mmeta: An R Package for Multivariate Meta-Analysis

Abstract

This paper describes the core features of the R package mmeta, whichimplements the exact posterior inference of odds ratio, relative risk, and risk difference given either a single 2 × 2 table or multiple 2 × 2 tables when the risks within the same study are independent or correlated.

1. Introduction

Epidemiological studies often involve comparisons between two populations with binary outcomes. Data from these studies are usually summarized by a single or multiple 2 × 2 tables. To quantify the association between an exposure and a certain disease, comparative measures between two risks, e.g., odds ratio (OR), relative risk (RR), and risk difference (RD), are frequently used. Bayesian approach has been widely applied to obtain the posterior distributions of these comparative measures that reflect evidence from the data and available prior knowledge. Bayesian inference on a single study based on a 2 × 2 table has been investigated by several researchers. Specifically, Nurminen and Mutanen (1987) derived the exact posterior distributions of OR, RR, and RD under independent beta prior distributions with integer hyper-parameters. Marshall (1988) extended the results of OR by using hypergeometric functions (Gauss 1813) to allow the hyperparameters being any positive numbers. Nadarajah and Kotz (2007) gave a formula for RD using Appell hypergeometric function. Chen and Luo (2011) corrected the formula by Nadarajah and Kotz (2007) and further simplified the formula to avoid divergence of the Appell hypergeometric function. Hora and Kelley (1983) and Hashemi et al. (1997) extended the results of Nurminen and Mutanen (1987) on RR to beta prior distributions with any positive hyperparameters.

Multiple 2 × 2 tables often arise in meta-analysis which combines statistical evidence from multiple studies. Two risks within the same study are possibly correlated because they share some common factors such as environment and population structure. For example, in genetic association studies, people in the same study are likely to live in the same community sharing similar environmental factors or similar ancestors (Lee 1996). Riley (2009) has showed via simulation studies that separate meta-analysis of correlated outcomes can lead to biased estimates of variances of the summary effect sizes. In contrast, multivariate meta-analysis summarizes simultaneously all outcomes of interest instead of conducting many separate univariate meta-analysis. Multivariate meta-analysis has recently drawn lots of attention (e.g., Reitsma et al. 2005; Chu and Cole 2006; Riley et al. 2007, 2008; Hamza et al. 2008). An excellent overview of multivariate meta-analysis can be found in Jackson et al. (2011) and Mavridis and Salanti (2012). In the multivariate meta analysis with a binary outcome and a categorical exposure, two modeling strategies have been commonly used: A bivariate general linear mixed effect model on the transformed proportions (Reitsma et al. 2005; Arends et al. 2008) and a bivariate generalized linear mixed effect model on the transformed risks (e.g., logit or probit transformations) (Van Houwelingen et al. 1993, 2002; Chu and Cole 2006; Chu et al. 2010). However, these two methods are based on the transformed proportions or the transformed risks and thus the interpretation is transformation dependent. Multivariate meta-analysis can be conducted using various software including STATA (STATA Inc. 2012), SAS (SAS Institute Inc. 2012), R (R Core Team 2012). Specifically, mvmeta command in STATA performs fixed- and random-effects multivariate meta-regression analysis. The SAS PROC MIXED was the first routines that popularized multivariate meta-analysis (Van Houwelingen et al. 2002). More recently, the SAS Macro METADAS was made available to fit bivariate meta-analysis models for diagnostic test accuracy studies (Takwoingi et al. 2008). R package metaSEM can be used to conduct univariate and multivariate meta-analysis using structural equation modeling (SEM) via the OpenMx package (Cheung 2012). I n addition, a new R package mvmeta (Gasparrini 2012) can perform fixed- and random-effects multivariate meta-analysis and meta-regression.

Instead of modeling the transformed proportions or transformed risks, we use a Sarmanov family of correlated beta prior distributions (referred to as Sarmanov beta prior distributions) (Sarmanov 1966) to model the risks directly; see for example, Chen et al. (2011). The correlation parameter can be intuitively interpreted as the correlation coefficient between risks. In addition, the Sarmanov beta prior distribution has the following advantages in modeling. First, it allows for both positive and negative correlations; second, it only needs specification of marginal distributions and the correlation parameter, which has important advantage in Bayesian inference because it is often easier to specify and interpret univariate prior than bivariate prior; third, it is pseudo-conjugate to binomial distributions, i.e., the Sarmanov beta prior distribution can be expressed as a linear combination of independent bivariate beta distributions (Lee 1996), which enables us to derive closed-form expressions of the exact posterior distributions for study-specific comparative measures. Such closed-form expressions offer computational convenience when the exact posterior distributions of the study-specific comparative measures are also of interest. We have used the Sarmanov beta prior distribution to make exact posterior inference of some comparative measures (e.g., OR, RR, and RD) (Chen et al. 2011, 2012, 2013). This paper describes the mmeta package as a collection of a new family of models different from those in the aforementioned packages. Specifically, the inference of the overall and study-specific comparative measures (i.e., OR, RR, and RD) are inferred under the Sarmanov beta prior distributions. The functions of the mmeta package have been written in R language, with some Fortran 77 routines which are interfaced through R. The package is built following the S3 formulation of R methods. The mmeta package (currently version 1.06) is available from the Comprehensive R Archive Network (CRAN) at http://cran.r-project.org/.

The paper is organized as follows. In Section 2 we outline the exact Bayesian posterior inference approach. We describe the features of two main functions in the mmeta package and the analysis of two real datasets in Section 3. In Section 4, we provide a brief discussion.

2. Theory of exact distributions

2.1. Models and inference on overall comparative measures

For the i-th study (i = 1,…, I, I is number of studies), let n_ji, y_ji and p_ji be the number of subjects, number of subjects experienced a certain event, and the risk of experiencing the event in the jth group (j = 1,…, J, J is number of groups), respectively. For simplicity, we consider the cases with two groups under comparison (i.e., J = 2) and the extension to cases with more than 2 groups is straightforward. We assume the following Bayesian hierarchical model. At the first stage, we assume that given the study-specific risks (p_1i, p_2i), y_1i and y_2i are independently distributed binomial variables, i.e

$$(y_{1i},y_{2i})|(n_{1i},n_{2i},p_{1i},p_{2i})~\text{Binomial}(y_{1i}|n_{1i},p_{1i})\times \text{Binomial}(y_{2i}|n_{2i},p_{2i}).$$ (1)

