Re-expressing coefficients from regression models for inclusion in a meta-analysis

Abstract

Meta-analysis poses a challenge when original study results have been expressed in a non-uniform manner, such as when regression results from some original studies were based on a log-transformed key independent variable while in others no transformation was used. Methods of re-expressing regression coefficients to generate comparable results across studies regardless of data transformation have recently been developed. We examined the relative bias of three re-expression methods using simulations and 15 real data examples where the independent variable had a skewed distribution. Regression coefficients from models with log-transformed independent variables were re-expressed as though they were based on an untransformed variable. We compared the re-expressed coefficients to those from a model fit to the untransformed variable. In the simulated and real data, all three re-expression methods usually gave biased results, and the skewness of the independent variable predicted the amount of bias. How best to synthesize the results of the log-transformed and absolute exposure evidence streams remains an open question and may depend on the scientific discipline, scale of the outcome, and other considerations.

Keywords: Conversion; Meta-analysis as topic; Regression analysis; Transformation.

Fig. 1

Plot of simulated values of y as a function of x from y = ln(x)(curve), along with slopes obtained by four methods (diagonal lines). The gold solid line represents a slope (β_Estimand) fitted with the model y = α + βx. The three dashed lines are estimates of β_Estimand obtained by the re-expression methods described in the text. Vertical lines indicate the first and third quartiles of the x-values. The intercepts of the diagonal lines have been adjusted to emphasize the similarity of the slopes in the interquartile range

Fig. 2

Plots of relative bias as a function of skewness (σ) in the exposure x, by type of estimator. Individual points represent the average result (n_sim = 2000) for each simulation scenario. A total of 890 of the possible 1,920,000 observations (960 scenarios × 2000 simulations) were not used in the calculation of the average results because β_estimand was < 0.0001 (essentially zero). Lines represent quadratic fits to the data for a specified prediction equation and set of values of independent variables (see text). Note that data have been artificially spread along the x-axis for visualization purposes, all actual x-values are the closest black vertical line (0.25, 0.45, 0.65, or 0.85). Figures A-C show points for 768 simulation scenarios (β_DGM > 0, see Figure S1 for plots including β_DGM < 0); Figure D shows points for a subset of scenarios (n = 32) chosen because they demonstrate differences among the estimator properties. A Rodriguez-Barranco estimator, B Dzierlenga estimator, and C Alternative estimator. D Shows all 3 estimators in the same plot with a subset of the data of the simulation data where β_DGM = 0.5, log base = 2 or 10, and median value = 0.5 or 8