STATISTICAL INTERACTIONS AND BAYES ESTIMATION OF LOG ODDS IN CASE-CONTROL STUDIES

Abstract

This paper is concerned with the estimation of the logarithm of disease odds (log odds) when evaluating two risk factors, whether or not interactions are present. Statisticians define interaction as a departure from an additive model on a certain scale of measurement of the outcome. Certain interactions, known as removable interactions, may be eliminated by fitting an additive model under an invertible transformation of the outcome. This can potentially provide more precise estimates of log odds than fitting a model with interaction terms. In practice, we may also encounter non-removable interactions. The model must then include interaction terms, regardless of the choice of the scale of the outcome. However, in practical settings, we do not know at the outset whether an interaction exists, and if so whether it is removable or non-removable. Rather than trying to decide on significance levels to test for the existence of removable and non-removable interactions, we develop a Bayes estimator based on a squared error loss function. We demonstrate the favorable bias-variance trade-offs of our approach using simulations, and provide empirical illustrations using data from three published endometrial cancer case-control studies. The methods are implemented in an R program, available freely at http://www.mskcc.org/biostatistics/~satagopj.

INTRODUCTION

One of the main objectives of case-control studies is to estimate the natural logarithm of disease odds (log odds) corresponding to the categorical levels of the risk factors of interest. This is an important epidemiologic parameter, which can facilitate the estimation of odds ratios and absolute risk of disease. It can also provide insights into the potential benefits of screening high-risk individuals [1]. This is generally done by estimating the effects of the individual risk factors using a logistic regression model. When multiple risk factors are examined, it becomes important to decide whether or not interactions between the risk factors must be included in the model to obtain accurate estimates of the log odds. The purpose of this paper is to develop an optimal estimation procedure for case-control studies when we wish to estimate the log odds corresponding to two risk factors, whether or not interactions are present.

The increasing ability to study multiple environmental and genetic factors for disease is contributing to a proliferation of research works in the evaluation of gene-gene and gene-environment interactions [2–4]. The rapidly growing interests in studies of novel treatments are contributing to investigations of gene-drug interactions [5–6]. A wide range of definitions is used to refer to the term interaction [7]. An additive model is defined as one in which the additive effects on the outcome of the various levels of one risk factor do not depend upon the levels of another risk factor. Whereas epidemiologists describe such a model as one having “additive interaction” [8], statisticians define interaction as a departure from an additive model [9,10]. This is commonly referred to as a statistical interaction. Throughout this paper we shall only use the word interaction in this statistical sense.

Certain interactions, referred to as removable interactions, may be eliminated via an invertible transformation of the outcome so that the resulting model is additive on the transformed scale [11]. The results can be back-transformed for clinical interpretation, and the interactions will reappear in the model upon back-transformation [7, 12]. When the disease trait is binary, a transformation corresponds to a link function [13]. When an interaction is removable, accurate and precise estimates of the log odds parameters can be obtained by fitting a parsimonious additive model under a suitable link function [13]. We define an accurate estimate as one having negligible bias, and a precise estimate as one having small standard error. In this paper, we first show that the Guerrero and Johnson [14] (abbreviated, GJ) link function is an appropriate transformation to additivity when an interaction under the logistic link is removable.

Not all interactions are removable. Non-removable interactions are also referred to as qualitative interactions [15, 16]. When a non-removable interaction exists, an additive model will not usually provide accurate estimates of the log odds, regardless of the choice of transformation, and interaction terms must be included in the model to obtain unbiased estimates of the log odds. However, in practical data analysis settings, we cannot know with certainty at the outset whether a non-removable interaction exists. In principle, we may conduct preliminary hypothesis tests for the existence of removable and non-removable interactions. Rather than trying to decide on significance levels for these tests, in this paper we develop a Bayes estimator for the log odds parameters by assuming a squared error loss function. We minimize the loss function subject to the condition that the resulting class of Bayes estimators includes minimax estimators in the limit.

This paper is organized as follows. In the Materials and Methods section, we first introduce some notations and describe the concept of removable interactions. Next, we describe the GJ link function and show that it is an appropriate link function to additivity when an interaction under the logistic link is removable. We also show that, when the model is additive under the GJ link, the logistic link function can result in a systematic departure from additivity. Thus, a suitable model under the logistic link function may be used to estimate the log odds when the model is additive under the GJ link. Since some interactions may be non-removable, we develop a Bayes estimation approach for obtaining precise estimates of log odds whether or not all interaction is removable. A main advantage of the proposed Bayes estimator is that it does not require preliminary hypothesis tests to determine whether an interaction exists and/or whether it is removable or non-removable, in order to decide how to estimate the log odds parameters. In the Results section, we first demonstrate the favorable bias-variance trade-offs of the Bayes estimator using simulations, and then illustrate our method using published data from three case-control studies of endometrial cancer [17–19]. These data represent three distinct types of interactions: some removable and some non-removable interaction [17], only removable interaction [18], and only non-removable interaction [19]. They help illustrate that the proposed Bayes estimation method gives similar estimates to what would have been otherwise found by testing separately for the presence of removable and non-removable interactions under some choice of significance levels for these tests. We have developed a computer program to implement the proposed methods, which we note in the section Computational Guidance for Practitioners, and conclude with a Discussion. We describe the main results in the paper and refer the reader to the Online Supplementary Material for technical details.

MATERIALS AND METHODS

Consider a case-control study with two risk factors X and Z having L₁ and L₂ levels, respectively, measured on each individual. Let N_ij and M_ij denote the number of affected cases and unaffected controls, respectively, having the i-th level of X and the j-th level of Z (i = 1, 2, …, L₁; j = 1, 2, …, L₂). Given the total number of individuals N_ij + M_ij in the (i,j)-th risk factor sub-class, N_ij is distributed as a binomial random variable with disease probability p_ij. We assume that there are no empty sub-classes. We generally fit a logistic regression model to case-control data, written as:

$$g_0\left(p_{ij}\right)=\mu +{\alpha }_{\mathrm{i}}+{\beta }_{\mathrm{j}}+{\gamma }_{\text{ij}},$$ (Equation 1)

where $g_0\left(p_{ij}\right)=log\left\{\frac{p_{ij}}{1-p_{ij}}\right\}$ is the logistic link function, μ is the baseline risk, α_i and β_j are the main effects of the i-th level of X and the j-th level of Z, respectively, and γ_ij is the effect of the interaction between the i-th level of X and the j-th level of Z, subject to the constraints ${\displaystyle \sum _{\mathrm{i}=1}^{{\mathrm{L}}_1}}{\alpha }_{\mathrm{i}}=0,{\displaystyle \sum _{\mathrm{j}=1}^{{\mathrm{L}}_2}}{\beta }_{\mathrm{j}}=0,{\displaystyle \sum _{\mathrm{i}=1}^{{\mathrm{L}}_1}}{\gamma }_{\text{ij}}=0$, and ${\displaystyle \sum _{\mathrm{j}=1}^{{\mathrm{L}}_2}}{\gamma }_{\text{ij}}=0$ for i = 1, …, L₁, and j = 1, …, L₂. The maximum likelihood estimates (MLEs) of the main and interaction effects can be obtained using the iteratively reweighted least squares method.