This conditional independence assumption is reasonable because y_1i and y_2i are calculated using subjects from different groups. To complete the Bayesian hierarchical model, we need to impose a parametric prior distribution on the study-specific risks (p_1i, p_2i). Here we consider a family of distributions first proposed by Sarmanov (1966), and later studied extensively by Cole et al. (1995), Lee (1996), Shubina and Lee (2004), Danaher and Hardie (2005), and Chen et al. (2011). The Sarmanov beta prior distribution is constructed such that the marginal distribution of the random effect in jth group p_ji is beta distribution with shape parameters (a_j, b_j) and the correlation coefficient between p_1i and p_2i is ρ (Sarmanov 1966; Lee 1996). Specifically, we denote beta(p; a, b) = {B(a, b)}⁻¹p^a−1(1−p)^b−1 where B(a, b) is a beta function defined by ${\int }_0^1{t}^{a-1}{(1-t)}^{b-1}dt$, μ_j = a_j/(a_j + b_j), and ${\delta }_j^2={\mu }_j(1-{\mu }_j)/(a_j+b_j+1)$. The joint prior distribution of the study-specific risks (p_1i, p_2i), referred to as Sarmanov beta prior distribution, is

$$(p_{1i},p_{2i})|(a_1,b_1,a_2,b_2,\rho )~g(p_1,p_2;a_1,b_1,a_2,b_2,\rho ),$$ (2)

where $g(p_1,p_2;a_1,b_1,a_2,b_2,\rho )=\text{beta}(p_1;a_1,b_1)\text{beta}(p_2;a_2,b_2)\{1+\rho \frac{(p_1-{\mu }_1)(p_2-{\mu }_2)}{{\delta }_1{\delta }_2}\}$.

With the Bayesian hierarchical model specified in (1) and (2), the log marginalized likelihood function for the unknown hyperparameters (a₁, b₁, a₂, b₂, ρ) is

$$\text{log}L(a_1,b_1,a_2,b_2,\rho )=\sum _{i=1}^n\text{log}\int \int \text{Pr}(y_{1i},y_{2i}|p_{1i},p_{2i})g(p_{1i},p_{2i};a_1,b_1,a_2,b_2,\rho )d{p}_{1i},d{p}_{2i}=\sum _{i=1}^n\text{log}[{\mathrm{P}}_{BB}(y_{1i};n_{1i},a_1,b_1){\mathrm{P}}_{BB}(y_{2i};n_{2i},a_2,b_2)\{1+\frac{\rho }{\sqrt{\frac{a_1{b}_1}{{(a_1+b_1)}^2(a_1+b_1+1)}}\sqrt{\frac{a_2{b}_2}{{(a_2+b_2)}^2(a_2+b_2+1)}}}(\frac{y_{1i}+a_1}{n_{1i}+a_1+b_1}-\frac{a_1}{a_1+b_1})·(\frac{y_{2i}+a_2}{n_{2i}+a_2+b_2}-\frac{a_2}{a_2+b_2}\left)\right\}],$$ (3)

where P_BB(y; n, a, b) is the probability mass function of a beta-binomial distribution, i.e.,

$$P_{BB}(y;n,a,b)=\left(\begin{array}{c}\hfill n\hfill \\ \hfill y\hfill \end{array}\right)\frac{B(y+a,n-y+b)}{B(a,b)}.$$

The last expression in (3) has been derived by Danaher and Hardie (2005) and an outline of derivation is provided in the Appendix for readers of interest. We refer to (3) as Sarmanov beta-binomial model. As a benefit of using Sarmanov beta prior distributions, the log marginalized likelihood function has a closed-form expression, which avoids numerical approximation of integrals. Hence the Bayesian hierarchical model specified in (1) and (2) has great computational advantage over commonly used multivariate generalized linear mixed effects models. When ρ = 0, the Sarmanov beta-binomial model reduces to the independent beta-binomial model, i.e., product of two beta-binomial distributions.

The hyperparameters (a₁, b₁, a₂, b₂, ρ) can be estimated by maximizing the log likelihood log L(a₁, b₁, a₂, b₂, ρ). We implement it through R (R Core Team 2012) with the optim function, which uses a quasi-Newton method with box constraints on the ranges of parameters. Denote (â₁,b̂₁, â₂,b̂₂, ρ̂) the maximum likelihood estimates based on the log likelihood function (3). We use delta method to obtain the variance of the overall comparative measures, namely the overall odds ratio estimate, $\widehat{\mathrm{O}\mathrm{R}}=\{{\mathrm{\mu ̂}}_2/(1-{\mathrm{\mu ̂}}_2)\}/\{{\mathrm{\mu ̂}}_1/(1-{\mathrm{\mu ̂}}_1)\}=({â}_2{\mathrm{b̂}}_1)/({â}_1{\mathrm{b̂}}_2)$, the overall relative risk estimate, $\widehat{\mathrm{R}\mathrm{R}}={\mathrm{\mu ̂}}_2/{\mathrm{\mu ̂}}_1=\{{â}_2/({â}_2+{\mathrm{b̂}}_2)\}/\{{â}_1/({â}_1+{\mathrm{b̂}}_1)\}$, and the overall risk difference estimate, $\widehat{\mathrm{R}\mathrm{D}}={\mathrm{\mu ̂}}_2-{\mathrm{\mu ̂}}_1=\{{â}_2/({â}_2+{\mathrm{b̂}}_2)\}-\{{â}_1/({â}_1+{\mathrm{b̂}}_1)\}$.

2.2. Inference on study-specific comparative measures

Denote the study-specific comparative measures OR_i = {p_2i/(1−p_2i)}/{p_1i/(1−p_1i)}, RR_i = p_2i/p_1i, and RD_i = p_2i − p_1i. The statistical evidence of these comparative measures from the ith study can be quantified by the posterior distributions, i.e., Pr(θ_i|data_i, a₁, b₁, a₂, b₂, ρ) where θi = OR_i, RR_i, or RD_i and data_i = (y_1i, n_1i, y_2i, n_2i). Note that the true values of the hyperparameters (a₁, b₁, a₂, b₂, ρ) are often unknown. One solution is to simply replace the hyperparameters by their estimates. Such an approach is called empirical Bayes method (Efron and Morris 1973, 1975; Gelman et al. 2004; Carlin and Louis 2009). The coverage property of the credible intervals using the empirical Bayes method has been investigated via simulation studies in Chen et al. (2012). The conclusion is that the credible interval without accounting for the uncertainty on the hyperparameters still performs reasonably well, when the number of studies is moderate (Chen et al. 2012).

