A Bayesian analysis offers a coherent method to incorporate information from prior research when estimating an association in an epidemiologic study. A Bayesian analysis may often be an improvement over one confined to information within a single study.^4,5 Informative priors for the parameters describing an exposure-disease association are often obtained from prior epidemiologic studies in comparable populations. When there is no previous epidemiologic research on human health effects of exposure to an agent, experimental or toxicologic evidence on whole organisms, tissues, cells, or molecules may be informative.

Experimental studies that use nonhuman animals or cell lines draw strength from their ability to control the exposures of interest, experimental conditions, and assessment of outcomes. However, use of evidence obtained from experimental studies to inform estimation of an association in an epidemiologic study may be complicated by differences in physiologic and pathologic responses across species⁶ and differences in the endpoints under study. For example, in radiation research, molecular and cellular studies often evaluate endpoints such as chromosomal aberrations, DNA strand breaks, or cell death, whereas epidemiologic studies often focus on cancer incidence or mortality.^7–9 Nonetheless, experimental research may provide useful information for specifying an informative prior for the parameters describing an exposure-disease association.

An order-constrained prior can provide an intuitive way of integrating toxicology results while avoiding the pitfalls of trying to directly apply effect estimates across species or outcomes. By utilizing an order constraint, the researcher imposes a structure to the relationship between the exposure and outcome of interest. For example, if we want to impose a monotonic order constraint for categories of exposure, β₁, β₂ and β₃, we could specify an order-constrained model such that β₁ ≤ β₂ ≤ β₃. In a Bayesian setting, which often relies on drawing large numbers of samples from the posterior distribution, we can impose that constraint by ensuring that each sample adheres to the specified ordering.^10,11

Order-constrained parameters have a history of use in the dose-response literature^10–12; however, their utility need not be limited to scenarios where the researcher is interested in specifying the direction and magnitude of a dose-response relationship for categories of a single exposure. Suppose that the investigator is studying a population exposed to agents X and Y, and is interested in disease, D, where Y is the agent of primary interest and X is another agent quantified using the same unit of measurement. For each agent we have experimental evidence regarding its association with outcome, Z, which might represent a suspected biologic marker (eg, a cellular transformation or change in disease biomarker) or induction of a tumor in an animal model. In a logistic regression setting, the associations between agents X and Y and disease endpoint Z might be described by two models of the form logit(Pr[Z = 1]) = α₀+α₁X and logit(Pr[Z = 1]) = α′₀+α′₁Y.

Suppose that the results from experimental research suggest that Y is more strongly associated with the outcome, Z, than X (eg, α′₁ > α₁). The investigator may incorporate such evidence by specifying an ordered-constrained prior informed by the rank ordering of the exposure effects regarding outcome Z when fitting a model for the outcome, D, in relation to the coexposures, X, Y. Such a model may take the form logit(Pr[D = 1]) = β₀ + β₁X + β₂Y, where the prior for β₁ may be vague, while the prior for the parameter of primary interest, β₂, reflects the ordered constrained prior assumption β₂ ≥ β₁.

A major distinction between order constraints and priors typical in Bayesian analysis is that the former assigns a probability of zero to those parameter values that do not adhere to the constraint. Therefore, a researcher who uses this approach should have a high degree of confidence in the evidence that informs such priors. When this is the case, specifying an order-constrained prior may yield substantial gains in estimation of the effect of the agent of primary interest.

Conditions for using an ordered-constrained prior are encountered in some important, interesting settings in occupational and environmental research. Considering investigations of the health effects of various congeners of polychlorinated biphenyl, where people are exposed to two or more congeners, exposure intensities vary (and are not perfectly correlated), and toxicologic data suggest prior expectations for differences in biologic effects between congener types. Studies of respiratory health effects associated with inhalation of asbestos fibers provide another setting in which these conditions may hold, because people are typically exposed to fibers of various dimensions, and toxicologic data suggest prior expectations or differences in biological effects as a function of fiber dimension. Although available human evidence may be insufficient to posit a prior for the association between the agent of primary concern and outcome of interest, toxicologic data may be informative regarding ordered constraints for two or more parameters.

We present an example utilizing an order-constrained prior to estimate the association between tritium exposure and leukemia mortality in a cohort of workers who were also exposed to gamma radiation at the Savannah River Site nuclear facility. The Savannah River Site has been identified as one of the largest US occupational cohorts with potential tritium exposure. Nonetheless, examination of the association with cancer risk is hindered by the fact that tritium is received at low levels and occurs with exposure to other occupational hazards.^13,14 Incorporation of prior knowledge via order-constrained priors was investigated as a method to stabilize risk estimates.

The Savannah River Site is a nuclear facility near Aiken, SC. Activities began in 1951, with the first production reactor going critical in December 1953. E. I. du Pont de Nemours and Company operated the site until March 31, 1989, when Westinghouse Savannah River Company took over.^15,16 Between 1950 and 1986, 21,204 people were known to have been hired by DuPont to work at the site. We restricted our cohort to those who worked at least 90 days, and had no history of employment at another Department of Energy facility.¹³ Additionally, workers were excluded if they were missing information on sex, date of birth, name, Social Security Number, or date of first hire. This leaves a cohort of 18,883 workers for whom individual, annual dose records have either been computed or estimated in previous research. Workers were followed through 2002 to obtain vital status information. In the current analysis, the outcomes, leukemia, and leukemia excluding chronic lymphocytic leukemia are based on International classification of disease in the United States codes (International classification of disease in the United States 9 codes 204–207; those with code 204.1 represent cases of chronic lymphocytic leukemia). Analyses excluding chronic lymphocytic leukemia are conducted because of potential differences in latency of chronic lymphocytic leukemia compared with acute and myeloid forms of leukemia.¹⁷

