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**|**HHS Author Manuscripts**|**PMC3892667

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- Summary
- 1. Introduction
- 2. Latent Multilevel Multidimensional Normal Regression Model
- 3. Prior Elicitation and Model Inference
- 4. Simulation Study of Potential efficiency Gains
- 5. Analysis of the Trial of Activity in Adolescent Girls Data
- 6. Discussion
- Supplementary Material
- References

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Stat Med. Author manuscript; available in PMC 2014 January 15.

Published in final edited form as:

PMCID: PMC3892667

NIHMSID: NIHMS505367

In this paper, we propose a Bayesian method for Group Randomized Trials (GRTs) with multiple observation times and multiple outcomes of different types. We jointly model these outcomes using latent multivariate normal linear regression, which allows treatment effects to change with time and accounts for 1.) intra-class correlation (ICC) within groups 2.) the correlation between different outcomes measured on the same subject and 3.) the over-time correlation (OTC) of each outcome. Moreover we develop a set of innovative priors for the variance components which yield direct inference on the correlations, avoid undesirable constraints, and allow utilization of information from previous studies. We illustrate through simulations that our model can improve estimation efficiency (lower posterior standard deviations) of ICCs and treatment effects relative to single outcome models and models with diffuse priors on the variance components. We also demonstrate the methodology using body composition data collected in the Trial of Activity in Adolescent Girls (TAAG).

In many public health studies, the goal is to evaluate interventions intended to improve the health of communities through education and promotion of healthy behaviors. For example, some studies have examined interventions to increase physical activity levels in a school [1] or a workplace [2] and some have aimed to increase cancer screening rates in underserved populations [3]. The Cluster or Group Randomized Trial (GRT) is regarded as the gold standard for evaluating interventions delivered to communities [4,5]. Groups, rather than individuals, are randomized to each treatment arm with randomization potentially stratified by factors believed to affect the outcome variable. In community-based research, GRTs are usually the only way to test an intervention without risk of bias due to contamination, which occurs when important components of the intervention find their way into the control condition. GRTs are also common in primary care research in which the goal is to change physician behavior and patient management in hopes of improving patient outcomes, such as secondary prevention of coronary heart disease [6].

The statistical implications of GRTs were first brought to light by Cornfield [7]. He identified two important penalties of this experimental design. First, members within the same group often have physical, social, or geographical connections with each other, so their responses are likely correlated. This correlation is called the Intra-class Correlation (ICC), and it is positive in most situations. Failure to account for the ICC in analysis can result in inflated type-I error [8-10] and failure to account for the ICC in sample size calculations can result in an under-powered trial [4,5,11]. The second penalty identified by Cornfield was limited degrees of freedom; the number of groups randomized to each condition is usually small, which limits the degrees of freedom available for hypothesis testing and for estimating between-group variation. For example, Varnell, Murray, and Baker [12] pointed out that in group randomized trials published between 1998 and 2002, 47% had fewer than 10 groups per condition. Thus, in a single study there is usually little information about the between-group variation and hence the ICC. In addition to these two penalties, nowadays studies collect data on multiple outcomes at several observation times. Such studies need a statistical model that can account for all these correlations: correlation due to clustering (the ICC), correlation between different outcomes measured on the same individual, and overtime correlation (OTC) of each outcome while allowing treatment effects to change with time.

Our work is motivated by the Trial of Activity in Adolescent Girls (TAAG), which was a school-based GRT that had all the above characteristics. The goal of TAAG was to promote physical activity among middle school girls. In the study, three different cross-sectional samples were obtained from each participating school: a sample of 6th grade girls was drawn at baseline and samples of 8th grade girls were drawn 2 and 3 years after the intervention was introduced. At each time point, multiple outcomes were measured on consented girls including percent body fat, activity level, BMI, and attitudes about PE class, as determined by questionnaire. A more detailed description of the design of the TAAG study is provided by Stevens et al. [1].

Most of the analyses of GRTs in the literature focus on single outcome models. Murray [4], Donner and Klar [5], and Hayes and Moulton [11] provided comprehensive reviews of frequentist approaches which use random effect models to account for ICC in cross-sectional and longitudinal analyses [4,5,11]. Bayesian hierarchical models have also been proposed for continuous [13] and binary [14,15] outcomes, though these papers only address cross-sectional data. When multiple highly correlated outcomes are collected on each subject, joint modeling seems more sensible than fitting separate single outcome models. Studies in the field of reproductive and developmental toxicology have confirmed the advantages of joint modeling. Gueorguieva and Agresti [19] showed that joint modeling of multiple outcomes (birth weight and presence of malformations in rat pups) can provide significant efficiency gains when the outcomes are highly correlated and the sample size is not too large. General models for mixtures of discrete and continuous outcomes have been studied by Catalano and Ryan [20] and Rochon [21] from a generalized estimating equation perspective, by Dunson [22] from a Bayesian perspective, and by Regan and Catalano [23] and Gueorguieva and Agresti [19] from a maximum likelihood perspective. Very recently, Daniels and Normand [24] and Dunson [25] proposed correlated probit models for joint modeling of longitudinal data with multiple outcomes.

