Standard Errors for the Percent of Goal Obtained

• In Event: Analysis and Meta-Analysis of Single-Case Experimental Data: Methodological Developments

Abstract

An effect size based on the percent of goal obtained (PoGO) was developed for single-case studies as a descriptive summary of effect that would align well with the effects seen in graphical displays, be easy to interpret, and allow for comparisons of intervention effects across cases and studies. However, if researchers want to average the effects across cases, provide a confidence interval for this average effect, or explore potential moderators of the effect, it would be helpful to have standard error estimates in addition to the point estimates. The purpose of this poster is to provide a method of estimating the standard errors of PoGO.
For each participant, the percent of goal obtained is defined as
PoGO = (β-α)/(γ-α),
and if temporal stability in each phase is assumed PoGO can be estimated by substituting the mean of the participant’s intervention observations for β, the mean of the participant’s baseline observations for α, and the goal for γ. To develop standard errors for PoGO we used the approximate formula for the standard error of a ratio of normal variables derived by Dunlap and Silver (1986). The approximate formula relies on estimates of the numerator of the ratio, the denominator of the ratio, and the error variances of each.
Assuming temporal stability of each phase and independence of observations, we estimated the error variance of the numerator of PoGO, s_(β-α)^2, as s_(β-α)^2= ((s_A^2)/n_A +(s_B^2)/n_B )
where s_A^2 is the variance of the participant’s baseline observations, s_B^2 is the variance of the participant’s intervention observations, n_A is the number of baseline observations, and n_B is the number of intervention observations. Because γ is a known quantity, we estimated the error variance in the denominator of PoGO as the error variance in the mean of the participant’s baseline observations, again assuming temporal stability and independence,
s_(γ-α)^2= (s_A^2)/n_A. Substituting these estimates into the formula yields the appropriate standard error, which will be presented in the power.
Dunlap and Silver (1986) showed through simulations that the approximation to the standard error of a ratio of normal variables was accurate when the variables were uncorrelated, which is consistent with our independence assumption, and when the denominator of the ratio was relatively large compared to its error variance (i.e., at least 6 times as large). We illustrate this method for estimating standard errors for PoGO using multiple-baseline studies of vocabulary learning from the published literature. We compare interval estimates of PoGO using these standard errors to estimates that can be obtained by transforming PoGO to a log-response ratio.

Authors

• Ella Patrona

• John M. Ferron, University of South Florida

• Howard Goldstein, University of South Florida