The primary type of external penetrating radiation exposure at the Savannah River Site was gamma rays. Neutrons were present in some areas, but constituted a small fraction of collective dose; therefore, we do not attempt to assess independent neutron effects.¹⁸ Penetrating forms of ionizing radiation were measured with film badges until 1970 and thermoluminescent dosimeters thereafter.¹⁶ Radiation doses from tritium intakes were estimated from urinalysis. We consider tritium and gamma doses independently in units of Gray (Gy).¹⁹ Estimated annual whole-body dose values in Sieverts (Sv)—the sum of gamma, tritium, and neutron doses—are available for all employment years; 6% of these records were estimated using a “nearby” method, previously described in Richardson et al.¹⁶ Annual whole-body dose estimates were utilized to derive annual tritium dose estimates for those person-years with missing information regarding the contribution of tritium to their cumulative annual whole-body dose. In total, there are 56.2 Gy of individual, annual tritium dose records, of which 4.3 Gy (7.7%) were estimated through use of a job-exposure matrix.²⁰ Imputed values of tritium dose were validated by combining gamma and tritium doses to recalculate the excess relative rate (RR)/10mSv whole-body radiation obtained by Richardson et al.¹³ Point estimates were the same in the dataset with imputed tritium dose values, with a modest gain in precision.

Dose-response relationships are estimated using an excess relative rate model of the form: RR=eαi(1+β1g+β2t) where e^αi indexes the baseline rate within stratum i, β represents the excess RR for a given exposure, g represents cumulative gamma radiation dose, and t represents cumulative tritium dose. Cumulative doses, expressed in mGy, are lagged 3 years from the date of death for leukemia cases and from the risk set time for controls. As in prior studies, we conducted nested case-control analyses by creating risk sets matched on age, sex, race, pay code, birth cohort, and employment status.¹³ All relevant confounders were matched in the study design, so no additional covariates are included in this analysis.

In a recent systematic review of animal and cellular studies, Little and Lambert²¹ conclude that a reasonable value for the biological effectiveness of an absorbed dose arising from tritium intake is two to three times that for an absorbed dose from external exposure to gamma radiation. In a more common Bayesian analysis, a researcher might incorporate the estimate of effect from previous work as a prior in the current analysis. Because these results are exclusive to leukemia events in animals and cancer precursors in animal and human cells, researchers may feel uncomfortable specifying a value for the effect (ie, the prior mean) based on these studies. However, although the outcomes of interest varied across studies, the direction of effect of tritium relative to gamma radiation dose is consistent based upon evidence from in vivo and in vitro studies^21–23; that is, the biologic effectiveness of tritium is always greater than that of gamma radiation. Therefore, we specify an order-constrained prior that β₂ ≥ β₁ ¹⁰ The order constraint allows us to specify that, although we are uncertain about the specific magnitude of the difference between β₂ and β₁, we are certain that tritium causes more biologic damage than gamma radiation. In a Markov chain Monte Carlo implementation of the analysis this translates to drawing samples of β₂ that are never less than β₁. The form of the distribution for the parameters is specified as normal but noninformative over admissible values, where β₁ ~ N(μ = 0, σ² = 100,000) and β₂ ~ N(μ = 0, σ² = 100,000), truncated below by β₁.

All analyses were conducted using SAS procedure Markov chain Monte Carlo (V 9.2, SAS Institute, Cary, NC). Posterior distributions are presented as 90% highest posterior density intervals for consistency with radiation epidemiology literature. All models were run three times as a diagnostic check, to ensure that results are not sensitive to starting values of the β₁ and β₂. In addition, to minimize simulation error, these models were run for 5 million iterations with a burn-in of 100,000 iterations, and all sampled values after the burn-in period were retained.

Table 1 documents the distribution of cases by categories of dose for tritium and gamma radiation. This illustrates the fact that tritium is received at lower levels than gamma radiation, for which the dose distribution is larger. The Pearson correlation of tritium and penetrating radiation is 0.64 for cases and 0.34 for cases and controls.

Table 2 presents Markov chain Monte Carlo analyses of leukemia and leukemia excluding chronic lymphocytic leukemia. When the model includes a noninformative prior for β₁ and no order constraint on β₂, the estimate of the excess RR/10mGy due to gamma radiation (β₁) for leukemia and leukemia excluding chronic lymphocytic leukemia are 0.053 (90% highest posterior density = −0.025 to 0.142) and 0.176 (0.011 to 0.375), respectively. The estimated excess RR/10mGy due to tritium (β₂) for leukemia and leukemia excluding chronic lymphocytic leukemia are 0.141 (−0.323 to 0.649) and −0.281 (−1.136 to 0.548), respectively. The Figure illustrates the relative weight of information based on kernel density estimates of posterior distributions for β₁ and β₂ when the order constraint is excluded.

When we integrate the order-constrained prior so that β₂≥ β₁, retaining a noninformative prior for β₁, estimates of the excess RR/10mGy due to gamma radiation for leukemia and leukemia excluding chronic lymphocytic leukemia are 0.034 (−0.031 to 0.110) and 0.082 (−0.017 to 0.206), respectively. The estimated excess RR/10mGy due to tritium for leukemia and leukemia excluding chronic lymphocytic leukemia are 0.298 (0.027 to 0.702) and 0.344 (0.049 to 0.817), respectively. The width of the confidence bounds indicates the change in precision that results from integration of the order-constrained prior. The posterior correlations between β₁ and β₂ when examining leukemia and leukemia excluding chronic lymphocytic leukemia are −0.0118 and 0.1635, respectively. Additionally, the Figure illustrates the shift in the distribution of β₂ relative to β₁, the latter of which is clearly better identified in this data.