Suppose there exists an alternative, but unknown, link function, denoted f(p_ij), under which the model is additive i.e., f(p_ij) = μ+ α_i + β_j. If g₀(.) is a linear function of f(.), then the model is also additive under g₀(.) i.e., there will be no interaction term in the model under the logistic link function. However, if g₀(.) is non-linear in f(.), then a quadratic approximation may provide a better fit to the data than an additive model under g₀(.): g₀(p_ij) = η₀ + η₁{f(p_ij)}+ η₂{f(p_ij)}², where η₀, η₁, and η₂ are unknown parameters. When this quadratic polynomial is monotonic in f(p_ij) in the range defined by the data, we obtain the approximation [10, 11, 13]:

$$g_0\left(p_{ij}\right)\approx \mu +{\alpha }_i+{\beta }_j+\theta \times {\alpha }_i\times {\beta }_j,$$ (Equation 2)

where θ is a scalar quantifying non-additivity of the model under the logistic link. In previous work [13] we have shown how to obtain MLEs of the parameters θ, μ, α_i, and β_j in Equation (2).

A comparison of Equations (1) and (2) shows that we can approximate the interaction contrasts as γ_ij ≈ θα_iβ_j when monotonicity holds. Thus, under monotonicity, there will be (L₁−1)×(L₂−1) − 1 fewer parameters in the model. In practical settings, we do not know whether this approximation is applicable since we do not know at the outset whether monotonicity holds. Here we propose to take advantage of potential monotonicity and write the interaction terms as:

$${\gamma }_{ij}=\theta {\alpha }_i{\beta }_j+e_{ij},$$ (Equation 3)

where the terms e_ij (i = 1, …, L₁; j = 1, …, L₂) can be interpreted as the error in representing γ_ij as θα_iβ_j when monotonicity does not hold. When e_ij = 0 for all i and j, and θ ≠ 0 in Equation (3), this would be an indication of monotonicity, and we say that the interaction is removable. When e_ij ≠ 0 for at least one i and j, this would be an indication of lack of monotonicity, and we say that there is non-removable interaction.

Our objective is to estimate the log odds. The following four scenarios arise depending upon whether an interaction exists and whether it is removable or non-removable:

• Scenario 1 (no interaction). When θ = 0 and e_ij = 0 for all i and j, there is no removable and no non-removable interaction i.e., there is no interaction at all (γ_ij = 0 for all i and j). Hence, the log odds summaries can be estimated using an additive model under the logistic link function i.e., using Equation (1) by setting γ_ij = 0 for all i and j.

• Scenario 2 (removable interaction). When θ ≠ 0 but e_ij = 0 for all i and j, there is removable interaction and no non-removable interaction. Hence, there exists a transformation to additivity. We anticipate the log odds summaries estimated using a transformation to additivity to be more precise (i.e., smaller standard error) than estimates based on Equation (1) since an additive model will have fewer parameters than Equation (1).

• Scenario 3 (non-removable interaction). When θ = 0 and e_ij ≠ 0 for at least one i and j, there is no removable interaction and there is only non-removable interaction. Therefore, a transformation to additivity is not feasible, and the log odds summaries must be estimated using the logistic regression model of Equation (1).

• Scenario 4 (both removable and non-removable interactions). When θ = 0 and e_ij ≠ 0 for at least one i and j, there are both removable and non-removable interactions. In this case, the log odds summaries may be estimated using Equation (1), or by making suitable use of the parametric form of γ_ij given by Equation (3). Since this latter approach takes advantage of modeling the removable component of the interaction suitably as θα_iβ_j, we anticipate this method would estimate the log odds summaries with better precision than Equation (1).

In practical settings, at the outset we do not know which of these four scenarios is applicable. One approach would be to test the null hypothesis H₀: γ_ij = 0 for all i = 1, 2, …, L₁−1 and j = 1, 2, …, L₂−1 against the alternative hypothesis H_A: γ_ij ≠ 0 for at least one (i,j) using a likelihood ratio statistic. We may conduct further hypothesis tests by evaluating the null hypothesis of no removable interaction H₀: θ = 0 against the alternative H_A: θ ≠ 0. We may also test the null hypothesis of no non-removable interaction H₀: e_ij = 0 for all i = 1, 2, …, L₁−1 and j = 1, 2, …, L₂−1 against H_A: e_ij ≠ 0 for at least one (i,j). [Test statistics for evaluating removable and non-removable interactions are given in the Online Supplementary Material.] The results of these hypothesis tests can be used to determine the specific scenario and estimate the log odds summaries using a suitable model.

While such preliminary testing procedures can be useful for selecting a method to estimate the log odds summaries, they rely on the choice of a significance level, can lead to inflated type I errors, and the parameter estimates may have poor precision [20,21]. Therefore, in this paper we develop a Bayes estimator of log odds by accounting for potential removable interaction, but without the need for conducting preliminary hypothesis tests. Before describing our proposed Bayes estimator, we show that the Guerrero and Johnson link function [14] is a suitable transformation to additivity when the interactions under the logistic link function are removable.

The Guerrero and Johnson link function

The Guerrero and Johnson (GJ) link function, indexed by a single transformation parameter λ_G, and denoted as g(p_ij, λ_G), is a Box-Cox transformation [22] of the disease odds, given by [14]:

$$g(p_{ij},{\lambda }_{\mathrm{G}})=\{\begin{array}{ll}\frac{\mathit{1}}{{\lambda }_{\mathrm{G}}}\left\{{\left(\frac{p_{ij}}{1-p_{ij}}\right)}^{{\lambda }_{\mathrm{G}}}-\mathit{1}\right\}\hfill & \text{if}{\lambda }_{\mathrm{G}}\ne 0\hfill \\ log\left(\frac{p_{ij}}{\mathit{1}-p_{ij}}\right)\hfill & \text{if}{\lambda }_{\mathrm{G}}=0\hfill \end{array}$$ (Equation 4)

The logistic link is a member of the GJ family when λ_G = 0. When λ_G ≠ 0, the GJ link is not symmetric (see Figure 1) since the disease risk approaches 1 (or 0) more rapidly than it approaches 0 (or 1). Further, the rate at which the disease risk approaches 1 (or 0) is higher under the GJ link than the logistic link. The identity link function is not a member of the GJ family since there is no value of λ_G under which g(p_ij, λ_G) = p_ij.

Figure 1.

