Human variability is a very important factor considered in human health risk assessment for protecting sensitive populations from chemical exposure. Traditionally, to account for this variability, an interhuman uncertainty factor is applied to lower the exposure limit. However, using a fixed uncertainty factor rather than probabilistically accounting for human variability can hardly support probabilistic risk assessment advocated by a number of researchers; new methods are needed to probabilistically quantify human population variability. We propose a Bayesian hierarchical model to quantify variability among different populations. This approach jointly characterizes the distribution of risk at background exposure and the sensitivity of response to exposure, which are commonly represented by model parameters. We demonstrate, through both an application to real data and a simulation study, that using the proposed hierarchical structure adequately characterizes variability across different populations.

To quantify health risk from chemical exposure faced by sensitive populations, human variability is an important factor to consider in human health risk assessment. Traditionally, to account for this variability, an interhuman uncertainty factor (up to 10) is applied to the 100(1 – ^{(1–5)} As agencies such as the U.S. Environmental Protection Agency (US EPA) move toward probabilistic risk assessment,^{(6–8)} new methods are needed to probabilistically quantify human variability.

We propose a Bayesian hierarchical model to quantify human variability. This is accomplished by jointly characterizing the distribution of risk at background exposure as well as the sensitivity of response to exposure using exposure-response data obtained from various populations (i.e., cohorts). Here, we define “population” as a cohort of individuals with different characteristics (e.g., gender, age, exposure, etc.) in a particular study. As this represents a cohort of individuals, such populations typically have less variability than the “target population” of interest (e.g., the entire U.S. population). In modeling the target population, the overall heterogeneity can be seen as originating from multiple sources: first, there is heterogeneity between individuals within each population, we call this interindividual variability; next, there is heterogeneity between studies, this is called interpopulation variability; and on top of that, there is variability across different populations, which is defined as interhuman (or human) variability describing the variability in the target population.

For the model, we assume that human variability in response to chemical exposure, especially in the low exposure range, is primarily affected by two factors: (1) individuals exposed at background exposure levels have different background risks, and (2) differences in genetics (e.g., resulting in different “detoxifying” metabolic rates), life style (e.g., co-exposures such as smoking that can exacerbate an individual’s response to a given chemical exposure), and other factors cause individuals to have different levels of sensitivity to the same exposure level, which affect individual risks. These causes of variability will also appear at interpopulation and interhuman levels. When we only have data at the population level (like most data reported in epidemiological studies), quantifying the interpopulation variability is an important first step to a full interhuman variability quantification.

To incorporate the interpopulation variability, we construct a Bayesian hierarchical model among different subpopulations for the parameters that represent background risk and response sensitivity. Most parametric exposure-response models for continuous data contain two to four parameters with different and varying biological relevance. For example, the simplest linear model has two parameters, intercept and slope, which well correspond to these two factors. However, the Power model and Hill model with more parameters also include equivalent parameters for the two factors. In this study, we demonstrate, through both an application to real data and a simulation study, that using Bayesian hierarchical structure on partial model parameters can quantify the distribution of the background risk and response sensitivity (represented by corresponding parameters) in various exposure-response models, and it is an adequate approach to quantify variability across different populations.

The article is organized as follows: in

We propose a Bayesian hierarchical model to quantify the variability and uncertainty in human populations with a focus on the low-exposure region. In particular, we fit an exposure-response model to data sets from different studies, where the subjects in the studies have the same or extremely similar endpoints and exposure metrics, and comparable external conditions, geographically, economically, etc., and build hierarchical structure over the parameters of interest (i.e., background risk and response sensitivity) to characterize their distributions across different populations. In the hierarchy, the higher-level distribution is used to estimate the distribution of relative risk at any given exposure level. The higher-level distributions describing the interpopulation heterogeneity will lead to heavier tailed distributions of relative risk, which will appropriately quantify the variability across populations and better estimate the interhuman variability.

