Effect Sizes (continued)

Hedges’ Correction for Small Sample Bias d overestimates effect size in small samples (< 10-15 total) Correction is I always use this correction as it never harms estimation. In SPSS COMPUTE D = ES*(1-(3/((4*(NT+NC))-9))).

Algebraic Equivalent: Between Groups t-test on raw posttest scores ,

Algebraic Equivalent: t-test for two matched groups, sample sizes, correlation between groups

Algebraic Equivalent: Two-group between-groups F-statistic on raw posttest scores (Data Set I)

Algebraic Equivalent: Multifactor Between Subjects ANOVA with Two Treatment Conditions Sums of Squares and Degrees of Freedom for all sources, and Marginal Means for Treatment Conditions Mean Squares and Degrees of Freedom for all sources, and Marginal Means for Treatment Conditions Sums of Squares and Degrees of Freedom for all sources, with Cell Means and Cell Sample Sizes Mean Squares and Degrees of Freedom for all sources, with Cell Means and Cell Sample Sizes Cell means, cell sample sizes, the F-statistic for the treatment factor, and the degrees of freedom for the error term F-statistics and degrees of freedom for all sources, sample size for treatment and comparison groups, where treatment factor has only two levels

Example: Sums of Squares and Degrees of Freedom for all sources, and Marginal Means for Treatment Conditions: Data Set II Row B1 B2 B3 Marginal A1 4.0 8.0 6.0 6.0 (3) (3) (3) (9) A2 10.0 2.0 12.0 8.0   Column 7.5 8.0 5.5 7.0 Marginal (6) (6) (6) (18) B1 B2 B3   A1 8 10 8 4 8 6 0 6 4 A2 14 4 15 10 2 12 6 0 9 Sum of Squares df Mean Square F Probability A 18.000 1 18.000 2.038 .179 B 48.000 2 24.000 2.717 .106 AB 144.000 2 72.000 8.151 .006 Residual 106.000 12 8.833 Total 316.000 17 18.588

Example: Sums of Squares and Degrees of Freedom for all sources, and Marginal Means for Treatment Conditions For a two group one factor ANOVA: For a two factor ANOVA: Which is the same as would have been obtained had Factor B not existed (with equal n per cell) ,

Algebraic Equivalent: Exact Probability and Sample Sizes If exact p value from t-test or two group F-test Use sample size to get df, which in turn allows you to get exact t statistic Then apply t-test method previously shown From Data Set I exact probability for t-test was p = .818. for df = 20-2 = 18, t = .2336 so d = -.104, same as before

Algebraic Equivalent: r to d To convert r to d uncorrected for small sample bias, using Data Set I: Which is the same as originally obtained using the standard formula for d

Converting d to r If already corrected for small sample bias: If not: where df = (n1 + n2 - 2).

Algebraic Equivalent: Raw Data Sometimes raw data is tabled as, say, Treatment group N = 10: A = 20%, B =20%, C = 30%, D = 20%, and F = 10% Comparison group N = 10: A = 10%, B = 20%, C = 20%, D = 30%, and F = 20% Create raw data as, say, A = 4, B = 3, C = 2, D = 1, and F = 0 treatment group is 4, 4, 3, 3, 2, 2, 2, 1, 1, 0 comparison group is 4, 3, 3, 2, 2, 1, 1, 1, 0, 0 Then d = .377