The shape of the Guerrero and Johnson (GJ) link function and the logistic link function. The link is shown for λ_G = −1.5 and 1.5. The logistic link corresponds to λ_G = 0. The horizontal axis shows the additive effect (say, A). The vertical axis plots the disease risk, calculated as B/(1+B), where B = exp(A) for the logistic link, and otherwise B = (1 + λ_GA)^1/λ_G.

An additive model under the GJ link is given by:

$$g\left(p_{ij},{\lambda }_{\mathrm{G}}\right)={\mu }^{\ast }+{\alpha }_{\mathrm{i}}^{\ast }+{\beta }_{\mathrm{j}}^{\ast },$$ (Equation 5)

where μ^* is the baseline effect, and ${\alpha }_{\mathrm{i}}^{\ast }$ and ${\beta }_{\mathrm{j}}^{\ast }$ are the main effects of the two risk factors on the GJ scale. In previous work [13] we have developed methods for obtaining MLEs of λ_G and the parameters of Equation (5). The following result establishes the GJ link function as an appropriate transformation to additivity when interactions under the logistic link are removable (proof given in the Online Supplementary Material).

Result

1. When the interaction between risk factors is removable under the logistic link function (i.e., Equation 2 holds with an equality sign instead of an approximation sign), there exists a link function, denoted g(p_ij), under which the model is additive; and this link function takes the form of the GJ link function given by Equation (4) under the boundary conditions g(0) = −1/λ_G and g(1) = 0.

2. Conversely, whenever the model is additive under the GJ link function (i.e., Equations 4 and 5 apply) but λ_G ≠ 0, and a quadratic polynomial in μ+α_i+β_j is strictly monotonic over the domain of values of α_i and β_j that fit the range of the data at hand, the logistic link function yields a systematic departure from the additive model. Further, the logistic regression model is given by Equation (2) with θ = −λ_G.

When the model is additive under the GJ link (i.e., Equations 4 and 5 hold), the log odds can be written as:

$$log\left(\frac{p_{ij}}{\mathit{1}-p_{ij}}\right)=\frac{\mathit{1}}{{\lambda }_{\mathrm{G}}}\times log\left\{\mathit{1}+{\lambda }_{\mathrm{G}}\times g\left(p_{ij},{\lambda }_{\mathrm{G}}\right)\right\}$$ (Equation 6)

Taken together, these observations suggest that when the interactions are removable under the logistic link function, we can fit an additive model under the GJ link, and plug the MLEs of the parameters into the right hand side of Equation (6) to estimate the log odds. This is equivalent to fitting the logistic regression model given by Equation (2), obtaining the MLEs of this model, and plugging these into the right hand side of Equation (2) to estimate the log odds.

Bayes’ estimator of log odds

Under scenarios 1 to 4 described above, the log odds can be modeled using Equation (1) or using an additive model under the GJ link (Equations 4, 5, and 6; equivalently, the logistic regression model of Equation 2). Let Y, a vector of length L₁×L₂, denote the log odds. Under the standard logistic regression model of Equation (1), a general from for the (i,j)-th element of Y is: Y_ij = μ + α_i + β_j + γ_ij. The MLE of Y, denoted Ŷ_MLE, can be obtained by using the MLEs of the main and interaction contrasts from a standard logistic regression model. When the design matrix of Equation (1) is of full rank, we have E(Ŷ_MLE) = Y i.e., Ŷ_MLE is an unbiased estimate of Y.

Let Y_GJ denote the log odds obtained via an additive model under the GJ link. From the results of the previous section, the (i,j)-th element of Y_GJ can be written as μ + α_i + β_j + θα_iβ_j. Let Ŷ_GJ denote the MLE of Y_GJ. Note that the estimation of Ŷ_MLE involves estimating L₁L₂ parameters (baseline risk, L₁−1 and L₂−1 main effects contrasts for risk factors X and Z, respectively, and (L₁−1)×(L₂−1) interaction contrasts). However, the estimation of Y_GJ involves estimating only L₁ + L₂ parameters (baseline risk, L₁−1 and L₂−1 main effects contrasts for the risk factors X and Z, respectively, and the scalar parameter θ = −λ_G) i.e., (L₁−1)×(L₂−1) − 1 fewer parameters. As a result of this parsimony, Ŷ_GJ is likely to have a smaller standard error than Ŷ_MLE.

Suppose there is no interaction at all so that the model is additive under the logistic link. This is equivalent to an additive model under the GJ link with λ_G = 0. Therefore, in principle, we may estimate the log odds using Ŷ_GJ either when the interaction between the risk factors is removable or when there is no interaction at all. Denote E(Ŷ_GJ) = Y_GJ. When all the interactions are removable, we have E(Ŷ_GJ) = Y_GJ = Y. Otherwise, Ŷ_GJ will be biased, thereby offsetting any precision gains that may be attained by fitting a parsimonious model (i.e., a model with fewer parameters). In this case, Y should be estimated using Ŷ_MLE.

In any practical setting, we do not know at the outset whether an interaction exists, and if so whether it is removable or non-removable. Here we propose a Bayes estimator for Y that does not require us to test for specific types of interactions, yet allows us to take advantage of model parsimony either when the interaction is removable or when there is no interaction at all. Using Equation (3) and the result from the previous section, we can write Y as:

$$\mathbf{Y}={\mathbf{Y}}_{\text{GJ}}+\mathbf{e}.$$ (Equation 7)

We know how to estimate Y_GJ via Equation (6) using an additive model under the GJ link function (equivalently, Equation 2). If we can estimate e, we can plug this into Equation (7) to estimate Y. We shall obtain an estimator, e_B, so that a desirable criterion is optimized. We choose the squared error loss as our criterion, which is given by the quadratic form (e − e_B)^T (e − e_B). Below we derive an admissible estimator of e by minimizing the risk function, which is the expected value of this loss function.

A crude estimate of e is given by ê = Ŷ_MLE − Ŷ_GJ. Relying on the asymptotic normality of the MLEs, given e, ê has an asymptotic N(e, Σ) distribution, where Σ can be estimated as Σ̂ = Var(Ŷ_MLE − Ŷ_GJ). Denote this normal distribution as π(ê | e) (we do not show the variance Σ in this notation). We shall identify admissible estimators e_B as functions of ê.

The risk function is defined as the expected value of the loss function with respect to π(ê | e), and is given by ∫(e − e_B)^T (e − e_B)π(ê | e)dê. This risk function is also known as the mean squared error (MSE). There are several classes of estimators that focus on minimizing this risk function. One such class consists of minimax estimators [23,24]. However, obtaining a minimax estimator is not straightforward in practical scenarios. Therefore, we propose to obtain a class of Bayes estimators such that certain minimax estimators are members of this class under limiting conditions.