The method assumes aggregated rather than individual data are available (a common situation in published epidemiological studies), and the data are from prospective cohort epidemiological studies consisting of count data. Such studies typically include exposure level, number of subjects, observed cases per exposure level, and adjusted relative risk. For such data, the mean value of observed cases is a product of adjusted expected number of cases and an exposure-response function, that is,
_{i}_{i}_{i}_{i}

The cohorts used in this study all have an internal referent (i.e., the group with the lowest exposure _{1}), and the expected values are themselves estimated from the same data as the model parameters. To account for this in the model, we define _{i}_{i}_{1}; _{1}. This forces the fitted relative risk at the reference group to be 1. In addition, because the denominator is a constant given estimated model parameters, _{i}

To model _{i}_{i}_{1} (expected value of the reference group). Consequently, we adopt a simplifying approach to quantify the relationship between _{i}_{1}. Assuming _{1} = _{1}, where _{1} represents the observed cases in the reference group, we further assume that the ratio of _{1} is equal to the ratio of _{i}_{1}, that is:

It is common to use the Poisson distribution to model count data (i.e., number of cases) observed in unit time. We assume that the observed cases in each exposure group follow a Poisson distribution parametrized by the mean value expressed by _{i}_{i}_{i}_{1} and parameters _{i}_{i}, o_{i}_{i}

We are particularly interested in applying a Bayesian hierarchical model to quantify exposure-associated variability in relative risk represented by the exposure-response model _{i}^{(9)} is used as an example to demonstrate the hierarchical modeling methodology in this study and has been reparameterized as ^{g}.” This is the same as the coefficient of the dose term in the Power model (i.e., ^{g}^{(10)} (PPP) can be used to indicate if the model has an adequate fit.

Bayesian methods have been widely applied in dose-response modeling research.^{(11–13)} In this study, we propose to characterize the interpopulation variability through the background risk (i.e., relative risk at background exposure) and response sensitivity (i.e., response rate), which corresponds to placing a hierarchical structure over parameters “

The Bayesian hierarchical structure over partial model parameters _{j}_{a}_{a}_{a}_{a}_{j}_{b}_{b}_{b}_{b}_{a}_{a}_{b}_{b}^{(14)} are calculated to compare alternative hierarchical models with regard to goodness of fit with adjustment for over-fitting. It is important to note that, in this study, both the PPP and WAIC are considered together when evaluating the models, and it is not intended to rule out any model solely based on any one of the indicators.

The posterior distribution of the parameters is sampled using MCMC simulation, and the population variability and uncertainty at any given exposure level is characterized by the distribution of relative risk calculated using the parameters’ posterior sample.

Instead of using the posterior sample of parameters of “_{j}_{j}_{a}_{a}, _{b}, and _{b} describing the distributions of “_{j}_{j}_{a} and _{a} (or _{b} and _{b}

The proposed approach is applied to a set of exposure-response data reported in epidemiological studies shown in ^{(15)} and Chen ^{(16)} These studies reported or estimated average arsenic concentration in drinking water, observed numbers of deaths, adjusted relative risk and person-years at risk (if available). Sohel ^{(15)} employed Cox proportional hazards models to estimate the mortality risks in relation to arsenic exposure and adjusted the values for potential confounders, including age, sex, socioeconomic status, and education. However, Chen ^{(16)} adjusted the relative risk values for age, sex, smoking status, and educational attainment. Both Sohel

As an important component in Bayesian analysis, prior distribution should be properly selected. In this study, we use uniform distribution as priors to set a fairly large range for each parameter (including hyper parameters) but without specifying preference on any values in the corresponding range (i.e., flat prior, all values for each parameter in the corresponding range are equally likely). For the parameters _{a} ~ Uniform(−1,1), _{a} ~ Uniform(0, 2), _{b} ~ Uniform(−3, 3), and _{b} ~ Uniform(0, 2). Based on Monte Carlo simulation, these settings allow parameter ^{−4} up to at least 1.3 × 10^{4} and ^{−6}, 3 × 10^{4}). Using uniform distribution as a prior for variance parameters (e.g., _{a}_{b}^{(10)} However, priors for _{1} in Equation (_{1} ~ Uniform(0,1000),

For the purpose of comparison, some alternative hierarchical structures were also employed and the results obtained are compared with the ones from the proposed hierarchical model on partial parameters. When hierarchical structure is only used for _{c} ~ Uniform(−3, 3), _{c} ~ Uniform(0, 2), _{g} ~ Uniform(−3, 3), and _{g} ~ Uniform(0, 2). Again, the main purpose of using uniform distribution as prior is to set a reasonable boundary on the parameters without giving any preferences. We also examined another flat prior option, normal distribution Normal(0,1000^{2}) with the same lower and upper bounds corresponding to the uniform distributions, and we obtained almost identical estimates for the quantities of interest.

