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One Factor Between Subjects Analysis of Variance Evaluating the F test Results Professor David Walsh Lecture 16 1 1 incomplete A1 outline 17 13 16 19 20 17 0 2 74 A2 outline 14 13 16 12 10 13 0 2 24 XA sA A3 detailed 15 10 11 12 12 12 0 1 87 nA 5 N 15 XG 14 0 2 Example Values MS SS 35 0 5 33 70 0 64 0 134 0 Source df Memory Cond Error Total 2 12 14 F 6 57 3 Sampling Distribution of F F 2 12 3 88 0 1 2 3 3 88 Skewed minimum 0 max Mode is 1 0 Expected value if H0 is true Critical region always in the positive tail MSA MSError 05 4 Finding Critical Value for F Table C 3 05 light type Table C 3 01 bold type F MS MS A Error df df AMS MS Error df df numerator deno min ator 5 Privitera C 3 Fcrit Values i r o t a n m o n e D m o d e e r F f o s e e r g r e D 6 Table C 3 05 probabilities F 2 12 F N D 1 2 Degrees of Freedom for Numerator 4 161 4 199 5 214 7 224 6 18 51 19 00 19 16 19 25 9 12 10 13 9 55 9 28 3 4 84 4 75 3 98 3 88 3 59 3 49 3 36 3 26 Degrees of Freedom for Denomin ator 1 2 3 11 12 7 5 Step Procedure 1 H0 A1 A2 A3 H1 not all s are equal 2 05 3 Set Decision Stage a use F test 3 groups b Sampling Distribution c critical value of F 2 12 3 88 05 d reject H0 if Fobs 3 88 8 4 Compute F ANOVA Source Memory Cond df 2 SS 70 0 12 14 64 0 134 0 Error Total 5 Decision Reject H0 accept H1 Fobs 2 12 6 57 Fcrit 2 12 3 88 F 6 57 MS 35 0 5 33 9 What we know when we reject H0 A1 A2 A3 We know not all s are equal However we don t know which means are different A1 A2 A1 A3 A2 A3 10 Post Hoc Comparisons When the ANOVA is significant conduct post hoc tests to determine which pair or pairs of group means significantly differ Post hoc test statistical procedure computed following a significant ANOVA to determine which pair or pairs of group means significantly differ These tests are necessary when a 2 3 or more levels of IV because multiple comparisons are needed Privitera Essential Statistics for the Behavioral Sciences 2e SAGE Publications 2019 11 Post Hoc Comparisons cont All post hoc tests control for experimentwise alpha Overall alpha level for multiple tests conducted on the same data Set for 05 for all tests Of the tests listed above Bonferonni is far too conservative Of the remaining tests Tukey s HSD is most conservative and Fisher s LSD is most liberal Privitera Essential Statistics for the Behavioral Sciences 2e SAGE Publications 2019 12 Following up the Overall F Test Also Called the Omnibus F Test Post hoc comparisons after the fact Tukey HSD test Honestly Significant Differences Computes a Critical Difference CD The minimum numerical difference between two treatment means that is statistically significant CD q MS Error n A Where q an entry from the table of the Studentized Range Statistic 13 Tukey HSD test CD q MS Error n A q studentized range statistics from Table C 4 of Privitera 1 Level of 2 Number of levels of IV a or of means compared 3 df for MSError MSError from omnibus or overall F nA of subjects in each treatment condition all the nA s must be equal 14 4 Compute F ANOVA Source Memory Cond df 2 SS 70 0 12 14 64 0 134 0 Error Total 5 Decision Reject H0 accept H1 Fobs 2 12 6 57 Fcrit 2 12 3 88 F 6 57 MS 35 0 5 33 15 Privitera C 4 q Values Range means being compared or a 16 Tukey HSD test for memory experiment q CD 77 3 MS Error n A 33 5 5 067 MSError 5 33 nA 5 q 3 77 3 78 for 05 a 3 Range in C 4 dfError 12 Two means are significantly different from one another 77 3 9 3 1 if their absolute difference is 3 9 units or more 17 Comparison XA1 vs XA2 17 0 13 0 XA1 vs XA3 17 0 12 0 XA2 vs XA3 13 0 12 0 Absolute value of comparison 4 0 5 0 1 0 Statistical hypotheses H0 A1 A2 H1 A1 A2 H0 A1 A3 H1 A1 A3 H0 A2 A3 H1 A2 A3 CD 3 9 Decisions Reject H0 Accept H1 Reject H0 Accept H1 Do not Reject H0 Do not Accept H1 18 Matrix Presentation of Tukey HSD Memory Cond A1 17 0 A2 13 0 4 0 A3 12 0 5 0 1 0 19 A1 17 0 A2 13 0 A3 12 0 Memory Cond p 05 CD 3 9 Post hoc Comparison 1 Necessary to determine where differences exist if a Significant omnibus F Test b Three or more levels of A obvious But 2 group F test is possible Item 3 below covers this 2 Not appropriate if omnibus F Test is not significant 3 Not necessary if only two levels of A and omnibus F Test is significant 20 What we know 1 The IV of drawing type produces causes differences in recall of common objects Overall F 6 57 p 0 05 Incomplete outline A1 caused better recall than outline A2 and detailed drawings A3 2 3 Outline A2 and Detailed A3 don t differ From Tukey HSD post hoc comparisons 21 What we don t know How much of the variance in recall can be predicted from knowledge of what condition of the IV subjects were in Strength of the Effect 22 One Factor Between Subjects Analysis of Variance Strength of Effects Professor David Walsh Lecture 16 2 23 Measuring Effect Size An ANOVA determines whether group means significantly vary in the population You can also determine the size of this effect by computing proportion of variance Proportion of variance estimates how much of the variability in the dependent variable can be accounted for by the levels of the factor or IV Two measures for proportion of variance for a between subjects design are Eta squared Omega squared Privitera Essential Statistics for the Behavioral Sciences 2e SAGE Publications 2019 24 Measuring Strength of Effect from ANOVA Summary Table 2 A Total SS SS 70 134 52 25 4 Compute F ANOVA Source Memory Cond df 2 SS 70 0 12 14 64 0 134 0 Error Total 5 Decision Reject H0 accept H1 Fobs 2 12 6 57 Fcrit 2 12 3 88 F 6 57 MS 35 0 5 33 26 Measuring Strength of Effect from ANOVA Summary …


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USC PSYC 274 - One-Factor Between-Subjects Analysis of Variance: Evaluating the F-test Results

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