Denote ψ(e) as a prior probability density of e. At this time we do not make any assumption about the specific form of ψ(.). Note that the likelihood function, which is closely related to π(ê | e), does not contain any information about ψ(e). The posterior probability density of e given ê, denoted π(e | ê), can be written as:

$$\pi \left(\mathbf{e}\mid \widehat{\mathbf{e}}\right)=\frac{\pi \left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)\psi (\mathbf{e})}{\int \pi \left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)\psi (\mathbf{e})\mathrm{d}\mathbf{e}}.$$

Thus, ∫π(e | ê)de = 1, regardless of the specific form of ψ(e). We can now define the posterior risk as the expectation of the loss function with respect to the posterior density of e, written as: ∫(e − e_B)^T (e − e_B)π(e | ê)dê. We shall identify Bayes estimators e_B that minimize the posterior risk. Clearly, these estimators depend upon both ê and the prior density ψ(.).

Taking the derivative of the posterior risk with respect to e_B and setting it equal to 0 shows that the e_B(ê; ψ) = E_ψ(e | ê), i.e, the posterior risk is minimized by the posterior mean of e. We have used the notation E_ψ(.) to denote that the specific form of the posterior mean depends upon the specific form of the prior density ψ(.). Thus, e_B(ê; ψ) is a class of Bayes estimators, and different members of this class can be obtained by specifying different prior densities ψ(.).

The risk function corresponding to this class of Bayes estimators is given by: ∫{e − ê_B(ê; ψ)}^T {e − ê_B (ê; ψ)} π(ê | e)dê. If this risk is a constant for a particular ψ(.), then the corresponding Bayes estimator is also a minimax estimator [23]. Therefore, to identify an admissible estimator of e, all we need to do is to identify a prior density ψ(.) under which the risk is a constant. Although it may not be always possible to identify such a prior density, we may be able to identify a ψ(.) that provides a Bayes estimator with constant risk under some limiting conditions.

Consider independent and identical N(0, σ²) prior distributions for the components of e. Denote I as an identity matrix. Then it is easy to see that the posterior density π(e | ê) is normal with mean E_ψ(e | ê) = σ² (Σ + σ²I)⁻¹ê and variance σ²Σ(Σ + σ²I)⁻¹. Different choices of σ² will give different Bayes estimators under this prior. For a given σ², the posterior mean is unique. Denoting $\mathbf{K}={\sigma }^2{\left(\mathbf{\sum }+{\sigma }^2\mathbf{I}\right)}^{-1}={\left(\frac{\mathbf{\sum }}{{\sigma }^2}+\mathbf{I}\right)}^{-1}$ and Trace{.} as the trace of a matrix, the risk function can be written as:

$$\begin{array}{l}\int {\left\{\mathbf{e}-{\widehat{\mathbf{e}}}_{\mathrm{B}}\left(\widehat{\mathbf{e}};\psi \right)\right\}}^{\top }\left\{\mathbf{e}-{\widehat{\mathbf{e}}}_{\mathrm{B}}\left(\widehat{\mathbf{e}};\psi \right)\right\}\pi \left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)\mathrm{d}\widehat{\mathbf{e}}\\ =\int {\left\{\mathbf{e}-\mathbf{K}\widehat{\mathbf{e}}\right\}}^{\top }\left\{\mathbf{e}-\mathbf{K}\widehat{\mathbf{e}}\right\}\pi \left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)\mathrm{d}\widehat{\mathbf{e}}\\ ={\mathbf{e}}^{\top }\left\{{(\mathbf{I}-\mathbf{K})}^{\top }(\mathbf{I}-\mathbf{K})\right\}\mathbf{e}+\text{Trace}\left\{{\mathbf{K}}^{\top }\mathbf{K}\mathrm{\sum }\right\}\end{array}$$ (Equation 8)

The last step follows from linear algebra results for quadratic forms:

$$\begin{array}{l}\mathrm{E}\left({\widehat{\mathbf{e}}}^{\top }{\mathbf{K}}^{\top }\mathbf{K}\widehat{\mathbf{e}}\mid \mathbf{e}\right)=\mathrm{E}\left({\widehat{\mathbf{e}}}^{\top }\mid \mathbf{e}\right){\mathbf{K}}^{\top }\mathbf{K}\mathrm{E}\left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)+\text{Trace}\left\{{\mathbf{K}}^{\top }\mathbf{K}\text{Var}\left(\widehat{\mathbf{e}}\mid \mathbf{e}\right)\right\}\\ ={\mathbf{e}}^{\top }{\mathbf{K}}^{\top }\mathbf{K}\mathbf{e}+\text{Trace}\left\{{\mathbf{K}}^{\top }\mathbf{K}\mathrm{\sum }\right\}.\end{array}$$

When σ² → ∞, we have K → I and the limit of the Bayes estimator E_ψ(e | ê)is ê. It follows from Equation (8) that the risk function then approaches Trace{Σ}, which is a constant with respect to e. Hence, ê is a minimax estimator of e when σ² → ∞. When σ² → 0, we have K → 0 and the limit of the Bayes estimator E_ψ(e | ê) is 0 and the risk function is e^Te. Note that e → 0 since it has an a priori normal distribution with mean 0 and variance σ² that goes to 0. Thus, the risk function approaches 0, which is a constant. Therefore, 0 is a minimax estimator of e when σ² → 0.

These observations show that an a priori normal distribution for e provides a class of estimators, given by E_ψ(e | ê) = σ² (Σ + σ²I)⁻¹ ê, that are minimax under the limiting conditions when σ² → ∞ and σ² → 0, respectively. Plugging this estimator into Equation (7), a Bayes estimator of Y from this class, denoted Ŷ_B, is given by:

$$\begin{array}{l}{\widehat{\mathbf{Y}}}_{\mathrm{B}}={\widehat{\mathbf{Y}}}_{\text{GJ}}+\widehat{\mathrm{E}}\left(\mathbf{e}\mid \widehat{\mathbf{e}}\right)\\ ={\widehat{\mathbf{Y}}}_{\text{GJ}}+{\widehat{\sigma }}^2{\left(\widehat{\mathrm{\sum }}+{\widehat{\sigma }}^2\mathbf{I}\right)}^{-1}\widehat{\mathbf{e}}\\ ={\widehat{\mathbf{Y}}}_{\text{MLE}}-\widehat{\mathrm{\sum }}{\left(\widehat{\mathrm{\sum }}+{\widehat{\sigma }}^2\mathbf{I}\right)}^{-1}\widehat{\mathbf{e}}\end{array}$$ (Equation 9)

The last step of Equation (9) follows from the identities ê = Ŷ_MLE − Ŷ_GJ and Σ̂(Σ̂ + σ̂²I)⁻¹ + σ̂² (Σ̂ + σ̂²I)⁻¹ =I. Here we have used the notations σ̂² and Σ̂ to denote estimated values of these variances. An empirical estimate of σ² can be obtained as ${\widehat{\sigma }}^2=\frac{{\widehat{\mathbf{e}}}^{\top }\widehat{\mathbf{e}}}{{\mathrm{L}}_1\times {\mathrm{L}}_2}$. The calculation of Σ̂ and an approximate formula for the variance of Ŷ_B are given in the Online Supplementary Material.

