PSY 216. Assignment 14. a. The mean differences among the levels of one factor are referred to as the main effect of that factor.

Transcription

1 Name: PSY 216 Assignment 14 a. The mean differences among the levels of one factor are referred to as the main effect of that factor. b. A(n) interaction between two factors occurs whenever the mean differences between individual treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors. c. When the effect of one factor depends on the different levels of a second factor, then there is a(n) interaction between the factors. d. When the results of a two-factor study are presented in a graph, the existence of nonparallel lines (lines that cross or converge) indicates a(n) interaction between the two factors. e. Problem 6 from the text The following matrix present the results from an independent-measures, two-factor study with a sample of n = 10 participants in each of the six treatment conditions. Note that one treatment mean is missing. Factor B Factor A B 1 B 2 B 3 A 1 M = 10 M = 20 M = 40 A 2 M = 20 M = 30 a. What value for the missing mean would result in no main effect for factor A? A1 = ( ) / 3 = A2 = ( X) / 3 = X = 20 b. What value for the missing mean would result in no interaction? A1,B1 - A2,B1 = = -10 A1,B2 - A2,B2 = = -10 A1,B3 - A2,B3 = 40 X = -10 X = Problem 7 from the text For the data in the following graph:

2 a. Is there a main effect for the treatment factor? Likely Yes. Treatment 1 = ( ) / 3 = 21 Treatment 2 = ( ) / 3 = 8.67 There is a large difference between Treatment 1 and Treatment 2 b. Is there a main effect for the age factor? Likely No 8 Years = ( ) / 2 = 15 9 Years = ( ) / 2 = Years = (27 + 3) / 2 = 15 There is a very small difference between 8 Years and 9 Years and 10 Years c. Is there an interaction between age and treatment? Likely Yes. The lines are not parallel. The difference between Treatment 1, 8 years (17) and Treatment 2, 8 years (13) = 4 is much smaller than the difference between Treatment 1, 10 years (27) and Treatment 2, 10 years (3) = Problem 16 from the text

3 The Preview section for this chapter described a two-factor study examining performance under two audience conditions (factor B) for high and low self-esteem participants (factor A). The following summary table presents possible results from the analysis of that study. Assuming that the study used a separate sample of n = 15 participants in each treatment condition (each cell), fill in the missing values in the table. (Hint: Start with the df values.) Source SS df MS Between treatments 67 _3_ Audience _ _ F = _4_ Self-esteem 29 _1 29_ F = _7.25_ Interaction _ _ F = 5.50 Within treatments _224 56_ 4 Total _291 59_ dftotal = N 1 ( 2 X 2 X 15) 1 = 59 dfaudience = KAudience 1 = 2 1 = 1 dfself-esteem = KSelf-esteem 1 = 2 1 = 1 dfaudience X Self-esteem = dfaudience X dfself-esteem = 1 X 1 = 1 dfbetween treatment = dfaudience + dfself-esteem + dfaudience X Self-esteem = = 3 dfwithin treatment = KAudience X KSelf-esteem X (n 1) = 2 X 2 X (15 1) = 56 = dftotal dfbetween treatments = 59 3 = 56 MSAudience X Self-esteem = FAudience x Self-esteem X MSWithin treatments = 5.50 X 4 = 22 SSWithin treatments = dfwithin treatment X MSwithin treatment = 56 X 4 = 224 SStotal = SSbetween treatment + SSwithin treatment = = 291 SSAudience X Self-esteem = MSAudience X Self-esteem X dfaudience X Self-esteem = 22 X 1 = 22 SSAudience =SSbetween treatments SSSelf-esteem SSAudience X Self-esteem = = 16 MSAudience = SSAudience / dfaudience = 16 / 1 = 16 MSSelf-esteem = SSSelf-esteem / dfself-esteem = 29 / 1 = 29 FAudience = MSAudience / MSWithin treatments = 16 / 4 = 4 FSelf-esteem = MSSelf-esteem / MSWithin treatments = 29 / 4 = Enter the data from problem 19 into SPSS. Use SPSS to answer the following question: Do the data indicate significant main effects and interaction? Give H0 and H1 for each hypothesis. Give α. Write a sentence or two in APA format and include a table that summarizes the results of the analysis.

4 The following data are from a two-factor study examining the effects of three treatment conditions on males and females. Factor A: Gender Male Female Treatments I II II M = 6 T = 18 SS = M = 3 T = 9 SS = M = 3 T = 9 SS = M = 12 T = 36 SS = M = 9 T = 27 SS = M = 15 T = 45 SS = 26 T male = 54 T female = 90 N = 18 G = 144 ΣX 2 = 1608 H0: μtx 1 = μtx 2 = μtx 3 H1: not H0 H0: μmales = μfemales H1: not H0 H0: there is no interaction H1: there is an interaction α =.05 Descriptive Statistics Treatment gender Mean Std. Deviation N 1.00 Male Female Total Male Female Total Male Female Total Total Male Female

