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Finite Mixture of Censored Linear Mixed Models for Irregularly Observed Longitudinal Data

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Abstract

Linear mixed-effects models are commonly used when multiple correlated measurements are made for each unit of interest. Some inherent features of these data can make the analysis challenging, such as when the series of responses are repeatedly collected for each subject at irregular intervals over time or when the data are subject to some upper and/or lower detection limits of the experimental equipment. Moreover, if units are suspected of forming distinct clusters over time, i.e., heterogeneity, then the class of finite mixtures of linear mixed-effects models is required. This paper considers the problem of clustering heterogeneous longitudinal data in a mixture framework and proposes a finite mixture of multivariate normal linear mixed-effects model. This model allows us to accommodate more complex features of longitudinal data, such as measurement at irregular intervals over time and censored data. Furthermore, we consider a damped exponential correlation structure for the random error to deal with serial correlation among the within-subject errors. An efficient expectation-maximization algorithm is employed to compute the maximum likelihood estimation of the parameters. The algorithm has closed-form expressions at the E-step that rely on formulas for the mean and variance of the multivariate truncated normal distributions. Furthermore, a general information-based method to approximate the asymptotic covariance matrix is also presented. Results obtained from the analysis of both simulated and real HIV/AIDS datasets are reported to demonstrate the effectiveness of the proposed method.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

We thank the editor, associate editor and three anonymous referees for their important comments and suggestions which lead to an improvement of this paper.

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001. Larissa A. Matos received support from FAPESP-Brazil (Grant 2020/16713-0).

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Authors and Affiliations

  1. Departamento de Estatística, Universidade Estadual de Campinas, Campinas, Brazil

    Francisco H. C. de Alencar & Larissa A Matos

  2. Department of Statistics, University of Connecticut, Storrs, USA

    Víctor H. Lachos

Authors
  1. Francisco H. C. de Alencar
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  2. Larissa A Matos
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  3. Víctor H. Lachos
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Correspondence to Larissa A Matos.

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Appendices

Appendix A. The derivatives of the expectations

Using the properties of conditional expectation, we obtain:

$$\begin{aligned}&\widehat{Z_{ij}\mathbf {y}}_i^{(k)} = \mathbb {E}(Z_{ij}\mathbf {y}_i|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}) = \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {y}_i))) \\&= \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbf {y}_i)) = \mathbb {E}(Z_{ij}|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)})\mathbb {E}(\mathbf {y}_i|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}, Z_{ij} = 1)\\&= \widehat{Z}_{ij}^{(k)}\widehat{\mathbf {y}}_i^{(k)},\\&\widehat{Z_{ij}\mathbf {y}_i^2}^{(k)} = \mathbb {E}(Z_{ij}\mathbf {y}_i\mathbf {y}_i^\top |\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}) = \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {y}_i\mathbf {y}_i^\top ))) \\&= \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i\mathbf {y}_i^\top |Z_{ij}}(\mathbf {y}_i)) = \mathbb {E}(Z_{ij}|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)})\mathbb {E}(\mathbf {y}_i\mathbf {y}_i^\top |\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}, Z_{ij} = 1)\\&= \widehat{Z}_{ij}^{(k)}\widehat{\mathbf {y}}_i^{2(k)},\\&\widehat{Z_{ij}\mathbf {b}}_{i}^{(k)} = \mathbb {E}(Z_{ij}\mathbf {b}_{i}|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}) = \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}))) \\&= \mathbb {E}(Z_{ij}|\mathbf {V}_i,\mathbf {C}_i, \widehat{\varvec{\theta }}_j^{(k)})\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}))\\&= \widehat{Z}_{ij}^{(k)}\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i})) = \widehat{Z}_{ij}^{(k)}\widehat{\mathbf {b}}_{i}^{(k)},\\&\widehat{Z_{ij}\mathbf {b}_{i}^2}^{(k)} = \mathbb {E}(Z_{ij}\mathbf {b}_{i}\mathbf {b}_{i}^\top |\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}) = \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}\mathbf {b}_{i}^\top ))) \\&= \mathbb {E}(Z_{ij}|\mathbf {V}_i,\mathbf {C}_i, \widehat{\varvec{\theta }}_j^{(k)})\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}\mathbf {b}_{i}^\top ))\\&= \widehat{Z}_{ij}^{(k)}\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}\mathbf {b}_{i}^\top )) = \widehat{Z}_{ij}^{(k)}\widehat{\mathbf {b}}_{i}^{2(k)},\\&\widehat{Z_{ij}\mathbf {y}_i\mathbf {b}_{i}^\top }^{(k)} = \mathbb {E}(Z_{ij}\mathbf {y}_i\mathbf {b}_{i}^\top |\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)}) = \mathbb {E}_{Z_{ij}|\mathbf {V}_i,\mathbf {C}_i}(Z_{ij}\mathbb {E}_{\mathbf {y}_i|Z_{ij}}(\mathbf {y}_i\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}^\top ))) \\&= \mathbb {E}(Z_{ij}|\mathbf {V}_i,\mathbf {C}_i,\widehat{\varvec{\theta }}_j^{(k)})\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbf {y}_i\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}^\top ))\\&= \widehat{Z}_{ij}^{(k)}\mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}(\mathbf {y}_i\mathbb {E}_{\mathbf {b}_i|\mathbf {y}_i,Z_{ij}}(\mathbf {b}_{i}^\top )) = \widehat{Z}_{ij}^{(k)}\widehat{\mathbf {y}_i\mathbf {b}}_{i}^{(k)}. \end{aligned}$$