Properties of the proposed Bayes estimator

In order to obtain Ŷ_B, it is not necessary to know whether an interaction exists or whether it is removable or non-removable. However, if there is an underlying interaction, Equation (9) suggests that Ŷ_B will have the following properties depending upon whether the interaction is removable or non-removable. When the interaction is removable, e will be negligible i.e., σ² will be near 0. Hence, Ŷ_B ≈ Ŷ_GJ. Since 0 is a minimax estimator of e in this limiting case, Ŷ_GJ is a minimax estimator of Y when the interaction is removable. When the interaction is non-removable, there is the implication that the true values of the components of e are not equal to 0. Equivalently, the prior variance σ² is not negligible. In the limiting case, as σ² → ∞, we have Ŷ_B ≈ Ŷ_MLE. Since ê is minimax in this limiting case, Ŷ_MLE is a minimax estimator of Y when the interaction is non-removable.

When 0 < σ² < ∞, π(e) is a proper prior i.e., $\underset{-\infty }{\overset{\infty }{\int }}\pi \left(\mathbf{e}\right)d\mathbf{e}=1$. For a given σ², the posterior mean, i.e., the Bayes estimator, is unique. Since a unique Bayes estimator under a proper prior is also admissible [24, Chapter 4.3], Ŷ_B is an admissible estimator when 0 < σ² < ∞. However, since σ² is unknown, estimation of this parameter may impact the admissibility property of the Bayes estimator. In the limit when σ² → 0 or σ² → ∞, we have an improper prior. A Bayes estimator under an improper prior can be inadmissible. In fact, in the limiting cases we obtain minimax estimators that are admissible only when (L₁−1)×(L₂−1) ≤ 2 [24, Chapter 4.5]. In the following section, we examine the properties of the proposed Bayes estimator using simulations.

Simulation plan

We conducted simulation studies to examine the bias-variance trade-offs of Ŷ_MLE, Ŷ_GJ, and Ŷ_B. We independently sampled case-control data with two ordinal risk factors, X with L₁ levels and Z with L₂ levels. We assumed a threshold model for disease risk in the following sense: there are thresholds C₁ and C₂ for the levels of X and Z, respectively, such that X and Z confer disease risk only if their levels exceed these values. Thus, the disease probability of each individual was assumed to follow a logistic regression model given by:

$$\mathit{logit}\left\{P\left(\mathit{disease}\mid X,Z\right)\right\}={\delta }_{\mathit{0}}+{\delta }_{\mathit{1}}\times I\left(X\ge C_{\mathit{1}}\right)+{\delta }_{\mathit{2}}\times I\left(Z\ge C_{\mathit{2}}\right)+{\delta }_{\mathit{12}}\times I\left(X\ge C_{\mathit{1}},Z\ge C_{\mathit{2}}\right),$$ (Equation 10)

where I(.) is the indicator function. The threshold type of model for generating disease risk is motivated by several practical scenarios where public health messages about disease risk are delivered based on thresholds or cutpoints for risk factors – for example, although every unit increase in body mass index (BMI) may contribute to an increase in the risk of endometrial cancer, public health messages about risk are generally quoted for categories such as normal, overweight and obese, based on relevant cutpoints for BMI.

We considered the following three settings:

1. 2×3 table: L₁ = 2 and L₂ = 3, with C₁ = 2 and C₂ = 3;

2. 2×5 table: L₁ = 2 and L₂ = 5, with C₁ = 2 and C₂ = 4; and

3. 5×5 table: L₁ = 5 and L₂ = 5, with C₁ = 4 and C₂ = 4.

For the risk factor prevalence, we assumed P(X ≥ C₁) = 0.10 = P(Z ≥ C₂). Given C₁, the specific levels of X were simulated with probability P(X=C₁) = P(X=C₁+1) = … = P(X=L₁) = 0.10/(L₁−C₁+1), and P(X=1) = P(X=2) = … = P(X = C₁−1) = 0.90/(C₁−1). The levels of Z were obtained in a similar manner. We considered the magnitude of δ₀ to be such that the disease prevalence was 0.10.

In previous work we showed that, given δ₁ and δ₂, an interaction is removable or non-removable when δ₁₂ falls in a certain range [13]. In particular, the data contain:

1. only removable interaction when δ₁₂ ≥ max{ −δ₁, −δ₂};

2. only non-removable interaction when δ₁₂ ≤ min{−δ₁, −δ₂}; and

3. some removable and some non-removable interactions when min{−δ₁, −δ₂} < δ₁₂ < max{−δ₁, −δ₂}.

Another type of interaction, known as compositional epistasis, occurs when the effect of a genetic marker at one locus is masked by a variant at another locus [25]. Thus, under compositional epistasis, we have δ₁ = 0 = δ₂ and δ₁₂ ≠ 0. We simulated data under the following parametric configurations:

1. Removable interaction: δ₁ = δ₂ = log(2) = 0.6931, δ₁₂ = {log(1.1), log(1.3), log(1.5), log(1.7), log(1.9), log(2), log(2.5), log(3)} i.e., δ₁₂ ≥ max{−δ₁, −δ₂};

2. Non-removable interaction: δ₁ = δ₂ = log(2) = 0.6931, δ₁₂ = {−log(2), −log(2.1), −log(2.3), −log(2.5), −log(2.7), −log(2.9), −log(3), −log(5), −log(7)} i.e., δ₁₂ ≤ min{−δ₁, −δ₂};

3. Some removable and some non-removable interactions: δ₁ = log(2), δ₂ = 0, and min{−δ₁, −δ₂} ≤ δ₁₂ ≤ max{−δ₁, −δ₂} i.e., δ₁₂ = {−log(1.25), −log(1.5), −log(1.75), −log(2), −log(2.25), −log(2.5), −log(2.75)}

4. Compositional epistasis: δ₁ = δ₂ = 0, δ₁₂ = {log(1.5), log(2), log(2.5), log(3), log(5), log(7)}.

We generated 1000 case-control data sets, each consisting of 1000 cases and 1000 controls, under each parametric configuration using the true model given by Equation (10). When analyzing the data sets, we assumed that we did not know the true model. To estimate Ŷ_B, we first estimated Ŷ_MLE using Equation (1) with L₁−1 and L₂−1 main effects contrasts for X and Z, respectively, and (L₁−1)×(L₂−1) interaction contrasts. Next, we estimated Ŷ_GJ using Equation (6) by fitting an additive model under the GJ link with L₁−1 and L₂−1 main effects contrasts for X and Z, respectively, and a scalar θ = −λ_G to represent the transformation parameter. Finally, we estimated Ŷ_B using Equation (9). We calculated the variances of these estimates and their MSEs as the sum of the variances of the log odds and the squared difference between the true (i.e., observed) and the estimated log odds, calculated for each risk factor sub-class and averaged over all the sub-classes. We calculated the root mean squared errors (RMSEs) as the square root of the MSEs. We summarized the interquartile range (IQR) and average of the RMSEs over the 1000 simulated data sets under each parametric configuration.

