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Likelihood-based inference for Tobit confirmatory factor analysis using the multivariate Student-t distribution

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Abstract

Factor analysis models have been one of the most popular multivariate methods for data analysis among psychometricians, behavioral and educational researchers. But these models, originally developed for normally distributed observed variables, can be seriously affected by the presence of influential observations and censored data. Motivated by this situation, in this paper we propose a likelihood-based estimation for a multivariate Tobit confirmatory factor analysis model using the Student-t distribution (t-TCFA model). An EM-type algorithm is developed for computing the maximum likelihood estimates, obtaining as a byproduct the standard errors of the fixed effects and the exact likelihood value. Unlike other approaches proposed in the literature, our exact EM-type algorithm uses closed form expressions at the E-step based on the first two moments of a truncated multivariate Student-t distribution with the advantage that these expressions can be computed using standard statistical software. The performance of the proposed methods is illustrated through a simulation study and the analysis of a real dataset of early grade reading assessment test scores.

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Notes

  1. This type of censoring scheme relies on the assumption that the time the task was not sufficient to better estimate the responses of the students

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Acknowledgments

The authors are grateful to the editor, associate editor and two anonymous reviewers for their valuable comments and suggestions that greatly improved this paper. We would also like to thank Dr. Jorge Bazán for supplying the EGRA data . The research of Luis M. Castro was supported by Grant FONDECYT 1130233 from the Chilean government and Grant 2012/19445-0 from FAPESP-Brazil. Denise Costa acknowledges support from CAPES-Brazil. Marcos Prates acknowledges support from CNPq and FAPEMIG-Brazil (Grant APQ-00570-13). The research of Victor Lachos was supported by FAPESP-Brazil (Grant 14/02938-9) and by CNPq-Brazil (Grant 305054/2011-2).

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Authors and Affiliations

  1. Department of Statistics, University of Concepción, Concepcion, Chile

    Luis Mauricio Castro

  2. Department of Statistics, Universidade Estadual de Campinas, Campinas, Brazil

    Denise Reis Costa & Victor Hugo Lachos

  3. Department of Statistics, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Marcos Oliveira Prates

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  1. Luis Mauricio Castro
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  2. Denise Reis Costa
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  3. Marcos Oliveira Prates
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  4. Victor Hugo Lachos
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Corresponding author

Correspondence to Denise Reis Costa.

Appendices

Appendix A: Proofs of Propositions

Proof of Proposition 1

The proof of \((i)\) is straightforward from Equation (1). The proof of \((ii)\) follows from Proposition 4 given in Arellano-Valle and Genton (2010) by setting \(\lambda =\tau =0\).

Proof of Proposition 2

If \(\mathbf{X}\sim t_p(\varvec{\mu },\varvec{\varSigma },\nu )\), then we can write

$$\begin{aligned} \left( \displaystyle \frac{\nu +p}{\nu +\delta }\right) ^rt_p(\mathbf{x}|\varvec{\mu },\varvec{\varSigma },\nu )=c_p(\nu ,r)t_p(\mathbf{x}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2r). \end{aligned}$$

Then, it follows that

$$\begin{aligned}&E\left\{ \displaystyle \left( \displaystyle \frac{\nu +p}{\nu +\delta }\right) ^r\mathbf{X}^{(k)}\right\} \\&\quad =c_p(\nu ,r)\frac{T_p(\mathbf{a}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2r)}{T_{p}(\mathbf{a}|\varvec{\mu },\varvec{\varSigma },\nu )} \times \\&\qquad E\left\{ \mathbf{X}^{(k)}|\mathbf{X}\le \mathbf{a}\right\} \\&\qquad \mathrm{and} \,\,\ \int _{\mathbf{w}\le \mathbf{a}}\mathbf{w}^{(k)}\displaystyle \frac{t_p(\mathbf{w}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2)}{T_{p}(\mathbf{a}|\varvec{\mu },\varvec{\varSigma },\nu )}d\mathbf{w}=\\&\qquad \displaystyle \frac{T_p(\mathbf{a}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2)}{T_{p}(\mathbf{a}|\varvec{\mu },\varvec{\varSigma },\nu )}\int _{\mathbf{w}\le \mathbf{a}}\displaystyle \frac{t_p(\mathbf{w}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2)}{T_{p}(\mathbf{a}|\varvec{\mu },\varvec{\varSigma }^*,\nu +2)}d\mathbf{w}, \end{aligned}$$

which concludes the proof.

