Fixed versus Mixed Effects Based Marginal Models for Clustered Correlated Binary Data: an Overview on Advances and Challenges

Abstract

In a cross-sectional cluster setup, the binary responses from the individuals in a cluster become correlated as they share a common cluster effect, whereas longitudinal responses from an individual those form a cluster become correlated as the present and past responses are likely to maintain a suitable dynamic relationship. In both cluster and longitudinal setups, the marginal means may or may not be specified as the function of regression effects/parameters only. In a cluster setup, this depends on the distributional assumption of the random cluster effects and in a longitudinal setup this depends on the form such as linear or non-linear dynamic relationships used to construct a conditional model. However, over the last four decades, many studies arbitrarily pre-specified the marginal means as the function of regression effects only under both cluster and longitudinal setups and accommodated correlations also using arbitrarily selected ‘working’ correlation structures. This paper makes a thorough in-depth review of these decades long binary correlation models for consistent and efficient estimation of the regression effects. Both progress and drawbacks of these works are presented clearly showing how the inconsistency can arise if the pre-specified marginal fixed model is used when in fact such a marginal fixed effects model does not exist. This is because, some of the conditional random effects models in a cluster setup produce mixed effect models for the marginal means, and conditional non-linear dynamic models in a longitudinal setup produce history based marginal recursive/dynamic models. As the practitioners in both cluster and longitudinal setups deal with large data sets, it is demonstrated for their benefits how one can use the GQL (generalized quasi-likelihood) estimation approach both in cluster and longitudinal setups. Furthermore, there exist many studies using the Bayesisn approach where unlike the aforementioned parametric correlation structure based inferences, the marginal mixed effects models have been used for inferences for correlated binary data without specifying their correlation structures, under both cluster and longitudinal setup. We also provide a brief review on this alternative approach.

Appendix: Higher Order Moments (up to order 4) for Clustered Binary Responses

To compute ${\Omega }_{i}(\boldsymbol {\beta },\sigma ^{2}_{\gamma })$ in Eq. 80 on top of var(Y_ij) and cov(Y_ij,Y_ik), we need the formulas for certain specific third and fourth order moments as follows.

Computation of var(Y _ij Y _ik)

This variance is computed as

$$ \begin{array}{@{}rcl@{}} \text{var}(Y_{ij}Y_{ik})&=&E[Y^{2}_{ij}{Y^{2}_{k}}]-[E(Y_{ij}Y_{ik})]^{2}\\ &=&E(Y_{ij}Y_{ik})-[E(Y_{ij}Y_{ik})]^{2}=\lambda^{BA}_{i,jk}[1-\lambda^{BA}_{i,jk}] =\omega^{BA}_{i,jjkk}, \end{array} $$

(98)

where $\lambda ^{BA}_{i,jk}$ is computed by Eq. 46.

Computation of $\text {cov}(Y_{ij},Y_{ik}Y_{i\ell })=\phi _{i,jk \ell } (\boldsymbol {\beta },\sigma ^{2}_{\gamma })$

Because

$$ \begin{array}{@{}rcl@{}} \text{cov}(Y_{ij},Y_{ik}Y_{i\ell})&=&E[Y_{ij}Y_{ik}Y_{i\ell}]-\mu^{BA}_{ij}\lambda^{BA}_{i,k \ell}, \end{array} $$

(99)

we need the formula for the third order moments, namely

$$ \begin{array}{@{}rcl@{}} &&E[Y_{ij}Y_{ik}Y_{i\ell}]=E_{\gamma_{i}}E[\{Y_{ij}Y_{ik}Y_{i\ell}\}|\gamma_{i}] \\ &&=E_{\gamma_{i}}\left[E(Y_{ij}|\gamma_{i})E(Y_{ik}|\gamma_{i})E(Y_{i\ell}|\gamma_{i})\right] \\ &=&\int p^{*}_{ij}(\boldsymbol{\beta},\gamma_{i}) p^{*}_{ik}(\boldsymbol{\beta},\gamma_{i})p^{*}_{i\ell}(\boldsymbol{\beta},\gamma_{i})g_{N}(\gamma_{i})d\gamma_{i}, \end{array} $$

(100)

where, for example, $p^{*}_{ij}(\boldsymbol {\beta },\gamma _{i})=\exp (\boldsymbol {x}^{\prime }_{ij}\boldsymbol {\beta } +\gamma _{i})/[1+\exp (\boldsymbol {x}^{\prime }_{ij}\boldsymbol {\beta }+\gamma _{i})],$ and $g_{N}(\gamma _{i}) \equiv [\gamma _{i} \sim N(0,\sigma ^{2}_{\gamma })].$ Similar to Eq. 46, this normal integration in Eq. 100 may be computed approximately by

$$ \begin{array}{@{}rcl@{}} &&\lambda^{BA}_{i,jk\ell}(\boldsymbol{\beta},\sigma^{2}_{\gamma}) \\ &=&{\sum}^{V}_{v_{i}=0}p^{*}_{ij}(\boldsymbol{x}_{ij};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i}))p^{*}_{ik}(\boldsymbol{x}_{ik};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i})) \\ & \times &p^{*}_{i\ell}(\boldsymbol{x}_{i\ell};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i}))\begin{pmatrix}V \\ v_{i}\end{pmatrix}(1/2)^{v_{i}}(1/2)^{V-v_{i}}, \end{array} $$

(101)

yielding

$$ \begin{array}{@{}rcl@{}} &&\phi^{BA}_{i,jk \ell} (\boldsymbol{\beta},\sigma^{2}_{\gamma})=\lambda^{BA}_{i,jk\ell}(\boldsymbol{\beta},\sigma^{2}_{\gamma}) -\mu^{BA}_{ij}\lambda^{BA}_{i,k \ell}. \end{array} $$

(102)

Computation of $\text {cov}(Y_{ij}Y_{ik},Y_{i\ell }Y_{im})=\omega _{i,jk\ell m} (\boldsymbol {\beta },\sigma ^{2}_{\gamma })$

By similar calculations as in Eq. 101, one obtains

$$ \omega^{BA}_{i,jk\ell m} (\boldsymbol{\beta},\sigma^{2}_{\gamma})=\lambda^{BA}_{i,jk\ell m}(\boldsymbol{\beta},\sigma^{2}_{\gamma}) -\lambda^{BA}_{i,jk}(\boldsymbol{\beta},\sigma^{2}_{\gamma})\lambda^{BA}_{i,\ell m}(\boldsymbol{\beta},\sigma^{2}_{\gamma}), $$

(103)

where

$$ \begin{array}{@{}rcl@{}} &&\lambda^{BA}_{i,jk\ell m}(\boldsymbol{\beta},\sigma^{2}_{\gamma}) \\ &=&{\sum}^{V}_{v_{i}=0}p^{*}_{ij}(\boldsymbol{x}_{ij};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i}))p^{*}_{ik}(\boldsymbol{x}_{ik};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i}))p^{*}_{i\ell}(\boldsymbol{x}_{i\ell};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i})) \\ & \times &p^{*}_{im}(\boldsymbol{x}_{im};\boldsymbol{\beta},\sigma_{\gamma} h(v_{i}))\begin{pmatrix}V \\ v_{i}\end{pmatrix}(1/2)^{v_{i}}(1/2)^{V-v_{i}}. \end{array} $$

(104)