Eight Simple Guidelines for Improved Understanding of Transformations and Nonlinear Effects

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Eight Simple Guidelines for Improved Understanding of Transformations and Nonlinear Effects © The Author(s) 2021 Published version Rönkkö, Mikko; Aalto, Eero; Tenhunen, Henni; Aguirre-Urreta, Miguel I. Rönkkö, M., Aalto, E., Tenhunen, H., & Aguirre-Urreta, M. I. (2022). Eight Simple Guidelines for Improved Understanding of Transformations and Nonlinear Effects. Organizational Research Methods, 25(1), 48-87. https://doi.org/10.1177/1094428121991907 2022 Regular Submission Eight Simple Guidelines for Improved Understanding of Transformations and Nonlinear Effects Mikko Ro ¨nkko ¨ 1 , Eero Aalto 2 , Henni Tenhunen 2 , and Miguel I. Aguirre-Urreta 3 Abstract Transforming variables before analysis or applying a transformation as a part of a generalized linear model are common practices in organizational research. Several methodological articles addressing the topic, either directly or indirectly, have been published in the recent past. In this article, we point out a few misconceptions about transformations and propose a set of eight simple guidelines for addressing them. Our main argument is that transformations should not be chosen based on the nature or distribution of the individual variables but based on the functional form of the relationship between two or more variables that is expected from theory or discovered empirically. Building on a systematic review of six leading management journals, we point to several ways the specification and interpretation of nonlinear models can be improved. Keywords transformations, generalized linear model, interaction, visualization, Poisson regression, logistic regression Nonlinear models are common in organizational research. These have been used, for example, for modeling diminishing returns (e.g., Cennamo, 2018), U-shape effects (e.g., Huang et al., 2018), S-shape effects (e.g., Brands & Fernandez-Mateo, 2017), or relative effects (York et al., 2018). A nonlinear model can be constructed by either (a) applying nonlinear transformations to variables before analysis or (b) including nonlinear transformations in the estimated model. The first approach is typically justified by stating that a statistical technique, typically ordinary least squares (OLS) 1 Jyva ¨skyla ¨University School of Business and Economics, Jyva ¨skyla ¨, Finland 2 Department of Industrial Engineering and Management, Aalto University, Espoo, Finland 3 Florida International University, College of Business Corresponding Author: Mikko Ro ¨nkko ¨, Jyva ¨skyla ¨University School of Business and Economics, P.O. Box 35, FI-40014 Jyva ¨skyla ¨, Finland. Email: [email protected] Organizational Research Methods 1-40 ªThe Author(s) 2021 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1094428121991907 journals.sagepub.com/home/orm regression, requires normally distributed data and that a transformation makes the distribution closer to normal (e.g., by reducing skewness). The second approach is typically justified by stating that a discrete (e.g., binary, count) dependent variable requires special modeling techniques, such as generalized linear models (GLMs). Logistic regression, probit regression, Poisson regression, and negative binomial regression are special cases of this approach. In this article, we explain why any justification that relies on the distribution of a variable is incorrect. Our article makes two key contributions. First, we show why transformation decisions, whether done as part of a model or applied manually before estimation, must consider the relationship between two variables instead of being based on the distribution of a single variable. Second, we discuss the interpretation of nonlinear models, an issue that is either overlooked or even explained incorrectly in recent guidelines (Becker et al., 2019; Blevins et al., 2015). The article is structured as a set of guidelines derived from a review of books and articles on econometrics (e.g., Angrist & Pischke, 2009; Greene, 2003; Wooldridge, 2002, 2013), sociology (e.g., Breen et al., 2018; Long & Mustillo, 2019; Mize, 2019), and data visualization (e.g., Breheny & Burchett, 2017; Cattaneo et al., 2019; Mitchell, 2012a) as well as from a review of recent articles from six leading management journals. The guidelines are summarized in Table 1. Each guideline is demonstrated using a publicly available data set, which allows easy replication and can also be used in teaching. The Supplemental Material available in the online version of the journal includes R and Stata code implementing the techniques we discuss, a web application for constructing the plots demonstrated in the article (https://mronkko.shinyapps.io/PredictionPlots/), and a set of short video lectures explaining the key concepts of the article (https://tinyurl.com/ nonlinearmodels). The Use of Transformations in Organizational Research To ground our article in current research practice, we reviewed the 2017 and 2018 volumes of Academy of Management Journal,Administrative Science Quarterly,Journal of Applied Psychology,Journal of Management,Strategic Management Journal,andPersonnel Psychology. The second and third authors read each volume in random order until 10 articles applying transformations were found or until all articles in the volume were read. The result is the list of 104 articles shown in Table 2. These articles were then coded in detail by the second and third authors, each of whom coded half of the articles, with the help of the first author. To assess reliability, 20 articles were cross-coded, producing an overall interrater reliability (Kraemer, 1980) of .75. The codes were further updated as the writing of the article progressed, further enhancing reliability. Of the studies that tested hypotheses using regression-type models, 66%applied at least one transformation. 1 Log transformation, used either manually or as a link function in a GLM model (e.g., Poisson and negative binomial models), was the most common approach. It was followed by logit and probit transformations used inlogisticandprobitregressionsaswellasin some ordered and categorical variable models. Power transformations, particularly the second power, were used for modeling U-shape effects. Table 3 shows the commonly used transformations. GLMs were the most common way to apply transformations, and therefore it is important to understand what GLMs do. 2 To this end, we take linear regression, shown in the first plot of Figure 1, as a starting point. In linear regression, the mean (or more precisely, the expected value) of the dependent variable depends linearly on the independent variables, and the data are assumed to be normally distributed around the population regression line. A GLM extends linear regression by 2Organizational Research Methods XX(X) Table 1.Summary of the Guidelines for Transformations and Nonlinear Effects. Guideline Explanation Guideline 1: Motivate transformations based on functional form, not distribution of the variables. Transformations alter the functional form between variables, changing what kind of effect is estimated. Nonnormality of variables is not a problem that needs to be addressed but a nonnormal dependent variable may indicate a nonlinear process that needs to be considered. Guidelines for building models with transformations Guideline 2: Hypotheses should state the form of the association when possible. A statistical model requires that a functional form is specified. This should ideally be done based on theory. Functional forms can be hypothesized with thought experiments or discovered empirically through diagnostics. Guideline 3: Never transform the dependent variable; use a GLM instead. A GLM model and a linear model with a transformed dependent variable are interpreted the same way. GLM is both methodologically and practically superior to transforming dependent variables. Guideline 4: Choose the GLM link function first, if used, and individual variable transformations later. A GLM link function determines not only how a single variable affects the dependent variable but how the variables work together. In linear models, independent variables combine additively, and in exponential models, they combine multiplicatively. Guideline 5: Choose the GLM distribution based on consistency, not model fit. GLM link function should be chosen based on the consistency of the curve, not the fit of the distribution. Use OLS for linear, Poisson QML for exponential, and logit QML for binary and fractional response models. Guideline 6: Do not default to power transformation when modeling