The joint models mentioned above are useful tools for analyzing complex cross-time multiple-outcome data structures; however, their priors don't reflect the nature of within group correlations in community intervention trials and don't address Corn-field's second penalty: limited degrees of freedom. In particular, the Bayesian models in these papers utilized diffuse or non-informative priors on the variance components, which can lead to U-shaped priors placing high prior probability on small and large ICCs [13]. In GRTs, ICCs are usually small (e.g., ≤ 0.05) and hence a prior which places high probability on large values may be inappropriate. In recent years, researchers have been encouraged to publish ICC estimates from GRTs, and as a result, a substantial number and variety of ICCs have become publicly available [16]. It is thus desirable to incorporate ICC estimates from previous studies using Bayesian [17] or meta-analytic methods [18], thereby addressing the small degree of freedom problem of GRTs.

In this paper, we propose a Bayesian method for joint modeling continuous and binary outcomes collected at multiple time points in a GRT. Similar to Daniels and Normand [24] we model the outcomes through a latent multivariate normal regression model with shared random effects between the different outcomes. The model allows treatment effects to vary with time and accounts for three types of correlations: 1.) ICC 2.) the correlation between different outcomes measured on the same subject and 3.) the over-time correlation (OTC) of each outcome. Moreover, we extend the method of Turner, Thompson, and Spiegelhalter [17] and propose an innovative prior structure for the variance components that directly places priors on the correlations; the priors reflect our knowledge about these correlations and allows the users to incorporate information from previous studies when they are available. In addition, our priors avoid undesirable constraints on the variance components that are imposed by some common methods and therefore yield unbiased estimates of the ICC, between outcome correlation, and OTC. We demonstrate through simulations the performance of our method under different ICCs and and between outcome correlations. We also apply our method to obesity and body fat data from the TAAG study. To construct an informative prior for the ICC, we draw information from 10 previous studies on relevant outcomes in similar populations, and use a hierarchical approach to group these ICCs by outcome rather than by study.

Our paper is organized as follows: In Section 2, we describe the latent multivariate normal regression model for longitudinal multiple outcome data from GRTs. In Section 3, we describe our proposed prior structure in particular, our new strategies for modeling the marginal variances in the latent normal model and for eliciting informative priors on the ICC based on information from previous studies. We next apply our method to simulated (Section 4) and TAAG data (Section 5) and we discuss the results and future work in Section 6.

We assume a GRT with a nested cross-sectional design as described by Murray [4]. Nested cross-sectional designs are used to examine the effect of an intervention on a large group of people. For example, the TAAG study used a nested cross-sectional design to determine if an educational program effectively increased physical activity levels within schools. In the basic nested cross-sectional design, independent random samples of subjects are drawn at baseline and follow-up. In this paper we will assume that each subject is only sampled once (at baseline or follow-up), which is often the case.

Let there be two observation times (*t*=1 for baseline, *t*=2 for follow-up) and two outcomes collected for each individual at each time point. Without loss of generality, we assume that the first outcome (*k*=1) is continuous and the second outcome (*k*=2) is binary. Suppose that there are two treatment conditions (control vs. intervention). Let there be *G* groups and assume that at time *t* (*t* = 1, 2), *m _{it}* members are randomly sampled from group

$${g}_{2}({z}_{ij2t)}=I({z}_{ij2t}>0)$$

for the binary outcome, where *I*(·) is the indicator function.

Our model for the mean of each underlying normal random variable is the following:

$${\mu}_{ijkt}={\beta}_{0kt}+{\beta}_{1kt}{\mathrm{trt}}_{i}+{b}_{ikt}+{c}_{ijt},$$

(2.1)

where *β*_{0kt} is a fixed intercept for the *k*th outcome at time *t*, trt* _{i}* is an indicator variable that equals 1 if group

$$\begin{array}{cc}\hfill {b}_{ik1}\sim & N(0,{\sigma}_{bk}^{2}),\hfill \\ \hfill \text{and}\phantom{\rule{1em}{0ex}}{b}_{ik2}\mid {b}_{ik1}\sim & N({\lambda}_{k}{b}_{ik1},(1-{\lambda}_{k}^{2}){\sigma}_{bk}^{2})\hfill \end{array}$$

(2.2)

where ${\sigma}_{bk}^{2}$ measures between-group variability in the *k*th outcome and Corr(*b _{ik}*

$${c}_{ijt}\sim N(0,{\sigma}_{c}^{2}).$$

(2.3)

This implies that the correlation coefficient between *z*_{ij1t} and *z*_{ij2t} (the two latent outcomes measured on the same subject at the same time) equals

$$\rho =\frac{{\sigma}_{c}^{2}}{\sqrt{({\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2})({\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2})}}.$$

(2.4)

Inferences are made from our model using a Bayesian approach. Bayesian methods have many well-known advantages over frequentist methods including the ability to perform exact inferences using Markov Chain Monte Carlo (MCMC) and the ability to incorporate information from previous studies through informative priors. The latter feature is especially pertinent to our research as we are interested in using published ICCs to address the small degrees of freedom problem in GRTs.