RESULTS

Simulation

The simulation results are illustrated in Figures 2 and 3. When the interaction was removable (i.e., there was no non-removable interaction), as expected Ŷ_GJ had smaller RMSE than Ŷ_MLE (Column 1 of Figure 2). When the interaction was non-removable (i.e., there was no removable interaction), as expected Ŷ_MLE had smaller RMSE than Ŷ_GJ (Column 2 of Figure 2). However, for large contingency tables (for example, a 5×5 table), the RMSE of Ŷ_MLE was larger than that of Ŷ_GJ for interaction effects of small magnitude. This is because obtaining Ŷ_MLE required estimation of 16 parameters even when the magnitude of the interaction effect was negligible, resulting in loss of efficiency. However, the RMSE of Ŷ_GJ increased rapidly and was close to that of Ŷ_MLE as the magnitude of the interaction effect increased. When the data contained some removable and some non-removable interactions, Ŷ_GJ had smaller RMSE than Ŷ_STD (Column 3 of Figure 2). Under compositional epistasis, Ŷ_GJ had higher RMSE than Ŷ_MLE, particularly when the magnitude of the interaction effect (δ₁₂) was large. In contrast, Ŷ_B had the smallest RMSE, for the most part, under all the scenarios, demonstrating the remarkable bias-variance trade off attained by the Bayes estimation method. The trade off was best realized for larger contingency tables, i.e., for larger values of (L₁−1)×(L₂−1).

Figure 2.

Simulation results showing the comparison of mean squared errors (MSE) of the three estimators Ŷ_STD (standard logistic; black line with “S”), Ŷ_GJ (additive GJ; purple line with “G”), and Ŷ_B (Bayes; red line with “B”) for data simulated under removable interactions (Column 1), non-removable interactions (Column 2), some removable and some non-removable interactions (Column 3), and compositional epistasis (Column 4), with the risk factors based on 2×3 (Row 1), 2×5 (Row 2), and 5×5 (Row 3) contingency tables.

Figure 3.

Simulation results of the interquartile ranges (IQRs) of the MSEs of Ŷ_STD (standard logistic; black line with “S”), Ŷ_GJ (additive GJ; purple line with “G”), and ŶB (Bayes; dashed and red line with “B”) for data simulated under removable interactions (Column 1), non-removable interactions (Column 2), some removable and some non-removable interactions (Column 3), and compositional epistasis (Column 4), with the risk factors based on 2×3 (Row 1), 2×5 (Row 2), and 5×5 (Row 3) contingency tables.

Figure 2 also illustrates the admissibility properties of the various estimators. Consider the first row of Figure 2, which corresponds to 2×3 contingency tables i.e., (L₁−1)×(L₂−1) = 2. We noted earlier that, in this case, Ŷ_GJ is minimax and admissible when the interaction is removable, and Ŷ_MLE is minimax and admissible when the interaction is non-removable. Thus, as expected, Ŷ_GJ and Ŷ_MLE had smaller MSE than Ŷ_B when the interactions were removable and non-removable, respectively (Columns 1 and 2 in Row 1 of Figure 2). There was only a modest increase in the RMSE of Ŷ_B, with the difference between the RMSEs of Ŷ_B and the minimax estimators falling between 0.001 and 0.028. For large contingency tables (for example, 5×5 tables), Ŷ_B had the smallest MSE under a wide range of parametric configurations considered. For example, in a 5×5 contingency table with removable interactions, the RMSEs of Ŷ_MLE, Ŷ_GJ, and Ŷ_B were in the range 0.60–0.62, 0.52–0.54, and 0.50–0.51, respectively, under various values of δ₁₂ considered in our simulations.

The interquartile ranges (IQRs) are shown in Figure 3. The RMSE of Ŷ_GJ had the largest IQR and that of Ŷ_MLE had the smallest IQR under all the scenarios considered, suggesting that the empirical distribution of the RMSE of Ŷ_GJ has a wider spread relative to the distributions of the RMSEs of Ŷ_MLE and Ŷ_B. The RMSE of Ŷ_B had IQRs closer to those of the RMSE of Ŷ_MLE. For example, in a 5×5 contingency table with removable interactions, the IQRs of the RMSEs of Ŷ_MLE, Ŷ_GJ, and Ŷ_B were in the range 0.04–0.05, 0.10–0.11, and 0.06–0.07, respectively.

In summary, these results suggest that our proposed Bayes estimator is a useful approach for estimating the log odds summaries, regardless of the type of interaction and regardless of the size of the contingency table.

Data applications – three case-control studies of endometrial cancer

Application 1: BMI, CYP19A1, and endometrial cancer

A case-control study within the Epidemiology of Endometrial Cancer Consortium [17] reported a significant interaction between SNP rs727479 in the CYP19A1 gene (two levels: CC, and AC or AA genotypes) and body mass index (BMI; three levels: normal, overweight and obese) among post-menopausal women (age ≥ 55 years) based on a logistic regression analysis. These data are shown in Table 1.

Table 1. Results for the Setiawan et al [17] data.

Columns 1 and 2 indicate the risk factor classes and jointly identify the sub-classes. Columns 3 and 4 provide the number of cases and controls. The estimated log odds and their standard errors (in parentheses) corresponding to Ŷ_STD, Ŷ_GJ, and Ŷ_B are given in Columns 5, 6, and 7, respectively. The last row shows the root mean squared errors of the three estimation methods. For the additive model under the GJ link, the transformation parameter is λ̂_G = −2.91.

BMI	rs727479	Cases	Controls	ŶSTD (SE)	ŶGJ(SE)	ŶB (SE)
Normal	CC	143	328	−0.8302 (0.1002)	−0.9339 (0.0505)	−0.8575 (0.1010)
Overweight	CC	78	175	−0.8081 (0.1361)	−0.6443 (0.0396)	−0.7613 (0.1284)
Obese	CC	72	101	−0.3385 (0.1542)	−0.3362 (0.1411)	−0.3579 (0.1531)
Normal	AC/AA	1004	2475	−0.9022 (0.0374)	−0.8848 (0.0343)	−0.8749 (0.0380)
Overweight	AC/AA	874	1456	−0.5104 (0.0428)	−0.5174 (0.0425)	−0.5572 (0.0426)
Obese	AC/AA	881	758	0.1504 (0.0495)	0.1503 (0.0495)	0.1699 (0.0496)
Root Mean Squared Error				0.098	0.106	0.102

For illustrative purposes, it will be useful to understand the properties of this interaction, although our proposed Bayes estimation method does not require preliminary tests for removable and non-removable interactions. Using the test statistics outlined in the Online Supplementary Material, there was significant evidence for removable and non-removable interactions between BMI and CYP19A1 at 5% significance level for each test (test statistic for removable interaction = 7.64, degrees of freedom = 1, p-value = 0.006; test statistic for non-removable interaction= 4.85, degrees of freedom = 1, p-value = 0.03). In the AC/AA genotype group, the observed log odds increased with increasing level of BMI. Although an increasing trend occurred in the CC genotype group, the normal and overweight individuals with CC genotype had fairly similar log odds. Further, the observed log odds were higher for the AC/AA genotypes relative to the CC genotypes among the overweight and obese individuals, but not among those with normal BMI. These observations suggest a lack of strict monotonicity, which is consistent with the presence of a non-removable interaction.