U-shape and other curvilinear effects; consider different alternatives instead. Power transformations (e.g., x 2 ) assume a specific kind of U-shape effect that may not be correct for the data. Consider exponential models and regression splines as alternatives for estimating nonlinear models. Guidelines for interpreting models with transformations Guideline 7: Interpret nonlinear effects by plotting, including confidence intervals and the data in the plot. Interpret nonlinear models by plotting adjusted predictions and clearly show the nature of nonlinearity in the plot. Include confidence bands to show the uncertainty of the estimates and either plot the data or use a contour plot to show that the chosen functional form fits well. Guideline 8: Do not infer moderation from an interaction effect but from a plot. Moderation hypotheses are ambiguous unless the form of the effect (e.g., relative, absolute) is stated in the hypothesis. Interpret moderation hypotheses of nonlinear models using plots even if the interaction term is nonsignificant. Note: GLM ¼generalized linear model; OLS ¼ordinary least squares; QML ¼quasi-maximum likelihood. Ro ¨nkko ¨et al. 3 Table 2. Overview of the Use of Transformations in the Reviewed Articles. Journal Year N Manual Transformations Generalized Linear Models IV Transformed DV Transformed Logarithm Power Square Root Inverse Destructive Other GLM Used Log Link Logit Link Probit Link Academy of Management Journal 2017107 2 5 40 0 0 0 632 1 2018105 2 7 100 008315 Administrative Science Quarterly 2017104 13200 0 07421 20188 3 2 4 0 0 0 0 0 6 0 5 1 Journal of Applied Psychology 20171011100 0 0 0 9171 2018105 0 0 50 0 0 17160 Journal of Management 2017107 3 4 40 0 10523 2 20189 3 3 4 10100311 1 Personnel Psychology 20173 0 0 0 0 0 0 0 0 3 120 20184 2 11110002020 Strategic Management Journal 2017100 0 0 00 0 0 010415 2018102 2 3 00 0 0 0 835 0 Total 104 39 17321811 1174 23 37 17 Note: Destructive transformations are transformations that destroy information and cannot be reversed with an inverse transformation; winsorization, where extreme values are replaced with specified percentiles of the variable, is an example of such a transformation. IV ¼independent variable; DV ¼dependent variable; GLM ¼generalized linear model. 4 Table 3. Potential Functional Forms in Organizational Research. Transformation y¼fb0þb1xðÞ Inverse Transformation y¼gb0þb1xðÞDefinitions Models, Interpretation, and Recommended Distribution for Consistency fxðÞ¼x gxðÞ¼x Linear regression Increasing xby 1is associated with b1 increase in y. Normal distribution fxðÞ¼log xðÞ gxðÞ¼ex Poisson and negative binomial regressions, survival models Increasing xby 1is associated with y increasing by eb1times (incidence rate ratio). Poisson distribution fxðÞ¼ ffiffiffix p gxðÞ¼x2 Second power commonly used for modeling U-shape effects No easy interpretation No recommended distribution fxðÞ¼1 x gxðÞ¼1 x No easy interpretation No recommended distribution fxðÞ¼log x 1x  gxðÞ¼ ex exþ1  Logistic, multinomial, ordered logistic, and beta regression Increasing xby 1is associated with the odds of yincreasing by eb1times (odds ratio). Bernoulli distribution fxðÞ¼F1xðÞ gxðÞ¼Fðx) Probit regression; Fis the standard normal cumulative distribution. No easy interpretation No recommended distribution Note: Inverse transformation reverses the original transformation so that x¼g(f(x)) and x¼f(g(x)). For example, if log(x)¼y,thene y ¼x. Ro ¨nkko ¨et al. 5 introducing (a) a link function that determines a curve that characterizes the mean of the dependent variable as a function of the independent variables and (b) a distribution that specifies how the values of the dependent variable are dispersed around the mean given by the curve. Importantly, the distribution is a conditional distribution specified for a specific combination of independent variables, not an unconditional distribution of the dependent variable overall. In the linear regression model shown in the first plot of Figure 1, the observations are always normally distributed around the regression line. That is, if we only look at observations that have the same xvalue (e.g., x¼1), that subset will be normally distributed (conditional distribution). However, the overall unconditional distribution of yis not and does not need to be normal. In the Poisson model shown in the second plot, yis always distributed as Poisson for any specific value of x(conditional distribution), but the overall unconditional distribution of yis not and does not need to be Poisson. As this example shows, the choice of GLM distribution cannot be justified by looking at the unconditional distributions. Linear regression and Poisson regression are both special cases of GLM, where the links are linear and logarithmic and distributions are normal and Poisson, respectively. Following the convention of using ffor the link function and gfor its inverse function, the linear model, a model with manual log transformation, and a GLM with a log link can be written as follows: Linear :y¼b0þb1xþuð1Þ Manual transformation :fyðÞ¼b0þb1xþuð2Þ GLM;presentation1 :y¼gb0þb1xðÞþuð3Þ GLM;presentation2 :fEy½ðÞ¼b0þb1xð4Þ where uis the error term representing variation around the line or curve. The linear function is a special case of GLM where the link function is the identity function f xðÞ¼xin which case Equations 1 through 4 are equivalent, and in practice, OLS would always be applied. Therefore, from now on, we will focus on discussing GLMs with nonlinear links. Figure 1. Comparison of linear model and Poisson model. Note: The red line shows how expected ydepends on xand how observations are distributed at x¼1, 2, 3, 4, and 5 (conditional distribution). Axes show kernel density plots of xand y(unconditional distribution). 6Organizational Research Methods XX(X) Equation 2 and Equation 4 are two different ways of applying the same transformation, which can also be understood by rewriting Equation 2 as y¼gb0þb1xþuðÞand comparing this against Equation 3. The difference between the two equations is that manual transformation is applied to the observed scores, but in GLM, the transformation is applied to the predicted values that approximate the observed scores. This has consequences on how well the model explains the data (see Guideline 3). But the choice between the techniques does not depend on any underlying theory and does not affect how the results should be interpreted, even though the techniques are not the same. We will now explain when transformations should be applied, how models with transformations should be specified, and how the results of these models should be interpreted. Guideline 1: Motivate Transformations Based on Functional Form, Not Distribution of the Variables Nonnormal dependent variables are a common justification for using transformations. Consider, for example, the following claim by Blevins et al. (2015): “The application of a linear regression model (LRM) is inappropriate for data with a count-based dependent variable (Cameron & Trivedi, 1998) and can result in inefficient, inconsistent, and biased regression models (Long, 1997)” (pp. 47–48). Similar statements were common in the reviewed articles; 55% 3 of the articles that applied manual transformations justified this decision based on the characteristics of variables, such as skewness. In 23%of the articles, no justification was given. The same pattern holds for articles that applied GLM models: 58%of these articles justified a GLM based on the characteristics of the dependent variable,followedby42%that did not provide a justification. 4 Claims that a nonnormal dependent variable would be problematic for OLS regression are incorrect and thus not valid reasons to apply transformations. In fact, the proofs that regression is unbiased (Wooldridge, 2013, Theorem 3.1) and consistent (Wooldridge, 2013, Theorem 5.1) do not need any assumptions about the distribution of the independent or dependent variables or about that of the error term. What is required is the following: (a) random sampling, (b) the relationships between the independent and dependent variables are linear in the population, (c) each independent variable adds unique variance to the model, and (d) there is no endogeneity (see also Angrist & Pischke, 2009, pp. 70–73). A nonnormal error term or even a discrete dependent variable, such as a count, is unproblematic for regression if these four assumptions hold (see also Villadsen & Wulff, 2020). Distributional assumptions are required in proofs of some OLS regression properties, but these assumptions are made only about the error term (conditional distribution) and not about the independent or dependent variables (unconditional distributions). The first assumption is that the variance of the error term is constant (homoskedasticity), which means that the observations are always equally spread out around the regression line. Homoskedasticity is required for the consistency of standard errors and efficiency of the OLS estimator. 