Models with Transformed Variables

To the Editor: Variables in regression models are frequently transformed to achieve homogeneity of variance, normality of errors, linearization of associations, or a more homogeneous distribution of predictors. Here, we show how to report and interpret effects in the original scale of the variables, in the case of linear, logistic, and Poisson regression models with logarithmic or power transformations. Strategies for identifying proper transformations can be found elsewhere.1–3 Consider the multiple linear regression model with no interaction terms where and can be logarithmic or power transformations of Y and/or X, respectively; X2, …, Xp are other explanatory variables; the unknown parameters β0 and β, β2, …, βp are the intercept of the model and the regression coefficients, respectively; and is the normally distributed error term. We assume that is normally distributed and that f() is bijective. In general, to compute summary measures of Y for a given value of X and the effect of X on Y, we set the value of K in such a way that the results can be interpreted as adjusted measures and adjusted effects (eAppendix A, https://links.lww.com/EDE/A879).4 Under these assumptions, applying the inverse transformation, , to equation (1) leads to a valid equation for the median of Y in the original scale.5 Under log transformation in the response variable, results can alternatively be interpreted in terms of the geometric mean (see eAppendix B, https://links.lww.com/EDE/A879, for formulas on adjusted measures under log transformations, and eAppendix C, https://links.lww.com/EDE/A879, for how to deal with the basis of the logarithm and other considerations). Regarding the effect of X on Y under log transformations in a linear regression model, it can be summarized in the original scale by a single number by appropriately using additive or relative changes, depending on which variables (X, Y, or both) have been transformed. Thus, for a log-transformed Y and an untransformed X, an additive change in X results in a relative change in the median or geometric mean of Y. For an untransformed Y and a log-transformed X, a relative change in X results in an additive change in the mean of Y. For log transformation in both Y and X, a relative change in X results in a percent change in the median or geometric mean of Y (eFigure 1 and eAppendix D, https://links.lww.com/EDE/A879). Under the logistic and the Poisson regression models (for a binary and a count response Y, respectively) with a log-transformed X, the effect of X on Y based on a given model can still be summarized by a single number which is independent of the values of the remaining explanatory variables included in the model, if any. Specifically, a relative change in X results in simple expressions for the odds ratio and for the percent change in the mean of the response, respectively. Formulas for all the effects described above are shown in the Table. See eAppendix F (https://links.lww.com/EDE/A879) for direct interpretations of the regression coefficient β, under certain conditions.TABLE: Interpretation and Size of the Adjusted Effect of X on Y Under Linear, Logistic, and Poisson Models with Log-transformed VariablesUnlike in the case of log transformations, under power transformations effects cannot be summarized by a single number by using appropriate additive or relative changes in X or Y (eAppendix G, https://links.lww.com/EDE/A879). In general, the additive and the relative change in the median of Y (equivalently, the mean, if Y is not transformed), associated with a change from to , depends on u1, u2, and the remaining explanatory variables in the model. Thus, the investigator is free to choose additive (using ) or relative (using or ) changes for X, along the range of X,6 and to interpret the resulting additive and relative changes in Y. In practice, we recommend creating tables with the 4 possible combinations, and then picking the one that results in the easiest interpretation (eAppendix G, https://links.lww.com/EDE/A879). See eAppendix H (https://links.lww.com/EDE/A879) for detailed illustrative examples on all transformations described here. We provide the package tlm for R (R Foundation for Statistical Computing, Vienna, Austria) that facilitates the computation, presentation, and interpretation of the effects described here (eAppendix J, https://links.lww.com/EDE/A879). The package and a user’s guide are available at http://cran.r-project.org/web/packages/tlm/. We also show how to use existing commands in the Stata software (Stata Corporation, College Station, TX) to obtain the same results (eAppendix J, https://links.lww.com/EDE/A879). ACKNOWLEDGMENTS We thank Natàlia Adell, Mercedes Medina-Ramón, and Margarita Triguero-Mas for their suggestions after a critical reading of the first draft. Jose Barrera-Gómez Xavier Basagaña Centre for Research in Environmental Epidemiology (CREAL) Barcelona, Spain Universitat Pompeu Fabra (UPF) Barcelona, Spain CIBER Epidemiología y Salud Pública (CIBERESP) Centro de Investigación Biomédica en Red Instituto de Salud Carlos III Madrid, Spain [email protected]