A Quasi-Likelihood/Generalized Estimating Equation Approach to Count Outcomes in Single-Case Experimental Designs

• In Event: 30.075 - The Meta-Analysis of Single-Subject Experimental Design Data: Methodological Challenges and Approaches

Abstract

We introduce the use of generalized estimating equations (GEE) for analyzing rates and counts in Single Case Experimental Design (SCED) situations. In a comprehensive review of the SCED literature, Shadish and Sullivan (2011) indicate that over 90% of the outcome measures used were some form of count or rate. Variables of these types are often poorly approximated by a Gaussian error distribution. For instance, it is not unusual for there to be a floor or ceiling effect in one of the phases of an experiment or for substantial heteroscedasticity across phases. Failing to account for these features can end up creating important biases in quantitative effect size measures derived from the model. We consider the use of quasi-likelihood to deal with both the over- or under-dispersion frequently found in these data along with serial correlation using GEE, which is a semi-parametric approach. In order to evaluate in a systematic way the performance of the quasi-likelihood GEE, a simulation study was performed in which a number SCED data having counts outcomes and heterogeneous data were simulated using a two parameter regression model (reflecting the baseline performance and the change in level between baseline and treatment performance).
The GENMOD procedure within SAS 9.4 (Copyright © 2015, SAS Institute Inc. SAS) was used because it allows estimating the parameters of interest using the quasi-likelihood GEE with scale correction. The following four design conditions were varied: The simulated series lengths (I) consisted of 20 or 40 measurements and (2) the treatment effect equaled (β_1) 0 or 1 on the natural logarithm scale. The autocorrelation coefficient (φ) was set to values going from -.50 to .50 (with increases of .10). For each condition 2,000 datasets were generated.
No systematic bias was found as all the values did not exceed Hoogland and Boomsma’s (1998) criterion of 5%. The mean squared error (MSE) varied between 0.088 (I = 40 and φ = -.1) and 0.416 (I = 20 and φ = -.5). A statistically significant effect of the autocorrelation on the MSE was found [F(10,85276) = 411.96, p <.0010, η_p^2 = .0449 ]. The MSE is not influenced by negative values for autocorrelation, however, when the autocorrelation is positive, the larger the autocorrelation, the larger the MSE. Unbiased values for the standard errors were obtained when the autocorrelation is between -0.2 and 0.2 for both when the number of measurements is set to 20 or 40. We found that the value of the autocorrelation [F(10,43) = 178.06, p < .0001, η^2 = .979] had a statistically significant effect on the CP95. The CP95 varied between 80.80% (I = 40 and φ = -0.5) and 98.70% (I = 40 and φ = 0.5). We found that the CP95 is too large when φ is smaller than -0.3 and too small when φ is larger than 0.1.
The GEE approach will be demonstrated using the Laski, Charlop and Schreibman (1988) dataset.

Authors

• Mariola Moeyaert, University at Albany

• Jay Verkuilen, City University of New York