Here, we describe the methodology associated with the bivariate and multivariate mixture models utilized in this exploration. These methods are implemented via the R package R2WinBUGS which calls WinBUGS from R [

32,

33,

34,

35]. All models reached convergence with two chains for the Markov chain Monte Carlo sampler running for a total burn-in of 45,000 iterations and a sample of 5000. Trace plots of the deviance and some random effect standard deviations, as well as model BUGS code, are included in the

supplemental materials.

We are interested in exploring the relationships between the two or three cancers in this bivariate or multivariate setting via a mixture model that was first introduced in Carroll et al. [

36] as a form of model selection. These methods were then further improved and tested via a simulation study in Lawson et al. [

2]. The general structure of this model for disease

$k$ in county

$i$ at time

$j$ is as follows:

where

${\alpha}_{0k}~Norm\left(0,{\tau}_{a}^{-1}\right)$ is an intercept,

${p}_{ik}^{h}$ is a mixture parameter, and

${M}_{ijk}^{h}$ is a mixture component. In the previous study, the set of mixture parameters and components was given by

$h=\left\{S,ST\right\}$, thus containing a spatial and spatio-temporal parts. However, the temporal effect appeared to be causing model fit issues with oral/pharynx in the multivariate setting as this parameter was shared between diseases. Thus, in this paper we wish to explore the benefit of either keeping the same mixture component structure and allowing the temporal effect to vary between diseases or introducing a third mixture component for temporal effects. Hence, the three alternative mixture definitions for both the univariate and bivariate/multivariate cases are displayed in

Table 2. The parameters displayed are such that

${X}_{i}$,

${X}_{j}$, and

${X}_{ij}$ are the

${i}^{th}$,

${j}^{th}$, and

$i{j}^{th}$ values of the spatial, temporal, and ST covariates, respectively,

$\beta ~N\left(0,{\tau}_{\beta}^{-1}\right)$ for the general case are the fixed effect parameter estimates,

${u}_{ik}~N\left(0,{\tau}_{uk}^{-1}\right)$ is the uncorrelated spatial random effect.

${v}_{i}~N\left(\frac{1}{{n}_{i}}{\displaystyle \sum}_{i~l}{v}_{l},\frac{1}{{n}_{i}{\tau}_{v}}\right)$ is a correlated spatial random effect following an intrinsic conditional autoregressive (CAR) model, where

${n}_{i}$ is the number of neighbors for county

$i$, and

$i~l$ indicates that the two counties

$i$ and

$l$ are neighbors (

$i\ne l$) [

37,

38].

${\gamma}_{j}~N\left({\gamma}_{j-1},{\tau}_{\gamma}^{-1}\right)$ is the temporal random effect modeled by a temporal random walk prior,

${\varphi}_{ijk}~N\left(0,{\tau}_{{\varphi}_{k}}^{-1}\right)$ is the uncorrelated ST interaction term, and all precisions are defined such that

${\tau}^{-1/2}~Unif\left(0,C\right)$. The random effects included here are typical of ST models such as the commonly applied Knorr-Held model [

13,

14]. Finally, the mixture parameter is made up of two parameters,

${z}_{i}$ and

${a}_{i}$ (univariate) or

${z}_{ik}$ and

${a}_{ik}$ (multivariate), that are correlated and uncorrelated in space, respectively. However, when three mixture components and parameters are needed, as in Alternative 2 (Alt2), constraints must be imposed such that the sum of the three mixture parameters is equaled to one. This is accomplished with the following structure:

$\mathrm{logit}\left({q}_{i}^{h}\right)={z}_{i}^{h}+{a}_{i}^{h}$ and

${p}_{i}^{h}={q}_{i}^{h}/{\displaystyle \sum}_{h}{q}_{i}^{h}$ or

$\mathrm{logit}\left({q}_{ik}^{h}\right)={z}_{ik}^{h}+{a}_{ik}^{h}$ and

${p}_{ik}^{h}={q}_{ik}^{h}/{\displaystyle \sum}_{h}{q}_{ik}^{h}$, where

$h=\left\{S,T,ST\right\}$.

The two correlated random effects,

${v}_{i}$ and

${\gamma}_{j}$, are defined such that they are common between diseases when there is no

$k$ subscript written. This is how information is shared between diseases of interest in these mixture models. Thus, Alternative 1 (Alt1) allows the temporally-correlated random effect to vary across diseases while the spatially-correlated random effect is shared amongst diseases. In the univariate setting, Alt1 is the same as F2PRED from the previous study [

2], and the multivariate version of F2PRED offered the best fit for oral/pharynx. On the other hand, Alt2 has the interpretation that the temporally correlated effect is common between diseases, but the relationship between disease and temporal effect is different depending on the cancer of interest through the addition of

${p}_{ik}^{T}$. Finally, Alternative 3 (Alt3) is like Alt1, but it allows for a shared temporal random effect that is scaled by a function which is related to the disease of interest. Alt3 was only applied in the bivariate and multivariate settings as an attempt to further improve model fits by offering even more temporal flexibility. The two functions, leading to Alt3a and Alt3b, are such that:

Thus, Alt3a offers a scaling coefficient that differs per disease to relate to the temporal random effect while Alt3b offers a power term that differs per disease for relating to the temporal random effect. These alternatives offer different ways of incorporating more model flexibility relating to the temporal random effect. For notation, a “U,” “B,” or “M” following Alt1, Alt2, Alt3a, or Alt3b indicates that these models are fit in the univariate (each disease fitted separately), bivariate (only considers oral/pharynx with lung), or multivariate setting.