Ordinal Frequency Rashmi Sinha Correlation: Pearson Regression CHI Square Some kinds of Regression
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 Assumptions of ANOVA Assume: Observations normally distributed within each population
Population variances are equal Homogeneity of variance or homoscedasticity Observations are independent 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. 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. 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 Logic--cont. Create a measure of variability among group means MSgroup Create a measure of variability within groups
MSerror 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 Grand mean = 3.78 Calculations
Start with Sum of Squares (SS) We need: SStotal SSgroups SSerror Compute degrees of freedom (df ) Compute mean squares and F Cont. Calculations--cont. SStotal ( X X .. ) 2 2
(1 3.78) 2 3 3.78 ... 1 3.78 216.444 SS groups n X j X .. 2 2
2 2 18 3.22 3.78 4.50 3.78 ... 1.89 3.78 18(7.364) 132.556 SSerror SStotal SS groups 216.444 132.556 83.889 2
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 Summary Table 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 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
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. 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 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. 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 h2 and w2 for Foa, et al. SS groups 507.8 .18 SStotal 2786.6 2 SS groups (k 1) MS error 507.8 3(55.6)
.12 SStotal MS error 2786.6 55.6 2 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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