The proposed methods are programed in R (Version 3.3.1) using RStan.^{(18)} The MCMC sampling process consisted three different Markov chains sampled for 20,000 iterations. The first half of each chain is disregarded as burn-in, which results in a posterior sample size of 30,000. The convergence of the MCMC sampling is judged by the potential scale reduction statistic, R̂ ^{(19)} provided in the output of RStan. The values of R̂ reported indicate that all chains converged.

The posterior distribution of relative risk estimates at 10 ^{g},

From the tables, we can find that the Hill model has higher PPP values (indicating a better fit) and WAIC values (posterior predictive density adjusted for overfitting, indicating a nonfavorable model selection) than the Power and linear models.

There are two main reasons for this result: (l) the Hill model with four parameters has a more flexible shape to fit the data than the Power and linear models, which do not fit the data well, and, (2) as the model has four parameters and there are a limited number of exposure groups, the variation in the posterior sample of the Hill model make the WAIC value higher than the Power and linear models. As demonstrated by the Power and Linear models, the distribution of estimated relative risk is wider (mainly the upper bound is higher) than the counterpart estimated from a single population in a study. As shown in

We propose a simulation study to test if the hierarchical structure on partial model parameters can adequately quantify the variability in relative risk with a focus on low exposures and investigate the effect of the number of studies on the estimate of population variability.

We again use the fundamental assumption that the human variability is jointly caused by two factors: the variability in risk at background exposure and in the sensitivity of human response, which can be explicitly represented by the two parameters in the linear model, the model chosen to serve as the “true” model for simulating data sets. As this study focuses on examining how well the proposed hierarchy can quantify the variability over the background and slope of the response, the linear model is chosen to avoid introducing uncertainty and variability represented by other model parameters. In the simulation study, some modeling assumptions and settings used in the above data analyses can be simplified and will be explained in detail in the description of the simulation study below.

The linear model ^{(15–17,20)} As the relative risk is a measurement calculated based on groups of subjects, each specified linear curve quantifies the relationship in a population (i.e., all subjects in a study) and the prescribed distributions of

The exposure level, relative risk, and observed cases are simulated to form simulation data sets. We assume that there are five exposure groups in the range of 0–400

To simulate relative risk, values of “_{i}_{i})/_{i}, is used to generate a randomized relative risk as

The third vector simulated is the observed number of cases in each exposure group. In epidemiological studies, the observed cases in each group are related to the total number of subjects and the relative risk of that exposure group. Therefore, we use the following steps to randomly generate the case numbers:

Generate 1,000 values from a lognormal distribution LogNormal(3.5, 1.5) (where 400

Bin these subjects into previously specified exposure groups, i.e., 0–10, 10–50, 50–100, 100–200, and 200–400 _{i}

Draw the number of cases randomly from a Poisson distribution with mean equal to _{i}_{i}_{i}.

The first step takes sampling variability into account. The number of values generated from the lognormal distribution (i.e., the sample size) is closely related to the sampling variability. The larger the sample size is, the more certain the proportion of the sample size of each group is (i.e., the smaller the sampling variability among exposure groups is). The reason to choose 1,000 is that it is close to the total number of observed cases in Sohel ^{(15)} one of the studies analyzed in

These three vectors together can form a complete simulation data set for exposure-response modeling, but the model approach is slightly different from the one stated in the previous section. The main reason for the difference is that the expected number of cases in each exposure group in the simulation study is independently generated rather than estimated based on the assumption that supports _{i}_{i}/rr_{i}′