We place a conventional diffuse prior on the fixed effects (*β*_{0kt}, *β*_{1kt})^{′}, that is,

$${({\beta}_{0kt},{\beta}_{1kt})}^{\prime}\sim {N}_{2}({\mu}_{{\beta}_{kt}},{\Sigma}_{{\beta}_{kt}})\phantom{\rule{1em}{0ex}}(k=1,2),$$

(3.1)

where *μ** _{kt}* is usually a vector of zeros and

We next consider modeling the variance of the latent *z _{ijkt}* conditional on the random effects, namely, ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$. In generalized linear regression models, when

$${\sigma}_{1}^{-2}\sim Ga({a}_{1}\u22152,{b}_{1}\u22152),$$

where *Ga*(*a, b*) denotes the Gamma distribution with mean *a/b* and variance *a/b*^{2}, and ${\sigma}_{2}^{2}$ is fixed at 1 for identifiability considerations. This approach is perfectly valid when the two outcomes are modeled separately, because setting ${\sigma}_{2}^{2}$ to any positive constant ${\sigma}_{0}^{2}$ only re-scales every term in the regression function for the binary outcome (*μ*_{ij2t}) by *σ*_{0}. However, when the two outcomes are modeled jointly, this approach places undesirable constraints on the variance components. Suppose that we set ${\sigma}_{2}^{2}$ to a positive constant ${\sigma}_{0}^{2}\ne 1$; consequently, the regression function for the binary outcome is re-scaled to *σ*_{0}*μ*_{ij2t} and the subject-specific random effect is re-scaled to *σ*_{0}*c _{ijt}* which has variance ${\sigma}_{0}^{2}{\sigma}_{c}^{2}$. Since

$$\rho =\frac{{\sigma}_{c}^{2}}{\sqrt{({\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2})({\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2})}}\le \mathrm{min}\left(\frac{{\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2}}{\sqrt{({\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2})({\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}}},\frac{{\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}}{\sqrt{({\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2})({\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2})}}\right)=\mathrm{min}\left(\frac{\sqrt{{\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2}}}{\sqrt{{\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}}},\frac{\sqrt{{\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}}}{\sqrt{{\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2}}}\right).$$

(3.2)

Therefore, when the two marginal variances ${\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2}$ and ${\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}$ are very different, this prior specification restricts the between-outcome correlation *ρ* to be a small value. In situations where the outcomes are highly correlated, this restriction will make the estimate of *ρ* significantly biased.

To solve this problem, we propose a new method for specifying priors for the error variances. We first standardize the continuous outcome by using ${z}_{ij1t}^{\ast}={z}_{ij1t}\u2215st{d}_{1}$, where *std*_{1} is the sample standard deviation of the continuous variable. Assuming that ${z}_{ij1t}^{\ast}$ satisfies the latent normal regression model (2.1), we then let

$${({\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2})}^{-1}\sim Ga({a}_{c1}\u22152,{b}_{c1}\u22152),$$

(3.3)

and fix ${\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}$ at 1. Since we standardize *z*_{ij1t} prior to analysis, ${\sigma}_{b1}^{2}+{\sigma}_{c}^{2}+{\sigma}_{1}^{2}$ should be close to 1, and so is the upper bound of *ρ*. Thus, our new prior reduces the constraint imposed on the range of *ρ* and in most cases should allow it to be supported on nearly the full range [0, 1]. It also provides unbiased estimates of the variance components which leads to unbiased estimates of the ICC.

When analyzing GRTs, it is crucial to accurately estimate the ICC. Under our latent normal regression model (2.1), the ICC for outcome *k* at each time point is

$$\begin{array}{cc}\hfill {\mathrm{ICC}}_{k}=& \mathrm{Corr}({y}_{ijkt},{y}_{i{j}^{\prime}kt})\phantom{\rule{1em}{0ex}}(j\ne {j}^{\prime})\hfill \\ \hfill =& \frac{{\sigma}_{bk}^{2}}{{\sigma}_{bk}^{2}+{\sigma}_{c}^{2}+{\sigma}_{k}^{2}}.\hfill \end{array}$$

(3.4)

The traditional prior structure is to put independent inverse-gamma priors on ${\sigma}_{bk}^{2}$, ${\sigma}_{c}^{2}$ and ${\sigma}_{k}^{2}$. As Spiegelhalter [13] pointed out, this structure implies a U-shaped prior on the ICC, with high prior probabilities assigned to ICCs close to zero and close to 1. However, in GRTs the ICCs are typically small (e.g., ICC ≤ 0.05 ) and hence priors which assign high weight to near perfect correlations don't make sense. Spiegelhalter proposed some more realistic priors for the ICC including a *Unif*(0, 1) prior, a shrinkage prior, and an informative beta prior whose hyperparameters are estimated from data collected in previous studies.

More recently, Turner, Thompson, and Spiegelhalter [17] proposed more structured methods of constructing informative priors for the ICC. They weighted the ICC estimates from previous studies by their relevance to the outcome in the current study, and then incorporated the information across several studies using a hierarchical structure. Turner et al.'s prior provides a very flexible structure for incorporating multiple relevant ICC estimates from several previous studies and increases the precision of the anticipated ICC. However, their model assumes that different outcomes from the same study are most related; we have found that among studies targeting the same type of group (e.g., school, worksite), the ICCs for the same outcome estimated in different studies tend to be more related than the ICCs for different outcomes from the same study. Also, in most cases the relevance weights used in Turner et al.'s priors will be difficult to choose and hence are bound to be arbitrary. Therefore, we modify the Turner et al. approach to group the ICC estimates by outcome rather than by study. We only include the most relevant outcome from each study and assign equal weights to each ICC estimate.