An important property of the Sarmanov beta prior distribution for p₁ and p₂ is that it can be written as a linear combination of independent bivariate beta distributions (Lee 1996),

$$g(p_1,p_2;a_1,b_1,a_2,b_2,\rho )={\upsilon }_1\text{beta}(p_1;a_1,b_1)\text{beta}(p_2;a_2,b_2)+{\upsilon }_2\text{beta}(p_1;a_1+1,b_1)\text{beta}(p_2;a_2,b_2)+{\upsilon }_3\text{beta}(p_1;a_1,b_1)\text{beta}(p_2;a+1,b_2)+{\upsilon }_4\text{beta}(p_1;a_1+1,b_1)\text{beta}(p_2;a_2+1,b_2),$$

where υ_k (k = 1,…, 4) are weights defined by υ₁ = 1 + ργ, υ₂ = υ₃ = −ργ, υ₄ = ργ, = γ = (μ₁μ₂)/(δ₁δ₂). After some algebra, the posterior distribution of p₁ and p₂ given data is also a linear combination of independent bivariate beta distributions,

$$\text{Pr}(p_1,p_2|{\text{data}}_i,a_1,b_1,a_2,b_2,\rho )={\omega }_1\text{beta}(p_1;{\alpha }_1,{\beta }_1)\text{beta}(p_2;{\alpha }_2,{\beta }_2)+{\omega }_2\text{beta}(p_1;{\alpha }_1+1,{\beta }_1)\text{beta}(p_2;{\alpha }_2,{\beta }_2)+{\omega }_3\text{beta}(p_1;{\alpha }_1,{\beta }_1)\text{beta}(p_2;{\alpha }_2+1,{\beta }_2)+{\omega }_4\text{beta}(p_1;{\alpha }_1+1,{\beta }_1)\text{beta}(p_2;{\alpha }_2+1,{\beta }_2),$$

where α_j = a_j + y_ji, β_j = b_j + n_ji−y_ji (j = 1, 2) and the weights ω_k (k = 1,…, 4) are defined as

$$\begin{array}{cc}\hfill {\omega }_1=\frac{{\upsilon }_{1}B({\alpha }_1,{\beta }_1)B({\alpha }_2,{\beta }_2)}{CB(a_1,b_1)B(a_2,b_2)},\hfill & \hfill {\omega }_2=\frac{{\upsilon }_{2}B({\alpha }_1+1,{\beta }_1)B({\alpha }_2,{\beta }_2)}{CB(a_1+1,b_1)B(a_2,b_2)},\hfill \\ \hfill {\omega }_3=\frac{{\upsilon }_{3}B({\alpha }_1,{\beta }_1)B({\alpha }_2+1,{\beta }_2)}{CB(a_1,b_1)B(a_2+1,b_2)},\hfill & \hfill {\omega }_4=\frac{{\upsilon }_{4}B({\alpha }_1+1,{\beta }_1)B({\alpha }_2+1,{\beta }_2)}{CB(a_1+1,b_1)B(a_2+1,b_2)},\hfill \end{array}$$

and the normalizing constant C is calculated as

$$C=\frac{{\upsilon }_{1}B({\alpha }_1,{\beta }_1)B({\alpha }_2,{\beta }_2)}{B(a_1,b_1)B(a_2,b_2)}+\frac{{\upsilon }_{2}B({\alpha }_1+1,{\beta }_1)B({\alpha }_2,{\beta }_2)}{B(a_1+1,b_1)B(a_2,b_2)}+\frac{{\upsilon }_{3}B({\alpha }_1,{\beta }_1)B({\alpha }_2+1,{\beta }_2)}{B(a_1,b_1)B(a_2+1,b_2)}+\frac{{\upsilon }_{1}B({\alpha }_1+1,{\beta }_1)B({\alpha }_2+1,{\beta }_2)}{B(a_1+1,b_1)B(a_2+1,b_2)}.$$

The exact posterior distributions of the comparative measures (i.e., OR, RR, and RD) under the Sarmanov beta prior distribution take the following generic form

$$f^{*}({\theta }_i;{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,\rho )={\omega }_{1}f({\theta }_i;{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2)+{\omega }_{2}f({\theta }_i;{\alpha }_1+1,{\beta }_1,{\alpha }_2,{\beta }_2)+{\omega }_{3}f({\theta }_i;{\alpha }_1,{\beta }_1,{\alpha }_2+1,{\beta }_2)+{\omega }_{4}f({\theta }_i;{\alpha }_1+1,{\beta }_1,{\alpha }_2+1,{\beta }_2).$$ (4)

If θ_i = OR_i, we have

$$f({\theta }_i;{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2)={\theta }_i^{-1-{\beta }_2}{\{B({\alpha }_1,{\beta }_1)B({\alpha }_2,{\beta }_2)\}}^{-1}B({\alpha }_1+{\alpha }_2,{\beta }_1+{\beta }_2)\times F({\alpha }_2+{\beta }_2,{\beta }_1+{\beta }_2;{\alpha }_1+{\alpha }_2+{\beta }_1+{\beta }_2;1-\frac{1}{{\theta }_i}),\text{for}{\theta }_i>0,$$ (5)

with F(·, ·; ·; ·) denotes the hypergeometric function Gauss (1813) defined by

$$F(\alpha ,\beta ;\gamma ;z)=\frac{1}{B(\beta ,\gamma -\beta )}{\int }_0^1{t}^{\beta -1}{(1-t)}^{\gamma -\beta -1}{(1-tz)}^{-\alpha }dt,\text{for}\gamma >\beta >0.$$

If θ_i = RR_i, we have

$$f({\theta }_i;{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2)=\{\begin{array}{cc}{\theta }_i^{{\alpha }_2-1}{\{B({\alpha }_1,{\beta }_1)B({\alpha }_2,{\beta }_2)\}}^{-1}B({\alpha }_1+{\alpha }_2,{\beta }_1)\times F(1-{\beta }_2,{\alpha }_1+{\alpha }_2;{\alpha }_1+{\alpha }_2+{\beta }_1;{\theta }_i)\hfill & \text{for}{\theta }_i\in [0,1)\hfill \\ {\theta }_i^{-{\alpha }_1-1}{\{B({\alpha }_1,{\beta }_1)B({\alpha }_2,{\beta }_2)\}}^{-1}B({\alpha }_1+{\alpha }_2,{\beta }_2)\times F(1-{\beta }_1,{\alpha }_1+{\alpha }_2;{\alpha }_1+{\alpha }_2+{\beta }_2;1/{\theta }_i)\hfill & \text{for}{\theta }_i\in [1,\infty )\hfill \end{array},$$ (6)