The estimated log odds are shown in Table 1 [See Supplementary Figure 1 for a visual representation of these estimates]. For these data, the model based on Equation (1) is a saturated model since we do not have any additional risk factors for consideration in these analyses. Therefore, Ŷ_MLE is the same as the observed log odds. For Ŷ_GJ, the estimated log odds for the normal and overweight BMI categories in the CC genotype group were not close to the observed values, possibly due to the non-monotonic interaction. The Bayes estimates Ŷ_B were closer to Ŷ_MLE, demonstrating shrinkage towards estimates based on the standard logistic regression. It is of interest that the standard errors of Ŷ_GJ were smaller than those of Ŷ_MLE, but the RMSE is a better indicator of estimate variability. All three estimators had fairly similar RMSE (0.098 for Ŷ_MLE, 0.106 for Ŷ_GJ and 0.102 for Ŷ_B) which is consistent with the results of the simulation study that shows that when monotonicity is not strict, as in compositional epistasis (Column 4 in Row 1 of Figure 2), the RMSEs of Ŷ_MLE and Ŷ_B are similar and are, for the most part, slightly smaller in magnitude than that of Ŷ_GJ.

Application 2: BMI, diabetes, and endometrial cancer

This case-control study [18] reported a significant interaction between BMI (three levels: normal, overweight, and obese) and diabetes (two levels: present and absent). The log odds increased with increasing level of BMI both in the absence and presence of diabetes (Table 2). Further, the log odds were higher in the presence of diabetes, regardless of the level of BMI. These observations illustrate strictly monotonic properties of the data. At 5% significance levels, there was significant evidence for a removable interaction (test statistic = 4.46, degrees of freedom = 1, p-value = 0.035) and no significant evidence for a non-removable interaction (test statistic = 0.280, degrees of freedom = 1, p-value = 0.60). Ŷ_GJ had smaller standard errors and smaller RMSE than Ŷ_MLE (RMSE = 0.153 and 0.199 for Ŷ_GJ and Ŷ_MLE, respectively). In the presence of diabetes, the standard errors of Ŷ_B for different levels of BMI were between those of Ŷ_MLE and Ŷ_GJ. In the absence of diabetes, the standard errors of all the estimators for different levels of BMI were small. The RMSE of Ŷ_B was 0.183, which lies between the RMSEs of Ŷ_GJ and Ŷ_MLE. This is also consistent with the results of the simulation study (Column 1 in Row 1 of Figure 2), which demonstrated that Ŷ_GJ, a minimax estimator when the interaction is removable and there is no non-removable interaction, is also admissible when (L₁−1)×(L₂−1) = 2, but Ŷ_B is still a useful estimator in the sense that its RMSE is smaller than that of Ŷ_MLE. [See Supplementary Figure 2 for a visual representation of Ŷ_MLE, Ŷ_GJ, and Ŷ_B.]

Table 2. Results for the Shoff and Newcomb [18] data.

Columns 1 and 2 indicate the risk factor classes. Columns 3 and 4 provide the number of cases and controls. The estimated log odds and their standard errors (in parentheses) corresponding to Ŷ_STD, Ŷ_GJ, and Ŷ_B are given in Columns 5, 6, and 7, respectively. The last row shows the root mean squared errors of the three estimation methods. For the additive model under the GJ link, the transformation parameter is λ̂_G = −1.07.

BMI	Diabetes	Cases	Controls	ŶSTD (SE)	ŶGJ (SE)	ŶB (SE)
Normal	Absent	373	1633	−1.4766 (0.0574)	−1.4827 (0.0558)	−1.4985 (0.0650)
Overweight	Absent	85	262	−1.1257 (0.1248)	−1.0917 (0.1053)	−1.0634 (0.1090)
Obese	Absent	178	253	−0.3516 (0.0978)	−0.3555 (0.0955)	−0.3921 (0.1061)
Normal	Present	20	81	−1.3987 (0.2497)	−1.2957 (0.0722)	−1.3768 (0.2471)
Overweight	Present	16	31	−0.6614 (0.3078)	−0.7912 (0.1826)	−0.7237 (0.2397)
Obese	Present	51	31	0.4978 (0.2277)	0.5063 (0.2232)	0.5383 (0.2264)
Root Mean Squared Error				0.199	0.153	0.187

Application 3: Tea intake, CYP19A1, and endometrial cancer

This case-control study, the Shanghai Endometrial Cancer Study [19], reported a significant interaction between tea intake (two levels: low and high intake) and CYP19A1 genotype based on the SNP rs1065779 (three levels: GG, GT, and TT genotypes). The observed log odds decreased monotonically with increasing number of the T allele among those with high tea intake (Table 3). However, there was no monotonic trend among individuals with low tea intake, suggesting that the interaction was not removable (test statistic for removable interaction = 1.02, degrees of freedom = 1, p-value = 0.31). Indeed the data contained evidence for significant non-monotonic interaction at 5% significance level (test statistic = 10.65, degrees of freedom = 1, p-value = 0.001). Hence, a transformation to additivity is not appropriate for these data. As expected, Ŷ_B was closer to the observed log odds than Ŷ_GJ, which also illustrates shrinkage towards Ŷ_MLE due to the presence of significant non-monotonic interaction. Further, Ŷ_B had RMSE comparable to that of Ŷ_MLE (RMSE = 0.129, 0.178, and 0.136 for Ŷ_MLE, Ŷ_GJ, and Ŷ_B, respectively). This is also consistent with the results of the simulation study (Column 2 in Row 1 of Figure 2), which showed that the minimax estimator Ŷ_MLE is also admissible when (L₁−1)×(L₂−1) = 2. [See Supplementary Figure 3 for a visual representation of Ŷ_MLE, Ŷ_GJ, and Ŷ_B.]

Table 3. Results for the Xu et al [19] data.