Proof of Proposition 3

If \(\mathbf{X}\sim t_p(\varvec{\mu },\varvec{\varSigma },\nu )\), and using the result given in Proposition 1-\((ii)\), we have

$$\begin{aligned}&\left( \displaystyle \frac{\nu +p}{\nu +\delta }\right) ^rt_{p_2}\left( \mathbf{x}_2|\varvec{\mu }_{2.1},\widetilde{\varvec{\varSigma }}_{22.1},\nu +p_1\right) \\&= \frac{d_p(p_1,\nu ,r)}{(\nu +\delta _1)^r}{t_{p_2}(\mathbf{x_2}|\varvec{\mu }_{2.1},\widetilde{\varvec{\varSigma }}^*_{22.1},\nu +p_1+2r)} \end{aligned}$$

and the proof concludes by noting that

$$\begin{aligned}&E\left\{ \left( \displaystyle \frac{\nu +p}{\nu +\delta }\right) ^r\mathbf{X}_2^{(k)}|\mathbf{X}_1\right\} =\frac{d_p(p_1,\nu ,r)}{(\nu +\delta _1)^r} \times \\&\displaystyle \frac{T_{p_2}(\mathbf{a}^{x_2}|\varvec{\mu }_{2.1},\widetilde{\varvec{\varSigma }}^*_{22.1},\nu +p_1+2r)}{T_{p_2}(\mathbf{a}^{x_2}|\varvec{\mu }_{2.1},\widetilde{\varvec{\varSigma }}_{22.1},\nu +p_1)}E\left\{ \mathbf{X}_2^{(k)}|\mathbf{X}_2\le \mathbf{a}^{x_2}\right\} , \end{aligned}$$

where \(\mathbf{X}_2\sim t_{p_2}\left( \varvec{\mu }_{2.1},\widetilde{\varvec{\varSigma }}^*_{22.1},\nu +p_1+2r \right) .\)

Appendix B: Details of the EM algorithm

First, we introduce Lemma 1, which used in our procedures. Its proof can be found in Arellano-Valle et al. (2005).

Lemma 1

Let \(\mathbf{Y}\sim N_p(\varvec{\mu },\varvec{\varSigma })\) and \( \mathbf{x}\sim N_q(\varvec{\eta },\varvec{\varOmega })\). So,

$$\begin{aligned}&\phi _p(\mathbf{y}|\varvec{\mu }+ \mathbf{A}x,\varvec{\varSigma })\phi _q(x|\varvec{\eta },\varvec{\varOmega }) \\&\quad = \phi _p(\mathbf{y}|\varvec{\mu }+ \mathbf{A}\varvec{\eta }, \varvec{\varSigma }+ \mathbf{A}\varvec{\varOmega }\mathbf{A}^{\top } )\\&\qquad \times \, \phi _q(x|\varvec{\eta }+ \varvec{\varLambda }\mathbf{A}^{\top }\varvec{\varSigma }^{-1}(\mathbf{y}-\varvec{\mu }-\mathbf{A}\varvec{\eta }),\varvec{\varLambda }), \end{aligned}$$

where \(\varvec{\varLambda }= (\varvec{\varOmega }^{-1} + \mathbf{A}^{\top }\varvec{\varSigma }^{-1}\mathbf{A})^{-1}\).