If θ_i = RD_i, we have

$$f({\theta }_i)=\mathrm{\Gamma }({\alpha }_1+{\beta }_1)\mathrm{\Gamma }({\alpha }_2+{\beta }_2)\{\begin{array}{c}\frac{{(-{\theta }_i)}^{{\beta }_1+{\beta }_2-1}{(1+{\theta }_i)}^{{\alpha }_1+{\beta }_2-1}}{\mathrm{\Gamma }({\beta }_1)\mathrm{\Gamma }({\alpha }_2)\mathrm{\Gamma }({\alpha }_1+{\beta }_2)}\times F_1({\beta }_2,{\alpha }_1+{\alpha }_2+{\beta }_1+{\beta }_2-2,1-{\alpha }_2,{\alpha }_1+{\beta }_2;1+{\theta }_i,1-{\theta }_i^2)\text{if}-1\le {\theta }_i\le 0,\hfill \\ \frac{{\theta }_i^{{\beta }_1+{\beta }_2-1}{(1-{\theta }_i)}^{{\alpha }_2+{\beta }_1-1}}{\mathrm{\Gamma }({\beta }_2)\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2+{\beta }_1)}\times F_1({\beta }_1,{\alpha }_1+{\alpha }_2+{\beta }_1+{\beta }_2-2,1-{\alpha }_1,{\alpha }_2+{\beta }_1;1-{\theta }_i,1-{\theta }_i^2)\text{if}0<{\theta }_i\le 1,\hfill \end{array}$$ (7)

where F₁ denotes the Appell function of the first kind defined by

$$F_1(a,b,b\prime ,c;x,y)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\frac{{(a)}_{m+n}(b)m{(b\prime )}_n{x}^m{y}^n}{{(c)}_{m+n}m!n!},\text{for}|x|<1,|y|<1,$$

and (c)_k = c(c + 1) ⋯ (c + k − 1) denotes the ascending factorial.

3. Using package mmeta

3.1. Package overview

The mmeta package has two major functions, i.e., multipletables and singletable. The function multipletables is to conduct inference based on multiple 2 × 2 tables. Specifically, the hyperparameters’ maximum likelihood estimates (â₁,b̂₁, â₂,b̂₂, ρ̂) and the inference on the overall comparative measures are obtained as described in Section 2.1. The posterior distributions of the study-specific comparative measures can be obtained either by the exact method as stated in Section 2.2 or by the sampling method based on Markov Chain Monte Carlo (MCMC) samples implemented in the R BRugs package, which is an interface to the OpenBUGS software. Based on the exact posterior distributions as (4), the posterior means of the study-specific comparative measures can be computed and the corresponding 95% equal tail credible intervals (the interval between the 2. 5% and 97.5% quantiles, referred to as 95% ET CI) and the 95% highest posterior density regions (referred to as 95% HDR) are calculated using the R Runuran package. The 95% ET CI and 95% HDR can be also obtained based on MCMC samples of the posterior distribution. The argument method can be either "exact" or "sampling" to control either the exact posterior distributions or the MCMC samples of the posterior distributions are used. The sampling method is implemented and set as default in this package because the current version of Gauss hypergeometric and Appell functions may diverge for some studies with extremely large numbers of subjects. To ensure reproducibility when the sampling method is used, the argument seed can be set to an integer from 1 to 14 (with default set to NULL) defining the state of the random number generator. This range restriction is from OpenBUGS (see the help file of BRugsFit function in BRugs manual for more details). Various plots can be generated by multipletables, which will be illustrated in the rest of this section. In contrast, the function singletable is to conduct exact posterior inference based on a single 2 × 2 table for the given prior distributions of risks. This function can be used as a sensitivity analysis tool to investigate the posterior distributions of the comparative measures under various pre-specified prior distributions. The details of the function singletable will be given in Section 3.5.

The arguments used in a call to the function multipletables() are multipletables <- function(data, measure, model = ”Sarmanov”, method = ”sampling”, alpha = 0.05, nsam = 10000)

We summarize next the main arguments of multipletables().

• data: A data frame that contains y1, n1, y2, n2, and studynames. The details of the data structure is described in Section 3.2.

• measure: A character string specifying a comparative measure. Options are "OR" (odds ratio), "RR" (relative risk), and "RD" (risk difference).

• model: A character string specifying the model. Options are "Independent" and "Sarmanov" (default). "Independent" is independent beta-binomial model. "Sarmanov" is Sarmanov beta-binomial model.

• method: A character string specifying the method. Options are "exact" and "sampling". "exact" is exact method. "sampling" (default) is a method based on MCMC samples of the posterior distribution obtained from the R BRugs package.

• alpha: A numeric value specifying the significant level. Default value is set to 0.05.

• nsam: A numeric value specifying the number of samples if method = "sampling". Default value is set to 10,000.

• seed: An integer from 1 to 14 defining the state of the random number generator. Default value is set to NULL.

3.2. Data structure

The structure of data in multipletables requires the input of a data frame with five columns, y1, n1, y2, n2, and studynames. The meanings of y1, n1, y2, and n2 can vary for different study designs. Users can define their own data frame to be used in multipletables. For example, a data frame named Bellamy based on a meta-analysis of the association between gestational diabetes mellitus and type 2 diabetes mellitus (Bellamy et al. 2009) can be defined as follows

R> y1 <- c(6628, 22, 0, 150, 1, 16, 7, 8, 0, 0, 0, 1, 0, 1, 7, 0, 0, 3, 18,
+     0)
R> n1 <- c(637341, 868, 39, 2242, 111, 783, 108, 489, 11, 435, 70, 61, 52,
+     39, 431, 35, 57, 47, 328, 41)
R> y2 <- c(2874, 71, 21, 43, 53, 405, 6, 13, 7, 23, 44, 21, 10,
+     15, 105, 10, 33, 14, 224, 5)
R> n2 <- c(21823, 620, 68, 166, 295, 5470, 70, 35, 23, 435, 696, 229, 28,
+     45, 801, 15, 241, 47, 615, 145)
R> studynames <- c("Feig 2008", "Lee H 2008", "Madarasz 2008",
+     "Gunderson 2007", "Vambergue 2008", "Lee 2007","Ferraz",
+     "Krishnaveni 2007", "Morimitsu 2007", "Jarvel 2006", "Albareda 2003",
+     "Aberg 2002", "Linne 2002", "Bian 2000", "Ko 1999", "Osei 1998",
+     "Damm 1994", "Benjamin 1993", "O'Sullivan 1964 and 1984",
+     "Persson 1991")
R> Bellamy <- data.frame(y1, n1, y2, n2, studynames = studynames,
+    stringsAsFactors = F)

There are two kinds of study design, i.e., retrospective (or case-control) study and prospective study (or clinical trial). In a case-control study, n1 and n2 are the numbers of subjects in the control and case groups, respectively, while y1 and y2 are the numbers of subjects with exposure in the control and case groups, respectively. measure = "OR", measure = "RR", and measure = "RD" correspond to the odds ratio, relative risk, and risk difference of exposure comparing the case group with the control group, respectively. In a prospective study, n1 and n2 are the numbers of subjects in the unexposed and exposed groups, respectively, while y1 and y2 are the numbers of subjects experienced a certain event in the unexposed and exposed groups, respectively. measure = "OR", measure = "RR", and measure = "RD" correspond to the odds ratio, relative risk, and risk difference of events comparing the exposed group with the unexposed group, respectively.

