Modeling biomarker ratios with gamma distributed components

Abstract

We propose a regression model termed "extended GB2 model", which is designed to analyze ratios of biomarkers in epidemiological and medical research. Typical examples of biomarker ratios are given by the LDL/HDL cholesterol ratio in cardiovascular research and the amyloid-β 42/40 ratio in dementia research. Unlike regression modeling with a log-transformed response, which is often used to describe ratio outcomes in observational studies, the extended GB2 model directly links the expectation of the untransformed biomarker ratio to a set of covariates. This strategy allows for a simple interpretation of the predictor-response relationships in terms of multiplicative increases/decreases of the expected outcome, similar to Poisson and Cox regression. In the theoretical part of the paper, we derive the log-likelihood of the proposed model, analyze its properties, and provide details on confidence intervals and hypothesis testing. We will also present the results of a simulation study demonstrating the robustness of the proposed modeling approach against model misspecification. The usefulness of the method is demonstrated by two applications on the aforementioned LDL/HDL cholesterol and amyloid-β 42/40 ratios. For this, we analyze data from a cohort study on kidney disease and from a large observational database on neurodegenerative diseases.

Parametric Quantile Beta Regression Model

“…Furthermore, the case where and are correlated gamma random variables (Weinhold et al . 2016) could be proposed as well, that is, a specific parametric quantile regression model to describe data of the type (and consequently for , see Berger et al ., 2019; Berger & Schmid, 2020) at different levels, when and are correlated. Work on these problems is currently in progress.…”

Flexible modeling of ratio outcomes in clinical and epidemiological research

“…Based on Kibble’s bivariate gamma distribution and by using results on gamma distributed variables derived by Nadarajah and Kotz 32 and Weinhold et al., 33 it has recently been shown by Berger et al. 21 that the ratio R follows an extended generalized beta distribution of the second kind (eGB2) with p.d.f. where γ:=λV/λU=E[U]/E[V] denotes the ratio of the two rate parameters.…”

“…As the mean ratio μ R can be written as the product of the ratio of means γ and the factor C(α,ρ) (that only depends on the two additional parameters α and ρ ), Berger et al. 21 proposed to set up a regression model by relating the ratio of means γ to the explanatory variables X1,…,Xp through the model equation where βγ:=(β0,γ,β1,γ,…,βp,γ)normal⊤ is a set of real-valued coefficients. Similar to Poisson and Cox regression models, the logarithmic transformation in equation (15) ensures that there are no restrictions on the coefficients.…”

“…20 For modeling the ratio of two correlated gamma distributed outcomes, Berger et al. 21 recently proposed the extended GB2 model . This model is characterized by three parameters, where one of the parameters is directly interpretable in terms of the Pearson correlation coefficient between the two outcomes.…”

A parametric quantile beta regression for modeling case fatality rates of COVID-19

“…. , X k ), providing a full parametric elegant quantile regression model; see Berger et al (2019) for a similar approach related to the ratio of two positively correlated biomarkers. We also propose a simple interpretation of the predictor-response relationship in terms of percentage increases/decreases of the quantile.…”