Our development of the prior for the ICC begins by specifying a hierarchical model for the ICC estimates from previous studies. Let ${\widehat{\mathrm{ICC}}}_{ks}$ denote the estimated ICC for outcome *k* (*k* = 1, 2) from the *s*th study (*s* = 1, . . . , *S _{k}*). Donner and Wells [27] showed that under large sample sizes

$${\widehat{\mathrm{ICC}}}_{ks}\sim N({\mathrm{ICC}}_{ks},V({\mathrm{ICC}}_{ks},{M}_{s},{G}_{s})),$$

(3.5)

where the variance, *V*(ICC* _{ks}*,

$$V({\mathrm{ICC}}_{ks},{M}_{s},{G}_{s})=\frac{2({M}_{s}-1)(1-{\mathrm{ICC}}_{ks}^{2})[1+({M}_{s}\u2215{G}_{s}-1){\mathrm{ICC}}_{ks}]}{{({M}_{s}\u2215{G}_{s})}^{2}({M}_{s}-{G}_{s})({G}_{s}-1)}.$$

(3.6)

We then assign a normal prior to the logit of the true ICC from each study:

$$logit\left({\mathrm{ICC}}_{ks}\right)\sim N({\mu}_{{\mathrm{ICC}}_{k}},{\sigma}_{{\mathrm{ICC}}_{k}}^{2})$$

(3.7)

where ${{\mu}_{\mathrm{ICC}}}_{k}\sim N({m}_{\mu k},{\sigma}_{\mu k}^{2})$ and ${\sigma}_{{\mathrm{ICC}}_{k}}^{-2}\sim Ga({{a}_{\mathrm{ICC}}}_{k},{b}_{{\mathrm{ICC}}_{k}})$. Finally, we assume that the ICC of outcome *k* in the current study (ICC* _{k}*) is drawn from the same distribution as ICC

$$\pi (logit\left({\mathrm{ICC}}_{k}\right)\mid \mathbf{I}{\widehat{\mathbf{CC}}}_{k})\propto \int \int \int N(logit\left({\mathrm{ICC}}_{k}\right);{{\mu}_{\mathrm{ICC}}}_{k},{\sigma}_{{\mathrm{ICC}}_{k}}^{2})\times \prod _{s=1}^{{S}_{k}}\left\{N(logit\left({\mathrm{ICC}}_{ks}\right);{\mu}_{{\mathrm{ICC}}_{k}},{\sigma}_{{\mathrm{ICC}}_{k}}^{2})N({\widehat{\mathrm{ICC}}}_{ks};{\mathrm{ICC}}_{ks},V({\mathrm{ICC}}_{ks},{M}_{s},{G}_{s}))\right\}\times N({\mu}_{{\mathrm{ICC}}_{k}};{m}_{\mu k},{s}_{\mu k}^{2})Ga({\sigma}_{{\mathrm{ICC}}_{k}}^{-2};{a}_{{\mathrm{ICC}}_{k}},{b}_{{\mathrm{ICC}}_{k}})\partial {\mu}_{{\mathrm{ICC}}_{k}}\partial {\sigma}_{{\mathrm{ICC}}_{k}}^{-2}\partial {\mathbf{ICC}}_{k}^{\ast}$$

(3.8)

where $\mathbf{I}{\widehat{\mathbf{CC}}}_{k}={({\widehat{\mathrm{ICC}}}_{k1},\dots ,{\widehat{\mathrm{ICC}}}_{k{S}_{k}})}^{\prime}$, ${\mathbf{ICC}}_{k}^{\ast}={({\mathrm{ICC}}_{k1},\dots ,{\mathrm{ICC}}_{k{S}_{k}})}^{\prime}$, *N*(*x*; *m, v*) denotes the pdf of the normal distribution with mean *m* and variance *v* evaluated at *x*, and *G*(*x*; *a, b*) denotes the pdf of the gamma distribution with mean *a/b* evaluated at *x*. To implement our prior, the values of *m _{μk}*, ${s}_{\mu k}^{2}$,

A similar method could be used to construct informative priors for the overtime correlation (OTC). As noted earlier, TAAG used a nested cross-sectional design with an independent sample measured in each group at baseline and at follow-up. In this design, the OTC is the correlation between measurements from members from the same group measured at different time points. In terms of our model, the OTC for outcome *k* is

$$\begin{array}{cc}\hfill {\mathrm{OTC}}_{k}=& \mathrm{Corr}({y}_{ijk1},{y}_{i{j}^{\prime}k2})\hfill \\ \hfill =& {\lambda}_{k}\cdot \frac{{\sigma}_{bk}^{2}}{{\sigma}_{bk}^{2}+{\sigma}_{c}^{2}+{\sigma}_{k}^{2}}\phantom{\rule{1em}{0ex}}(j=1,\dots ,{m}_{i1};{m}_{i1};{j}^{\prime}=1,\dots ,{m}_{i2}).\hfill \end{array}$$

(3.9)

Unfortunately, we are unaware of any estimates of group-specific OTCs and hence instead of using informative priors on the OTCs, we assign non-informative *Unif*(0, 1) priors to λ_{1} and λ_{2}; the autocorrelations between the random group effects.