We have provided two example datasets, i.e., colorectal based on a meta-analysis of case-control studies and withdrawal based on a meta-analysis of clinical trials. In Sections 3.3 and 3.4, we illustrate the working of the package with the help of these two example datasets.

3.3. Example: colorectal dataset

The dataset colorectal consists of data from twenty published case-control studies of the N-acetyltransferase 2 (NAT2) acetylation status and colorectal cancer risk. NAT2 is a low-penetrance gene that regulates metabolizing enzymes. The activity of the enzymes is classified as rapid and slow acetylators. Ye and Parry (2002) investigated the association between rapid NAT2 acetylator status (event) and colorectal cancer (case) by conducting a meta-analysis based on twenty published case-control studies from January 1985 to October 2001. The data are summarized in Table 3.3. We define the odds ratio as the ratio of odds of having rapid NAT2 acetylator status comparing those with colorectal cancer to those without. The colorectal dataset example takes around 3 minutes and 1 minute to run using exact and sampling methods (10,000 samples), respectively.

To start analyzing the dataset, we first load the mmeta package and the colorectal dataset.

R> library("mmeta")
R> data("colorectal")

The colorectal dataset has the following structure

R> str(colorectal)

'data.frame':       20 obs. of  5 variables:
  $ y1        : num 10 19 13 40 13 92 33 151 50 34 …
  $ y2        : num 27 27 23 49 20 14 33 112 96 32 …
  $ n1        : num 41 45 41 96 28 205 36 329 112 96 …
  $ n2        : num 49 49 43 109 44 34 36 234 202 103 …
  $ studynames: chr "Ilett" "Ilett1" "Wohlleb" "Ladero" …

The function multipletables is called to conduct exact posterior inference of the odds ratios.

R> multiple.OR <- multipletables(data = colorectal, measure = "OR",
+    model = "Sarmanov", method = "exact")
R> summary(multiple.OR)

Model: Sarmanov Beta-Binomial Model
Overall odds ratio
Mean: 1.1
95% CI:[0.704, 1.718]

Maximum likelihood estimates of parameters:
a1 = 3.108, b1 = 2.914, a2 = 3.942, b2 = 3.361, rho = 0.125
Likelihood ratio test for within-group correlation (H0: rho = 0):
chi2: 3.152; p value: 0.08
Study-specific odds ratio :
           Mean lower bound upper bound
Ilett     3.475       1.401       7.330
Ilett1    1.719       0.754       3.460
Wohlleb   2.414       1.016       5.033
Ladero    1.184       0.662       1.937
Rodriguez 1.077       0.418       2.327
Lang      0.979       0.466       1.794
Oda       1.156       0.272       3.266
Shibuta   1.099       0.781       1.504
Bell      1.151       0.712       1.765
Spurr     0.885       0.484       1.498
Hubbard   0.848       0.598       1.150
Welfare   1.003       0.652       1.479
Gil       1.284       0.785       1.981
Chen      0.829       0.557       1.188
Lee       1.051       0.671       1.583
Yoshika   0.900       0.288       2.144
Potter    0.979       0.697       1.340
Slattery  1.928       1.686       2.191
Agundez   1.204       0.773       1.795
Butler    1.046       0.649       1.607
Overall   1.100       0.704       1.718

The likelihood ratio test of H₀ : ρ = 0 yields a p value of 0.08 with χ² test statistic being 3.152. The estimates of the hyperparameters, the estimated mean and the 95% confidence interval (CI) of the overall odds ratio are provided. In addition, the posterior means and the 95% credible intervals (CI) of all study-specific odds ratios are given. If the argument model = "Independent", the independent Beta-Binomial model is fitted to the dataset. If the argument method = "sampling", Monte Carlo sampling implemented in BRugs is used to obtain the posterior inference. The function xtable can convert the R object multiple.OR to an xtable object, which can then be printed as a LaTeX or HTML table.

R> multiple.OR.table <- xtable(multiple.OR)
R> print(multiple.OR.table)
R> print(multiple.OR.table, type = "html")

The argument type can be either "latex" (default) or "html" to control whether to print the LaTeX or HTML table.

The forest plot with the 95% CI of the overall odds ratio and the 95% CIs of the study-specific odds ratios as shown in Figure 1 can be obtained using the plot function with the argument type = "forest".

R> plot(multiple.OR, type = "forest", addline = 1)

Figure 1.

Forest plot of 20 study-specific and the overall odds ratios with 95% CIs.

The argument addline is to add a blue dotted reference line to the plot. If the argument file is specified, (e.g., file = "multiple_OR_forest"), the plot will be saved as "./mmeta/multiple_OR_forest.eps", where "./" denotes the current working directory and the directory mmeta is created automatically if it does not exist. The argument select (e.g., select = 1:4) can be set to select multiple target studies to be displayed. If the argument ciShow = TRUE (by default), the numbers of the posterior means and the CIs will be displayed at the right side of the forest plot. Many standard R plotting arguments (e.g., ylim, xlim) can be set in the plot function. Please refer to the help file for more details. The posterior density functions of some target studies can be overlaid as shown in Figure 2 with the argument type = "overlap".

R> plot(multiple.OR, type = "overlap", select = c(4, 14, 16, 20))

Figure 2.

Posterior distributions of study-specific odds ratios for four selected studies.

Figure 2 displays the overlaid posterior density functions of odds ratios for four selected studies, i.e., studies 4 (Ladero et al. 1991), 14 (Chen et al. 1998), 16 (Yoshioka et al. 1999), and 20 (Butler et al. 2001). Such plot provides an useful visualization of statistical evidence on association contributed from individual studies. Figure 2 shows that while in Chen et al. (1998), most of the density of the odds ratio is between 0.5 and 1.2 (mean: 0.829, 95% CI: 0.557, 1.189), the density shifts to the right in Butler et al. (2001) with the majority of the density laying between 0.6 and 1.5 (mean: 1.047, 95% CI: 0.649, 1.610). In the studies of Ladero et al. (1991) and Yoshioka et al. (1999), the density curves are more spread out because of their relatively smaller study sample sizes (mean: 1.185, 95% CI: 0.662, 1.948, and mean: 0.903, 95% CI: 0.288, 2.152, respectively).