5 The failure of this assumption is referred to as heteroskedasticity, and transformations have been proposed as a way to address this problem (Cohen et al., 2003, Section 6.4.1; Rosopa et al., 2013). However, this is ineffective 6 and has the side effect of converting the originally linear model into a nonlinear one, which may not be the ideal representation of the phenomenon. Heteroskedasticity-consistent (robust) standard errors (Angrist & Pischke, 2009, Chapter 8; Wooldridge, 2002, Sections 4.2.3, 19.2.3, 2013, Section 8.2) are a superior solution because they are effective even when the form of heteroskedasticity is unknown, retain the original model, and are available in commonly used statistical software (StataCorp, 2017b, Section 20.22; Zeileis, 2006). Indeed, 46%of the reviewed articles using transformations applied robust standard errors, and not a single article used a transformation to deal with heteroskedasticity. Ro ¨nkko ¨et al. 7 The second assumption, normality of the error term, is only required for the proof that the t statistics used for calculating the pvalues of the regression coefficients and the Fstatistic that is used for the overall model test follow their reference distributions in small samples (Wooldridge, 2013, Theorem 4.2). But this assumption is mostly irrelevant for applied research because the large sample behavior of OLS regression, which does not depend on the normality of the error term, starts to kick in at sample sizes well under 100 (Wooldridge, 2013, Section 5.2). Example 1: Coin Throws. We now demonstrate that a nonnormal dependent variable is not problematic for linear regression. The data set shown in Table 4 contains two variables about a fair coin: Throws indicates how many times the coin was tossed and tails how many tails were counted. Table 5 shows the results of using these data in (a) linear regression and (b) Poisson regression, which is often used with count variables in organizational research (Blevins et al., 2015), to study how the expected number of tails depends on the number of throws. The linear regression model gives the correct answer: The expected number of tails is half the number of throws. In other words, following the standard interpretation of linear regression, for each additional unit of throws, we should expect an increase of half a unit of tails. How should the substantially smaller coefficient of 0.022 from the Poisson model be interpreted? Organizational researchers struggle with this question, and many articles simply check the pvalue and state the presence of a relationship, leaving the coefficients themselves uninterpreted. The problem is not limited to organizational research given that the same concern was expressed in the classic book by Long (1997): “Unfortunately, all too often when these models are used, the substantive meaning of the parameter is incompletely explained, incorrectly explained, or simply ignored. Sometimes only the statistical significance or possibly the sign is mentioned” (p. xxiii). The recently published guidelines serve as examples: Blevins et al. (2015) omitted the interpretation of the coefficients altogether, whereas Becker et al. (2019, p. 853) give the correct interpretation but incorrectly claim that the interpretation of a GLM would be different from estimating a comparable model using a transformed dependent variable. Table 4. Coin Throw Data Used in an Example. Throws 7 25 8 33 37 511415792474168295957275204897 Tails 4 11 3202024 7 84312 39 9 46 57 5135 41122244 Table 5. Linear Regression and Poisson Regression Models Predicting the Number of Tails Using the Coin Throw Data. 12 Linear Regression Poisson Regression (Intercept) 0.058 1.965*** (0.620) (0.134) Throws 0.504*** 0.022*** (0.015) (0.002) R 2 .968 Log likelihood –62.497 Note: N¼20. Standard errors in parentheses. Model 1uses heteroskedasticity-consistent (robust) standard errors. ***p<.001. 8Organizational Research Methods XX(X) Figure 4. Residual versus fitted plots and added-variable plots of regression of income and log of income on education and women using the Prestige data. Note: The blue curve is the best fitting second-order polynomial (U-shape) for residual versus fitted plot and the best fitting line for added-variable plots. 15 the resulting scatterplots were sent to 15 researchers asking them to identify the functional form. Out of the 300 evaluations, 139 (46%) incorrectly thought that the functional form was linear; just 54 (18%) identified the U-shape correctly, in 64 cases (21%) the researchers simply could not make a call, and in the remaining 43 cases (14%), the researchers inferred an incorrect nonlinear functional form. To summarize, the fact that organizational researchers do not explicitly state the functional form of the relationship as part of the hypotheses but nevertheless apply nonlinear models for their testing cannot be seen as a methodological problem (Becker et al., 2019) but is rather a consequence of the imprecision of management theory or of the way it is tested (Cortina, 2016; Edwards & Berry, 2010). Ideally, researchers should state the expected functional form in the hypotheses based on what kind of effect makes theoretical sense. Table 3 shows commonly used functional forms and can be used as a reference. If such precision is not afforded by theory, the functional form should be chosen based on how well the model appears to fit the data in a visual inspection or by comparing models using fit statistics when appropriate. This is a holistic evaluation that unfortunately cannot be condensed into a simple rule. If functional form is determined empirically, it is important to report this as an exploratory analysis (Hollenbeck & Wright, 2017). Indeed, introducing a functional form and its explanation to a theory can be considered an important theoretical contribution (Jaccard & Jacoby, 2020, p. 41). Even if a researcher cannot explain why the identified functional form exists, such observation can inspire future work designed to understand why a given functional form was observed. Guideline 3: Never Transform the Dependent Variable; Use a GLM Instead Transforming a dependent variable is common in the reviewed empirical articles. As shown in Table 2, 17 articles applied a transformation to a dependent variable. Of these, 16 (15%) used log transformation, and one article applied winsorization (Souder & Bromiley, 2017). Yet analyzing a transformed dependent variable is not an ideal approach for two reasons. First, to interpret the results, predictions from the model need to be converted back to the original metric. In the case of the log transformation, this happens by using the exponential function (e.g., Eggers & Kaul, 2018). However, the mean of the logarithm does not equal the logarithm of the mean, and this applies to all nonlinear transformations. This is demonstrated in Figure 5 showing the distributions of incomes for womenand men-dominated occupations. Comparing the original means against the means of logged incomes that are back-transformed to the original metric shows that the transformed variables underestimate the means. The effect is stronger for men-dominated occupations, which leads to underestimating the difference between the two groups. Second, transforming the dependent variable requires awkward workarounds. Perhaps the most common way is to add þ1 before taking logs (e.g., Clement et al., 2018; Gomulya et al., 2017; Vasudeva et al., 2018). The justification for the procedure is that the logarithm is not defined for zero and works differently for numbers more or less than one, and if all values were initially nonnegative, the transformation makes all values one or greater (Kline, 2011, pp. 63–64; Wooldridge, 2013, p. 193). Another, albeit less common, workaround addresses the bias caused by the backtransformation by using a correction (e.g., Balen et al., 2019; Duan, 1983; Wooldridge, 2013, pp. 212–215). These workarounds are required because a linear model of a transformed dependent variable, although common, is simply not the ideal way to model the dependent variable as a nonlinear function of the independent variables (Villadsen & Wulff, 2020). A GLM with a log link avoids the problems caused by log transforming the dependent variable. Instead of manually transforming the dependent variable and estimating log yðÞ¼b0þb1xþu (Equation 2), the same transformation can be applied in a GLM and estimating log E y½ðÞ¼b0þb1x(Equation 4). In particular, Poisson regression produces consistent estimates 16 Organizational Research Methods XX(X) for the exponential model regardless of the distribution of the error term and can even be used for noncount data (Cameron & Trivedi, 1998, Chapter 3; Gourieroux et al., 1984; Silva & Tenreyro, 2006; Wooldridge, 2010, Chapters 18.2–18.3, 2013, Chapter 17.3). This point was made eloquently by Stata’s founder William Gould (2011) in a nontechnical blog post. When Poisson regression is applied to a noncount variable, the likelihood statistics will be incorrect. For this reason, this approach is referred to as a quasi-maximum likelihood (QML) estimation. Because the QML likelihood values are not proper likelihoods, they cannot be used for model testing, but robust standard errors can be employed for correct inference. To demonstrate that Poisson QML works better than manually transforming the dependent variable, we applied both techniques to the Prestige data set, as shown in Models 2 and 3 in Table 7 and graphically in Figure 6. The plot shows two curves, one based on predictions from the GLM model and another using the transformed dependent variable, back-transformed to the original scale, following current recommendations (e.g., Dawson, 2014). The GLM curve goes through the middle of the data and is thus a better fit than the curve using transformed dependent variable, which tends to underestimate the mean of the data, particularly for larger values. As an additional advantage, commonly used statistical software will automatically plot a GLM model correctly without the need to specify a back-transformation manually. In the reviewed articles, none of the articles that applied a log transformation to the dependent variable considered a GLM model with a log link as a substitute. 8 Similarly, the articles that used a GLM model with a log link (Poisson or negative binomial regressions in the review) did not use the nonlinearity of the link as a justification and hence did not consider the log transformation as an alternative. Thus, the interchangeability of a GLM and a manual transformation does not appear to be common knowledge among organizational researchers, although there are some examples outside the reviewed articles that note the general applicability of GLM models (Dahlander et al., 2016, p. 289). Figure 5. Distribution of income for menand women-dominated occupations and the effect of transformation on estimates of mean. Ro ¨nkko ¨et al. 17 To summarize, GLM is both statistically and practically the superior alternative to transforming the dependent variable. The predicted curves go closer to the middle of the data, modern statistical software automatically constructs the adjusted prediction plots correctly, and the need for awkward workarounds (e.g., when a dependent variable can take the value of zero) is eliminated. Yet using OLS regression with transformed dependent variables can be useful for diagnostic purposes, as demonstrated in the previous guideline. The differences between these two techniques are summarized in Table 8. Although it has been known for some time in the technical statistical literature that Poisson regression can be used for this purpose, its uptake in research practice has been slow. A possible reason for this is the institutionalized idea that these models should be used (only) for counts (Blevins et al., 2015). As noted by Nichols (2010): “If you decide on a log link, you may want to call your model ‘GLM with a log link,’ rather than a ‘Poisson’ QMLE—some older reviewers believe Poisson regression is only for counts” (p. 20). Guideline 4: Choose the GLM Link Function First, if Used, and Individual Variable Transformations Later Because GLMs are preferable to transforming dependent variables, specifying nonlinear models reduces to two decisions: whether to use a GLM and whether to transform any of the independent variables. The first decision is more consequential because it determines how the independent variables together influence the dependent variable. We will now turn to this, which requires understanding the difference between additive (linear) and multiplicative (exponential) models. For simplicity, we focus on the case of two independent variables, x 1 and x 2 , and only for the linear and exponential cases: Linear:y¼b0þb1x1þb2x2þuð5Þ Figure 6. Effects of education on income estimated with regression of transformed dependent variable and Poisson quasi-maximum likelihood (QML) regression. 18 Organizational Research Methods XX(X) ExponentialðGLMÞ:y¼eb0þb1x1þb2x2þu ¼eb0eb1x1eb2x2þuð6Þ An important difference between the two models is that in the linear model, the effects of x 1 and x 2 are added together, whereas in the exponential model (e.g., Poisson regression), they are multiplied together. That is, in the linear model, the effect of changing x 1 is always the same regardless of the current values of x 1 and x 2 . However, in the exponential model, the effect of x 1 is proportional to the predicted y, thus depending on both x 1 and x 2. Thedifference between the additive linear and multiplicativeexponential models was largelyignored in the reviewed articles. For example, Botelho and Abraham (2017) used four dependent variables: number of views, number of comments, and two ratings of online recommendations. All four variables were used to test thesame hypotheses; an additive model was used forthe two ratings variables, but when explaining the number of views and number of comments, the model was multiplicative. The use of a multiplicative model for one set of variables and an additive model for the other set of variables was neither noted nor explained in the article. More generally, out of the 40 reviewed articles that applied a GLM model with a log link or used a log-transformed dependent variable, just one (York et al., 2018) explicitly noted that the effects should be interpreted as multiplicative instead of additive, and one other (Dutta, 2017) noted that the effects in such models are relative to the current level. The choice between additive and multiplicative models should be driven by theory. For example, if a study investigated the effects of a nationwide policy (x 1 ) on the number of new companies founded (y), the effect should not be the same for all countries but be relative to the size of the population (x 2 ), suggesting a multiplicative model. In other words, implementing the policy in a larger country should produce more new companies than implementing the policy in a smaller country. In contrast, an additive model would be appropriate, for example, if one modeled the effects of R&D grants (x 1 ) on the number of patents a firm receives (y). If two firms of different size (x 2 ) receive a similarly sized grant and there are no large economies of scale in R&D, both firms should get an equal amount of R&D work done using the grant and thus receive roughly comparable Table 8. Comparison of Linear Regression Model With Transformed Dependent Variable and Generalized Linear Model. Linear Regression With Transformed Dependent Variable Generalized Linear Model Mathematical presentation fyðÞ¼b0þb1xþuy¼gb 0þb1xðÞþu or fEy½ðÞ¼b0þb1x Interpretation of coefficients Nonlinear interpretation depending on the transformation (see Table 2) Nonlinear interpretation depending on the transformation (see Table 2) Bias of predictions Predictions are biased because the mean of a transformation is not a transformation of the mean. Predictions are unbiased. Support for values on the boundary (e.g., zero in log transformation) Awkward workarounds required Supported Software support for plotting Back-transformation needs to be specified manually to the plotting command. Correct functional form plotted automatically Software support for diagnostics Very broad, ordinary least squares regression diagnostics More limited, typically specific link-distribution combinations supported Ro ¨nkko ¨et al. 19 number of patents from the work. Therefore, the effect should be additive with other variables that determine how many patents a company receives. In some scenarios, it may be desirable to combine linear and exponential effects. For example, if one models the total number of publications of a researcher as a function of time (x 2 ), an individual’s overall productivity can be exponential as a function of skills (x 1 ), but the number of publications (y) increases linearly, and not exponentially, over time. Here, the effects of skills and time are clearly multiplicative, but the effect of time itself is linear and not exponential. The