ANOVA continued Chapter 10
Zettergren (003) School adjustment in adolescence for previously rejected, average, and popular children. Effect of peer reputation on academic performance and school adjustment IV or Factor = Peer reputation 3 levels: rejected, average, popular (based on ) 3 rd and 4 th grade students ranked every classmate (same gender) in the order they wanted them to stay with the class if they were to move to a smaller room and not everyone could go DV = Academic ability (8 th grade) DV = Attitudes toward school (8 th grade)
Zettergren (003) results
ANOVA: Partitions the Variance Total Variance Between Treatment Variance Within Treatment Variance 1. Treatment effects. Error F = Between variance ---------------------- Within variance Error
ANOVA: Example summary table Source df SS MS F Between 30 15 11.8* Within 1 16 1.33 Total 14 46 * Significant at.01 level F (, 1) = 11.8, p <.01 MS between SS df between between MS within SS df within within F MS MS between within
Anova: Definitional formulas Between groups SS (sums of squares) Sum of squared deviations from each group s mean from grand mean multiplied by the number of Ss in group Within groups SS Sum of squared deviations of each score from group mean Total SS Sum of squared deviations of each score from the grand mean [( M g M G ( X M g ) ( X MG) ) n ]
ANOVA formulas: Sum of Squares (SS) Total SS: sum of squared deviations from grand mean G SS TOTAL X N Within SS = mean SS within SS = sum of squared deviations from grp ( X g ) ( X g ) n Between SS = sum of squared deviations of grp mean from grand mean T G SS between ( ) n N SS total = SS between + SS within Where G = grand (overall) mean Where N = total number of scores g Where T = sum of scores for group Where n = number of scores in group
ANOVA formulas: SS Total SS = SS = 106 30 /15 = 46 Within SS = SS = 6 + 6 + 4 = 16 Between SS = = 5 /5 + 0 /5 + 5 /5 30 /15 = 30 SS total = SS between + SS within 46 = 30 + 16 G X Temp Cond N SS between T G ( ) n N 1 3 0 4 1 X = 1 3 106 3 6 G=30 1 3 0 N=15 0 4 0 k=3 T 1 =5 T =0 T 3 =5 SS 1 =6 SS =6 SS 3 =4 n 1 =5 n =5 n 3 =5 M 1 =1 M =4 M 3 =1
Vitamin C Study: Year 1 = # of cold symptoms, Year = # cold symptoms with treatment Factor: Group Placebo Low dose of Vitamin C High dose of Vitamin C Dependent variable: Difference in cold symptoms from year 1 to year Hypotheses
DIFF Boxplot of Vitamin C data 0 10 1 3 0-10 -0 N = 10 10 10 pl acebo low hi Vitamin C Treatment
Vitamin C: Data X Report Report DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 3. 50 10 4. 143 35 -.10 10 4. 067-1 -.00 10 5. 477-0 -. 0 30 5. 18-6 DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 7. 7000 10 47. 3406 77.00 19. 3000 10 4. 3977 193.00 31. 0000 10 1. 75408 310.00 6. 0000 30 30. 87014 780.00 T 1 = T = T 3 = G= SS 1 = SS = SS 3 = x = n 1 = n = n 3 = N= M 1 = M = M 3 = k=
Vitamin C: Data Report Report DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 3. 50 10 4. 143 35 -.10 10 4. 067-1 -.00 10 5. 477-0 -. 0 30 5. 18-6 DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 7. 7000 10 47. 3406 77.00 19. 3000 10 4. 3977 193.00 31. 0000 10 1. 75408 310.00 6. 0000 30 30. 87014 780.00 T 1 = 35 T = -1 T 3 = -0 G= -6 SS 1 = SS = SS 3 = x = n 1 = 10 n = 10 n 3 = 10 N= 30 M 1 = 3.5 M = -.1 M 3 = -.0 k= 3
