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Analysis of Differences

Concept Map For Statistics as taught in IS271 (a work in progress). Correlation: Pearson. One Predictor. Regression. Analysis of Relationships. Multiple Predictors. Multiple Regression. Interval Data. Independent Samples t-test. Independent Groups. Between Two Groups.

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Analysis of Differences

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  1. Concept Map For Statistics as taught in IS271 (a work in progress) Correlation: Pearson One Predictor Regression Analysis of Relationships Multiple Predictors Multiple Regression Interval Data Independent Samples t-test Independent Groups Between Two Groups Repeated Measures t-test Dependent Groups Analysis of Differences Independent Samples ANOVA Independent Groups Type of Data Repeated Measures ANOVA Between Multiple Groups Dependent Groups Correlation: Spearman Nominal / Ordinal Data Ordinal Regression CHI Square Frequency Some kinds of Regression Rashmi Sinha

  2. Analysis of Variance or F test ANOVA is a technique for using differences between sample means to draw inferences about the presence or absence of differences between populations means. • The logic • Calculations in SPSS • Magnitude of effect: eta squared, omega squared

  3. Assumptions of ANOVA • Assume: • Observations normally distributed within each population • Population variances are equal • Homogeneity of variance or homoscedasticity • Observations are independent

  4. Assumptions--cont. • Analysis of variance is generally robust to first two • A robust test is one that is not greatly affected by violations of assumptions.

  5. Logic of Analysis of Variance • Null hypothesis (Ho): Population means from different conditions are equal • m1 = m2 = m3 = m4 • Alternative hypothesis: H1 • Not all population means equal.

  6. Lets visualize total amount of variance in an experiment Total Variance = Mean Square Total Between Group Differences (Mean Square Group) Error Variance (Individual Differences + Random Variance) Mean Square Error F ratio is a proportion of the MS group/MS Error. The larger the group differences, the bigger the F The larger the error variance, the smaller the F

  7. Logic--cont. • Create a measure of variability among group means • MSgroup • Create a measure of variability within groups • MSerror

  8. Logic--cont. • Form ratio of MSgroup /MSerror • Ratio approximately 1 if null true • Ratio significantly larger than 1 if null false • “approximately 1” can actually be as high as 2 or 3, but not much higher

  9. Grand mean = 3.78

  10. Calculations • Start with Sum of Squares (SS) • We need: • SStotal • SSgroups • SSerror • Compute degrees of freedom (df ) • Compute mean squares and F Cont.

  11. Calculations--cont.

  12. Degrees of Freedom (df ) • Number of “observations” free to vary • dftotal = N - 1 • N observations • dfgroups = g - 1 • g means • dferror = g (n - 1) • n observations in each group = n - 1 df • times g groups

  13. Summary Table

  14. When there are more than two groups • Significant F only shows that not all groups are equal • We want to know what groups are different. • Such procedures are designed to control familywise error rate. • Familywise error rate defined • Contrast with per comparison error rate

  15. Multiple Comparisons • The more tests we run the more likely we are to make Type I error. • Good reason to hold down number of tests

  16. Bonferroni t Test • Run t tests between pairs of groups, as usual • Hold down number of t tests • Reject if t exceeds critical value in Bonferroni table • Works by using a more strict level of significance for each comparison Cont.

  17. Bonferroni t--cont. • Critical value of a for each test set at .05/c, where c = number of tests run • Assuming familywise a = .05 • e. g. with 3 tests, each t must be significant at .05/3 = .0167 level. • With computer printout, just make sure calculated probability < .05/c • Necessary table is in the book

  18. Magnitude of Effect • Why you need to compute magnitude of effect indices • Eta squared (h2) • Easy to calculate • Somewhat biased on the high side • Formula • See slide #33 • Percent of variation in the data that can be attributed to treatment differences Cont.

  19. Magnitude of Effect--cont. • Omega squared (w2) • Much less biased than h2 • Not as intuitive • We adjust both numerator and denominator with MSerror • Formula on next slide

  20. h2 and w2 for Foa, et al. • h2 = .18: 18% of variability in symptoms can be accounted for by treatment • w2 = .12: This is a less biased estimate, and note that it is 33% smaller.

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