Power analysis - reliability and sample size

Reliability is defined as the ratio of true score variance to observed score variance.

Reliability: rho(X) = sigma(T)^2 / sigma(X)^2 = 1 - sigma(E)^2/sigma(X)^2

sigma(T)^2 is the variance of the true score

sigma(X)^2 is the variance of the observed score

sigma(E)^2 is the variance of the error score

sigma(T) = sigma(X)*sqrt(rho(X))

sigma(X) = sigma(T)/sqrt(rho(X))

true effect size: d(T) = (mu_1 - mu_0)/sigma(T)

observed effect size: d(X) = (mu_1 - mu_0)/sigma(X) = d(T)*sqrt(rho(X))

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Ref: Kanyongo et al. (2017) Reliability and Statistical Power: How Measurement Fallibility Affects Power and Required Sample Sizes for Several Parametric and Nonparametric Statistics

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Interactions between reliability, sample size and effect size