We can calculate Cohen’s f for the ANOVA, and Cohen’s dz for the contrasts: The standard deviation is 0.9, and the correlation between the dependent measurements is 0.7. Based on pilot data, we expect the means (on a 7 point validated mood scale) are 3.8, 4.2, and 4.3. We ask people what they mood is when their alarm clock wakes them up, when they wake up naturally on a week day, and when they wake up naturally on a weekend day. Imagine we have a within-subject experiment with 3 conditions. Therefore, the relation between dz and d is \(\sqrt\) The relation, as Cohen (1988, formula 2.3.7) explains, is: In turn, because the standardized effect size is the mean difference divided by the standard deviation of the difference scores, the correlation has an effect on the standardized mean difference in a within design, Cohen’s dz. This has an effect on the standard deviation of the difference scores. Whereas in an independent t-test the two observations are uncorrelated, in a within design the observations are correlated. The higher the correlation, the larger the relative benefit of within designs, and whenever the correlation is negative (up to -1) the relative benefit disappears. If the correlation is 0, a within-subject design needs half as many participants as a between-subject design (e.g., 64 instead 128 participants), simply because every participants provides 2 datapoints. The extent to which this reduces the sample size compared to a between-subject design depends on the correlation ( r) between the two dependent variables, as indicated by the 1-r part of the equation. The division by 2 in the equation is due to the fact that in a two-condition within design every participant provides two data-points. The sample size needed in a two-group within-design (NW) relative to the sample needed in two-group between-designs (NB), assuming normal distributions, and ignoring the difference in degrees of freedom between the two types of tests, is (from Maxwell, Delaney, and Kelley ( 2004), p. 561, formula 45): One difference between an independent t-test and a dependent t-test is that an independent t-test has 2(n-1) degrees of freedom, while a dependent t-test has (n-1) degrees of freedom. Within designs can have greater power to detect differences than between designs because the values are correlated, and a within design requires less participants because each participant provides multiple observations. As Cohen ( 1988) writes, “The Z subscript is used to emphasize the fact that our raw score unit is no longer X or Y, but Z,” where Z are the difference scores of X-Y. If we want to perform an a-priori power analysis, we are asked to fill in the effect size dz. To illustrate the effect of correated observations, we start by simulating data for a medium effect size for a dependent (or paired, or within-subject) t-test. 15.8.1 Reproducing Brysbaert Variation 1: Changing Correlation.Appendix 2: Direct Comparison to MOREpower.Appendix 1: Direct Comparison to pwr2ppl.15.3 Conclusion on a Binary RCT with Interim Analyses.15.2 Binary RCT with a Interim Analysis.15.1.5 Conclusions on the Binary RCT Analysis.15.1.2 Step 1: Create a Data Generating Function.15.1 A Clincal Trial with a Binary Outcome.15 Beyond Superpower II: Custom Simulations.14 Beyond Superpower I: Mixed Models with simr.13.4 Equivalence and non-superiority/-inferiority tests.12.2.1 MANOVA or Sphericity Adjustment?.12.2 Violation of the sphericity assumption.12.1 Violation of Heterogeneity Assumption.11.5 Code to Reproduce Power Curve Figures.11.4 Increasing correlation in on factor decreases power in second factor.11.3 Explore increase in correlation in moderated interactions.11.2 Explore increase in effect size for cross-over interactions.11.1 Explore increase in effect size for moderated interactions.10.2 Two-way Between Subject Interaction.4.4.2 Examine variation of means and correlation.4.2.2 Three within conditions, medium effect size.4.2.1 The relation between Cohen’s f and Cohen’s d.4.1.1 Two conditions, medium effect size.3.2.3 Two conditions, large effect size.3.2.2 Four conditions, medium effect size.3.2.1 Three conditions, small effect size.3.2 Effect Size Estimates for One-Way ANOVA.2.1.6 Specifying the standard deviation.2.1.1 Specifying the design using design.1.6.2 Post hoc power is merely a transformation of your obtained p value.1.6.1 The sample effect size is not the population effect size.1.4 Sample effect size vs. population effect size.
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