds)-1 df factor = (#conds)-1 df error =(a)(b)(n-1) df error =(a)(b)(n-1) F is the ratio of these two estimates of population variance. F is the ratio of these two estimates of population variance. Critical values of F are found using F(df group, df error ), Critical values of F are found using F(df group, df error ),

Example

Effect Size Each effect has its own partial eta squared. Each effect has its own partial eta squared. Eta squared for gender is simply the ratio: Eta squared for gender is simply the ratio: 42% of the variability in aggression is explained by participant gender. 42% of the variability in aggression is explained by participant gender.