Chapter 14 Repeated Measures and Two Factor Analysis of Variance PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau

14.1 Overview Analysis of Variance –Evaluated mean differences of two or groups Complex Analysis of Variance –Samples are related not independent (Repeated measures ANOVA) –More than one factor is tested (Factorial ANOVA, here Two-Factor)

14.2 Two-Factor ANOVA Factorial designs –Consider more than one factor –Joint impact of factors is considered. Three hypotheses tested by three F-ratios –Each tested with same basic F-ratio structure

Main effects Mean differences among levels of one factor –Differences are tested for statistical significance –Each factor is evaluated independently of the other factor(s) in the study

Structure of the Two-Factor Analysis Three distinct tests –Main effect of Factor A –Main effect of Factor B –Interaction of A and B A separate F test is conducted for each

Interactions between factors The mean differences between individuals treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors H 0 : There is no interaction between Factors A and B H 1 : There is an interaction between Factors A and B

Interpreting Interactions Dependence of factors –The effect of one factor depends on the level or value of the other Non-parallel lines (cross or converge) in a graph –Indicate interaction is occurring Typically called the A x B interaction

Figure 14.2 Graph of group means with and without interaction

Table 14.7, p. 436

Table 14.4, p. 429

Table 14.5, p. 431

Table 14.6, p. 433

Two Stages of the Two-Factor Analysis of Variance First stage –Identical to independent samples ANOVA –Compute SS Total, SS Between treatments and SS Within treatments Second stage –Partition the SS Between treatments into three separate components, differences attributable to Factor A, to Factor B, and to the AxB interaction

Figure 14.3 Structure of the Two- Factor Analysis of Variance

Stage One of the Two-Factor Analysis of Variance

Stage 2 of the Two Factor Analysis of Variance This stage determines the numerators for the three F-ratios by partitioning SS between treatments

Degrees of freedom for Two-Factor ANOVA df total = N – 1 df within treatments = Σdf inside each treatment df between treatments = k – 1 df A = number of rows – 1 df B = number of columns– 1 df error = df within treatments – df between subjects

Mean squares and F-ratio for the Two-Factor ANOVA

Table 14.8, p. 439

Table 14.9, p. 443

Effect Size for Two-Factor ANOVA η 2, is computed as the percentage of variability not explained by other factors.

Figure 14.4 Sample means for Example 14.3

Assumptions for the Two-Factor ANOVA The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests 1.The observations within each sample must be independent of each other 2.The populations from which the samples are selected must be normally distributed 3.The populations from which the samples are selected must have equal variances (homogeneity of variance)

Learning Check If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____. either the main effect for factor A or the main effect for factor B is also significant A neither the main effect for factor A nor the main effect for factor B is significant B both the man effect for factor A and the main effect for factor B are significant C the significance of the main effects is not related to the significance of the interaction D

Learning Check - Answer If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____. either the main effect for factor A or the main effect for factor B is also significant A neither the main effect for factor A nor the main effect for factor B is significant B both the man effect for factor A and the main effect for factor B are significant C the significance of the main effects is not related to the significance of the interaction D

Learning Check Decide if each of the following statements is True or False. Two separate single-factor ANOVAs provide exactly the same information that is obtained from a two-factor analysis of variance. T/F A disadvantage of combining 2 factors in an experiment is that you cannot determine how either factor would affect the subjects' scores if it were examined in an experiment by itself. T/F

Answer The two-factor ANOVA allows you to determine the effect of one variable controlling for the effect of the other. False Either main effect can be examined in a Oneway ANOVA, but the Two-Factor ANOVA provides more information, not less. False