PSYCHOLOGY 210 Dr. Jennifer Kunz Module 9: two-factor ANOVA – Chapter 14 The two-factor Analysis of Variance (ANOVA) is a statistic that allows us to compare two or more independent variables or...

ANOVA Psych Stats 210


PSYCHOLOGY 210 Dr. Jennifer Kunz Module 9: two-factor ANOVA – Chapter 14 The two-factor Analysis of Variance (ANOVA) is a statistic that allows us to compare two or more independent variables or factors, each having at least two levels. Again, we are comparing the differences in means between treatments, through a comparison of variances. Here, in its simplest version, four variances are computed and compared: the variability between the condition or levels for factor 1, the variability between the conditions or levels for factor 2, the variability associated with the interaction of variables 1 and 2, and the average variability within the sample conditions which is sometimes considered our error variance or variability due to chance. In this module you will calculate a two-factor ANOVA by hand as you see in Chapter 14. We have discussed these topics in lecture and in the textbook. Please review these ideas. A researcher is interested in determining the effects of drug use on the number of sick days taken. One hypothesis is that alcohol users will have a higher frequency of sick days than marijuana users. A second hypothesis is that females will have a higher sick day rate than males. Finally, it is possible that the two variables will interact, and the researcher predicts that women who drink alcohol will have the greatest frequency of sick days. Is there enough evidence to support any of these hypotheses? Use the data provided in the table below to compute a two-factor ANOVA using the steps below. Below are the null and alternative hypotheses. a. Main Effect for Factor A (gender): H0: females = males H1: µfemales > µmales b. Main Effect for Factor B (Drug): H0: alcohol = marijuana H1: µalcohol > µmarijuana c. Interaction: H0: There is NO interaction between gender and drug H1: There is an interaction between gender and drug, such that women who drink alcohol will have the greatest number of sick days. STAGE 1: The first stage of a two-factor analysis separates the total variability into two components: between-treatments and within-treatments. Show your work. (.5pt./question) 1. Calculate SStotal: ΣX2 – (G2/N) = 2. Calculate dftotal: N-1= 3. Calculate SSwithin: ∑SSeach treatment 4. Calculate dfwithin: ∑dfeach treatment 5. Calculate SSbetween: Σ(T2/n) – (G2/N) OR SSTotal - SSWithin = 6. Calculate dfbetween: k-1 STAGE 2: The second stage of analysis determines the numerators for the F-ratio. Specifically, this stage determines the between-treatments variance for factor A, factor B, and the interaction. Show your work. (.5pt/question) 7. Calculate SSA: Σ(T2row/nrow) – G2/N 8. Calculate dfA: # of rows -1 9. Calculate SSB: Σ(T2col/ncol) – G2/N 10. Calculate dfB: # of columns – 1 11. Calculate SSAxB: SSbetween-SSA-SSB 12. Calculate dfAxB: dfbetween- dfA – dfB 13. Calculate MSwithin: SSwithin/dfwithin 14. Calculate MSA: SSA/dfA 15. Calculate MSB: SSB/dfB 16. Calculate MSAxB: SSAxB/dfAxB Finally, calculate the three F-ratios: 17. FA = MSA/MSwithin 18. FB = MSB/MSwithin 19. FAxB= MSAxB/MSwithin Are these F-ratios significant at the .05 level? Find the critical F values for each hypothesis and make a statistical decision. (hint: the df for both factors and the interaction are 1, 36) 20. Originals prepared by Pamela M. Schaefer, Ph.D.