Lastly, we put a noninformative prior on the between-outcome correlation, *ρ*, that is *ρ* ~ *Unif*(0, 1). Of course, if previous data on *ρ* are available, one could replace the *Unif*(0, 1) prior with an informative beta prior. Compared with the conventional method that places an inverse gamma prior on ${\sigma}_{c}^{2}$, our prior is much easier to interpret and provides more direct estimates of the between outcome correlation.

To summarize, the joint prior on our variance components is as follows:

$$\pi \left({\sigma}_{1}^{2}+{\sigma}_{b1}^{2}+{\sigma}_{1}^{2},{\mathrm{ICC}}_{1},{\mathrm{ICC}}_{2},\rho ,{\lambda}_{1},{\lambda}_{2}\right)=Ga({({\sigma}_{1}^{2}+{\sigma}_{b1}^{2}+{\sigma}_{1}^{2})}^{-1};{a}_{c1}\u22152,{b}_{c1}\u22152)\times \pi (logit\left({\mathrm{ICC}}_{1}\right)\mid \mathbf{I}{\widehat{\mathbf{CC}}}_{1})\pi (logit{(\mathrm{ICC}}_{2})\mid \mathbf{I}{\widehat{\mathbf{CC}}}_{2})\times I(0,\le \rho ,{\lambda}_{1},{\lambda}_{2}\le 1),$$

(3.10)

where $\pi (logit\left({\mathrm{ICC}}_{k}\right)\mid \mathbf{I}{\widehat{\mathbf{CC}}}_{k})$ is our informative prior given in Equation 3.8 and *I*(·) denotes the indicator function. As mentioned in Section 3.2.1, ${\sigma}_{2}^{2}+{\sigma}_{b2}^{2}+{\sigma}_{c}^{2}$ is fixed at 1.

Under the above model specification, we design a Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the parameters. Our algorithm consists of both Gibbs and Metropolis-Hastings steps. The detailed conditional distributions may be found in Appendix A of the Supplementary Materials. Under mild regularity conditions, samples from this MCMC algorithm converge to a stationary distribution that is the joint posterior. We can then obtain parameter estimates using the posterior summaries of these samples. Sample R code for implementing our method is available upon request from the corresponding author.

We performed a simulation study to investigate the efficiency gains provided by our method. We simulated data with the same structure as the Trial of Activity in Adolescent Girls – nested cross-sectional design, two treatment groups, two time points (baseline and follow-up). The underlying normal random variables for the continuous and binary outcomes were generated from normal distributions with variance 1 and mean

$$\begin{array}{cc}\hfill {\mu}_{ijk1}=& -1+tr{t}_{i}+{b}_{ik1}+{c}_{ij1}\hfill \\ \hfill {\mu}_{ijk2}=& 2tr{t}_{i}+{b}_{ik2}+{c}_{ij2},\hfill \end{array}$$

which means that the true effect of the intervention on the change in each outcome was 1. Random group effects (*b _{ikt}*) were generated assuming that λ

Each dataset was analyzed using three methods:

- Latent Multilevel Multidimensional Normal Regression Model (LMMNRM) with informative priors on the ICCs (our method).
- LMMNRM with
*Unif*(0, 1) priors on the ICCs. - Separate latent multilevel normal regression models for each outcome with informative priors on the ICCs. Note that this method is synonymous with implementing Method 1 fixing all
*c*= 0._{ijt}

To construct informative priors on the ICC in Methods 1 and 3, we simulated ICCs from 10 previous studies (ICC_{1}, . . . , ICC_{10}) using (3.7) and their sample estimates $({\widehat{\mathrm{ICC}}}_{1},\dots ,{\widehat{\mathrm{ICC}}}_{10})$ from (3.5) with each study consisting of 20 groups and 50 subjects per group. These sample estimates were then used to construct the informative priors for the ICC of the binary and continuous outcomes (Equation 3.8). In each method, the following diffuse priors were specified for the hyperparameters *μ*_{ICC}_{k} and ${\sigma}_{{\mathrm{ICC}}_{k}}^{-2}$:

$$\begin{array}{cc}\hfill {\mu}_{{\mathrm{ICC}}_{k}}\sim & N(0,10,000)\hfill \\ \hfill {\sigma}_{{\mathrm{ICC}}_{k}}^{-2}\sim & Ga(0.001,0.001).\hfill \end{array}$$

We ran our MCMC for a total of 80,000 iterations following a burn-in of 20,000 and saved every fifth value to thin the chain.

Table 1 summarizes the results from our simulation study. The treatment effect referred to in the table is the difference in the pre-post change in the continuous and binary outcome across treatment arms, which is equivalent to the difference in the time-dependent intervention effects in our model, namely, *β*_{112}–*β*_{111} for the continuous outcome and *β*_{122} – *β*_{121} for the binary outcome. In each simulation scenario, the biases of the treatment effect estimates were small and similar across methods. This result was expected; due to the identity link for the mean, the marginal effects of treatment provided by Method 3 are identical to the effects provided by Methods 1 and 2 conditional on *c _{ijt}* (the subject-specific effect). Also, since the historical ICC estimates used in our informative prior were drawn from a distribution centered on the true ICC (i.e., the prior is consistent with the data), we don't expect our informative prior to introduce bias in any of the parameter estimates.