Vitamin C: Data Report Report DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 3. 50 10 4. 143 35 -.10 10 4. 067-1 -.00 10 5. 477-0 -. 0 30 5. 18-6 DIFF Vitamin C Treatment plac ebo low hi Total Mean N Std. Dev iation Sum 7. 7000 10 47. 3406 77.00 19. 3000 10 4. 3977 193.00 31. 0000 10 1. 75408 310.00 6. 0000 30 30. 87014 780.00 T 1 = 35 T = -1 T 3 = -0 G = -6 SS 1 = 154.5 SS = 148.9 SS 3 = 70 x = 780 n 1 = 10 n = 10 n 3 = 10 N= 30 Within SS SS X (X g ) g ng SS SS 77 193 (35) 10 ( 1) 10 M 1 = 3.5 M = -.1 M 3 = -.0 k= 3 SS 310 ( 0) 10
Vitamin C: Data T 1 = 35 T = -1 T 3 = -0 G = -6 SS 1 = 154.5 SS = 148.9 SS 3 = 70 x = 780 n 1 = 10 n = 10 n 3 = 10 N= 30 M 1 = 3.5 M = -.1 M 3 = -.0 k= 3 Total SS: SS X G N SS 780 ( 6) 30 Within SS: SS: 154.5 + 148.9 + 70 = 573.4 Between SS: T n SS between G N = 778.8 35 1 0 6 SS between ( ) = 05.4 10 10 10 30
Vitamin C: Data T 1 = 35 T = -1 T 3 = -0 G = -6 SS 1 = 154.5 SS = 148.9 SS 3 = 70 x = 780 n 1 = 10 n = 10 n 3 = 10 N= 30 M 1 = 3.5 M = -.1 M 3 = -.0 k= 3 df between = k 1 df within = N - k MS MS between within SS df SS df between between within within MS between MS within 05.4 573.4 7 10.7 1.37 F MS MS between within 10.7 F 4.836 1.37 Critical F @.05 = 3.35, @.01 = 5.49
ANOVA summary table SPSS v. Write-up ONEWAY ANOVA DIFF Between Groups Within Groups Total Sum of Squares df Mean Square F Signif icance 05.400 10.700 4. 836.016 573.400 7 1. 37 778.800 9 Source df SS MS F Between 05.4 10.7 4.84* Within 7 573.4 1. Total 30 778.8 * Significant at the.0 level
Vitamin C: Conclusions A one-way ANOVA was conducted to examine the hypothesis that different types of vitamin C treatment have a differential effect on cold symptoms compared to prior years without the treatment. It was found that the number of colds were significantly different for the placebo (M = 3.5), low dose (M = -.1), and high dose (M = -.0) groups, F(, 7) = 4.8, p <.05.
Post Hoc Tests If find a significant F: there is at least 1 mean that is different Post-tests examine which means are and are not significantly different Compare means at a time (pair-wise comparisons) Type I error: divide alpha among all tests need to do Planned comparisons: based on predictions Tukey s HSD Scheffe test (numerator is for MS between for only the two treatments you want to compare) Bonferroni
Effect size How much of the variability in DV is attributed to IV? Effect size for ANOVA: eta-squared (ή ) SS SS Between Total
Self-esteem study: Self-Esteem Descriptor (SED) at 5, 7, 9, 11, 13 160 140 3 10 11 3 3 100 11 80 60 1 5 11 5 8 6 40 0 0-0 N = 5 5 5 5 5 Self-esteem at age 5 Self-esteem at age 9 Self-esteem at age 1 Self esteem at age 7 Self-esteem at age 1
