Linear Maximum Likelihood Regression Analysis for Untransformed Log-Normally Distributed Data

Abstract

Medical research data are often skewed and heteroscedastic. It has therefore become practice to log-transform data in regression analysis, in order to stabilize the variance. Regression analysis on log-transformed data estimates the relative effect, whereas it is often the absolute effect of a predictor that is of interest. We propose a maximum likelihood (ML)-based approach to estimate a linear regression model on log-normal, heteroscedastic data. The new method was evaluated with a large simulation study. Log-normal observations were generated according to the simulation models and parameters were estimated using the new ML method, ordinary least-squares regression (LS) and weighed least-squares regression (WLS). All three methods produced unbiased estimates of parameters and expected response, and ML and WLS yielded smaller standard errors than LS. The approximate normality of the Wald statistic, used for tests of the ML estimates, in most situations produced correct type I error risk. Only ML and WLS produced correct confidence intervals for the estimated expected value. ML had the highest power for tests regarding β<sub>1</sub>

Regression Models for Log-Normal Data: Comparing Different Methods for Quantifying the Association between Abdominal Adiposity and Biomarkers of Inflammation and Insulin Resistance

“…Thus ML LN will have a higher power and for lognormal data the probability of detecting a true explanatory variable is higher. The smaller interval lengths of ML LN corroborate the results of a previous simulation study [ 11 ].…”

“…, absolute effects) it is natural to use a method that can estimate the absolute effects in the model selection process. We also wanted the method that was expected to have a high power, and based on previous studies, [ 11 ], ML LN was expected to have higher power than e.g., WLS and LS lin .…”

“…The method GLM N , does not, however, take into account the stochastic variation due to estimating . Therefore we also used a maximum likelihood method (ML LN , see [ 11 , 12 ]) based on the likelihood function of the log-normal distribution: where . The estimates , , …, and are found using iterations, for example the Newton-Raphson iteration used here [ 13 ].…”

A mixed effects model for analyzing area under the curve of longitudinally measured biomarkers with missing data

“…In this case, a linear form of the model rather than a log‐linear form, as the one studied in this paper, would be more appropriate. A maximum likelihood (ML)‐based approach has been studied for the univariate case, 26 which may be extended to the repeated measure case and applied to draw inference on the absolute effect of the covariate on the AUC of the response variable. It will be interesting to investigate the relative performance of such a model compared with the existing linear mixed model 12 …”

Random intercept and linear mixed models including heteroscedasticity in a logarithmic scale: Correction terms and prediction in the original scale

“…Some extensions to address heteroscedasticity in linear regression models with a logarithmic scale have also been proposed, e.g. [ 2 , 16 ]. Other authors have compared the logarithmic transformation in linear models with other type of models in different applications, e.g.…”