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Mixtures of factor analyzers with covariates for modeling multiply censored dependent variables

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Abstract

Censored data arise frequently in diverse applications in which observations to be measured may be subject to some upper and lower detection limits due to the restriction of experimental apparatus such that they are not exactly quantifiable. Mixtures of factor analyzers with censored data (MFAC) have been recently proposed for model-based density estimation and clustering of high-dimensional data in the presence of censored observations. In this paper, we consider an extended version of MFAC by considering regression equations to describe the relationship between covariates and multiply censored dependent variables. Two analytically feasible EM-type algorithms are developed for  computing maximum likelihood estimates of model parameters with closed-form expressions. Moreover, we provide an information-based method to compute asymptotic standard errors of mixing proportions and regression coefficients. The utility and performance of our proposed methodology are illustrated through a simulation study and two real data examples.

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Acknowledgements

We are grateful to the Editor, Associate Editor, and two referees for their valuable comments and suggestions on the earlier version of this paper. W.L. Wang and T.I. Lin would like to acknowledge the support of the Ministry of Science and Technology of Taiwan under Grant Nos. MOST 107-2628-M-035-001-MY3 and MOST 107-2118-M-005-002-MY2, respectively. L.M. Castro acknowledges support from Grant FONDECYT 1170258 and Millennium Science Initiative of the Ministry of Economy, Development and Tourism, Grant “Millenium Nucleus Center for the Discovery of Structures in Complex Data” from the Chilean government.

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Authors and Affiliations

  1. Department of Statistics, Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taichung, 40724, Taiwan

    Wan-Lun Wang

  2. Department of Statistics, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

    Luis M. Castro

  3. Millennium Nucleus Center for the Discovery of Structures in Complex Data, Santiago, Chile

    Luis M. Castro

  4. Centro de Riesgos y Seguros UC, Pontificia Universidad Católica de Chile, Santiago, Chile

    Luis M. Castro

  5. Institute of Statistics, National Chung Hsing University, Taichung, 402, Taiwan

    Wan-Chen Hsieh & Tsung-I Lin

  6. Department of Public Health, China Medical University, Taichung, 404, Taiwan

    Tsung-I Lin

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  1. Wan-Lun Wang
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  2. Luis M. Castro
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  3. Wan-Chen Hsieh
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  4. Tsung-I Lin
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Correspondence to Tsung-I Lin.

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Appendices

Appendices

A. Proof of Proposition 1

To evaluate the conditional expectation of the complete-data log-likelihood function given the observed data \((\varvec{v}_j,\varvec{c}_j)\) and the current estimates \(\hat{{\varvec{\varTheta }}}^{(k)}\) in the EM algorithm, we need the following conditional moment involving latent variables \(\varvec{y}_j^{\mathrm{c}}\) and \(\varvec{u}_{ij}\):

$$\begin{aligned}&E \{ (\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i-\varvec{B}_i \varvec{u}_{ij})(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i-\varvec{B}_i \varvec{u}_{ij})^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1,\hat{{\varvec{\varTheta }}}^{(k)}\} \\&\quad = E\{(\varvec{y}_j \varvec{y}_j^\top -\varvec{y}_j {\varvec{\beta }}_i^\top \varvec{X}_j^\top -\varvec{y}_j \varvec{u}_{ij}^\top \varvec{B}_i^\top -\varvec{X}_j {\varvec{\beta }}_i \varvec{y}_j^\top +\varvec{X}_j {\varvec{\beta }}_i {\varvec{\beta }}_i^\top \varvec{X}_j^\top \\&\qquad {+}\varvec{X}_j {\varvec{\beta }}_i \varvec{u}_{ij}^\top \varvec{B}_i^\top -\varvec{B}_i \varvec{u}_{ij} \varvec{y}_j^\top {+}\varvec{B}_i \varvec{u}_{ij} {\varvec{\beta }}_i^\top \varvec{X}_j{+}\varvec{B}_i \varvec{u}_{ij} \varvec{u}_{ij}^\top \varvec{B}_i^\top ) \mid \varvec{v}_j, \varvec{c}_j,z_{ij}{=}1 ,\hat{{\varvec{\varTheta }}}^{(k)}\}. \end{aligned}$$
  1. (a)

    Making use of the fact that \(\varvec{y}_j=\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}}\), we have

    $$\begin{aligned} E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)= & {} E\{(\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}})\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\= & {} \varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} + \varvec{C}_j^\top E(\varvec{y}_j^{\mathrm{c}} \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1) \\= & {} \varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} +\varvec{C}_j^\top E(\varvec{W}_{ij}^{\mathrm{c}}), \end{aligned}$$

    where \(\varvec{W}_{ij}^{\mathrm{c}}\) is a random vector following the distribution in (4).