Our method (Method 1) consistently provided the most efficient treatment effect estimates, though the effects on the binary outcome were more pronounced. The effects of joint modeling on efficiency were greatest when the two outcomes were strongly correlated; in Scenarios 1 and 3 (*ρ* = 0.8), the average posterior standard deviation of the treatment effect on the binary outcome was between 7-12% lower for our method compared to modeling the outcomes separately (Method 3) but only 2% lower in Scenarios 2 and 4 (*ρ* = 0.5). The finding that the benefits of joint modeling on efficiency 1.) is greatest for the binary outcome and 2.) increases with the between outcome correlation is consistent with the results reported by Gueorguieva and Agresti [19].

The informative prior on the ICC improved the efficiency of both the ICC and treatment effect estimates. In the case of the ICC, the improvement was dramatic. When Method 1 was implemented, the informative prior reduced the posterior standard deviation of the ICC of the binary outcome by approximately 80% in Scenarios 1 and 2 and approximately 70% in Scenarios 3 and 4 relative to Method 2, where a uniform prior was placed on the ICC. Similar results were observed for the ICC of the continuous outcome. The informative prior also improved the efficiencies of the treatment effect estimates, especially for the binary outcome. The greatest effect was observed in Scenario 1 (ICC = 0.1, *ρ* = 0.8) where our method exhibited a 13% reduction in the average posterior standard deviation of the treatment effect on the binary outcome relative to the joint model with a *Unif*(0, 1) prior on the ICC (Method 2). In contrast, in Scenarios 2-4 the reduction in the posterior standard deviation of the effect on the binary outcome was only 2-5%. The differences in effect of the informative prior on efficiency of treatment effect estimates make sense when one thinks about the magnitude of the between group variance $\left({\sigma}_{bk}^{2}\right)$ relative to

$$\mathrm{Var}({z}_{ijkt}\mid {c}_{ijt})={\sigma}_{bk}^{2}+{\sigma}_{k}^{2};$$

the residual variance after accounting for the between outcome correlation (*ρ*). Since our method fixes $\mathrm{Var}\left({z}_{ij2t}\right)={\sigma}_{b2}^{2}+{\sigma}_{c}^{2}+{\sigma}_{2}^{2}$ at 1 and re-scales *z*_{ij1t} to ${z}_{ij1t}\u2215\sqrt{\mathrm{Var}\left({z}_{ij1t}\right)}$, we have ${\sigma}_{b2}^{2}={\mathrm{ICC}}_{2}$, ${\sigma}_{c}^{2}=\rho $, and

$$\mathrm{Var}({z}_{ij2t}\mid {c}_{ijt})=1-\rho .$$

In Scenario 1, the between group variability explains half of the residual variance and thus the informative prior results in a considerable reduction in the posterior standard deviation of the treatment effect. On the other hand, in Scenarios 2, 3, and 4, the between group variability only accounts for 20, 5, and 2% of the residual variance respectively and thus the informative prior has a negligible effect on the efficiency of the treatment effect.

As mentioned in Section 3.2.1, part of the novelty of our method is our priors for the residual error variances. The common practice of placing independent inverse gamma priors on the variance components imposes undesirable restrictions in joint models and may lead to biased estimates of the ICCs and the between outcome correlation *ρ*. To illustrate this deficiency, we took the 50 data sets simulated under Scenario 3 (*ρ* = 0.8, ICC = 0.01), divided the continuous outcome by 10 (to make the variances of *z*_{ij1t} and *z*_{ij2t} different), and analyzed the data using a joint model with informative priors but set ${\sigma}_{2}^{2}$ to 1 and placed Ga(0.001,0.001) priors on ${\sigma}_{c}^{-2}$ and ${\sigma}_{1}^{-2}$; i.e., the common approach used for latent normal regression models. Under these new priors, the treatment effect estimates remained the same (the average posterior means were within 3% of the true value), but the ICCs were considerably biased (average posterior means of ICC_{1} and ICC_{2} were 0.007 and 0.006 respectively) as was *ρ* (average posterior mean = 0.095). The bias of *ρ* is understandable since according to Equation 3.2, this method constrains *ρ* to be less than or equal to 0.1. As seen in the results for Methods 1 and 2 (Table 1), our prior structure for the variance components successfully fixed this problem, yielding unbiased estimates for both the ICCs and *ρ*.

TAAG was a multi-center GRT which examined a two year intervention targeting schools, community agencies, and girls intended to increase physical activity levels. A total of 36 schools were enrolled; 18 schools were randomly assigned to the intervention and 18 were assigned control. Random samples of 6th grade girls were drawn at baseline (average of 48 per school) and random samples of 8th grade girls were drawn two years (average of 97 per school) and three years (average of 115 per school) later.

We illustrated our methodology through application to data collected at baseline and three year follow-up. We chose the three year follow-up instead of the two year follow-up for our analysis because the TAAG study found an effect of intervention on activity levels (the primary outcome) at the second follow-up but not the first [29]. Two groups (one per arm) were lost after the first follow-up leaving 34 groups for the analysis. The intervention effects of interest were those on change in average percent body fat and obesity rate over this time interval. The latter outcome was measured by a binary variable which equaled 1 if a girl's BMI was greater than the 95th percentile for her age and 0 otherwise. No covariates were included in our analysis.