To compute the expectation terms above, it is important to note that,

$$\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i {\mathop {\sim }\limits ^{\text {ind.}}} TN_{n_i}(\mathbf {X}_i\varvec{\beta }_j, \sigma ^2_j\mathbf {E}_{ij}; \mathbb {A}), \text {and}$$
$$\mathbf {b}_{i}|\mathbf {y}_i, Z_{ij} {\mathop {\sim }\limits ^{\text {ind.}}} N_q(\varvec{\varphi }_{ij}(\mathbf {y}_i - \mathbf {X}_i\varvec{\beta }_j), \varvec{\Lambda }_{ij}),$$

where \(TN_{n_i}(\cdot ;\mathbb {A})\) denotes the truncated normal distribution in the interval \(\mathbb {A}\), \(\varvec{\varphi }_{ij} = \frac{1}{\sigma ^2_j}\varvec{\Lambda }_{ij}\mathbf {U}_i^\top\) and \(\varvec{\Lambda }_{ij} = \left( \mathbf {D}^{-1} + \frac{1}{\sigma ^2_j}\mathbf {U}_i^\top \mathbf {E}_{ij}^{-1}\mathbf {U}_i\right) ^{-1}\). So, the expectation terms are given by

$$\begin{aligned}&\widehat{\mathbf {b}}_{i}^{(k)} = \widehat{\varvec{\varphi }}_{ij}^{(k)}(\widehat{\mathbf {y}}_i^{(k)} - \mathbf {X}_i\widehat{\varvec{\beta }}^{(k)}_ j),\\&\widehat{\mathbf {b}}_{i}^{2(k)} = \widehat{\varvec{\Lambda }}_{ij}^{(k)} + \widehat{\varvec{\varphi }}_{ij}^{(k)}\left( \widehat{\mathbf {y}}_{i}^{2(k)} - \widehat{\mathbf {y}}_{i}^{(k)}\widehat{\varvec{\beta }}_j^{\top (k)}\mathbf {X}_i^\top - \mathbf {X}_i\widehat{\varvec{\beta }}_j^{(k)}\widehat{\mathbf {y}}_{i}^{\top (k)} + \mathbf {X}_i\widehat{\varvec{\beta }}_j^{(k)}\widehat{\varvec{\beta }}_j^{\top (k)}\mathbf {X}_i^\top \right) \widehat{\varvec{\varphi }}_{ij}^{\top (k)},\\&\widehat{\mathbf {y}_i\mathbf {b}}_{i}^{(k)} = \mathbb {E}_{\mathbf {y}_i|Z_{ij},\mathbf {V}_i,\mathbf {C}_i}[\mathbf {y}_i(\mathbf {y}_i - \mathbf {X}_i\varvec{\beta }_j)^\top \varvec{\varphi }_{ij}^\top ] = (\widehat{\mathbf {y}}_i^{2(k)}-\widehat{\mathbf {y}}_i^{(k)}\widehat{\varvec{\beta }}^{\top (k)}_j\mathbf {X}_i^\top )\widehat{\varvec{\varphi }}_{ij}^{\top (k)}. \end{aligned}$$

As mentioned earlier, the E-step reduces to the computation of \(\widehat{\mathbf {y}}_{i}^{(k)}\) and \(\widehat{\mathbf {y}}^{2(k)}_{i}\), that is, the mean and the second moment of a multivariate truncated Gaussian distribution, and can be determined in closed form (Vaida and Liu , 2009; Matos et al. , 2016).