  2. (b)

    Similarly,

    $$\begin{aligned}&E(\varvec{y}_j \varvec{y}_j^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)\\&\quad = E\{(\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}})(\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}})^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad = E\{(\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} \varvec{v}_j^{\mathrm{o^\top }} \varvec{O}_j+\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} \varvec{y}_j^{\mathrm{c^\top }} \varvec{C}_j+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}} \varvec{v}_j^{\mathrm{o^\top }} \varvec{O}_j\\&\qquad +\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}} \varvec{y}_j^{\mathrm{c^\top }} \varvec{C}_j) \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\} \\&\quad = \varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} \varvec{v}_j^{\mathrm{o^{\top }}} \varvec{O}_j +\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}} E^\top (\varvec{W}_{ij}^{\mathrm{c}}) \varvec{C}_j +\varvec{C}_j^\top E(\varvec{W}_{ij}^{\mathrm{c}}) \varvec{v}_j^{\mathrm{o^{\top }}} \varvec{O}_j\\&\qquad +\varvec{C}_j^\top E(\varvec{W}_{ij}^{\mathrm{c}} \varvec{W}_{ij}^{\mathrm{c^{\top }}}) \varvec{C}_j. \end{aligned}$$
  3. (c)

    Using the law of iterated expectations, we calculate

    $$\begin{aligned} E(\varvec{u}_{ij} \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1 )= & {} E\{E(\varvec{u}_{ij} \mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\= & {} E[\varvec{B}_i^\top {\varvec{\varSigma }}_i^{-1} (\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1]\\= & {} {\varvec{\varGamma }}_i^\top \{E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j)-\varvec{X}_j {\varvec{\beta }}_i\}, \end{aligned}$$

    where \({\varvec{\varGamma }}_i={\varvec{\varSigma }}_i^{-1}\varvec{B}_i\).

  4. (d)

    Likewise,

    $$\begin{aligned}&E(\varvec{y}_j \varvec{u}_{ij}^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1 )\\&\quad = E\{E(\varvec{y}_j \varvec{u}_{ij}^\top \mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j,z_{ij}=1)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad = E\{\varvec{y}_j(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top {\varvec{\varSigma }}_i^{-1}\varvec{B}_i \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad = E\{(\varvec{y}_j \varvec{y}_j^\top -\varvec{y}_j {\varvec{\beta }}_i^\top \varvec{X}_j^\top ) \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}{\varvec{\varSigma }}_i^{-1}\varvec{B}_i\\&\quad = \{E(\varvec{y}_j \varvec{y}_j^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1){\varvec{\beta }}_i^\top \varvec{X}_j^\top \}{\varvec{\varGamma }}_i. \end{aligned}$$
  5. (e)

    A standard calculation gives

    $$\begin{aligned}&E(\varvec{u}_{ij} \varvec{u}_{ij}^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)\\&\quad = E\{E(\varvec{u}_{ij} \varvec{u}_{ij}^\top \mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j,z_{ij}=1)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad = E\{E(\varvec{u}_{ij} \mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j,z_{ij}=1)E(\varvec{u}_{ij}^\top \mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j,z_{ij}=1)\\&\qquad + \mathrm{cov}(\varvec{u}_{ij}\mid \varvec{y}_j,\varvec{v}_j,\varvec{c}_j,z_{ij}=1)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad = E\{\varvec{B}_i^\top {\varvec{\varSigma }}_i^{-1}(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top {\varvec{\varSigma }}_i^{-1} \varvec{B}_i\\&\qquad + (\varvec{I}_q-\varvec{B}_i^\top {\varvec{\varSigma }}_i^{-1}\varvec{B}_i) \mid \varvec{v}_j,\varvec{c}_j ,z_{ij}=1\}\\&\quad = {\varvec{\varGamma }}_i^\top \{E(\varvec{y}_j \varvec{y}_j^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1){\varvec{\beta }}_i^\top \varvec{X}_j^\top \\&\qquad -\varvec{X}_j {\varvec{\beta }}_i E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)^\top +\varvec{X}_j {\varvec{\beta }}_i{\varvec{\beta }}_i^\top \varvec{X}_j^\top \}{\varvec{\varGamma }}_i+ {\varvec{\varOmega }}_i, \end{aligned}$$

    where \({\varvec{\varOmega }}_i=\varvec{I}_q-\varvec{B}_i^\top {\varvec{\varSigma }}_i^{-1} \varvec{B}_i\).