As in the simulation study, the TAAG data were analyzed using the following methods: 1.) the latent multilevel multidimensional normal regression model with informative priors on the ICCs 2.) LMMRM with Unif(0, 1) priors on the ICCs and 3.) separate latent multilevel normal regression models fit to each outcome. In Methods 1 and 3, informative priors for the ICCs were constructed using ICC estimates from previous studies. Ideally, the historical estimates used in constructing the prior should be from the same type of group as the application, but estimates from schools were unavailable for the outcomes we considered. Among the available ICC estimates, we selected those from groups similar to schools with respect to size and diversity of group members; this left us with estimates from 10 previous studies, among which 3 used worksite as the group of interest while the remaining 7 reported clinic or physician as the group of interest. From each study, we chose the ICC corresponding to the outcome most relevant to obesity or percent body fat, such as the ICCs for BMI, body fatness, weight, whether or not a subject had a BMI greater than 30, and whether or not a subject was overweight. The ICC estimates and their references are provided in Table 2. It turned out that in each study the ICC estimate most relevant to percent body fat was also the ICC estimate most relevant to obesity and hence we used the same informative prior for the ICC of each outcome. The resulting prior distribution for the ICC was right-skewed with mean 0.03 and median 0.02 (Figure 1). The remaining priors were identical to those used in our simulations. For each method, we ran the MCMC for 100,000 iterations following a burn-in of 20,000, and we saved the results from every fifth iteration to thin the chain. We also repeated our analysis using a subset of 12 schools from California and Minnesota (6 schools/state) to determine if improvements in efficiency were more evident in this smaller sample.

Informative Prior on the ICC used in the TAAG Analysis. Distribution was estimated by first drawing values of *μ*_{ICC}_{k} and ${\sigma}_{{\mathrm{ICC}}_{k}}^{2}$ from their full conditional posteriors at each iteration of the MCMC and then drawing values of *logit*(ICC_{k}) **...**

Table 3 provides the posterior means, standard deviations, and 95% credible intervals of the treatment effects for Methods 1-3. As in the simulation study, the term treatment effect refers to the difference in pre-post change in each outcome across treatment arms. While there were some differences in the posterior means across methods, the overall conclusions were the same; the intervention caused an increase in both average percent body fat and obesity rate though the 95% credible intervals all contain zero which is suggestive of an insignificant treatment effect. These results are consistent with those in Webber et al. [29]. They analyzed the complete TAAG data set using an ANCOVA model applied to group-level means and found that percent body fat and BMI were slightly higher in the intervention arm at the three-year followup, though these differences were insignificant.

Although the conclusions about the intervention effect did not differ with method, their efficiencies differed considerably for the binary outcome. The posterior standard deviation for the treatment effect on obesity rate was smallest when the two outcomes were jointly modeled and when there was an informative prior on the ICC (Method 1) and largest when the outcomes were modeled separately (Method 3). For this data set, joint modeling provided a greater benefit than informative priors: the posterior standard deviation of the treatment effect on obesity obtained from Method 1 (joint model, informative prior on ICC) was 22% smaller than the posterior standard deviation from Method 3 (separate models, informative prior on ICC) but only 4% smaller than the posterior standard deviation from Method 2 (joint model, uniform prior on ICC). This is because the correlation between outcomes was large (posterior mean of *ρ* ≈ 0.95 as seen in Table 1) and the informative prior had a relatively modest effect on the posterior standard deviation of the ICC of the binary variable (24% difference in the standard deviation between Methods 1 and 2). The modest impact of the informative prior on the standard deviation of the ICC was due in part to the large sample size. Hence, to investigate the influence of sample size more thoroughly, we repeated our analysis using a subsample of 12 TAAG schools from California and Minnesota. As seen in Table B-1 in the Appendix, the posterior standard deviations of the ICC obtained from Method 1 were approximately half the posterior standard deviations from Method 2, that is our informative priors on the ICC improved the estimation efficiency (defined in terms of the ratio of variances) by approximately 300%. As a result, the informative prior caused a greater reduction in the posterior standard deviation of the treatment effect on obesity; the posterior standard deviation under Method 1 was 7% lower than the value from Method 2.

Another interesting finding from our analysis of the TAAG data was that the ICC estimates varied with method. While the posterior means of the percent body fat ICC were similar across the three methods, the posterior mean of the obesity ICC was considerably higher when the outcomes were modeled separately. It is well known that regression adjustment for covariates can often reduce between group variability, thereby decreasing the ICC [4]. Thus, it could be that the reason why joint modeling decreased the obesity ICC is that the subject-specific random effects (*c _{ijt}*) accounted for missing covariates which varied considerably by group. Joint modeling did not impact the ICC in the simulations because the only between group covariate was treatment arm, which was included in the model.

