Statistical Methods for Analyzing Collapsibility in Regression Models

Abstract

We give a unified treatment of statistical methods for assessing collapsibility in regression problems, including some possible extensions to the class of generalized linear models. Terminology is borrowed from the contingency table area where various methods for assessing collapsibility have been proposed. Our procedures, however, can be motivated by considering extensions, and alternative derivations, of common procedures for omitted-variable bias in linear regression. Exact tests and interval estimates with optimal properties are available for linear regression with normal errors, and asymptotic procedures follow for models with estimated weights. The methods given here can be used to compareβ1 and β2 in the common setting where the response function is first modeled asXβ1(reduced model) and then asXβ2+Zγ(full model), withZ a vector of covariates omitted from the reduced model. These procedures can be used in experimental settings (X= randomly asigned treatments,Z= covariates) or in nonexperimental settings where two models viewed as alternative behavioral or structural explanations are compared (one model withX only, another model withX andZ).

Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods

“…These include the steps mentioned in Baron and Kenny (1986) and Judd and Kenny (1981) and the joint significance test of ␣ and ␤ described in MacKinnon et al (2002), which do not include explicit methods to compute confidence limits. There are other methods to compute confidence limits for the indirect effect based on the standard error of the difference in coefficients, -Ј (e.g., Allison, 1995b ;Clogg, Petkova, & Cheng, 1995;Clogg, Petkova, & Shihadeh, 1992;Olkin & Finn, 1995) but these methods perform similar to the traditional z test described in this article, in part because a normal distribution for the indirect effect is assumed.…”

How do dynamic capabilities transform external technologies into firms’ renewed technological resources? – A mediation model

“…First, the Sobel test is commonly used (Sobel, 1982) as a product of coefficients approach (MacKinnon et al, 2002). Second, the Clogg and Freedman test is commonly used as a difference in coefficients approach, (Clogg, Petkova, & Shihadeh, 1992;Freedman & Schatzkin, 1992).…”

“…The last step involves demonstrating that when the mediator and the independent variable are used simultaneously to predict the dependent variable, the previously significant path between the independent and dependent variables will be reduced. Alternative tests for mediation effect exist and will be used in this study as well (Sobel, 1982;Clogg, Petkova, & Shihadeh, 1992;Freedman & Schatzkin, 1992).…”

Integrating VBM into the General Linear Model with voxelwise anatomical covariates

“…8) can be used to examine the effect of including voxelwise covariates and whether the full model is needed (e.g. [Clogg et al, 1992]), or an optimization technique such as a "backward" or "stepdown" approach [Neter et al, 1996;Rao and Toutenburg, 1999] could be employed to determine the benefit of including voxelwise covariates. However, an unresolved issue is how this information should be used.…”