The “true” distribution of relative risk at a given exposure is characterized by Monte Carlo simulation. From the distribution _{10 μg/L}, can be calculated. These 100 million values are further used as the median in the lognormal distribution to simulate 100 million relative risk _{10 µg/L} for each of the three situations: no randomness (NR), small randomness (SR), and large randomness (LR). We believe that the 100 million samples can quite accurately characterize the distribution of the relative risk, and therefore the median and 95th percentile estimated from the 100 million sample are used as “true” value. For example, at 10

The simulated data sets from the known variability in relative risk are used to test how accurately the proposed hierarchical model can quantify the variability, especially in the low-exposure region (i.e., 10 _{est} represents the 50th (median) or 95th percentile of the estimated distribution of relative risk, and the _{true} represents the corresponding true values calculated in the previous section. Thus, positive log ratio means that the estimated value is larger than the true value, and negative log ratio means that the estimated value is smaller. If the log ratio is zero, then the estimated value is equal to the true value. The closer to zero the log ratio is, the more accurate the estimate is.

The mean and standard deviation of the log ratio of the median and 95th percentile based on the 500 repetitions for each model/scenario combination are listed in

The results show that as the number of studies included increased, both the median and 95th percentile estimates are closer to the true value, and the variance in the estimates decrease. The majority of the log ratios are positive, indicating that the estimated values are typically larger than the true value. For the 95th percentile, higher estimated value is acceptable because the higher value will usually lead to a more conservative regulation on the exposure. As shown in both

As suggested by both the simulation study and application to real data, the proposed hierarchical model on partial model parameters can be used to adequately quantify variability across different populations.

The proposed hierarchical model on partial parameters focuses on quantifying the variability via quantifying the distribution in risk at background exposure and the sensitivity of response, which are the two main factors that are mostly relevant to the variability in responses at low exposures. However, we need to note that the sensitivity of response in the Hill model (Equation (

The available exposure groups in the studies being analyzed has a very important impact on model selection and the variability quantification. As suggested by the data analysis results in

The results suggest that as the number of studies increases, the uncertainty in estimates decreases. When the number of studies is small, the estimated distribution is wider than the true distribution. From a conservative perspective, a limited number of studies will not hurt risk assessment. Theoretically, to include more studies is always beneficial for increasing the accuracy of estimation, but the improvement in accuracy is clearly reduced when the number of studies included reaches five. This suggests that to balance the cost and the effectiveness of reducing uncertainty and improving accuracy, five studies might be a good option. However, we need to note that we assumed each data set had five exposure groups in the simulation study, so it should be expected that more studies are needed when fewer exposure groups are contained in each study.

Prior and hyper prior distribution can have potentially large impacts on the distribution estimation, that is, wider prior distribution may lead to wider distribution of the relative risk estimates. In this study, we examined two possible options of flat prior, the uniform distribution and truncated normal distribution with very large variance, to set a boundary on the model parameters. Because such settings on boundaries were kept identical in various situations and scenarios we examined, the results of comparison are robust to the prior specifications. However, priors should be carefully selected and tested for sensitivities in practice.

Finally, we also need to note that what has been focused on in this study is the interpopulation variability, which is only a part of the interhuman variability. When data are only available at the population level like the examples we employed in article, the hierarchical model is useful to probabilistically quantify the variability among different populations. However, if more detailed information regarding pharmacokinetic and/or pharmacodynamics in humans is available, variability at the individual level also should be incorporated and quantified.^{(3)} One source of uncertainty not considered in this study is exposure uncertainty, which is an important but difficult factor in exposure-response assessment using epidemiological data. In our next study, we will focus on investigating how exposure uncertainty will influence the variability and uncertainty in risk estimates.

This research is partially supported by Indiana University School of Public Health Developmental Research Grants for Pre-Tenure Faculty. The authors thank Drs. Jeffrey S. Gift and Woodrow Setzer and anonymous reviewers for their comments on earlier versions of the article.