GLM framework provides two alternative ways of modeling this combination of effects: Linear with interaction:y¼b0þb1ex1þb2x2þb3x2ex1þuð7Þ Exponential:y¼eb0þb1x1þb2log x2 ðÞ þu ¼eb0eb1x1x2b2þuð8Þ We demonstrate these specifications in Figure 7, where we estimate a linear effect of women (x 2 ) and exponential effect of education (x 1 ) on income using the Prestige data set. The first specification is problematic because the shape of the exponential effect ex1is not estimated from the data. That is, b1ex1þb3x2ex1is always about 22,000 times larger 9 when estimated at 6 years of education (minimum) compared to 16 years of education (maximum). Moreover, the steepness of the effect of education is the same for all levels of women, and women mostly affect the base level from which the exponential growth starts, as shown in the first plot of Figure 7. Indeed, none of the reviewed articles used an exponential transformation on the independent variables. The second specification is more attractive, but adding log(x 2 ) is not enough to model a linear effect 10 ; we must further constrain the coefficient b 2 to 1, producing: Exponential with exposure:y¼eb0þb1x1þlog x2 ðÞ þu ¼eb0eb1x1x2þuð9Þ A logged variable constrained this way is referred to as an exposure variable, and it enters the regression equation as a multiplier, thus modeling an effect that is directly proportional to the Figure 7. Comparison of two approaches for combining linear and exponential effects. 20 Organizational Research Methods XX(X) exposure variable (Cameron & Trivedi, 2009, p. 559). The use of exposure variables is not common in organizational research, but they are common in, for example, epidemiology, where researchers might be interested in which factors affect the number of cases of a disease in a country. Because countries vary in their population numbers, each country has a different number of people that could be potentially exposed to a disease and become a case. In this case, we would use the number of people in a country as an exposure variable, and the total expected number of cases would be individual risk (eb0eb1x1in Equation 9) multiplied by the size of the population used as an exposure variable (x 2 in Equation 9). In an organizational context, these models present a compelling alternative to control for scale effects that are commonly controlled for by using a ratio of the focal variable and the scale variable (e.g., return on assets ¼net income / total assets) as a dependent variable (Certo et al., 2020). The exposure model is visualized in the second plot of Figure 7. In this model, the magnitude of the exponential increase (shape of the curve) is estimated from the data, producing a more gradual estimate where both the base level and the magnitude of increase due to education vary as a function of the share of women. To summarize, the first decision when specifying a model is to determine whether the variables in the model are combined additively or multiplicatively. In the linear, additive effects model, the effect of each independent variable is in absolute terms not dependent on any other variables in the model. In the exponential, multiplicative effect model (e.g., Poisson regression), the effects are always relative to the current values of all independent variables, typically expressed as percentages. In practice, the choice is whether to use a GLM and which link function to use, decisions that have been traditionally made based on the distribution of the dependent variable. Instead, we argue that theory should play a much more important role in this decision. The thought experiment strategies explained earlier can be useful here as well. Guideline 5: Choose the GLM Distribution Based on Consistency, Not Model Fit The specification of a GLM involves two key decisions: choosing the link function for the relationship between the independent variable and the expected value of the dependent variable and the (conditional) distribution for the dependent variable. Importantly, the distribution concerns the conditional distribution for a specific set of predictor variables, not the unconditional distribution that one would analyze in the data preparation and screening stage of research (see Figure 1). Of these, the first decision is much more important because it determines which functional form is estimated; the second decision only influences whether the chosen form is estimated correctly. Unfortunately, the exact opposite decision process has been entrenched in the discipline given that current guidelines take the (unconditional) distribution as a starting point and largely omit discussing the transformation (e.g., Blevins et al., 2015). The same is true in empirical applications of GLMs; 58%of the reviewed articles that applied a GLM justified the model choice based on the distribution of the dependent variable, 42%did not justify the model choice, and just one article (Mata & Alves, 2018) focused on the link function when choosing a model. As discussed earlier and demonstrated with the coin throw example, this practice can lead to choosing a functional form that has a poor fit with the studied phenomenon. How should the (conditional) distribution be chosen? Because distributions represent variation of the dependent variable due to variables other than the ones in the model, the choice of a distribution can be difficult to motivate based on theory. Hence, the question becomes which distribution should be applied if the chosen distribution may, in fact, be incorrect? GLM models are estimated with maximum likelihood estimation, which has been proven to be consistent and asymptotically efficient if the model is specified correctly (Lehmann & Casella, 1998, pp. 443–450; Wooldridge, 2002, Chapter 13), including the correct (conditional) distribution for the dependent variable. Unfortunately, maximum likelihood estimation is generally not robust to misspecification of the distribution Ro ¨nkko ¨et al. 21 and becomes inconsistent (Wooldridge, 2002, Chapter 13). But fortunately, there are exceptions to this rule. A case in point is linear regression, which assumes a normal distribution of the error term but works well with any distribution if the model is otherwise specified correctly, as previously discussed. With exponential models, the Poisson distribution has been proven to produce consistent estimates regardless of the actual distribution and can be even used for noncount data (Cameron & Trivedi, 1998, Chapter 3; Gourieroux et al., 1984; Silva & Tenreyro, 2006; Wooldridge, 2010, Chapters 18.2–18.3, 2013, Chapter 17.3). If the model uses the logit curve, we can use the Bernoulli distribution for the same effect (Wooldridge, 2002, Section 19.4.2), and the data do not need to be binary but can also be fractions (i.e., in the [0, 1] range). In these cases, the estimates are QML, and heteroskedasticity-consistent (robust) standard errors must be used. The current practice of choosing between Poisson regression and negative binomial regression is to estimate both models and choose the best fitting one with a likelihood ratio test (e.g., Blevins et al., 2015). To understand why this rule is potentially problematic, we need to consider the four different scenarios in Table 9. Negative binomial regression is a safe choice and may also be more efficient when the distribution of the dependent variable (conditionally on the independent variables) is Poisson or Poisson with overdispersion. In other cases, negative binomial regression can be inconsistent and should not be used. The problem with using the likelihood ratio test to determine which distribution to use is that the test does not provide evidence that the negative binomial distribution is correct but only that the data are more likely to come from a negative binomial distribution than from a Poisson distribution. Thus, the use of negative binomial distribution should be limited to scenarios where there is a strong theoretical reason to believe that that distribution is indeed correct, and the Poisson distribution should be used instead as a default alternative due to its more general consistency. Although none of the articles in our review justified the distribution based on proven consistency, other articles have done so (e.g., Carnahan & Somaya, 2013, p. 1587; Wu, 2011, p. 936). To summarize, when using a GLM, a researcher needs to choose a link function that describes how the independent variables are related to the expected value of the dependent variable and a (conditional) distribution for the dependent variable around the expected value. The first decision should be based on theory or, if theory does not provide enough