Self-esteem: Between subject ONEWAY Descriptives SED 1.00.00 3.00 4.00 5.00 Total 95% Conf idence Interval for Mean N Mean Std. Dev iation Std. Error Lower Bound Upper Bound Minimum Max imum 5 33. 8800 7. 9170 5. 58340. 3564 45. 4036. 00 106.00 5 7. 6000 35. 35180 7. 07036 13. 0075 4. 195 1. 00 138.00 5 9. 6000 31. 4906 6. 9841 16. 6007 4. 5993 3. 00 17.00 5 9. 9600 34. 86340 6. 9768 15. 5691 44. 3509 1. 00 15.00 5 16. 0800 16. 95071 3. 39014 9. 0831 3. 0769 1. 00 66. 00 15 7. 440 30. 0181. 70133. 0773 3. 7707 1. 00 138.00 ONEWAY ANOVA SED Between Groups Within Groups Total Sum of Squares df Mean Square F Signif icance 4539.088 4 1134.77 1.54.9 108567.4 10 904.79 113106.5 14
Self-esteem: Within subject Descriptive Statistics Self -esteem at age 5 Self esteem at age 7 Self -esteem at age 9 Self -esteem at age 11 Self -esteem at age 13 Mean Std. Dev iation N 33. 88 7. 917 5 7. 60 35. 35 5 9. 60 31. 49 5 9. 96 34. 863 5 16. 08 16. 951 5 Tests of Withi n-subjects Effects Measure: MEASU RE_1 Sourc e AGE Error(AGE) Sphericity Assumed Greenhouse-Geisser Huy nh-feldt Lower-bound Sphericity Assumed Greenhouse-Geisser Huy nh-feldt Lower-bound Ty pe III Sum of Squares df Mean Square F Sig. 4539.088 4 1134.77 4. 771.001 4539.088. 989 1518.796 4. 771.004 4539.088 3. 461 1311.589 4. 771.003 4539.088 1. 000 4539.088 4. 771.039 834. 51 96 37.859 834. 51 71. 77 318.355 834. 51 83. 058 74.9 834. 51 4. 000 951.438
Self-esteem: Planned contrasts Paired Sampl es Statistics Pair 1 Pair Pair 3 Pair 4 Self -esteem at age 5 Self esteem at age 7 Self -esteem at age 5 Self -esteem at age 9 Self -esteem at age 5 Self -esteem at age 11 Self -esteem at age 5 Self -esteem at age 13 Std. Error Mean N Std. Dev iation Mean 33. 88 5 7. 917 5. 583 7. 60 5 35. 35 7. 070 33. 88 5 7. 917 5. 583 9. 60 5 31. 49 6. 98 33. 88 5 7. 917 5. 583 9. 96 5 34. 863 6. 973 33. 88 5 7. 917 5. 583 16. 08 5 16. 951 3. 390 Paired Samples Test Pair 1 Pair Pair 3 Pair 4 Self -esteem at age 5 - Self esteem at age 7 Self -esteem at age 5 - Self -esteem at age 9 Self -esteem at age 5 - Self -esteem at age 11 Self -esteem at age 5 - Self -esteem at age 13 Mean Std. Dev iation Paired Dif ferences 95% Conf idence Interv al of the Std. Error Dif f erence Mean Lower Upper t df Sig. (-tailed) 6.8 18.311 3.66-1.8 13.84 1.715 4.099 4.8.868 4.574-5.16 13.7.936 4.359 3.9.546 4.509-5.39 13.3.869 4.393 17.80 0.738 4.148 9.4 6.36 4.9 4.000
Self-esteem write-up Within-subject design A longitudinal study was conducted on self-esteem. A repeatedmeasures ANOVA was conducted over five time periods; five years old (M = 33.88, SD = 7.9), seven years old (M = 7.60, SD = 35.35), nine years old (M = 9.60, SD = 31.49), 11 years old (M = 9.96, SD = 34.86), and 13 years old (M = 16.08, SD = 16.95). A significant effect of age was found, F (4, 96) = 4.77, p =.001. Post-hoc tests were performed comparing the youngest age (five years old) with each of the other ages (7, 9, 11, and 13 years). One significant result was found. Self-esteem at age five (M = 33.88, SD = 7.9) was significantly different compared to self-esteem at age 13 (M = 16.08, SD = 16.95), t(4) = 4.9, p <.001. This suggests that self-esteem remains stable from age five until age 11, and then declines at age 13.
Repeated-measures ANOVA 3 1 0 5 10 15 rating