The posterior density functions of these target studies can be viewed in a side-by-side manner as in Figure 3 if the argument type = "sidebyside", where both the prior and posterior distributions are displayed.

R> plot(multiple.OR, type = "sidebyside", select = c(4, 14, 16, 20),
+    ylim = c(0, 2.7), xlim = c(0.5, 1.5))

Figure 3.

Posterior distributions of study-specific odds ratios for four studies.

Figure 3 displays the prior and posterior distributions of the study-specific odds ratios for these four target studies. Such plot is useful to investigate the difference between the prior and posterior distributions, hence the strength of contribution from individual studies.

3.4. Example: withdrawal dataset

Tricyclic antidepressants are effective in preventing headaches and have become a standard modality in headache prevention. To investigate the efficacy and related adverse effects of tricyclic antidepressants in the treatment of headaches, Jackson et al. (2010) reported a meta-analysis based on multiple clinical trials from year 1964 to year 2009. Among several outcomes of interest, proportion of withdrawal during a trial is paid special attention because it is a very important measure of adverse effects and it plays a critical role in drug safety. One question of interest is whether the probability of withdrawing due to adverse effects is increased by the tricyclic treatment compared with the placebo. This can be measured by relative risk (defined as the ratio of risks of withdrawal comparing those in the tricyclic treatment group to those in the placebo group). The numbers of withdrawals due to adverse effects in sixteen clinical trials are summarized in Table 3.4. The withdrawal dataset example takes around 2 minutes and 1 minute to run using exact and sampling methods (10,000 samples), respectively.

To start analyzing the dataset, we first load the withdrawal dataset.

R> data("withdrawal")

The available data have the following structure

R> str(withdrawal)
'data.frame':       16 obs. of  5 variables:
$ y1         : num 0 9 8 13 10 22 2 8 4 4 …
$ n1         : num 40 27 53 29 34 48 18 21 49 38 …
$ y2         : num 1 4 8 14 15 9 3 12 7 10 …
$ n2         : num 40 16 47 56 44 53 18 26 105 36 …
$ studynames : chr "Bendtsen 1996" "Canepari 1985" "Couch 1976"
+    "Diamond  1971" …

The function multipletables is called to conduct exact posterior inference of relative risks.

R> multiple.RR <- multipletables(data = withdrawal, measure = "RR",
+     model = "Sarmanov")
R> summary(multiple.RR)

Model: Sarmanov Beta-Binomial Model
Overall Relative risk
Estimate: 1.263
95% CI:[0.82, 1.943]

Maximum likelihood estimates of hyperparameters:
a1 = 2.042, b1 = 7.408, a2 = 1.943, b2 = 5.179, rho = 0.093
Likelihood ratio test for within-group correlation (H0: rho = 0):
chi2: 0.207; p value: 0.65
Study-Specific Relative risk:
                  Mean lower bound upper bound
Bendtsen 1996     2.844      0.260       13.298
Canepari 1985     0.909      0.331       1.847
Couch 1976        1.261      0.502       2.623
Diamond 1971      0.678      0.363       1.162
Gobel 1994        1.272      0.655       2.278
Holroyd 2001      0.454      0.228       0.783
Indaco 1988       1.717      0.395       5.228
Jacobs 1972       1.359      0.681       2.519
Lance 1964        0.915      0.287       2.340
Landemark 1990    2.524      0.935       5.912
Loldrup 1989      6.169      3.684       10.444
Mathew 1981       1.013      0.621       1.564
Morland 1979      1.522      0.420       3.998
Noone 1980        1.324      0.517       2.934
Pfaffenrath 1994  1.331      0.842       2.008
Vernon 2009       3.350      0.479       14.053
Overall           1.263      0.820       1.943

The likelihood ratio test of H₀ : ρ = 0 yields a p value of 0.65 with χ² test statistic being 0.207. The estimates of the hyperparameters, the estimated mean and the 95% CI of the overall 16 relative risk are provided. In addition, the posterior mean and 95% CI of each study-specific relative risk are given. The forest plot with the confidence interval of the overall relative risk and the credible intervals of the study-specific relative risks as shown in Figure 3.4 can be obtained using the plot function with the argument type = "forest".

R> plot(multiple.RR, type = "forest", addline = 1)

The posterior density functions of some target studies can be overlaid as shown in Figure 3.4 with the argument type = "overlap".

R> plot(multiple.RR, type = "overlap", select = c(3, 8, 14, 16))

Figure 3.4 shows that while Couch et al. (1976) and Noone (1980) have most of the density of relative risk less than 3 (mean: 1.263, 95% CI: 0.502, 2.633, and mean: 1.326, 95% CI: 0.517, 2.944, respectively), the density shifts to the right in Jacobs (1972) (mean: 1.361, 95% CI: 0.681, 2.526). The density curve of the study of Vernon et al. (2009) (mean: 3.431, 95% CI: 0.479, 14.210) is more spread out because of the relatively small study sample size.

Moreover, the posterior density function of each target study can be viewed in a side-by-side manner as in Figure 3.4 if the argument type = "sidebyside", where both the prior and posterior distributions are displayed.

R> plot(multiple.RR, type = "sidebyside", select = c(3, 8, 14, 16),
+    ylim = c(0, 1.2), xlim = c(0.4, 3))

Figure 3.4 displays that the study specific odds ratios have very different posterior distributions under the same prior distributions.

Because the estimated relative risk for the study of Loldrup et al. (1989) (mean: 6.176, 95% CI: [3.684, 10.460]) is much larger than those for the other studies, it can be potentially influential to the analysis results. To evaluate the influence of this study, we remove it and reanalyze by calling the function multitables.

R> multiple.RR.sens <- multipletables(data = withdrawal[-11,], measure = "RR",
+    model = "Sarmanov")
R> summary(multiple.RR.sens)

The full results output is omitted here due to space limit. The likelihood ratio test of zero correlation coefficient results in p value of 0.40 with χ² test statistics being 0.707. Although the study of Loldrup et al. (1989) slightly changes the overall and study-specific relative risk estimates, it is not influential because the overall relative risk estimates are not significant with or without it.

If the risk difference is of interest, we define it as the difference of risks of withdrawal comparing those in the treatment group to those in the placebo group. The function multipletables is called to conduct posterior inference of the risk differences.

