Parametric models for biomarkers based on flexible size distributions

Abstract Recent advances in social science surveys include collection of biological samples. Although biomarkers offer a large potential for social science and economic research, they impose a number of statistical challenges, often being distributed asymmetrically with heavy tails. Using data from the UK Household Panel Survey, we illustrate the comparative performance of a set of flexible parametric distributions, which allow for a wide range of skewness and kurtosis: the four‐parameter generalized beta of the second kind (GB2), the three‐parameter generalized gamma, and their three‐, two‐, or one‐parameter nested and limiting cases. Commonly used blood‐based biomarkers for inflammation, diabetes, cholesterol, and stress‐related hormones are modelled. Although some of the three‐parameter distributions nested within the GB2 outperform the latter for most of the biomarkers considered, the GB2 can be used as a guide for choosing among competing parametric distributions for biomarkers. Going “beyond the mean” to estimate tail probabilities, we find that GB2 performs fairly well with some disparities at the very high levels of glycated hemoglobin and fibrinogen. Commonly used linear models are shown to perform worse than almost all the flexible distributions.

A growing literature analyses the effect of socioeconomic position on the conditional mean of biomarkers (e.g., Davillas et al., 2017, Jürges et al., 2013. However, biomarkers create several statistical modeling challenges as they often have skewed distributions with heavy tails (Jones, 2017). Furthermore, errors are likely to be heteroskedastic and responses to covariates may be nonlinear. Existing studies have estimated linear regression models using ordinary least squares (OLS) on raw or log-transformed biomarkers (Jürges et al., 2013) and alternative inherently nonlinear specifications, such as the generalized linear models (Davillas et al., 2017). Although OLS on log rather than on levels might improve performance by reducing skewness, re-transformation to the raw scale-as health policymakers require-is highly challenging, requiring knowledge of the degree and form of heteroscedasticity (Jones, Lomas, & Rice, 2014). Although the generalized linear model family deals with heteroskedasticity, it fails to explicitly account for skewness and kurtosis, imposing potential bias and efficiency losses (Jones et al., 2014).
Our paper contributes to the literature on modeling biomarkers by comparing the performance of a set of more flexible parametric distributions, the generalized beta of the second kind (GB2), the generalized gamma (GG), and their nine nested and limiting cases; we use nationally representative UK data on commonly used blood-based biomarkers for inflammation, diabetes, cholesterol, and stress-related hormones (Carrieri & Jones, 2017). The GG and GB2 allow for a wide range of skewness and kurtosis to better accommodate the biomarker data generation processes; these distributions have been proposed for fitting heavily skewed outcomes (e.g., health care costs; Jones et al., 2014), to which biomarkers share similar distributional features. For comparison purposes, linear regression models using OLS are also estimated. Given that different biomarkers exhibit different distributions, identifying GB2 as a discriminatory tool among competing distributions might be useful for health researchers. Going "beyond the mean", we also explore to what extent the GB2 and its nested cases that exerted the best goodness of fit (for each biomarker) regarding the whole distribution also perform well to predict tail probabilities.

| METHODS
The three-parameter GG distribution has been introduced as robust alternative to common estimation techniques for asymmetric data (Manning, Basu, & Mullahy, 2005). More recently, Jones, Lomas, and Rice (2014) have suggested adding further flexibility based on the four-parameter GB2 distribution. GB2 allows for a wider range of skewness and kurtosis, choosing among its several special or nested cases, whereas GB2's extra flexibility may also enhance performance (Jones et al., 2014).
The GG distribution has a density function and conditional expectation that take the form: where γ = |κ| −2 , z = sign (κ){ln(y) − μ}, u = γ exp (|κ|z), μ = x ′ β, and Γ(.) is the gamma function. Parameters κ and σ are the shape parameters (Manning et al., 2005). The GG nests the gamma (κ = σ), Weibull (κ = 1), exponential (κ = 1, σ = 1), and lognormal (κ = 0) distributions. The four-parameter GB2 distribution adds further flexibility and has a probability density function and conditional mean of and where b = exp (x ′ β) and B(.) and Γ(.) are the beta and gamma functions (Jones et al., 2014). Parameter a influences kurtosis and p and q the skewness of the distribution. We also estimate the nested and limiting cases of GB2: the three-parameter beta of the second kind (B2) [a = 1], Singh-Maddala (SM) [p = 1], and Dagum [q = 1]; the two-parameter Fisk [p = q = 1], and Lomax [p = a = 1]. GG itself is also a limiting case of the GB2, where b = q 1/a β and q → ∞ (Jones et al., 2014). We also estimate linear regression models using OLS. The restrictions imposed by each of the special and limiting cases within the GG and GB2 are evaluated using Wald and likelihood ratio (LR) tests. To assess the comparative performance of beta-with gamma-family distributions (being limited cases and not a linear restriction of a parameter), we compare Akaike (AIC) and Bayesian (BIC) information criteria across all models (Jones et al., 2014).

