Tutorials in Quantitative Methods for Psychology (Jul 2022)
Point and interval estimates for a standardized mean difference in paired-samples designs using a pooled standard deviation.
Abstract
A standardized mean difference using a pooled standard deviation with paired samples (d_p; paired-pooled design) can be compared directly to a d_p from an independent samples design, but the unbiased point estimate g_p and confidence interval (CI) for d_p cannot unless the population correlation \rho between the scores is known in the paired-pooled design, which it rarely is. The \rho is required to calculate the degrees of freedom \nu for the design, and \nu is necessary to calculate the g_p and CI. If a variable sample correlation is substituted for \rho the \nu is only approximate and the sampling distribution for d_p is unknown. This article uses simulations to compare the characteristics of the unknown distribution to the noncentral t distribution as an approximation and provides empirically-derived regression equations to compensate for the bias in the approximated CI computed using the noncentral t distribution. The result is an approximate but much more accurate coverage of the CI than previously available. Tables are supplied to assist sample size planning and computer programs are provided for computations. These results are experimental and tentative until the actual distribution can be discovered. The regularity of the deviation in coverage that allows the compensation to work encourages that search.
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