R> multiple.RD <- multipletables(data = withdrawal, measure = "RD",
+     model = "Sarmanov", seed=3)
R> summary(multiple.RD)

Model: Sarmanov Beta-Binomial Model
Overall Risk difference
Estimate: 0.057
95% CI:[−0.049, 0.162]

Maximum likelihood estimates of hyperparameters:
a1 = 2.042, b1 = 7.408, a2 = 1.943, b2 = 5.179, rho = 0.093
Likelihood ratio test for within-group correlation (H0: rho = 0):
chi2: 0.207; p value: 0.65
Study-Specific Risk difference:
                   Mean lower bound upper bound
Bendtsen 1996      0.021     −0.063       0.113
Canepari 1985     −0.041     −0.255       0.182
Couch 1976         0.024     −0.109       0.162
Diamond 1971      −0.136     −0.324       0.048
Gobel 1994         0.054     −0.129       0.235
Holroyd 2001      −0.232     −0.386      −0.075
Indaco 1988        0.048     −0.145       0.245
Jacobs 1972        0.093     −0.142       0.320
Lance 1964        −0.023     −0.122       0.062
Landemark 1990     0.147     −0.012       0.310
Loldrup 1989       0.591      0.509       0.666
Mathew 1981       −0.004     −0.127       0.118
Morland 1979       0.042     −0.138       0.228
Noone 1980         0.055     −0.196       0.309
Pfaffenrath 1994   0.060     −0.040       0.159
Vernon 2009        0.134     −0.142       0.420
Overall            0.057     −0.049       0.162

The forest plot, the overlaid, and side-by-side plots of the posterior density functions for risk difference can also be obtained by using the plot function with the argument type = "forest", type = "overlap", and type = "sidebyside", respectively.

Note that the study of Loldrup et al. (1989) can be influential on the analysis results because the estimated risk difference (mean: 0.590, 95% CI: [0.508, 0.666]) is much larger than those in other studies. To evaluate the sensitivity of the inference on this study, we remove it and reanalyze the dataset by call the function multitables.

R> multiple.RD.sens <- multipletables(data = withdrawal[-11, ], measure = "RD",
+    model = "Sarmanov")
R> summary(multiple.RD.sens)

The full results output is omitted here due to space limit. The likelihood ratio test of zero correlation coefficient results in p value of 0.40. Although the study of Loldrup et al. (1989) slightly changes the overall and study-specific risk difference estimates, it is not influential because the overall risk difference estimates are not significant with or without it.

3.5. Using singletable function

When the inference on a specific study based on a single 2 × 2 table is of interest, the function singletable can be used as a sensitivity analysis tool to conduct the exact posterior inference on comparative measures under various prior distributions. The arguments used in a call to the function singletable() are

• singletable <- function(y1, n1, y2, n2, measure, model = ”Sarmanov”, method = ”sampling”, alpha = 0.05, nsam = 10000)

We summarize the main arguments of singletable().

• y1, n1: Integers indicating the number of events and total subjects in group 1.

• y2, n2: Integers indicating the number of events and total subjects in group 2.

measure: A character string specifying a comparative measure. Options are "OR" (odds ratio), "RR" (relative risk), and "RD" (risk difference).

model: A character string specifying the model. Options are "Independent" and "Sarmanov" (default). "Independent" is independent beta-binomial model. "Sarmanov" is Sarmanov beta-binomial model.

method: A character string specifying the method. Options are "exact" and "sampling". "exact" is exact method. "sampling" (default) is a method based on MCMC samples of the posterior distribution obtained from the R BRugs package.

a1, b1, a2, b2: Numeric values specifying the hyperparameters of the beta prior distributions for groups 1 and 2.

rho: A numeric value specifying correlation coefficient for the Sarmanov beta prior distribution. Default value is set to 0.

alpha: A numeric value specifying the significant level. Default value is set to 0.05.

nsam: A numeric value specifying the number of samples if method is sampling. Default value is set to 10000.

seed: An integer from 1 to 14 defining the state of the random number generator. Default value is set to NULL.

To illustrate the use of the function singletable, we consider the study by Ladero et al. (1991) in the colorectal dataset with y₁ = 40, n₁ = 96, y₂ = 49, and n₂ = 109. The function singletable is called to conduct exact posterior inference of the odds ratio under four different prior distributions, i.e., Jeffreys prior distribution (a₁ = b₁ = a₂ = b₂ = 0.5), Laplace prior distribution (a₁ = b₁ = a₂ = b₂ = 0.5), and two Sarmanov correlated prior distributions with strong positive and negative prior correlations (a₁ = b₁ = a₂ = b₂ = 0.5, ρ = 0.5, −0.5). The results are listed below.

R> single.OR.Jeffreys <- singletable(a1 = 0.5, b1 = 0.5, a2 = 0.5, b2 = 0.5,
+     y1 = 40, n1 = 96, y2 = 49, n2 = 109, model = "Independent",
+     measure = "OR", method = "exact")
R> summary(single.OR.Jeffreys)

Measure: Odds ratio
Model: Independent Beta-Binomial Model
Mean: 1.189
Median: 1.143
95% ET CI: [0.656, 1.988]
95% HDR CI: [0.588, 1.87]

R> single.OR.Laplace <- singletable(a1 = 1, b1 = 1, a2 = 1, b2 = 1, y1 = 40,
+     n1 = 96, y2 = 49, n2 = 109, model = "Independent", measure = "OR",
+     method = "exact")
R> summary(single.OR.Laplace)

Measure: Odds ratio
Model: Independent Beta-Binomial Model
Mean: 1.188
Median: 1.142
95% ET CI: [0.658, 1.994]
95% HDR CI: [0.594, 1.876]

R> single.OR.Sar1 <- singletable(a1 = 0.5, b1 = 0.5, a2 = 0.5, b2 = 0.5,
+     rho = 0.5, y1 = 40, n1 = 96, y2 = 49, n2 = 109, model = "Sarmanov",
+     measure = "OR", method = "exact")
R> summary(single.OR.Sar1)

Measure: Odds ratio
Model: Sarmanov Beta-Binomial Model
Prior: Sarmanov
Mean: 1.188
Median: 1.141
95% ET CI: [0.656, 1.982]
95% HDR CI: [0.594, 1.868]

R> single.OR.Sar2 <- singletable(a1 = 0.5, b1 = 0.5, a2 = 0.5, b2 = 0.5,
+     rho = −0.5, y1 = 40, n1 = 96, y2 = 49, n2 = 109, model = "Sarmanov",
+     measure = "OR", method = "exact")
R> summary(single.OR.Sar2)

Measure: Odds ratio
Model: Sarmanov Beta-Binomial Model
Prior: Sarmanov
Mean: 1.191
Median: 1.144
95% ET CI: [0.656, 1.995]
95% HDR CI: [0.588, 1.878]

The corresponding prior and posterior distributions of the odds ratio under four prior distributions are shown in Figure 3.5 using the plot function with the argument type="overlap".