As in the simulation study, we also analyzed the TAAG data (34 groups) using a joint model with an informative prior on the ICC and the common approach of setting ${\sigma}_{2}^{2}=1$ and assigning diffuse inverse gamma priors on ${\sigma}_{c}^{2}$ and ${\sigma}_{1}^{2}$. We did not standardize percent body fat prior to this additional analysis. As in the simulation study, the treatment effect estimates (0.031 for percent body fat and 0.055 for obesity) were similar to the estimates from Methods 1-3, but the ICC estimates (posterior mean = 0.001 and 0.002 for percent body fat and obesity, respectively) were very different. However, unlike the simulation study, the posterior mean of *ρ* (0.948) was similar to the values in Methods 1 and 2; this was because the total variances of *z*_{ij1t} and *z*_{ij2t} were similar (posterior means = 76.235 and 72.923, respectively) and hence, according to Equation 3.2, the upper bound of *ρ* was close to 1. It is interesting to note that even though the between outcome correlation was unaffected, the ICCs were deflated relative to Methods 1-3 because, as explained in Section 3.2.1, fixing ${\sigma}_{2}^{2}$ at 1 imposes undesirable restrictions on the variance components of both the continuous and binary outcome.

In this paper, we developed a joint model for binary and continuous outcomes collected in Group Randomized Trials. We have also extended previous work by Turner et al. [17] to place an informative prior on the ICC of each outcome which, in combination with joint modeling, improves efficiency, particularly in the case of the binary outcome. While the work presented was limited by one data example and a simulation study consisting of a limited number of scenarios, the gains in efficiency we observed are consistent with previous work. In a simulation study, Gueorguieva and Agresti [19] reported that joint modeling resulted in a 17% reduction in the standard error of the treatment effect on the binary outcome when the true correlation between the continuous and binary outcome was 0.8; similarly, we found that when *ρ* = 0.8 in our simulations, joint modeling caused a 7-12% reduction in the posterior standard deviation of the effect of treatment on the binary outcome. In an analysis of data from a previous GRT, Turner et al. [17] found that placing an informative prior on the ICC caused a 10% reduction in the posterior standard deviation of the intervention effect; in our simulations we found that the informative prior reduced the average posterior standard deviation of the treatment effect on the binary outcome by a maximum of 13% across the scenarios considered. While a 12-13% reduction in the standard deviation may sound modest, it could be very beneficial in a GRT. For example, when designing a study, if we know that we'll be using an analysis method that decreases the standard deviation of the treatment effect by 12% relative to a univariate mixed model, this could mean a reduction in sample size of one group per arm which would significantly reduce study costs in many GRTs, particularly those targeting large groups of people (e.g., counties).

Our simulation studies demonstrated the behavior of our method under different levels of ICC and between outcome correlation. As in Gueorguieva and Agresti [19], we found that joint modeling causes the greatest improvement in efficiency when the two outcomes are strongly correlated. In the case of the binary outcome, we found that the informative prior causes the greatest improvement in efficiency when a substantial proportion of the residual variance is due to between group variability (or ICC); to the best of our knowledge, we are the first to present such a finding. Thus, our recommendations for future use are as follows: when the between outcome correlation between a continuous and binary outcome is strong we recommend joint modeling. If the ICC is large (particularly for the binary variable), we recommend extending the approach to include an informative prior on the ICC, otherwise a uniform prior on the ICC is probably sufficient if all one cares about is improving the efficiency of the treatment effect estimates. Weakly correlated outcomes could be analyzed using our method but the results would probably be similar to univariate models for each outcome, and hence the benefit is probably too small relative to the added computational complexity.

Evaluation of model fit is an important part of any modeling exercise. Unfortunately, we are unaware of a formal method for evaluating the fit of a joint model for a binary and continuous response. Future research should consider developing a method for measuring goodness of fit of our model which allows direct comparison to competing models including univariate models for each outcome and models with non-informative priors.

In this paper, we focused our efforts on nested cross-sectional GRTs. Another design commonly used is the nested cohort design where members in each group are followed over time. Potentially, our methodology could be extended to handle these designs with the major modification being that the model for the mean must include random subject-specific effects for each outcome in addition to shared random effects across outcomes. We also focused on joint modeling of binary and continuous outcomes; while these are probably the two most common types of outcomes in GRTs, ordinal, count, and survival outcomes are also collected and hence joint modeling of these outcomes presents an interesting area of future research.

Finally, our model examined the effects of treatment on change in each outcome (post-pre) which is a common approach in GRTs, but not the only option. For example, Webber et al. [29] analyzed the TAAG data using a two-stage approach; in the first stage, a mixed model was used to regress each outcome on fixed time and ethnicity effects and random school and school-by-time interaction effects. This model was used to generate ethnically-adjusted pre- and post-test means for each group which were inserted into a second stage ANCOVA model in which the adjusted post-test mean was regressed on treatment group and the adjusted pre-test mean. Future research should examine the potential of joint modeling and informative priors within two-stage and other group-level analyses.

The project described was supported by Award Number UL1RR025755 from the National Center for Research Resources (Xu, Pennell, and Lu) and Award Number DMS-09-07070 from the National Science Foundation (Xu). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Center For Research Resources or the National Institutes of Health. TAAG was funded by the following cooperative agreements from the National Heart, Lung and Blood Institute: U01 HL066855 (Tulane University); U01HL066845 (University of Minnesota); U01HL066852 (University of South Carolina); U01HL066853 (University of North Carolina at Chapel Hill); U01HL066856 (San Diego State University); U01HL066857 (University of Maryland); U01HL066858 (University of Arizona).

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