A recentering approach for interpreting interaction effects from logit, probit, and other nonlinear models

Research Summary: Strategic management has seen numerous studies analyzing interaction terms in nonlinear models since Hoetker’s (2007) best-practice recommendations and Zelner’s (2009) simulation-based approach. We suggest an alternative recentering approach to assess the statistical and economic importance of interaction terms in nonlinear models. Our approach does not rely on making assumptions about the values of the control variables; it takes the existing model and data as is and requires fewer computational steps. The recentering approach not only provides a consistent answer about statistical meaningfulness of the interaction term at a given point of interest, but also helps to assess the effect size using the template that we offer in this study. We demonstrate how to implement our approach and discuss the implications for strategy researchers. This article is protected by copyright. All rights reserved. 2 Managerial Summary: In industry settings, the relationship between multiple corporate strategyrelated inputs and corporate performance is often nonlinear in nature. Furthermore, such relationships tend to vary for different types of firms represented within the broader population of firms in a given industry. It is thus imperative for managers to know how to take nonlinear relationships between related business factors into account when they make strategic decisions. We suggest a simple and easily implementable way of assessing and interpreting interactions in a nonlinear setting, which we term a recentering approach. We demonstrate how to apply our approach to a strategic management setting. This article is protected by copyright. All rights reserved. 3 INTRODUCTION Interaction terms are frequently modeled in strategic management research in order to evaluate the effect of one explanatory variable on the response variable given the magnitude of another explanatory variable (e.g., the relationship between corporate strategy-related inputs and management performance outcomes varies depending on the internal and external business environments). Assessing and interpreting interaction terms becomes more complicated when models are nonlinear. Unlike linear models where the effect of a one-unit change in a covariate on the outcome variable (i.e., marginal or partial effect) is constant over the whole range of the covariate given the level of the other covariates in the model, the same effect in nonlinear models relies on the values of all other covariates in the model (Ai & Norton, 2003; Norton, Wang, & Ai, 2004). Given the frequency with which strategic management researchers have encountered interaction terms in nonlinear models (see, e.g., Shook, Ketchen, Cycyota, & Crockett, 2003; Hoetker, 2007), we will argue and show by way of mathematical proof and empirical analysis that there is room for another methodological option for achieving simplicity and consistency of interpretation of those interaction terms. In strategic management research, Hoetker (2007) recommended a set of best practices for the use of logit and probit models, including interpreting interaction terms. To further improve the assessment of statistical meaningfulness and interpretation of logit and probit results, Zelner (2009, p. 1336) suggested “a simulation-based technique developed by King, Tomz, and Wittenberg (2000)” and argued for the benefits of this technique over the conventional calculus-based method known as 1 For an introduction to the simulation method, see Krinsky and Robb (1986, 1990, 1991). See Greene (2018, pp. 647-648, p. 752) for an instructive discussion on the simulation-based method and the specific method of Krinsky and Robb. This article is protected by copyright. All rights reserved. 4 the delta method (Zelner, 2009, pp. 1341-1342) proposed by Dorfman (1938). In particular, Zelner proposed (i) calculating and interpreting a difference in predicted probabilities associated with discrete changes in key predictor values (known as the cross-partial derivative or cross-difference, which measures how the marginal effect of one variable changes when the other variable in the interaction term changes) and (ii) testing whether the difference in predicted probabilities is different from zero by constructing a confidence interval (C.I.) around the estimated quantity and finding out if the interval contains zero. This simulation-based technique argued for by Zelner (2009) requires user-written Stata commands ‘CLARIFY’ and ‘intgph’ (Tomz, Wittenberg, & King, 2003; Zelner, 2009). This simulation approach, however, must by inherent definition include the researcher picking assumed values for all the control variables in the model in order to generate output about whether an interaction effect is statistically meaningful. To address this concern, we propose and recommend a recentering approach, which focuses on the main independent variable at a point of theorymotivated interest. The recentering approach does not require assumed values for any of the control variables, takes the data and model as is, is computationally simpler and is easier to implement with one simple mathematical transformation as seen below. Last but not least, our approach enables one to assess, with the help of the template we provide in this study, the effect size of the interaction term 2 For an analysis comparing the delta method and the simulation method, see Krinsky and Robb (1990). For an additional description of the delta method, see Rothenberg (1984) and Horowitz (2001). For a separate way to implement the simulation method, see the NLOGIT software package and its WALD command. 3 Greene (2010) notably raised issues with this common practice of computing the cross-partial derivative and testing the interaction effect in nonlinear models. He pointed out the difficulty of interpreting the interaction effect measured by the cross-partial derivative in nonlinear models given the relationships among the variables. This article is protected by copyright. All rights reserved. 5 in a nonlinear model. Overall, the recentering approach we propose gives researchers an additional option to consider when assessing the interaction effect in nonlinear models. Our recentering approach is based on a recentered regression where one or both variables involved in the interaction term are centered at a value of interest―whether it be at the sample mean, sample median, sample 75 percentile value, sample 25 percentile value, or any other theory-driven value. That is, every value of the variable being interacted in the data set is deducted by the same value of the researcher’s interest. For ease of explication and comparison, we begin our discussion below by showing our simulation results from three logit model examples used in prior research. We then illustrate our recentering approach where we first show the link to generalized linear models and discuss the benefits of using the log of odds ratio in assessing and interpreting interaction effects in nonlinear models. Next, we present our mathematical proof that concisely illustrates why the recentering method provides a simple and consistent identification process. We then show how the recentering approach can help researchers assess the effect size of the interaction term in a nonlinear setting beyond its statistical meaningfulness. There, we provide a table that researchers can easily consult to evaluate the relationship between the odds ratio and Cohen’s d (Cohen, 1988), a widely accepted measure for assessing the effect size in the field of statistics and in the behavioral and health sciences. Lastly, we demonstrate the steps to implement the recentering approach. We conclude by discussing the benefits of using the recentering approach in comparison to the simulation-based approach. THE SIMULATION-BASED APPROACH This article is protected by copyright. All rights reserved. 6 For ease of explication and comparison across studies, we utilize three well-specified logit models (Models IV A, VI and V) and the data (N=469) used in Leiblein and Miller (2003) as our examples. The three logit models specified in equation (2) below take the form of the following population logistic model of the binary outcome variable Y with the vector of independent variables X ≡ ( 1 i X , . , X ):