guidance, on an empirically identified relationship. The choice of which distribution to use in the model affects the robustness and efficiency of the analysis. In models that use a log link, Poisson distribution has been proven to produce consistent estimates, and the same has been proven for the Bernoulli distribution in the case of a logit link. Given that consistency is the most important feature of Table 9. Consistency and Efficiency of Poisson Regression and Negative Binomial Regression in Four Different Scenarios of Conditional Distribution. Conditional distribution of the dependent variable Poisson regression Negative binomial regression Conditional distribution is exactly Poisson. Consistent, efficient Consistent, inefficient Conditional distribution is Poisson-like with overdispersion. Consistent, inefficient, SEs may be inconsistent Consistent, efficient Conditional distribution is Poisson-like with underdispersion. Consistent, inefficient, SEs may be inconsistent Inconsistent Conditional distribution is not Poisson-like Consistent, SEs inconsistent Inconsistent Note: For proofs, see Wooldridge (2002, Sections 19.2.2, 19.3.1). Conditional distribution refers to the conditional distribution of the dependent variable for a specific set of independent variables, not the unconditional distribution. 22 Organizational Research Methods XX(X) estimation results, these two distributions should be preferred over negative binomial (log link) or gamma (logit) distributions unless there are strong theoretical reasons to believe that the chosen distribution holds for the data. A likelihood ratio test between two models does not provide this information. See Villadsen and Wulff (2020) for an excellent discussion on the distribution choice. Guideline 6: Do Not Default to Power Transformation When Modeling U-Shape and Other Curvilinear Effects; Consider Different Alternatives Instead Adding squared independent variables to a model is currently the dominant practice for modeling nonlinear relationships in organizational research; out of the 18 (17%) articles that stated some kind of nonlinear hypothesis, just one did not use a squared independent variable (Bamberger et al., 2018). Out of the 19 (18%) studies that did use a power transformation, just two did not state a nonlinear hypothesis (DeOrtentiis et al., 2018; Hou et al., 2017), but even these articles used power terms to model nonlinearity. 11 Thus, there is a strong convention that nonlinear effects should be tested by adding power terms to the model and that power terms should be used only for that purpose. The almost exclusive use of power terms is problematic because it can lead to incorrect inference of a U-shape effect. Power terms assume a specific form of nonlinearity (parabola opening up or down) and can lead to incorrect inference of a U-shape effect if this form does not fit the data well. For example, fitting a parabola to a data set where y¼log(x) would erroneously indicate the presence of a U-shape (first up, then down, or the other way) effect. An opposite incorrect conclusion of an increasing trend could be made if yfirst increases gradually but then decreases steeply (Simonsohn, 2018, Figures 2, 3). We show this effect in the first panel of Figure 8, which shows a binned scatterplot and second-order polynomial curve produced by the binsreg command and an exponential curve fitted with Poisson QML regression for reference. Because the inflection point of the U-shape curve falls within the range of the data (Haans et al., 2016), we would conclude that education has a negative effect on income for about the first 8 years of schooling, which does not sound realistic. Figure 8. Binned scatterplot matrix, spline regression, and fractional polynomial regression for discovering functional form between income and education. Note: The hollow circles indicate the data, and the solid red circles are the binned scatterplot. Ro ¨nkko ¨et al. 23 Although the exponential model is clearly a better alternative for modeling the nonlinear effect of education on income, it is not a universal solution; the exponential form is a specific form, which may not fit all data well and cannot be used to test U-shape effects the same way that power terms can be used. Regression splines provide an alternative approach for modeling nonlinear effects. Here, the idea is to fit a linear model that has one or more knots, where the direction of the regression line changes. An in-depth explanation of spline regression is beyond the scope of this article, but we refer the reader to Edwards and Parry (2018) for details. Of the reviewed articles, Kim and Rhee (2017) and Rawley et al. (2018) used splines with fixed knots to capture nonlinearities in the data. Simonsohn (2018) proposed a variant of this technique and an accompanying test. This technique was used by Shin and Grant (2019) in a recent publication. Another alternative is the use of fractional polynomials (Nikolaeva et al., 2015; Royston & Sauerbrei, 2008; Sauerbrei et al., 2006), but this strategy was not applied in any of the reviewed articles. The second plot of Figure 8 shows a spline regression with two knots estimated from the data. We chose to use two knots because we assumed that the effects of education probably increase after primary education and further increase at the university level. However, this assumption was not supported empirically because there is just one clear change in direction in the plot. This suggests that the effect of education is steady for primary and secondary education, after which the effect increases steeply. The spline model also fits the higher education occupations much better than either the exponential or polynomial curve. To summarize, the currently dominant practice of estimating nonlinear effects in organizational research is to add a second power of the independent variable to the model. Although this approach is useful in detecting whether nonlinearity exists, it may not be ideal for detecting what kind of nonlinearity is present in the data and can lead to incorrect interpretations. Instead of the routine application of power terms, we suggest that researchers also consider other transformations, such as the log transformation of the dependent variable or a GLM or by using regression splines, which provide a flexible alternative for modeling nonlinear effects. Guidelines for Interpreting Models With Transformations Our review indicated that nonlinear models were commonly interpreted incompletely or even incorrectly. This is a major problem because the form and strength of the relationship between the focal variables should be of prime theoretical interest (Edwards & Berry, 2010). Therefore, we will now present two guidelines for improving the interpretation and reporting practices. Guideline 7: Interpret Nonlinear Effects by Plotting, Including Confidence Intervals and the Data in the Plot The interpretation of a linear model is straightforward: The regression coefficient gives the expected increase in the dependent variable as an independent variable increases by one unit, and this effect is constant through the range of the independent variable. The case of nonlinear models is more complicated because changing an independent variable by one unit produces a different absolute change in the dependent variable depending on the current value of the dependent variable, the independent variable, or both. Consequently, several different interpretation techniques were applied in the reviewed articles, as summarized in Table 10. The most common strategy was to simply look at the pvalues and the sign of the coefficient to infer the direction of the effect, ignoring both the strength and form of the relationship. The second most common approach was to use prediction plots, but this is mostly explained by the convention of doing so with models containing interactions or powers as independent variables. Only one article that did not use powers or interactions plotted the effects (Eggers & Kaul, 2018). The remaining articles used direct interpretations, 24 Organizational Research Methods XX(X) shows why plotting the data is helpful in choosing a model: The linear model clearly extrapolates the data and underpredicts the incomes of higher education occupations, particularly those with more women, and hence does not fit as well as the nonlinear model. Why is an interaction effect not required to infer moderation? The reason is that in the exponential model, the effects are multiplicative instead of additive, as noted before. The lack of a significant interaction term thus does not mean a moderation effect would not exist in absolute terms (Russell & Dean, 2000); it is simply presented in an alternative form, as Figure 11 shows. Similarly, if the appropriate model for the data is an exponential model, then fitting a linear model can produce significant interactions that did not exist in the exponential model. This is a form of confounding interactions with nonlinearity (see also Cortina, 1993). The same logic also applies to logistic models, where a moderation effect can exist even if there is no interaction in the model, as shown in McCann and Folta (2011, Figure 2). In some instances, the logit and linear models can even produce interaction coefficients of different signs (Ganzach et al., 2000). Plotting the effects can also resolve such apparent contradictions. Indeed, 51 (66%) of the reviewed articles that tested moderation hypotheses did so by plotting, although in eight cases, this was done incorrectly by visualizing a nonlinear model as a set of straight lines instead of using the curves that the model implied. Should we interpret Figure 11 as supporting a moderation effect? Ideally, moderation hypotheses should be stated in a way that leaves no room for ambiguity (see also Gardner et al., 2017). If we assume that the effect of education on income is relative to the current income level, we could state an alternative, more precise moderation hypothesis: Hypothesis 2a: The positive relative effect of education on income is moderated by percentage of women; the effect is stronger for occupations with fewer women. The results shown in Table 7 and Figure 11 would not support Hypothesis 2a. Even though the absolute increase of income is greater for men-dominated occupations, the results do not demonstrate that education would increase income differently between men and women. Both experience the same relative effect, and the differences in the absolute effect are simply due to men earning more regardless of the education level. To summarize, nonlinear models neither hide nor produce spurious moderation results, although an incorrect interpretation can do so. Testing a moderation hypothesis requires a comprehensive interpretation, preferably through plotting, instead of just checking the significance of the interaction term in the model. A key problem in testing moderation is that the main hypothesis and the moderation hypothesis are typically imprecise, lacking the expected functional form. This creates ambiguity because it is possible that moderation exists when considering absolute effects but not when considering relative effects, as our example shows. It is also possible that the effects are both moderated but in different directions. To resolve these issues, researchers should assess their models more comprehensively and interpret results by looking at the shape of the curves in a plot and comparing how well they match the data. Conclusions Our review of the methodological literature and empirical practice clearly indicate that organizational researchers should rethink their use of transformations. Decisions to use nonlinear models (Blevins et al., 2015) or transformations (Aguinis et al., 2019; Becker et al., 2019) should not be based on the distributions of individual variables but should be based on the expected functional form of the relationship between independent and dependent variables, which should in turn be based on theory. As such, transformations are not something that should be relegated to the data preparation stage (Aguinis et al., 2019; Becker et al., 2019) but should be considered as an integral Ro ¨nkko ¨et al. 31 part of the research design, on par with the choice of independent variables in the models. Against this background, the recent claims that the use of transformed variables for testing hypotheses would be problematic are simply incorrect (Aguinis et al., 2019; Becker et al., 2019; Maula & Stam, 2020). If the hypothesis does not specify a functional form, then any monotonic form should do (Edwards & Berry, 2010). Unfortunately, despite recent calls for more precise theorizing, the current theories still provide limited support for determining the functional form (Cortina, 2016; Edwards & Berry, 2010; Ferris et al., 2012). Our article highlights three key concerns. First, transformations are often applied either unnecessarily or at least with incorrect justification, such as relating to nonnormality of variables or because the dependent variable is a count. Second, transformations are not always applied where they should be. This is evident in the insufficient consideration the functional form is given when framing hypotheses and in how omitting regression diagnostics that would provide evidence of a given functional form appears to be the rule rather than the exception. Third, the nonlinear models that transformations create are often interpreted incorrectly. We hope that these guidelines and empirical examples will improve future research practice. The Supplemental Material available in the online version of the journal provides example Stata and R code and a web-based tool for users of other statistical software that should be helpful for researchers interested in adopting these techniques. Finally, we wish to point out two problems discovered during our review that are not directly related to transformations and nonlinear models but are nonetheless worth discussing. First, many articles document their analytical steps incompletely, leaving readers guessing how a statistic was calculated based on the modeling results. To reduce such ambiguity, researchers should fully disclose their analytical procedures, including publishing analysis files as supplements to the articles. Fortunately, the field seems to be moving in this direction as prominent journals are making efforts to increase transparency in the empirical research process (Aguinis et al., 2018; Antonakis, Banks, et al., 2019; Banks et al., 2019; Bergh & Oswald, 2020; Chen, 2018; Grand et al., 2018). Second, there are several misunderstandings relating to the role of transformations in statistical analysis in the organizational sciences. These myths may have been caused by the use of transformations without providing supporting citations (Becker et al., 2019). However, the loose citation practice also extends to methodological guidelines that often cite books such as Cameron and Trivedi (1998), Long (1997), Cohen et al. (2003), or Greene (2003) without providing a page, chapter, or section number. The loose citation practice is problematic because it makes it difficult for readers to verify whether the citation supports the presented practice given that some of these books are difficult to navigate unless one is an expert in quantitative methods. Loose citation practice also easily translates into loose argumentation. For example, the claim that regression analysis requires normally distributed data or that the error term must be normally distributed are strictly incorrect and misleading, respectively. The normality assumption pertains to the error term and is only relevant in very small samples. We believe that more rigorous citation and use of econometrics books focusing on what these books prove or demonstrate would contribute significantly to avoiding these misconceptions. Acknowledgments This document was prepared using the StatTag software (Welty et al., 2016). The software is cited as per the license conditions. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. 32 Organizational Research Methods XX(X) Funding The author(s) received no financial support for the research, authorship, and/or publication of this article: This research was supported in part by a Grant from the Academy of Finland (Grant 311309). ORCID iD Mikko Ro¨nkko¨https://orcid.org/0000-0001-7988-7609 Henni Tenhunen https://orcid.org/0000-0003-2151-5161 Supplemental Material Supplemental material for this article is available online. Notes 1. The percentage is calculated by first calculating the frequencies for each journal volume and then taking a mean of these frequencies to adjust for differences in the number of articles in the journals. This number is substantially larger than the 38%found by Becker et al. (2019) because they only reviewed manual transformations and not generalized linear models (GLMs). In our sample, a manual transformation was applied in 31%of cases. The difference between these two figures is probably attributable to differences in how the numbers were calculated. When screening the articles, we excluded all latent variable models, survival models, and articles without hypotheses from the total count. 2. Readers that are entirely new to GLMs or need a refresher can also watch the set of video lectures that we provide as a supplement for the article at https://tinyurl.com/nonlinearmodels. 