Cal Poly Pomona PSY 307 - Chapter 16 – One-Factor Analysis of Variance (ANOVA) (23 pages)

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Chapter 16 – One-Factor Analysis of Variance (ANOVA)



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Chapter 16 – One-Factor Analysis of Variance (ANOVA)

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Pages:
23
School:
California State Polytechnic University, Pomona
Course:
Psy 307 - Statistics for the Social Sciences
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PSY 307 Statistics for the Behavioral Sciences Chapter 16 One Factor Analysis of Variance ANOVA Fisher s F Test ANOVA Ronald Fisher Testing Yields in Agriculture X1 X2 X1 ANOVA Analysis of Variance ANOVA a test of more than two population means One Way ANOVA only one factor or independent variable is manipulated ANOVA compares two sources of variability Two Sources of Variability Treatment effect the existence of at least one difference between the population means defined by IV Between groups variability variability among subjects receiving different treatments alternative hypothesis Within groups variability variability among subjects who receive the same treatment null hypothesis F Test If the null hypothesis is true the numerator and denominator of the F ratio will be the same F random error random error If the null hypothesis is false the numerator will be greater than the denominator and F 1 F random error treatment effect random error Difference vs Error Difference on the top and the error on the bottom Difference is the variability between the groups expressed as the sum of the squares for the groups Error is the variability within all of the subjects treated as one large group When the difference exceeds the variability the F ratio will be large F Ratio F MSbetween MSwithin MS SS df SS is the sum of the squared differences from the mean F Ratio F MSbetween MSwithin MSbetween treats the values of the group means as a data set and calculates the sum of squares for it MSwithin combines the groups into one large group and calculates the sum of squares for the whole group Testing Hypotheses If there is a true difference between the groups the numerator will be larger than the denominator F will be greater than 1 Writing hypotheses H0 1 2 H1 H0 is false 3 Formulas for F Description in words of what is being computed Definitional formula uses the SS described in the Witte text Computational formula used by Aleks and in examples in class Formula for SStotal SStotal is the total Sum of the Squares It is the sum of the squared deviations of scores around the grand mean SStotal X Xgrand 2 SStotal X2 G2 N Where G is the grand total and N is its sample size Hours of Sleep Deprivation SS between SS within 0 24 48 0 3 6 4 6 8 2 6 10 Grand Total 6 15 24 G 45 6 2 3 15 2 3 24 2 3 45 2 9 36 3 225 3 576 3 0 2 4 2 2 2 3 2 6 2 6 2 8 2 10 2 0 16 4 9 36 36 64 100 above 22 2025 9 above 54 Formula for SSbetween SSbetween is the between Sum of the Squares It is the sum of the squared deviations for group means around the grand mean SSbetween n X Xgrand 2 SSbetween T2 n G2 N definition computation Where T is each group s total and n is each group s sample size Formula for SSwithin SSwithin is the within Sum of the Squares It is the sum of the squared deviations for scores around the group mean SSwithin X Xgroup 2 SSwithin X2 T2 n definition computation Where T is each group s total and n is each group s sample size Degrees of Freedom dftotal N 1 dfbetween k 1 The number of all scores minus 1 The number of groups k minus 1 dfwithin N k The number of all scores minus the number of groups k Checking Your Work The SStotal SSbetween SSwithin The same is true for the degrees of freedom dftotal dfbetween dfwithin Calculating F Computational SSbetween T2 G2 n N Where T is the total for each group and G is the grand total SSwithin SStotal X2 T2 N X2 G2 N F Distribution Common retain null Critical value Look up F critical value in the F table using df for numerator and denominator Rare reject null Effect Sizes Effect size 2 for ANOVA is the squared curvilinear correlation The effect size 2 is the amount of variance in the dependent variable explained by the independent variable To calculate effect size divide the SSbetween by SStotal Interpreting Cohen s rule of thumb 2 If 2 approximates 01 effect size is small If 2 approximates 06 effect size is medium If 2 approximates 14 or more effect size is large Effect size is especially important when large samples sizes are used ANOVA Assumptions Assumptions for the F test are the same as for the t test Underlying populations are assumed to be normal with equal variances Results are still valid with violations of normality if All sample sizes are close to equal Samples are 10 per group Otherwise use a different test Cautions The ANOVA presented in the text assumes independent samples With matched samples or repeated measures use a different form of ANOVA The sample sizes shown in the text are small in order to simplify calculations Small samples should not be used


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