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A marginalized multilevel model for bivariate longitudinal binary data

A Publisher Correction to this article was published on 12 September 2019

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Abstract

This study considers analysis of bivariate longitudinal binary data. We propose a model based on marginalized multilevel model framework. The proposed model consists of two levels such that the first level associates the marginal mean of responses with covariates through a logistic regression model and the second level includes subject/time specific random intercepts within a probit regression model. The covariance matrix of multiple correlated time-specific random intercepts for each subject is assumed to represent the within-subject association. The subject-specific random effects covariance matrix is further decomposed into its dependence and variance components through modified Cholesky decomposition method and then the unconstrained version of resulting parameters are modelled in terms of covariates with low-dimensional regression parameters. This provides better explanations related to dependence and variance parameters and a reduction in the number of parameters to be estimated in random effects covariance matrix to avoid possible identifiability problems. Marginal correlations between responses of subjects and within the responses of a subject are derived through a Taylor series-based approximation. Data cloning computational algorithm is used to compute the maximum likelihood estimates and standard errors of the parameters in the proposed model. The validity of the proposed model is assessed through a Monte Carlo simulation study, and results are observed to be at acceptable level. Lastly, the proposed model is illustrated through Mother’s Stress and Children’s Morbidity study data, where both population-averaged and subject-specific interpretations are drawn through Emprical Bayes estimation of random effects.

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  • 12 September 2019

    Unfortunately, due to a technical error, the articles published in issues 60:2 and 60:3 received incorrect pagination. Please find here the corrected Tables of Contents. We apologize to the authors of the articles and the readers.

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Acknowledgements

Authors would like to thank to Assoc. Prof. Dr. Alexander de Leon for suggesting the use of data cloning computational algorithm in this study. Gul Inan would like to thank to The Scientific and Technological Research Council of Turkey (TUBITAK) for funding her studies through 2211-A and 2219 Fellowship Programmes. Authors would also like to thank the editor and two anonymous reviewers for their constructive comments which improved the quality of the paper.

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Correspondence to Gul Inan.

Appendix

Appendix

Detailed calculations of \(\Delta _{{itj}}\)

In probability theory, it is known that any conditional expectation can be written in terms of marginal expectation. This implies that the integration of conditional probability \(Pr(Y_{itj}=1|\mathbf {X}_{it},b_{it})\) over the distribution of random effect results in marginal probability \(Pr(Y_{itj}=1|\mathbf {X}_{it})\) for random effects models for longitudinal binary data, as given follows:

$$\begin{aligned} \begin{aligned} E(Y_{itj}|\mathbf {X}_{it})&=\int E(Y_{itj}|\mathbf {X}_{it},b_{it} )f(b_{it})db_{it} \\ \Rightarrow Pr(Y_{itj}=1|\mathbf {X}_{it})&= \int Pr(Y_{it}=1|\mathbf {X}_{it},b_{it})f(b_{it})db_{it}, \end{aligned} \end{aligned}$$
(14)

where \(f(b_{it})\) is a univariate normal distribution with mean 0 and variance \(Var(b_{it})\). Substituting marginal and conditional probabilities given in Eq. 1, which are the first and second levels of the proposed model, into Eq. 14 gives the following expression:

$$\begin{aligned} \frac{exp(\mathbf {X}_{it}\varvec{\beta }_{j})}{1+exp(\mathbf {X}_{it}\varvec{\beta }_{j})}&=\int ^{+\infty }_{-\infty } \Phi (\Delta _{itj}(\mathbf {X}_{it})+ b_{it})f(b_{it}) db_{it}. \end{aligned}$$
(15)

Since \(b_{it}\sim N(0,Var(b_{it}))\), it is possible to write \(b_{it}=\sqrt{Var(b_{it})}z_{i}\), where \(z_{i}\sim N(0,1)\). Following Griswold (2005), define a \(W_{i}\sim N(0,1)\), where \(W_{i}\perp z_{i}\) and the symbol \(\perp \) denotes independence. Then it is easy to show that

$$\begin{aligned} \begin{array}{ll} \frac{W_{i}}{\sqrt{Var(b_{it})}}-z_{i}&\sim N(0,1 + Var(b_{it})^{-1}). \end{array} \end{aligned}$$
(16)

The right-hand side of Eq. 15 can be rewritten as follows:

Then Eq. 15 can be rewritten as follows:

$$\begin{aligned} \frac{exp(\mathbf {X}_{it}\varvec{\beta }_{j})}{1+exp(\mathbf {X}_{it}\varvec{\beta }{j})}&=\Phi \left( \frac{\Delta _{itj}(\mathbf {X}_{it})}{\sqrt{1+ Var(b_{it})}}\right) . \end{aligned}$$
(17)

Solving Eq. 17 for \(\Delta _{itj}\) provides a closed-form solution for it such that

$$\begin{aligned} \Delta _{itj}= \Phi ^{-1} \left( \frac{exp(\mathbf {X}_{it}\varvec{\beta }_{j})}{1+exp(\mathbf {X}_{it}\varvec{\beta }_{j})}\right) \sqrt{1+ Var(b_{it})}, \end{aligned}$$
(18)

\(\forall \)i, t and j, where \(\Delta _{itj}\) is an explicit function of both marginal regression parameter (i.e., \(\varvec{\beta }_{j}\)) and variance of random effect (i.e., Var(\(b_{it}\))).