R> par(mfrow = c(2,2))
R> plot(single.OR.Jeffreys, type = "overlap", xlim = c(0.04, 0.3),
+     ylim = c(0, 15), main = "Jefferys Prior")
R> plot(single.OR.Laplace, type = "overlap", xlim = c(0.04, 0.3),
+     ylim = c(0, 15), main = "Laplace Prior")
R> plot(single.OR.Sar1, type = "overlap", xlim = c(0.04, 0.3),
+     ylim = c(0, 15),
+     main = expression(paste("Sarmanov Prior ", rho, " = 0.5")))
R> plot(single.OR.Sar2, type = "overlap", xlim = c(0.04, 0.3),
+     ylim = c(0, 15),
+     main = expression(paste("Sarmanov Prior ", rho, " = −0.5")))

As shown in Figure 3.5, the posterior distributions under all prior distributions share similar pattern of having most of weights on odds ratios between 0.5 and 2. This leads to similar credible intervals under all prior distributions although the Sarmanov beta prior distributions impose relatively strong prior correlations between p₁ and p₂ (ρ = −0.5 or 0.5).

4. Conclusion

In this paper, we present an overview of the mmeta package to conduct the exact posterior inference of the odds ratio, relative risk, and risk difference based on multiple studies or a single study of two populations with binary outcomes. The theory used for modeling fitting is summarized briefly, and the two major functions (multipletables and singletable) of the package are described in details. Practical use of the mmeta package is illustrated with two real examples of meta-analysis based on multiple 2 × 2 tables and one real example of a single 2 × 2 table. As a future research direction, we would expand the functionality of this package to conduct mega-regression analysis using the Sarmanov beta prior distributions as illustrated in Chen et al. (2011) and Chen et al. (2012). Moreover, we have investigated many available non-commercial algorithms and software to compute the hypergeometric function and the Appell function. To the best of our knowledge, we cannot find one that provides stable computation for the studies with extremely large numbers of subjects. Developing a robust algorithm for computation of these two functions are also part of our future research.

Figure 4.

Forest plot of 16 study-specific and the overall relative risk with 95% credible intervals.

Figure 5.

Posterior distributions of study-specific relative risk for four studies.

Figure 6.

Posterior distributions of study-specific relative risks for four studies.

Figure 7.

Prior and posterior distributions of odds ratio under Jeffreys prior distribution, Laplace prior distribution, and Sarmanov prior distributions (ρ = 0.5 and ρ = −0.5).

Table 1.

Data from a meta-analysis (Ye and Parry 2002) of case-control studies on the association between rapid N-acetyltransferase 2 (NAT2) acetylator status (event) and colorectal cancer risk (cases).

Author	Cases		Control
	no. events	no. individuals	no. events	no. individuals
Ilett	27	49	10	41
Ilett	27	49	19	45
Wohlleb	23	43	13	41
Ladero	49	109	40	96
Rodriguez	20	44	13	28
Lang	14	34	92	205
Oda	33	36	33	36
Shibuta	112	234	151	329
Bell	96	202	50	112
Spurr	32	103	34	96
Hubbard	100	275	140	343
Welfare	73	174	74	174
Gil	44	114	68	201
Chen	81	212	96	221
Lee	156	216	134	187
Yoshika	99	106	95	100
Potter	228	527	88	200
Slattery	931	1624	807	1963
Agundez	60	120	119	258
Butler	156	200	162	209

Table 2.

Data from a meta-analysis of sixteen studies on the association between withdrawal due to the adverse effects and the tricyclic treatment in Jackson et al. (2010). no. events: Number of individuals who withdrew from the study. no. individuals: Number of individuals who started the study.

Author	Treatment		Control
	no. events	no. individuals	no. events	no. individuals
Bendtsen 1996	1	40	0	40
Canepari 1985	4	16	9	27
Couch 1976	8	47	8	53
Diamond 1971	14	56	13	29
Gobel 1994	15	44	10	34
Holroyd 2001	9	53	22	48
Indaco 1988	3	18	2	18
Jacobs 1972	12	26	8	21
Lance 1964	7	105	4	49
Langemark 1990	10	36	4	38
Loldrup 1989	222	306	11	98
Mathew 1981	23	86	26	94
Morland 1979	4	23	3	23
Noone 1980	6	16	5	15
Pfaffenrath 1994	35	133	26	128
Vernon 2009	2	7	0	5

Appendix: Derivation of Equation (3)

For simplicity of notation, we suppress the index “i”. After some algebra, we can show

$$\int \text{Binomial}(y|n;p)\text{beta}(p;a,b)\{1+c(p-\mu )\}dp=P_{BB}(y|n;a,b)\{1+C(\frac{y+a}{n+a+b}-\mu \left)\right\}.$$

Denote μ_j = a_j/(a_j + b_j) and ${\sigma }_j^2={\mu }_j(1-{\mu }_j)/(a_j+b_j+1)(j=1,2)$. We have

$$\int \int \text{Pr}(y_1|n_1;p_1)\text{Pr}(y_2|n_2;p_2)\text{beta}(p_1;a_1,b_1)\text{beta}(p_2;a_2,b_2)·\{1+\frac{\rho }{{\sigma }_1{\sigma }_2}(p_1-{\mu }_1)(p_2-{\mu }_2)\}d{p}_{1}d{p}_2=\int \text{Pr}(y_2|n_2;p_2)\text{beta}(p_2;a_2,b_2)\int \text{Pr}(y_1|n_1;p_1)\text{beta}(p_1;a_1,b_1)·\{1+\frac{\rho }{{\sigma }_1{\sigma }_2}(p_1-{\mu }_1)(p_2-{\mu }_2)\}d{p}_{1}d{p}_2=\int \text{Pr}(y_2|n_2;p_2)\text{beta}(p_2;a_2,b_2)P_{BB}(y_1|n_1;a_1,b_1)·\{1+\frac{\rho }{{\sigma }_1{\sigma }_2}(\frac{y_{1i}+a_1}{n_{1i}+a_1+b_1}-{\mu }_1)(p_2-{\mu }_2)\}d{p}_2=[{\mathrm{P}}_{BB}(y_{1i};n_{1i},a_1,b_1){\mathrm{P}}_{BB}(y_{2i};n_{2i},a_2,b_2)\{1+\frac{\rho }{{\sigma }_1{\sigma }_2}(\frac{y_{1i}+a_1}{n_{1i}+a_1+b_1}-{\mu }_1)·(\frac{y_{2i}+a_2}{n_{2i}+a_2+b_2}-{\mu }_2)\left\}\right].$$