B. Proof of Proposition 2

We use Proposition 2 to solve the following conditional expectations in the AECM algorithm:

$$\begin{aligned}&E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i-\varvec{B}_i \varvec{u}_{ij})(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i-\varvec{B}_i \varvec{u}_{ij})^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1, \hat{{\varvec{\varTheta }}}^{(k)}\} \\&\quad = E\{(\varvec{y}_j-\varvec{X}_j{\varvec{\beta }}_i)(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1,\hat{{\varvec{\varTheta }}}^{(k)}\}\\&\qquad -E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)\varvec{u}_{ij}^\top \varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1,\hat{{\varvec{\varTheta }}}^{(k)}\}\\&\qquad -E\{\varvec{B}_i\varvec{u}_{ij}(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1,\hat{{\varvec{\varTheta }}}^{(k)}\}\\&\qquad +E(\varvec{B}_i \varvec{u}_{ij} \varvec{u}_{ij}^\top \varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1,\hat{{\varvec{\varTheta }}}^{(k)}). \end{aligned}$$
  1. (a)

    Making use of the fact that \(\varvec{y}_j=\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}}\), we have

    $$\begin{aligned} \mathrm{cov}(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)= & {} \mathrm{cov}[(\varvec{O}_j^\top \varvec{v}_j^{{\mathrm{o}}}+\varvec{C}_j^\top \varvec{y}_j^{\mathrm{c}}) \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1]\\= & {} \varvec{C}_j^\top \mathrm{cov}(\varvec{W}_{ij}^{\mathrm{c}})\varvec{C}_j \triangleq {\varvec{\varLambda }}_{ij}. \end{aligned}$$
  2. (b)

    A standard calculation gives

    $$\begin{aligned}&E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\} \\&\quad =E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}^\top \\&\qquad +\mathrm{cov}(\varvec{y}_j\mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)\\&\quad = \{E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-\varvec{X}_j {\varvec{\beta }}_i\}\{E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-\varvec{X}_j {\varvec{\beta }}_i\}^\top +{\varvec{\varLambda }}_{ij}. \end{aligned}$$
  3. (c)

    By the law of iterated expectations, we calculate

    $$\begin{aligned}&E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)\varvec{u}_{ij}^\top \varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad =E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)E(\varvec{u}_{ij}^\top \mid \varvec{y}_j)\varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad =E\{(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}{\varvec{\varGamma }}_i \varvec{B}_i^\top . \end{aligned}$$
  4. (d)

    A standard calculation gives

    $$\begin{aligned}&E(\varvec{B}_i \varvec{u}_{ij} \varvec{u}_{ij}^\top \varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1) \\&\quad =E\{\varvec{B}_i E(\varvec{u}_{ij} \varvec{u}_{ij}^\top \mid \varvec{y}_j )\varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1\}\\&\quad =E[\varvec{B}_i \{E(\varvec{u}_{ij} \mid \varvec{y}_j)E(\varvec{u}_{ij}^\top \mid \varvec{y}_j )+\mathrm{cov}(\varvec{u}_{ij}\mid \varvec{y}_j)\}\varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1] \\&\quad =E[\varvec{B}_i\{{\varvec{\varGamma }}_i^\top (\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)(\varvec{y}_j-\varvec{X}_j {\varvec{\beta }}_i)^\top {\varvec{\varGamma }}_i+{\varvec{\varOmega }}_i\}\varvec{B}_i^\top \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1]\\&\quad =\varvec{B}_i( {\varvec{\varGamma }}_i^\top [\{E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-\varvec{X}_j {\varvec{\beta }}_i\}\\&\qquad \times \{E(\varvec{y}_j \mid \varvec{v}_j,\varvec{c}_j,z_{ij}=1)-\varvec{X}_j {\varvec{\beta }}_i\}^\top +{\varvec{\varLambda }}_{ij}]{\varvec{\varGamma }}_i+{\varvec{\varOmega }}_i ) \varvec{B}_i^\top . \end{aligned}$$

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Wang, WL., Castro, L.M., Hsieh, WC. et al. Mixtures of factor analyzers with covariates for modeling multiply censored dependent variables. Stat Papers 62, 2119–2145 (2021). https://doi.org/10.1007/s00362-020-01177-1

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