Taylor series-based derivation of marginal correlation function

The formula for the marginal correlation between \(Y_{itj}\) and \(Y_{it^{\prime }j^{\prime }}\), \(Corr\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) \) is given as follows:

$$\begin{aligned} \begin{array}{ll} Corr\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right)&=\frac{Cov\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) }{\sqrt{Var(Y_{itj})} \sqrt{Var(Y_{it^{\prime }j^{\prime }})} } \quad \forall i, t\,\text {and}\,j, \end{array} \end{aligned}$$
(19)

where \(Cov\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) \) is covariance between \(Y_{itj}\) and \(Y_{it^{\prime }j^{\prime }}\) and \(Var(Y_{itj})\) is variance of \(Y_{itj}\).

For any model including random effects, the formulation for \(Corr\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) \) in Eq. 19 relies on conditional expectation, variance, and covariance formulas. The marginal covariance \(Cov\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) \) can be formulated in terms of conditional expectation and covariance as follows (Rudary 2009):

$$\begin{aligned} Cov\left( Y_{itj},Y_{it^{\prime }j^{\prime }}\right) \!\!=\!E\left( Cov\left( Y_{itj},Y_{it^{\prime }j^{\prime }}|b_{it},b_{it^{\prime }}\right) \right) \!+\!Cov\left( E\left( Y_{itj}|b_{it}\right) , E\left( Y_{it^{\prime }j^{\prime }}|b_{it^{\prime }}\right) \right) , \end{aligned}$$
(20)

where \(E\left( Cov\left( Y_{itj},Y_{it^{\prime }j^{\prime }}|b_{it},b_{it^{\prime }}\right) \right) =E\left( E[(Y_{itj}-\mu _{itj})\times (Y_{it^{\prime }j^{\prime }}-\mu _{it^{\prime }j^{\prime }})]|b_{it}, b_{it^{\prime }}\right) \)\(=E\left( E\left( Y_{itj}-\mu _{itj}\right) \times E\left( Y_{it^{\prime }j^{\prime }}-\mu _{it^{\prime }j^{\prime }}\right) \right) =0\), with \(E\left( Y_{itj}|b_{it}\right) =\mu _{itj}\), since \( Y_{itj}|b_{it}\) and \(Y_{it^{\prime }j^{\prime }}|b_{it^{\prime }}\) are independent of each other, which is also known as the conditional independence assumption, and \(Cov\left( E\left( Y_{itj} |b_{it} \right) , E\left( Y_{it^{\prime }j^{\prime }} |b_{it^{\prime }} \right) \right) = Cov\left( \mu _{itj},\mu _{it^{\prime }j^{\prime }} \right) \). Then \(Cov\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right) \) in Eq. 20 can be rewritten as follows:

$$\begin{aligned} \begin{array}{ll} Cov\left( Y_{itj}, Y_{it^{\prime }j^{\prime }}\right)&=Cov\left( \mu _{itj},\mu _{it^{\prime }j^{\prime }} \right) . \end{array} \end{aligned}$$
(21)

The marginal variance \(Var(Y_{itj})\) can be formulated in terms of conditional expectation and variance as follows:

$$\begin{aligned} \begin{array}{ll} Var(Y_{itj})&=E\left( Var\left( Y_{itj}|b_{it}\right) \right) + Var\left( E\left( Y_{itj}|b_{it}\right) \right) , \end{array} \end{aligned}$$
(22)

where \(E\left( Var\left( Y_{itj}|b_{it}\right) \right) =E\left( \mu _{itj}(1-\mu _{itj}) \right) \) and \(Var\left( E\left( Y_{itj}|b_{it}\right) \right) =Var\left( \mu _{itj}\right) \). Then \(Var(Y_{itj})\) in Eq. 22 can be rewritten as follows:

$$\begin{aligned} \begin{array}{ll} Var(Y_{itj})&=E\left( \mu _{itj}(1-\mu _{itj})\right) + Var\left( \mu _{itj}\right) . \end{array} \end{aligned}$$
(23)

Then \(Corr\left( Y_{itj},Y_{it^{\prime }j^{\prime }}\right) \) in Eq. 19 can be rewritten based on Eqs. 21 and 23 as follows:

(24)

Taylor series-based approximations for \(E(\mu _{itj}(1-\mu _{itj})), Var(\mu _{itj})\) and \(Cov(\mu _{itj}, \mu _{it^{\prime }j^{\prime }})\)

Equation 24 requires a closed-form solution for \(E(\mu _{itj}(1-\mu _{itj}))\), \(Var(\mu _{itj})\) and \(Cov(\mu _{itj}, \mu _{it^{\prime }j^{\prime }})\). However, unless the link function in \(\mu _{itj}\) is an identity function, a closed-form solution for these expressions cannot be obtained (e.g., for binary responses). In this sense, following Goldstein and Rasbash (1996), Vangeneugden et al. (2010), and Vangeneugden et al. (2011), these expressions can be approximated by a first-order Taylor series expansion around \(b_{it}=0\).

