A New Approach to Logistic Regression: Using the von Bertalanffy Equation as a Link Function

Abstract

Binary logistic regression relies on the symmetric logit link, whose fixed inflection point cannot adapt to asymmetric response mechanisms and so may yield miscalibrated probabilities when the true response curve is asymmetric-a problem that can be particularly consequential in rare event or imbalanced applications. This study proposes a flexible, asymmetric link derived from the von Bertalanffy growth equation and tests a single, calibration-oriented hypothesis: estimating an asymmetric link improves the calibration of the predicted probabilities under asymmetric response mechanisms while leaving discrimination unchanged. The single shape parameter nu (the reparameterized von Bertalanffy shape, nu = m - 1; the growth curve baseline parameter is absorbed into the intercept and is not separately identifiable) is estimated jointly with the regression coefficients by L2-penalized maximum likelihood, giving a coherent generalized linear model rather than a post hoc re-thresholding of logistic scores; the logistic link is nested exactly at nu = 1 (m = 2), so the model reduces to ordinary logistic regression whenever the logit is adequate. The link was benchmarked on identical out-of-sample partitions against four competitors-logistic regression at the 0.5 cut-off, threshold-optimized logistic regression, ridge-penalized logistic regression at the same penalty (which separates the link from the regularization it requires), and Stukel's generalized logistic model-across a correctly specified logistic mechanism and two asymmetric mechanisms (a complementary log-log mechanism and a von Bertalanffy-type best-case reference), three class imbalance ratios (0.10, 0.25, and 0.50) and five sample sizes (100-10,000), for 45,000 iterations in total. Paired differences were summarized by the Hodges-Lehmann median difference, its 95% confidence interval, and the rank-biserial correlation, with Benjamini-Hochberg control of multiplicity. The results supported the hypothesis. At moderate-to-large sample sizes, and increasingly as prevalence approached 0.50, the proposed link achieved lower Brier score and Log-Loss values than the three logistic competitors under both asymmetric mechanisms-for example, a median Log-Loss reduction against the matched-penalty ridge model of about 0.004 at balanced prevalence and the largest sample (95% confidence interval excluding zero; rank-biserial approximate to -1)-an advantage small in absolute size but, at the larger sample sizes, consistent across essentially every replication, and increasing with sample size and prevalence. Discrimination showed no practically important differences at moderate-to-large sample sizes: the area under the ROC curve and overall accuracy were essentially the same across the five methods, so the flexible link improves the quality of the probabilities without degrading class separation. Under the correctly specified logistic mechanism, the method showed no material deterioration, which is consistent with its exact nesting of the logistic model. The contribution is a growth curve-motivated asymmetric link that improves probability calibration under response asymmetry while preserving discrimination and recovering the logistic model when it is adequate.