November 11, 2009Statistical Methods II Two-Factor ANOVA Factorial ANOVA chapter 13- one-way ANOVA chapter 14- factorial ANOVA factorial ANOAs simultaneously examine the relationship of several factors to the DV going to examine two-factor independent measures equal n designs reason 2 for not using multiple t-test Example ?It?s hot!? ?No, it?s humid!? we want to see if heat or humidity relates to performance on exam 3 3 levels for heat, one at 70, 80, 90 2 levels for humidity: low and high independent variable level of heat and level of humidity dependent variable performance on exam in addition to finding out if heat or humidity affect you, we are interested in the relationship between the effect that heat and humidity might together have on you we have 2 factors heat and humidity 3x2 factorial between-subjects ANOVA 3 factors in heat 2 factors in humidity Factorial ANOVA major difference from one-way ANOVA is that factorial ANOVAs address multiple statistical tests specifically two-way factorial ANOVA has 3 separate statistical questions and tests that are all conducted simultaneously 2 main effect and 1 interaction test f-test are still of same form: effect/error Main effects obvious tests are called main effects 2 main effects in a two-way factorial ANOVA these test if there are population mean difference among the levels of one factor, holding the other factor constant average across the levels of the other factor independent of the effects of the other factor Marginal mean tables often we use a marginal mean table to list our results a marginal mean table will have 2 types of means marginal means cell means main effects test difference among MARGINAL means Main effect the test of the main effect has the same form as for the rest of the ANOVA tests we use an F-test between group variability (due to marginal means) divided by within as the marginal means get more discrepant the between group variability goes up the within group variability is based on variability within each cell now, not each level each cell is considered a ?group? for the within-group error now we have a second F-test that will examine the different among the humidity difference means same between group MS/ within MS null and alternative hypotheses for humidity? For heat? humidity null- there is no difference between the 2 levels of humidity and test scores heat null- there is no difference between the 3 levels of heat and test scores Interaction the last test is a test to determine if there is any between-group variability not explained by the main effects we call this an interaction a significant interaction means that the effect of one factor differs across the levels of the other an interaction reflects a dependency between the two factors often the most interesting in psychology No Interaction Effect of humidity was consistent effect of heat was consistent Interaction Is there a difference between high and low humidity on test scores? depends on what level of heat is there a difference between levels of heats on test scores? depends on what level of humidity Interaction notice that an interaction test is itself INDEPENDENT of the main effect tests you can have any combination of significant main effects and interactions significant interaction does not mean that the main effects are not significant all tested simultaneously interaction are best understood through graphs if the lines are parallel, there is no interaction Interaction tests main effects test marginal means interaction tests are a little tricky interaction test if cell means difference for one factor differ across levels of the other null hypothesis is that there is no interaction/ alternative is that there is an interaction 70 degrees 80 degrees 90 degrees Low humidity High humidity 70 80 90 Low humidity M=85 M=80 M=75 M=80 High humidity M=75 M=70 M=65 M=70 M=80 M=75 M=70 70 80 90 Low humidity M=85 M=80 M=75 M=80 High humidity M=75 M=70 M=65 M=70 M=80 M=75 M=70 70 80 90 Low humidity M=80 M=80 M=80 M=80 High humidity M=80 M=70 M=60 M=70 M=80 M=75 M=70