Columns 1 and 2 indicate the risk factor classes. Columns 3 and 4 provide the number of cases and controls. The estimated log odds and their standard errors (in parentheses) corresponding to Ŷ_STD, Ŷ_GJ, and Ŷ_B are given in Columns 5, 6, and 7, respectively. The last row shows the root mean squared errors of the three estimation methods. For the additive model under the GJ link, the transformation parameter is λ̂_G = 0.01.

rs1065779	Tea intake	Cases	Controls	ŶSTD (SE)	ŶGJ (SE)	ŶB (SE)
GG	Low	211	226	−0.0687 (0.0957)	0.0636 (0.0847)	−0.0205 (0.0994)
GT	Low	382	322	0.1709 (0.0757)	0.0981 (0.0693)	0.1144 (0.0759)
TT	Low	126	153	−0.1942 (0.1203)	−0.2193 (0.1056)	−0.1858 (0.1194)
GG	High	117	90	0.2624 (0.1402)	−0.0196 (0.1019)	0.2142 (0.1452)
GT	High	148	171	−0.1445 (0.1123)	0.0149 (0.0909)	−0.0880 (0.1117)
TT	High	45	65	−0.3677 (0.1939)	−0.3027 (0.1235)	−0.3760 (0.1925)
Root Mean Squared Error				0.129	0.178	0.136

COMPUTATIONAL GUIDANCE FOR PRACTITIONERS

In this paper we have developed a Bayes estimator for log odds, which can be calculated using the following steps:

1. Obtain Ŷ_MLE as the right hand side of Equation (1) by plugging in the MLEs of that model.

2. Obtain Ŷ_GJ as the right hand side of Equation (6) by plugging in the MLEs of an additive model under the GJ link function.

3. Obtain Σ̂ and σ̂² (formula given in Online Supplementary Material).

4. Obtain Ŷ_B using Equation (9).

5. Obtain their standard errors (formula given in Online Supplementary Material).

We have prepared a computer program to implement these methods using the R programming language (http://cran.r-project.org). This program, along with instructions for use, can be downloaded freely from the first author’s academic web page: (https://www.mskcc.org/biostatistics/~satagopj). This program takes as input the case-control status and the values of the two risk factors of interest. The output contains Ŷ_MLE, Ŷ_GJ, and Ŷ_B, their standard errors and RMSEs.

DISCUSSION

There is a large body of literature on estimating interaction effects and on conducting a hypothesis test for the presence of a significant interaction for binary traits [2–4]. In contrast, the emphasis of our work is on accurate and precise estimation of the log odds parameters. Our work is based on the thesis that, when there is removable interaction under the logistic link function, the model relating the binary disease trait and the risk factors is additive on some scale of risk. We have shown that the GJ link function is an appropriate scale for additivity. Under the GJ link function, disease risk increases (or decreases) at a higher rate than that under an additive logistic link function (Figure 1). When this happens, it means that interaction terms will be needed in a logistic regression model to obtain a better fit to the data. In contrast, disease risk may be characterized accurately and more parsimoniously using an additive model under the GJ link. This would provide a better fitting model, and would facilitate accurate and more precise estimation of epidemiologic parameters such as log odds, especially in the extremes of the risk distribution, using fewer parameters. To attain this, we have developed a Bayes estimator that exploits model parsimony while simultaneously accounting for potential non-removable interactions. Our simulations show that this method has remarkable bias-variance trade-off under a wide range of parametric configurations and is, hence, a valid method for use in practical settings.

All our empirical examples are case-control data from 2×3 contingency tables. Even in this small contingency table setting, the proposed Bayes estimation approach has good bias-variance trade-off. This can also be observed in the second example [18], where the data exhibit strict monotonicity properties, suggesting that the interaction between BMI and diabetes in this study is removable. When diabetes is present, the sample size in this data set is modest for all three levels of BMI. Even in this setting, the RMSE of the Bayes approach is intermediate between those of the additive GJ model and the standard logistic model, though considerable reduction in RMSE would be attained for larger contingency tables as seen in our simulations.

The topic of interaction continues to garner much attention in epidemiology. There is a long-standing debate as to whether interactions should be examined under a logistic link (referred to in epidemiology as multiplicative interaction) or under an identity link, i.e., on the scale of disease risk (referred to as additive interaction), since the latter is anticipated to be more relevant from a public health perspective [8]. To some extent, this has also led to an anticipation that additive interactions can help understand biological interactions underpinning disease risk. However, biological interactions can occur regardless of the presence of a statistical interaction [12, 26]. Further, the absence of a statistical interaction on one scale of the outcome may imply its presence on another scale. Therefore, finding a parsimonious model, even if it is not on the logistic or the identity scale, should be useful for obtaining practically relevant insights about the risk factors in relation to disease etiology [27, 28].

We set out to obtain an admissible estimator of log odds by minimizing the expected value of the squared error loss function, i.e., the risk function. This would provide minimax estimators, but minimizing the risk function is not straightforward. Therefore, we minimized the posterior risk function to obtain Bayes estimators. Our efforts to obtain a Bayes estimator that has a constant risk in the limit resulted in a normal prior distribution for the parameters e. This turns out to be a conjugate prior when the MLE ê has a normal distribution, as expected asymptotically. This choice of a normal prior is not motivated by its property of being a conjugate prior. Instead, it is a natural choice of prior to obtain a class of Bayes estimators that include minimax estimators as its members in the limit [23, 24]. It is not necessary to know what the parameters of this normal distribution are, and we estimate the prior variance empirically from our observed data. Although our proposed Bayes estimator is based on the squared error loss function, other criteria (for example, the absolute difference between e and e_B) may also be considered. A comprehensive evaluation of other optimality criteria is outside the scope of this paper.

This methodology can be extended to accommodate models adjusting for additional variables. We briefly outline this here. If we are solely interested in the main effect of an additional variable, but not in its interaction with the other risk factors, then the main effect, denoted δ_k, for this additional variable can be added to the right hand sides of Equations (1) and (2). In other words, if the interaction between the risk factors of interest is removable, then the effects of these risk factors can be represented parsimoniously as the logarithm of their additive effects, as shown in the right hand side of Equation (6) using arguments pertaining to transformation to additivity. The main effect of the new variable can be added to the right hand side of Equation (6). Further approximation would lead to Equation (2). However, suppose we are also interested in the interaction between this new variable and the other risk factors. If all the pair-wise interactions are removable, then the corresponding interaction effects may be written parsimoniously as θα_iβ_j, θα_iδ_k, θβ_jδ_k etc. A comprehensive evaluation of the operating characteristics of these extensions, including evaluation of higher order interactions, will be pursued elsewhere.

Whether the precision gains of our proposed Bayes estimator of log odds leads to better risk prediction remains an open question. Further research is needed to measure and evaluate the discriminatory or predictive performance of our proposed approach, and to quantify the statistical significance of the improvements in the performance. Another important use of the log odds parameters (equivalently, odds ratios) is for projecting the benefits of interventions in screening studies. Further research is also needed to evaluate the accuracy of the projected benefits of interventions based on the log odds estimated via our proposed method.