ANOVA continued Chapter 11
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
group Between-Treatment Variance 3 X 3 1 X 4 Includes systematic variance plus error variance 1 X1 0 5 10 15 rating
Anova: Definitional formulas Between groups SS (sums of squares) Sum of squared deviations from each group s mean (g) from grand mean (G) multiplied by the number of Ss (n) in group Within groups SS Sum of squared deviations of each score from group mean (g) Total SS Sum of squared deviations of each score from the grand mean (G) [( M g M G ( X M g ) ( X MG) ) n ]
ANOVA computational 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 1 3 ( T) ( G) ( ) n N 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: ( G SS X 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 = 3.37, @.01 = 5.53
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) How to reduce Type I error 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: divide alpha among all tests need to do
Effect size How much of the variability in DV is attributed to IV? Effect size for ANOVA: eta-squared (η ) 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: 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 Total df = N 1 Within df = N k Between df = k 1 SED Between Groups Within Groups Total ONEWAY ANOVA 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 Total df = 15 1 = 14 Within df = 15 5 = 10 Between df = 5 1 = 4
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
One-way v. Repeated ANOVA F treatment effect chance/err or chance/err or F MS MS between error One-way ANOVA chance/error = Between subject individual differences For overall sample For each group Within subject experimental error Repeated ANOVA chance/error= Between subject sampling error (only for overall sample) Within subject experimental error Advantage to remove individual differences that can mask effect
Repeated-measures ANOVA One-way or independent-measures ANOVA w/o individual differences error F treatment effect chance/err or chance/err or F MS MS between error Advantage: remove individual differences that can mask treatment effect
Structure of data sets One-way v. Repeated ANOVA Group Data 1 5 1 67 1 33 59 4 56 3 5 3 49 3 53 Ss Test1 Test Test3 1 5 59 5 67 4 49 3 33 56 53
Pain Relief The effect of drug treatment on the amount of time (in seconds) a stimulus is endured.
Pain relief by subject 8 7 6 5 4 3 1 0 Placebo DrugA DrugB DrugC
MSerror
The partitioning of degrees of freedom for a repeated-measures experiment N = total # scores n = # of participants k = # of conditions
Compute df For N = 0; k = 4; n = 5 df total = N 1 0 1 = 19 df between = k 1 4 1 = 3 df within = N k 0 4 = 16 df between subjects = n 1 5 1 = 4 df error = df within df between subjects = 16 4 = 1
The partitioning of sum of squares (SS) for a repeated-measures analysis of variance
Repeated-measures Anova: Definitional formulas Between treatment SS (sums of squares) Sum of squared deviations from each group s mean from grand mean multiplied by the number of Ss in group [( M M ) n] Within groups SS Sum of squared deviations of each score from group mean Participant (between subject) SS Sum of squared difference scores from the mean of each participant across the conditions and the grand mean, multiplied by the number of conditions [( M M Total SS g Sum of squared deviations of each score from the grand mean G ( X M g ) P G ) k ( X MG) ]
Calculate MS and F-ratio MS between SS df between between MS between 50 3 16.67 MS error F SS df MS MS error error between error MS error F 16.67 0.67 8 1 0.67 4.88 Critical F @.05 = 3.49, @.01 = 5.95 F (3, 1) = 4.88, p <.01
ANOVA summary table: Repeated-measures 50 3 16.67 4.88* 3 16 4 4 8 8 1 19 0.67 Significant at p <.01
Effect size How much of the variability in DV is attributed to IV? Effect size for ANOVA: eta-squared (η ) % of variance among scores attributed to the IV SS SS Between Total 4539.088 113106.5 0.04 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 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: Within subject Descriptive Statistics Mean Std. Dev iation N Self -esteem at age 5 33. 88 7. 917 5 Self esteem at age 7 7. 60 35. 35 5 Self -esteem at age 9 9. 60 31. 49 5 Self -esteem at age 11 9. 96 34. 863 5 Self -esteem at age 13 16. 08 16. 951 5 df total = N 1 df between = k 1 df within = N k df between subjects = n 1 df error = df within df between subjects 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 Tests of Withi n-subjects Effects 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 df total = 15 1 = 14 df between = 5 1 = 4 df within = 15 5 = 10 df between subjects = 5 1 = 4 df error = 10 4 = 96