First-order Taylor series expansions of \(E(\mu _{itj}(1-\mu _{itj}))\) and \(Var(\mu _{itj})\) around \(b_{it}=0\) give the following approximations, respectively:

$$\begin{aligned} E(\mu _{itj}(1-\mu _{itj}))\simeq (\mu _{itj}(1-\mu _{itj})|_{b_{it}=0}) \quad \text{ and } \end{aligned}$$
(25)
$$\begin{aligned} Var(\mu _{itj})&\simeq \left( \frac{\partial \mu _{itj}}{\partial b_{it}}|_{b_{it}=0}\right) ^2 Var(b_{it}). \end{aligned}$$
(26)

In a similar fashion, a first-order Taylor series expansion for \(Cov(\mu _{itj}, \mu _{it^{\prime }j^{\prime }})\) around \(b_{it}=0\) and \(b_{it^{\prime }}=0\) gives the following approximation:

$$\begin{aligned} \begin{array}{ll} Cov\left( Y_{itj},Y_{it^{\prime }j^{\prime }}\right)&\simeq \left( \frac{\partial \mu _{itj}}{\partial b_{it}}|_{b_{it}=0}\right) Cov\left( b_{it},b_{it^{\prime }}\right) \left( \frac{\partial \mu _{it^{\prime }j^{\prime }}}{\partial b_{it^{\prime }}}|_{b_{it^{\prime }}=0}\right) . \end{array} \end{aligned}$$
(27)

For the proposed model in Eq. 1, \(\mu _{itj}= Pr(Y_{itj}=1|\mathbf {X}_{it},b_{it}) = \Phi (\Delta _{itj}(\mathbf {X}_{it})+ b_{it})\). Then Eqs. 25, 26 and 27 can be rearranged, respectively, as follows:

$$\begin{aligned} \begin{array}{ll} E(\mu _{itj}(1-\mu _{itj}))&{}\simeq (\mu _{itj}(1-\mu _{itj})|_{b_{it}=0}) \\ &{}= \Phi (\Delta _{itj}(\mathbf {X}_{it})) (1-\Phi (\Delta _{itj}(\mathbf {X}_{it}))), \end{array} \end{aligned}$$
(28)
$$\begin{aligned} \begin{array}{ll} Var(\mu _{itj})&{}\simeq \left( \frac{\partial \mu _{itj}}{\partial b_{it}}|_{b_{it}=0}\right) ^2 Var(b_{it})\\ &{}=(\phi (\Delta _{itj}(\mathbf {X}_{it})))^2 Var(b_{it}), \end{array} \end{aligned}$$
(29)

and

$$\begin{aligned} \begin{array}{ll} Cov\left( Y_{itj},Y_{it^{\prime }j^{\prime }}\right) &{}\simeq \left( \frac{\partial \mu _{itj}}{\partial b_{it}}|_{b_{it}=0}\right) Cov\left( b_{it},b_{it^{\prime }}\right) \left( \frac{\partial \mu _{it^{\prime }j^{\prime }}}{\partial b_{it^{\prime }}}|_{b_{it^{\prime }}=0}\right) \\ &{}=\phi (\Delta _{itj}(\mathbf {X}_{it}))Cov\left( b_{it},b_{it^{\prime }}\right) \phi (\Delta _{it^{\prime }j^{\prime }}(\mathbf {X}_{it})), \end{array} \end{aligned}$$
(30)

where \(Var(b_{it})\) and \(Cov(b_{it},b_{it^{\prime }})\) are the corresponding variance and covariance components in \(\Sigma _{i}\).

For the proposed model in Eq. 1, \(Corr(Y_{itj},Y_{it^{\prime }j^{\prime }})\) can be expressed through Eqs. 2830 as follows:

(31)

\(\forall \)i, t and j, where \(v_{itj}=\Phi (\Delta _{itj}(\mathbf {X}_{it})) (1-\Phi (\Delta _{itj}(\mathbf {X}_{it})))\), \(v_{itj^{\prime }}=\Phi (\Delta _{itj^{\prime }}(\mathbf {X}_{it})) (1-\Phi (\Delta _{itj^{\prime }}(\mathbf {X}_{it})))\), \(\phi (.)\) is the probability density function of standard normal distribution.

R code for data cloning

The R code for model fitting for Sect. 3 is as follows:

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Inan, G., Ilk, O. A marginalized multilevel model for bivariate longitudinal binary data. Stat Papers 60, 601–628 (2019). https://doi.org/10.1007/s00362-016-0840-1

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Keywords

  • Bivariate binary responses
  • Covariance matrix decomposition
  • Data cloning
  • Multilevel models
  • Multiple correlated random effects