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Relative Variation, Variance Heterogeneity, and Effect Size

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by Relative Variation, Variance Heterogeneity, and Effect Size Andrew R. Gilpin & Helen C. Harton Number Crunchers, April 7, 1998 Homogeneity of Variance as Assumption for Tests on Means Robustness of t, F Unequal n’s, non-normal data are troublesome Variance as Dependent Variable

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Relative Variation, Variance Heterogeneity, and Effect Size

Andrew R. Gilpin & Helen C. Harton

Number Crunchers, April 7, 1998

homogeneity of variance as assumption for tests on means
Homogeneity of Variance as Assumption for Tests on Means

Robustness of t, F

Unequal n’s, non-normal data are troublesome

variance as dependent variable
Variance as Dependent Variable

Selection bias

Differential influences between groups

Learning

Attitudinal shifts

tests on homogeneity of variance
Tests on Homogeneity of Variance

Fisher’s F

Levene’s ANOVA procedure (ANOVA on transformed scores)

Miscellaneous other approaches

Box

Cochran

Hartley

O’Brien

experimental effect size
Experimental Effect Size

Cohen’s d

Glass’s g

Hedges’ h

Pooled Variance Issue

relative variation
Relative Variation

Pearson’s Coefficient of Variation

Means are often proportional to standard deviations

Psychophysics research (Weber/Fechner Law)

Physical size

implications of homogeneity of relative variation for h vs g
Implications of Homogeneity of Relative Variation for h vs. g

Pooled variance estimate based on smaller variance (and mean) will underestimate actual variance; pooled variance estimate based on larger variance (and mean) will overestimate actual variance.

Distorted pooled variance will cause h to depart from g

simulation design
Simulation Design

10,000 simulated experiments per cell

9 Populations (normal, 8 real radically non-normal)

9 Sample sizes (5,5), (25,25), (100,100), (5,25), (5,100), (25,100), (25,5), (100,5), (100,25)

3 Coefficient of Variation (V=.1, V=.2, V=.3)

6 Nominal g sizes: 0.0, 0.5, 1.0, 1.5, 2.0, 2.5

simulation dependent variables
Simulation: Dependent variables

Mean observed h

Proportion (of 10,000) significant for =.05

Fisher’s F for variance heterogeneity

Levene’s F (t) for variance heterogeneity

projected sample sizes are distorted
Projected Sample Sizes Are Distorted

Noncentrality parameter for independent-groups, equal N t-test

For power (1-) = .80, =2.80 and N=15.68/d2

Estimated distortion from Normal population, (25,25)

comparison of estimated sample sizes
Comparison of Estimated Sample Sizes

Assumes N1=N2=N, Power=.80

general conclusions
Variance heterogeneity is implied by homogeneity of relative variation

Use g rather than h if means are related to standard deviations

General Conclusions
suggestions anyone
Suggestions, Anyone?

How common is variance heterogeneity?

How common is proportionality of means and standard deviations?

Other?

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