| DATA
The UK Household Panel Study (UKHLS) is a large, nationally representative UK study. At UKHLS Wave 2, participants from its predecessor, the British Household Panel Survey, were also incorporated. Non-fasted blood samples were collected, after the UKHLS Wave 2 interview for the original UKHLS respondents and, at Wave 3, for the British Household Panel Survey sample. Pooling biomarker data from UKHLS Waves 2 and 3 (2010-2013) resulted in a potential sample of 13,107 respondents.
Four biomarkers are used. Fibrinogen is an inflammatory biomarker, with higher values linked to cardiovascular morbidity and all-cause mortality risks (Davillas et al., 2017). Glycated hemoglobin (HbA1c) is a diagnostic biomarker for diabetes. The ratio of total cholesterol to high-density lipoprotein cholesterol is used as a marker for fatty substances in the blood. Dehydroepiandrosterone sulfate (DHEAS) is a steroid hormone and one of the mechanisms through which psychosocial stressors might affect health (Vie, Hufthammer, Holmen, Meland, & Breidablik, 2014). Given our focus on the comparative performance of parametric distributions regarding goodness of fit, rather than explore potential effects from covariates, a parsimonious set of covariates is used: polynomials of age (cubic or quartic depending on the biomarker used), gender, and their interactions to allow for flexible gender effects ( Figure A1). 1 4 | RESULTS Figure 1 presents the distribution of biomarkers (descriptive statistics in Table A1). Fibrinogen has a symmetric distribution but with fat tails (Figure 1). HbA1c is much more skewed (skewness statistic of 4.2 compared with zero for normal data) with long right-hand tails and excess kurtosis (31.15 vs. 3 for normal data; Table A1). The cholesterol ratio and DHEAS also exhibit long right-hand tails and high kurtosis. Table 1 contains restriction tests for the nested and limiting cases within the GG and GB2. Across all biomarkers, we find no evidence in support of any of the special cases within the GG distribution. For fibrinogen, we are unable to reject the null hypothesis of the restriction being valid for the SM model. Our results for HbA1c do not support any of the nested distributions. For the cholesterol ratio, both the LR and Wald tests favor the B2 distribution. Although the Wald   test also fails to reject the null hypothesis for SM, this is not confirmed by the LR test; this disparity reflects the wide confidence intervals for GB2's p parameter (which include both 1, satisfying the SM restriction, and 0; Table A2).
Our results for DHEAS favor the SM distribution. Table 2 shows that AIC and BIC results are in accordance with the tests of Table 1. For all biomarkers, linear regressions estimated by OLS perform worse than each of the four-and three-parameter and most of the more parsimonious distributions. For fibrinogen, GB2 and SM perform best according to AIC and BIC criteria, with the latter showing the best performance. GB2 outperforms all the competing distributions regarding HbA1c. Although the B2 and SM distributions exhibit the best performance for the cholesterol ratio and DHEAS, GB2 is ranked the second best. Figure 2 presents the conditional tail probabilities (at k equal to 10th, 25th, 50th, 75th, and 90th quantile) and spike plots of the actual-fitted difference (bias) for the GB2 distribution, and its nested cases exerted the best performance for each biomarker (Table 2). Specifically, 20-quantiles of the fitted values from these models are used to split the sample to calculate within-quantiles means of actual [P(y > k)] and predicted [P(y > k| X)] probabilities.
There are limited differences in the predictive ability of the more parsimonious distributions compared with GB2, confirming previous evidence that a flexible distribution is not a substitute for finding the correct distribution (Jones et al., 2014). GB2 performs reasonably well at predicting tail probabilities, although there are some disparities at the very high fibrinogen levels (90th quantile) and HbA1c above the pre-diabetes threshold (HbA1c ≥ 42).

| CONCLUSION
We illustrate the comparative performance of a set of more flexible parametric distributions: the GB2, GG, and their nested and limiting cases for a set of biomarkers. Although some of the three-parameter distributions nested within the GB2 (mainly the B2 and SM) outperform the latter in most of the biomarkers considered, GB2 can be used as a guide for choosing among competing distributions; a potentially useful message for applied researchers given that different biomarkers follow different distributions. The linear models estimated by OLS are dominated by almost all the competitive models. GB2 performs well at predicting biomarkers' tail probabilities, although with some disparities at the very high levels of fibrinogen and HbA1c.