Appendix B. Empirical information matrix

The elements of \(\widehat{s}_i\) in Equation (13) are given by

$$\begin{aligned}&\widehat{s}_{i,\pi _j} = \frac{\widehat{\mathbf {Z}}_{ij}}{\widehat{\pi }_j} - \frac{\widehat{\mathbf {Z}}_{ig}}{\widehat{\pi }_g},\\&\widehat{s}_{i,\varvec{\beta }_{j}} = (\widehat{s}_{i,\beta _{j1}}, \ldots , \widehat{s}_{i,\beta _{jp}}) = \frac{1}{\widehat{\sigma ^2_j}}\left[ \mathbf {X}_i^\top \widehat{\mathbf {E}}_{ij}^{-1}(\widehat{\mathbf {Z}}_{ij}\widehat{\mathbf {y}}_i - \widehat{\mathbf {Z}}_{ij}\mathbf {U}_i\widehat{\mathbf {b}}_{i}) - \widehat{\mathbf {Z}}_{ij}\mathbf {X}_i^\top \widehat{\mathbf {E}}_{ij}^{-1}\mathbf {X}_i\widehat{\varvec{\beta }}_j \right] ,\\&\widehat{s}_{i,\sigma ^2_{j}} = - \frac{\widehat{Z}_{ij}n_i}{2\widehat{\sigma ^2_j}} + \frac{1}{2\widehat{\sigma ^4_j}}\left[ \mathrm {tr}\left( \widehat{a}_{ij}^{(k)} \widehat{\mathbf {E}}_{ij}^{-1} \right) - 2\widehat{\varvec{\beta }}_j^\top \mathbf {X}_i^{\top }\widehat{\mathbf {E}}_{ij}^{-1}(\widehat{{Z}}_{ij}\widehat{\mathbf {y}}_{i} - \widehat{{Z}}_{ij}\mathbf {U}_i\widehat{\mathbf {b}}_{i}) \right. \\&+ \left. \widehat{{Z}}_{ij}\widehat{\varvec{\beta }}^\top _j\mathbf {X}_i^{\top } \widehat{\mathbf {E}}_{ij}^{-1}\mathbf {X}_i\widehat{\varvec{\beta }}_{j} \right] ,\\&\widehat{s}_{i,\varvec{\alpha }_{j}} = (\widehat{s}_{i,\alpha _{j1}}, \ldots , \widehat{s}_{i,\alpha _{jr}}) = -\frac{1}{2}\left\{ \widehat{{Z}}_{ij}\mathrm {tr}\left[ \widehat{\mathbf {D}}^{-1}_j\dot{\mathbf {D}}_j(r)\widehat{\mathbf {D}}^{-1}_j(\widehat{\mathbf {D}}_{j} - \widehat{\mathbf {b}}_{i}^2)\right] \right\} ,\\&\widehat{s}_{i,\varvec{\phi }_{j}} = (\widehat{s}_{i,\phi _{j1}}, \widehat{s}_{i,\alpha _{j2}}) = -\frac{1}{2}\widehat{{Z}}_{ij}\mathrm {tr}(\widehat{\mathbf {E}}_{ij}^{-1}\dot{\mathbf {E}}_{ij}(s)) + \frac{1}{2\widehat{\sigma ^2_j}}\left[ \mathrm {tr}\left( \widehat{a}_{ij}^{(k)}\widehat{\mathbf {E}}_{ij}^{-1}\dot{\mathbf {E}}_{ij}(s)\widehat{\mathbf {E}}_{ij}^{-1}\right) \right. \\&+ \left. \widehat{{Z}}_{ij}\widehat{\varvec{\beta }}^\top _j\mathbf {X}_i^{\top } \widehat{\mathbf {E}}_{ij}^{-1}\mathbf {X}_i\widehat{\varvec{\beta }}_{j} - 2\widehat{\varvec{\beta }}^\top _j\mathbf {X}_i^{\top }\widehat{\mathbf {E}}_{ij}^{-1}\dot{\mathbf {E}}_{ij}(s)\widehat{\mathbf {E}}_{ij}^{-1}(\widehat{{Z}}_{ij}\widehat{\mathbf {y}}_{i} - \widehat{{Z}}_{ij}\mathbf {U}_i\widehat{\mathbf {b}}_{i}) \right] , \end{aligned}$$

where \(\left. \dot{\mathbf {D}}_j(r) = \frac{\partial \mathbf {D}_j}{\partial \alpha _{jr}}\right| _{\alpha =\widehat{\alpha }}\), \(r = 1, 2, \ldots , \dim (\varvec{\alpha }_j)\); and \(\left. \dot{\mathbf {E}}_{ij}(s) = \frac{\partial \mathbf {E}_{ij}}{\partial \phi _{js}}\right| _{\phi =\widehat{\phi }}\), \(s = 1, 2\). For the DEC structure, the derivatives are given by

$$\begin{aligned}&\frac{\partial \mathbf {E}_{ij}}{\partial \phi _{j1}} = |t_{ik} - t_{ih}|^{\phi _{j2}}\phi _{j1}^{|t_{ik} - t_{ih}|^{\phi _{j2}}-1},\\&\frac{\partial \mathbf {E}_{ij}}{\partial \phi _{j2}} = |t_{ik} - t_{ih}|^{\phi _{j2}}\log (|t_{ik} - t_{ih}|)\log (\phi _{j1})\phi _{j1}^{|t_{ik} - t_{ih}|^{\phi _{j2}}}. \end{aligned}$$

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de Alencar, F.H.C., Matos, L.A. & Lachos, V.H. Finite Mixture of Censored Linear Mixed Models for Irregularly Observed Longitudinal Data. J Classif 39, 463–486 (2022). https://doi.org/10.1007/s00357-022-09415-x

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