The derivatives of the function \(Q(\varvec{\theta }|\varvec{\theta }^{(k)}\) with respect to \(\varvec{\beta }\), \(\varvec{\varLambda }\) and \(\varvec{\varPsi }\) leads to

$$\begin{aligned} \displaystyle \frac{\partial Q(\varvec{\theta }|\varvec{\theta }^{(k)})}{\partial \varvec{\beta }}&= - \sum ^n_{i=1}\left[ -\widehat{u_i\mathbf{y}_i}^{(k)}\,\mathbf{X}_i^{\top }+\widehat{u_i}^{(k)}\varvec{\beta }\mathbf{X}_i^{\top }\mathbf{X}_i \right. \\&\left. +\,\varvec{\varLambda }\widehat{u_i\mathbf{z}_i}^{(k)}\mathbf{X}_i \right] ,\\ \displaystyle \frac{\partial Q(\varvec{\theta }|\varvec{\theta }^{(k)})}{\partial \varvec{\varLambda }}&= -\sum ^n_{i=1}\left[ \widehat{u_i\mathbf{z}_i}^{(k)} \varvec{\beta }^{\top }\mathbf{X}_i^{\top }-\widehat{u_i\mathbf{y}_i\mathbf{z}_i^{\top }}^{(k)} \right. \\&\left. +\,\varvec{\varLambda }\widehat{u_i\mathbf{z}_i\mathbf{z}_i^{\top }}^{(k)}\right] \\ \displaystyle \frac{\partial Q(\varvec{\theta }|\varvec{\theta }^{(k)})}{\partial \varvec{\varPsi }}&= \sum ^n_{i=1}\left[ \varvec{\varPsi }^{-1}-\varvec{\varPsi }^{-2}\widehat{B_i}^{(k)}\right] , \end{aligned}$$

where

$$\begin{aligned} \widehat{B}^{(k)}_i&= \mathrm{tr}(\widehat{u_i\mathbf{y}_i\mathbf{y}_i^{\top }}^{(k)})- \widehat{u_i\mathbf{y}}_i^{\top (k)}\mathbf{X}_i\varvec{\beta }-\mathrm{tr}(\widehat{u_i\mathbf{y}_i \mathbf{z}_i^{\top }}^{(k)}\varvec{\varLambda })\\&-\varvec{\beta }^{\top }\mathbf{X}_i^{\top }\widehat{u_i\mathbf{y}_i}^{(k)}+\varvec{\beta }^{\top } \mathbf{X}_i^{\top }\widehat{u_i}^{(k)}\mathbf{X}_i\varvec{\beta }\\&+\varvec{\beta }^{\top }\mathbf{X}_i^{\top }\varvec{\varLambda }\widehat{u_i\mathbf{z}_i}^{(k)}- \mathrm{tr}(\widehat{u_i\mathbf{y}_i\mathbf{z}_i^{\top }}^{(k)}\varvec{\varLambda }^{\top })\\&+\widehat{u_i\mathbf{z}_i}^{\top (k)}\varvec{\varLambda }^{\top }\mathbf{X}_i\varvec{\beta }+ \mathrm{tr}(\widehat{u_i\mathbf{z}_i\mathbf{z}_i^{\top }}^{(k)}\varvec{\varLambda }^{\top }\varvec{\varLambda }). \end{aligned}$$

The solution of these derivatives at zero gives the estimates of the MLE presented in (8)–(10).

Appendix C: Complementary results of the simulation study

Figures 10 and 11 present the absolute bias and the MSE of \(\lambda _{42}\), \(\varPsi _{33}\) and \(\varPsi _{44}\).

Fig. 10
figure 10

Simulated data. Absolute bias and MSE for the parameter \(\lambda _{42}\) under scenario 3

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Fig. 11
figure 11

Simulated data. Absolute bias and MSE for the parameters \(\varPsi _{33}\) (first row) and \(\varPsi _{44}\) (second row) under scenario 3

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Castro, L.M., Costa, D.R., Prates, M.O. et al. Likelihood-based inference for Tobit confirmatory factor analysis using the multivariate Student-t distribution. Stat Comput 25, 1163–1183 (2015). https://doi.org/10.1007/s11222-014-9502-0

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