The Bayesian hierarchical structure over all four model parameters extended from _{j}_{c}_{c}_{c}_{c}_{j}_{g}_{g}_{g}_{g}_{c}_{c}_{g}_{g}

Mean and Standard Deviation of the Log Ratio Estimated from the Hill Model with Hierarchical Structure on All Parameters

Median | 95th Percentile | ||||||
---|---|---|---|---|---|---|---|

2 Studies | 5 Studies | 8 Studies | 2 Studies | 5 Studies | 8 Studies | ||

NR | Hill-all | −0.0147 | −0.0037 | −0.0026 | 0.9611 | 0.2439 | 0.1346 |

(0.0293) | (0.0218) | (0.0167) | (0.2705) | (0.1869) | (0.1423) | ||

SR | Hill-all | −0.0234 | −0.0128 | −0.0119 | 0.9807 | 0.2555 | 0.1272 |

(0.0315) | (0.0226) | (0.0172) | (0.2732) | (0.1911) | (0.1441) | ||

LR | Hill-all | −0.0433 | −0.0484 | −0.0525 | 1.1954 | 1.1302 | 1.1651 |

(0.0546) | (0.0403) | (0.0303) | (0.6287) | (0.7414) | (0.7194) |

The Bayesian hierarchical structure with specifications for the priors. Here, Unif(

The boxplot for the ratio of median and 95th percentile estimated from the Hill model, Power model, and linear model to the true median and 95th percentile, respectively, for the situation with no randomness (NR) in the simulated relative risk values.

The boxplot for the ratio of median and 95th percentile estimated from the Hill model, Power model, and linear model to the true median and 95th percentile, respectively, for the situation with small randomness (SR) in the simulated relative risk values.

The boxplot for the ratio of median and 95th percentile estimated from the Hill model, Power model, and linear model to the true median and 95th percentile, respectively, for the situation with large randomness (LR) in the simulated relative risk values.

The boxplot for the ratio of median and 95th percentile estimated from the two hierarchical structures (H-ab and H-all) of the Hill model to the true median and 95th percentile, respectively, for the three situations with none, small, and large randomness (NR, SR, and LR) in the simulated relative risk values.

CVD Mortality and Exposure Data Used

Exposure Group | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Sohel ^{(15)}
| |||||

Arsenic water concentration ( | 3.8 | 25.8 | 90.6 | 214.7 | 427.5 |

Observed CVD deaths | 129 | 153 | 476 | 388 | 152 |

Adjusted relative risk | 1 | 1.03 | 1.16 | 1.23 | 1.37 |

Person-years | NR | NR | NR | NR | NR |

Chen ^{(16)}
| |||||

Arsenic water concentration ( | 3.7 | 35.9 | 102.5 | 265.7 | – |

Observed CVD deaths | 43 | 51 | 41 | 63 | – |

Adjusted relative risk | 1 | 1.21 | 1.24 | 1.46 | – |

Person-years | 20064 | 19109 | 18699 | 19380 | – |

_{i}′ in _{i}/arr_{i}, where _{i} and arr_{i} are the observed case numbers and adjusted relative risk value in this table, respectively.

Relative Risk at 10

Hill Model | Power Model | Linear Model | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5th | 50th | 95th | WAIC | 5th | 50th | 95th | WAIC | 5th | 50th | 95th | WAIC | |

H-ab | 1.000 | 1.118 | 1.418 | 73.07 | 1.006 | 1.086 | 1.631 | 65.93 | 1.008 | 1.259 | 2.520 | 67.16 |

H-0 | 1.000 | 1.118 | 1.330 | 73.44 | 1.006 | 1.019 | 1.035 | 65.11 | 1.003 | 1.006 | 1.009 | 66.25 |

H-a | 1.000 | 1.132 | 1.309 | 73.22 | 1.004 | 1.026 | 1.406 | 65.57 | 1.001 | 1.007 | 1.152 | 66.36 |

H-b | 1.000 | 1.089 | 1.315 | 72.44 | 1.007 | 1.034 | 1.298 | 65.26 | 1.007 | 1.122 | 2.075 | 67.13 |

H-all | 1.000 | 1.021 | 3.392 | 74.54 | 1.002 | 1.100 | 16.47 | 65.95 | – | – | – | – |