3. The percentages do not always add up to 100%because one article could present several justifications. 4. Justifying transformations based on a nonlinear relationship between the variables was less common except for studies that applied power transformation to test a U-shape effect. Of the 42%of the articles applying a manual transformation because of expected nonlinear form, just one article provided a theoretical justification for a transformation other than power transformation (Cennamo, 2018, p. 3048); in this case, the log transformation was justified due to decreasing marginal value of game titles on a video game platform. Additionally, Zhang et al. (2017) reported that they “used the log transformation of [popularity] to adjust for skewness and to capture the nonlinear impact of increasing industry popularity” (p. 1373). Although Zhang et al. noted that they modeled a nonlinear relationship, we did not code this as a theoretical justification because neither the form of nonlinearity nor why such form was expected was reported in the article. In the articles that applied GLMs, just 9%did so based on nonlinear functional form expected based on theory and 4%based on empirically identified nonlinearity. 5. The efficiency of an ordinary least squares (OLS) estimator is sometimes captured in the acronym BLUE: Best Linear Unbiased Estimator (Wooldridge, 2013, Section 3.5). This means that among all possible unbiased linear estimators, OLS has the smallest variance. In practice, this means that OLS produces the most precise estimates. If the form of the heteroskedasticity is known, one can obtain more precise results by using weighted least squares (WLS). However, this is rarely the case in applied research, and using a heteroskedastic form estimated from the same data set leads to biased regression estimates (Wooldridge, 2013, Section 8.4). Perhaps for this reason and the fact that the efficiency difference between WLS and OLS is sometimes small, WLS estimation has seldom been applied in organizational research. 6. For example, the log transformation would reduce the problems caused by the “megaphone opening right” (Cohen et al., 2003, Figure 4.4.5B) form of heteroskedasticity but would not do much for the “barrel shape” (Cohen et al., 2003, Figure 4.4.5C) or other forms of heteroskedasticity. 7. For example, Ouyang et al. (2018) hypothesized the following: “There is a curvilinear relationship between matched favor giving and favor givers’ social status, such that the overall positive relationship is attenuated at higher levels of matched favor giving” (p. 617). 8. Eggers and Kaul (2018) reported in a footnote that a negative binomial model, which uses log link, produced similar results to their main analysis with a log-transformed dependent variable—as the Ro ¨nkko ¨et al. 33 technique should. However, they did not explain why log transformation instead of a GLM with a log link was used in the study. Leahey et al. (2017) applied log transformation and linear model to two of their dependent variables and a GLM with log link to a third but did not report whether these were considered as alternatives or why the two different modeling approaches were used. 9. This follows from the calculation rules of exponentials b1þb3x2 ðÞe16=b1þb3x2 ðÞb1e6¼e16=e6¼ e166¼e10 ¼22;026:47. 10. Villadsen and Wulff (2020) provide an illustration how the shape of the effect changes with different values of b 2 in their Figure 2, Panel B. 11. DeOrtentiis et al. (2018) used a power term to test for nonlinear effects of time and compared that to a linear model. Hou et al. (2017) investigated the moderating effect of CEO tenure on the effect of pay on performance on firm performance and added a square of CEO tenure to the model because previous research has suggested a curvilinear relationship. 12. For example, if the coefficient is 0.3, the effect of a þ1 unit change is exp(.3) 1 ¼1.35 or þ35%, and the effect of –1 unit change is exp(.3) 1 ¼0.74 or –26%. 13. This bias is different from the omitted variable bias in linear models, which only occurs if the omitted variables are correlated with the included independent variables, leading to endogeneity (Antonakis, Bendahan, et al., 2014; Wooldridge, 2013, pp. 88–92). However, it also occurs when the omitted variables are uncorrelated with the included ones (Breen et al., 2018). 14. For logit and probit models, this approach eliminates the bias due to omitted variables (Cramer, 2007; Wooldridge, 2002, Section 15.7.1). 15. We note that using a line may be appropriate in specific circumstances. For example, if the main independent variable is binary (e.g., an experimental manipulation), the estimated predictions are two discrete values, there is no functional form involved, and linking the points with a line is the simplest possible approach (e.g., Tu et al., 2018, Note 3). Busenbark et al. (2017, p. 2502) explicitly noted that the actual model-based predictions would be more useful for interpretation than their figure showing linear predictions. 16. Although most articles did not report which predictions were calculated, we inferred which approach was used by checking the software defaults. For example, Stata’s margins command calculates adjusted predictions by default. 17. Soda et al. (2018) presented a scatterplot of the data in one of their plots, and Hornstein and Zhao (2018) applied a binned scatterplot. 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The conditional importance of prior ties: A group-level analysis of venture capital syndication. Academy of Management Journal,60(4), 1360-1386. https://doi.org/ 10.5465/amj.2013.1088 Ro ¨nkko ¨et al. 39 Author Biographies Mikko Ro ¨nkko ¨is an associate professor of entrepreneurship at Jyva¨skyla¨ University School of Business and Economics (JSBE) and a docent at Aalto University School of Science in the field of statistical methods in management research, where he also completed his PhD. His current research interests are growth entrepreneurship and quantitative research methods in management research. He is a department editor in Journal of Operations Management handling methodological articles and on the editorial boards of Organizational Research Methods and Entrepreneurship Theory and Practice. He runs a research methods focused YouTube channel at https://www.youtube.com/mronkko. In the past Mikko has also been an entrepreneur. Eero Aalto is a doctoral candidate in industrial engineering and management at Aalto University. His primary research interests are in the areas of platform economy, digital strategy, industry change, innovation policy, and research methods. Eero’s work has been published in Journal of World Business. He has also written several research reports for the Research Institute of the Finnish Economy, and has a book chapter in Collaborative Value Co-creation in the Platform Economy. Henni Tenhunen is a doctoral candidate in industrial engineering and management at Aalto University. Her research interests are healthcare operations management, mixed methods research, patient experience, and evaluation of digital health technologies. Henni’s work has been published in healthcare management journals and health technology conferences. She is also project manager of a service engineering research project, which aims to create a virtual care management system. Miguel I. Aguirre-Urreta is an associate professor in the Department of Information Systems and Business Analytics, College of Business, at Florida International University. Before joining FIU, he was on the faculty at Texas Tech University and DePaul University. Miguel received his PhD in Information Systems from the University of Kansas, and his MBA in Information Systems and Finance from the Kelley School of Business at Indiana University. His undergraduate degree is in public accounting from the Universidad de Buenos Aires, in Argentina. Before entering academia, he was an auditor with Arthur Andersen and an external reporting analyst with Movicom Bellsouth. Miguel is interested in quantitative research methods, computer self-efficacy, human -computer interaction, technology acceptance and diffusion, and formal modeling and theory development. He is also a past chair of the AIS Special Interest Group on Human-Computer Interaction. His research has appeared or is forthcoming at MIS Quarterly, Information Systems Research, Psychological Methods, Communications of the AIS, Research Synthesis Methods, IEEE Software, Measurement, and The DATA BASE for Advances in Information Systems, among others. 40 Organizational Research Methods XX(X)