5 Descriptive Statistics Treatment gender Mean Std. Deviation N 1.00 Male Female Total Male Female Total Male Female Total Total Male Female Total Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model a Intercept treatment gender treatment * gender Error Total Corrected Total a. R Squared =.789 (Adjusted R Squared =.702) 1. Treatment 95% Confidence Interval Treatment Mean Std. Error Lower Bound Upper Bound gender

6 95% Confidence Interval gender Mean Std. Error Lower Bound Upper Bound Male Female Treatment * gender 95% Confidence Interval Treatment gender Mean Std. Error Lower Bound Upper Bound 1.00 Male Female Male Female Male Female Dependent Variable Tukey HSD Multiple Comparisons Mean 95% Confidence Interval (I) Treatment (J) Treatment Difference (I-J) Std. Error Sig. Lower Bound Upper Bound * * * * Based on observed means. The error term is Mean Square(Error) = *. The mean difference is significant at the.05 level. Table 1 shows the means and standard deviations for all treatment conditions. The 2 X 3 analysis of variance (ANOVA) revealed a significant main effect of treatment, F(2, 12) = 15.75, MSerror = 8.00, p =.000, η 2 =.724. Tukey multiple comparisons revealed that treatments I and II were reliably different, p =.008, as were treatments I and III, p =.000. However, treatments II and III were not reliably different, p =.199. The ANOVA revealed a significant main effect of gender, F(1, 12) = 9.00, p =.0, η 2 =.429. The ANOVA failed to reveal a significant interaction of treatment and gender, F(2, 12) = 2.25, p =.148, η 2 =.273.

7 Table 1 Means and Standard Deviations for each Condition Treatment I Treatment II Treatment III Marginal M sd M sd M sd M sd Males Females Marginal Problem 19 by hand: Male Factor B Treatments I II III Total ΣX = 9 ΣX 2 = 41 SS = / 3 = 14 M = 3 Female ΣX = 18 ΣX 2 = 134 SS = / 3 = 26 M = ΣX = 27 ΣX 2 = 251 SS = / 3 = 8 M = ΣX = = 54 ΣX 2 = =426 SS = / 9 = 102 M = 6 ΣX = 9 ΣX 2 = 35 SS = /3=8 M = 3 Total ΣX = = 18 ΣX 2 = = 76 SS = /6=22 M =3 ΣX = 36 ΣX 2 = 446 SS = / 3 = 14 M =12 ΣX = = 54 ΣX 2 = = 580 SS = /6=94 M = 9 ΣX = 45 ΣX 2 = 701 SS = / 3 =26 M = 15 ΣX = = 72 ΣX 2 = = 952 SS = /6=88 M = 12 ΣX = = 90 ΣX 2 = =82 SS = / 9 = 282 M = 10 ΣX = = 144 ΣX 2 = = 1608 SS Total = /18=456 SS Between Treatments = n * SS M n is the number of scores per condition ΣM = = 48 ΣM 2 = = 504 SS M = / 6 = 120 SS Between Treatments = 3 * 120 = 360 SS Factor A = n A * SS MA ΣM A = = 16 ΣM A 2 = = 136 SS MA = / 2 = 8 SS Factor A =9 * 8 = 72 SS Factor B = n B * SS MB ΣM B = = 24

8 ΣM B 2 = = 234 SS MB = / 3 = 42 SS Factor B =6 * 42 = 252 SS Factor A X Factor B = SS Between Treatments SS Factor A SS Factor B = = 36 SS Within Treatment = Σ(SS each condition) = = 96 SS Total = 456 = SS Between Treatments + SS Within Treatment = df Factor A = # levels of Factor A 1 = 2 1 = 1 df Factor B = # levels of Factor B 1 = 3 1 = 2 df Factor A X Factor B = SS Factor A * SS Factor B = 1 X 2 = 2 df Within Treatment = Σ(df for each condition) = (3 1) + (3 1) + (3 1) + (3 1) + (3 1) + (3 1) + (3 1) = 12 df Total = N 1 = 18 1 = 17 = df Factor A + df Factor B + df Factor A X Factor B + df Within Treatment = = 17 MS Factor A = SS Factor A / df Factor A = 72 / 1 = 72 MS Factor B = SS Factor B / df Factor B = 252 / 2 = 126 MS Factor A X Factor B = SS Factor A X Factor B / df Factor A X Factor B = 36 / 2 = 18 MS Within Treatment = SS Within Treatment / df Within Treatment = 96 / 12 = 8 F Factor A = MS Factor A / MS Within Treatment = 72 / 8 = 9 F Factor B = MS Factor B / MS Within Treatment = 126 / 8 = F Factor A X Factor B = MS Factor A X Factor B / MS Within Treatment = 18 / 8 = 2.25 η 2 Factor A = SS Factor A / (SS Total SS Factor B SS Factor A X Factor B) = 72 / ( ) =.43 η 2 Factor B = SS Factor B / (SS Total SS Factor A SS Factor A X Factor B) = 252 / ( ) =.72 η 2 Factor A X Factor B = SS Factor A X Factor B / (SS Total SS Factor A SS Factor B) = 36 / ( ) =.27