A Data Augmentation Prior in Fractional Polynomial Generalized Linear Models

In microbial and chemical risk assessments, careful dose-response modeling is emphasized because a target risk level (probability of infection or illness) is often in the low benchmark response range of 1% to 10%. To address model uncertainty at low doses and enhance diversity and flexibility for model-averaging, a set of fractional polynomial dose-response models can be considered. However, elicitation of an informative prior in Bayesian approach is difficult in those models because their parameters are not interpretable. This paper illustrates a method of elicitating informative prior known using data augmentation prior. INTRODUCTION In microbial risk assessment (MRA), an infectious dose for given risk level p ( 100 ID p ) is the dose that corresponds to 100p % of infection (or illness). Statistically, 100 ID p is the dose that satisfies 100 ( = 1|ID )= p P Y p , where Y is a binary random variable with Y = 1 indicating infection. Assuming a monotonic doseresponse relationship, 100 ID p is unique, and the monotonicity is assumed in our discussion. In chemical risk assessment, a benchmark dose for given risk level p ( 100 BMD p ) is similarly defined. Often, an allowed risk level for population is extremely small, and an accurate estimation of such a small risk of an environmental hazard requires sufficiently large samples. In MRA, a target p often ranges from 0.01 to 0.10, but an experimental range is mostly well above p. In this regard, a downward extrapolation is commonly practiced based on the inference of 100 ID p . Although multiple dose-response models are able to fit the data in an experimental range, estimation of 100 ID p can be very sensitive to a chosen model because each dose-response model behaves uniquely at low doses. In this regard, applications of model-selection criteria and model-averaging methods have been considered by several authors [1-4]. In this manuscript we focus on prior elicitation in fractional polynomial models for Bayesian inference of 100 ID p or 100 BMD p . For the estimation of a BMD, Namata et al. (2008) used seconddegree fractional polynomial (FP) dose-response models to enhance diversity and flexibility in a model space for modelaveraging. They showed that careful modifications of generalized linear models (GLMs) could effectively extend a model space for model-averaing. In this manuscript an FP model modified from a GLM is shortened as FP-GLM. In Bayesian framework, [5] and [6] applied Bayesian model averaging (BMA), proposed by [7] using diffuse priors in estimation of a BMD. Though experienced investigators may have more abundant prior knowledge than merely a flat prior, it seems difficult to properly elicit prior knowledge through the FP-GLMs because the model parameters are not easily interpretable. Furthermore, it is difficult to a priori guess a plausible range of the parameter space as the powers of fractional polynomial vary. To address this difficulty, we apply the method of a data augmentation prior (DAP) as discussed by several authors [8-10]. Tractable prior elicitation in dose-response studies have been practiced from the past. [11] re-parameterized the two parameters in a logistic regression model. In the context of MRA, the first parameter corresponds to the 100 ID p , and the second parameter corresponds to 1 1 = ( = 1| ) d P Y d θ where d1 is the minimum dose level in an experiment. It is a useful technique in a GLM, but the re-parameterization is not as simple in a FP-GLM as in a GLM. [12] used a DAP in a logistic regression for a Bayesian adaptive design for Phase I clinical trials. In this manuscript we describe the procedure for the use of a DAP in FP-GLMs. In addition, we provide some examples to explain how a DAP controls the prior precision of ID10 in any FPGLM, which is not easily achieved by a direct prior specification on the model parameters. A roadmap of our discussion follows. Section 2 includes a brief description of FP-GLMs and introduces the method of a DAP in FP-GLMs. Section 3 provides examples of prior elicitation in dose-response models to highlight the benefits of using a DAP. In Section 4 we apply a DAP to Campylobacter jejuni dataset, and in Section 5 we conclude with brief discussion. Central Moon et al. (2014) Email: JSM Math Stat 1(1): 1005 (2014) 2/7 STATISTICAL METHOD Fractional polynomial generalized linear models We let d > 0 denote a dose, and let θd be the probability of infection (or illness) at dose d. Further, we let F (⋅) and F -1(⋅) denote a cumulative distribution function (CDF) and its inverse function, respectively. Then, a GLM is given by ( ) 1 1 2 = , d d F x θ β β − + (1) where 1 < < β −∞ ∞ , 2 > 0 β , and = log( ) ( , ) d x d ∈ −∞ ∞ . For a FPGLM, we let = log( 1)> 0 d x d + so that 0 d x → as 0 d → and d x →∞ as d →∞ . Then, a second-degree FP-GLM is given by ( ) ( ) 1 1 2 1 2 ( )= , P P d d d F x x θ β β − + (2) where 1 < 0 β , 2 > 0 β , 1 < 0 P , and 2 > 0 P . By the four inequalities, we have the linear predictor such that ( ) ( ) 1 2 0 1 2 lim = P P d d d x x β β →   + −∞     and ( ) ( ) 1 2 1 2 lim = . P P d d d x x β β →∞   + ∞     The general form of second-degree FPs in Equation 2 is modified from the original definition of mth degree FPs. The modification satisfies essential properties of a monotonic doseresponse model. That is, by the definition of a CDF, 0 d θ → as 0 d → and 1 d θ → as d →∞ . [13] suggested that { } 1 2 1 2 ( , ): = 2, 1, 0.5, = 0.5,1,2,3 P P P P − − − is sufficient for practical purpose, and we can create twelve FPGLMs from an original GLM. In this manner, a model space easily enhances its diversity and flexibility for model-averaging. We may consider several link functions including logistic, probit, complementary log-log, and gumbel links. In particular, if we denote the linear predictor by ( ) ( ) 1 2 1 2 , P P d d d x x η β β ≡ + the respective CDFs are 1 ( )=(1 ) F e η η − − + , ( )= ( ) F η η Φ , ( )= 1 exp( ) F eη η − − , or ( )= exp( ) F e η η − − , where ( ) Φ ⋅ denotes the CDF of (0,1) N . A more detail discussion about the FP-GLMs is given by Namata et al. (2008). Data augmentation prior Instead of a direct prior elicitation on model parameters β1 and β2, a DAP induces the prior distribution of (β1, β2) through the prior distribution of probabilities of infection at two arbitrary doses, say 1 0 < d d − . Denoting the probability of infection at dj by = ( = 1| ) d j j P Y d θ for = 1,0 j − , we independently model : Beta( , ) j j j a b θ . Then, we transform 1 0 ( , ) d d θ θ − to 1 2 ( , ) β β using the Jacobian transformation. We define θj as d j θ . The independent Beta priors yield the joint density function 0 1 1 1 0 = 1 ( , ) ( ) (1 ) . a b j j j j j f θ θ θ θ − − − − ∝ − ∏ Since the linear predictor in a GLM is 1 2 j d x η β β ≡ + , the chain rule applies as 1 ( ) ( ) ( ) = = = j j j j j k d j k k j k j F F F x θ η η η η β β η β η − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ For k = 1,2, and the Jacobian determinant is