S-1 | 1.000 | 1.086 | 1.273 | – | 1.004 | 1.017 | 1.032 | – | 1.002 | 1.005 | 1.008 | – |

S-2 | 1.000 | 1.091 | 1.506 | – | 1.004 | 1.027 | 1.067 | – | 1.002 | 1.010 | 1.021 | – |

Posterior Predictive

Hill Model | Power Model | Linear Model | ||||
---|---|---|---|---|---|---|

S-1 | S-2 | S-1 | S-2 | S-1 | S-2 | |

H-ab | 0.219 | 0.201 | 0.067 | 0.116 | 0.143 | 0.061 |

H-0 | 0.333 | 0.234 | 0.066 | 0.088 | 0.704 | 0.805 |

H-a | 0.122 | 0.077 | 0.062 | 0.109 | 0.100 | 0.089 |

H-b | 0.223 | 0.211 | 0.059 | 0.027 | 0.139 | 0.180 |

H-all | 0.313 | 0.141 | 0.071 | 0.097 | – | – |

Single | 0.499 | 0.613 | 0.008 | 0.109 | 0.136 | 0.167 |

^{l}^{Pred,l}^{l}^{pred} represent, respectively, the observed data (e.g., case numbers) and predicted case numbers, which were generated using the posterior sample of estimated parameters. Under a Bayesian framework, the probability can be numerically approximated by counting the number of sets of posterior samples that satisfy the inequality out of the entire posterior sample space. Using such a method, a very large or very small PPP means that it is very likely to see a discrepancy in predicted data, further indicating a poor fitting. Therefore, a PPP value within the range from 0.05 to 0.95 indicates an adequate fit. For the linear model, the H-ab model is the same as the H-all model, so the PPP values are not reported for H-all.

Mean and Standard Deviation of the Log Ratio Estimated using Various Number of Studies

Median | 95th Percentile | ||||||
---|---|---|---|---|---|---|---|

2 Studies | 5 Studies | 8 Studies | 2 Studies | 5 Studies | 8 Studies | ||

NR | Hill | −0.0042 | −0.0019 | −0.0019 | 0.4429 | 0.2469 | 0.1471 |

(0.0321) | (0.0219) | (0.0166) | (0.2088) | (0.1894) | (0.1415) | ||

Power | 0.0003 | 0.0015 | 0.0011 | 0.4899 | 0.2469 | 0.1471 | |

(0.0338) | (0.0225) | (0.0172) | (0.2147) | (0.1950) | (0.1451) | ||

Linear | 0.0014 | 0.0020 | 0.0014 | 0.4891 | 0.2453 | 0.1463 | |

(0.0334) | (0.0226) | (0.0171) | (0.2127) | (0.2004) | (0.1455) | ||

SR | Hill | −0.0113 | −0.0102 | −0.0106 | 0.4372 | 0.2127 | 0.1180 |

(0.0344) | (0.0225) | (0.0173) | (0.2307) | (0.1930) | (0.1454) | ||

Power | −0.0064 | −0.0064 | −0.0071 | 0.4905 | 0.2373 | 0.1345 | |

(0.036) | (0.0235) | (0.0179) | (0.2339) | (0.2002) | (0.1502) | ||

Linear | −0.0073 | −0.0069 | −0.0074 | 0.4720 | 0.2304 | 0.1312 | |

(0.0338) | (0.0224) | (0.0171) | (0.2110) | (0.2028) | (0.1475) | ||

LR | Hill | −0.0167 | −0.0160 | −0.0178 | 0.0952 | −0.0596 | −0.1253 |

(0.0770) | (0.0526) | (0.0401) | (0.5262) | (0.3083) | (0.2232) | ||

Power | 0.0056 | −0.0011 | −0.0060 | 0.3010 | 0.0388 | −0.0617 | |

(0.0763) | (0.0518) | (0.0390) | (0.4876) | (0.2974) | (0.2220) | ||

Linear | −0.0142 | −0.0142 | −0.0153 | 0.2108 | −0.0145 | −0.0998 | |

(0.0377) | (0.0265) | (0.0200) | (0.2221) | (0.2074) | (0.1638) |