A Generalization of the Familial Longitudinal Binary Model to the Multinomial Setup

Abstract

When repeated binary responses along with time dependent covariates are collected over a short period of time from the members of a large number of independent families, there exits a well developed binary dynamic mixed logit (BDML) model to analyze such familial longitudinal binary data. As far as the inferences are concerned, this BDML model has been fitted by using the generalized quasi-likelihood (GQL) and the well known maximum likelihood (ML) methods. There are however situations in practice where categorical/multinomial responses with more than two categories are repeatedly collected from all members of the family. However, the analysis for this type of familial longitudinal multinomial data is not adequately addressed in the literature. We offer two main contributions in this paper. First, for the analysis of familial longitudinal multinomial data, we propose a multinomial dynamic mixed logit (MDML) model as a generalization of the BDML model and derive the basic properties such as non-stationary mean, variance and correlations for the repeated multinomial responses. Next, to understand these basic properties, we develop step by step likelihood estimating equations for the parameters involved in these properties. The relative asymptotic efficiency performance of the ML and GQL approaches is examined through a simulation study based on repeated binary responses, for example, from a large number of independent families each consisting of two members, causing both familial and longitudinal correlations. Also, a real life example on repeated multinomial data analysis is considered as an illustration.

Appendix

1.1 Algebras for the Likelihood Estimating Equation (32) for β

To obtain the formula for $\frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \beta }$ in (31), we first compute the derivatives with respect to category based location regression parameters. Thus,

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \beta _{c}^{{\ast}}} & =& \pi _{(ij1)c}^{{\ast}}[1 -\pi _{ (ij1)c}^{{\ast}}]x_{ ij1} \\ \frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \beta _{k}^{{\ast}}} & =& -[\pi _{(ij1)c}^{{\ast}}\pi _{ (ij1)k}^{{\ast}}]x_{ ij1},{}\end{array}$$

(65)

yielding

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \beta } & =& \left (\begin{array}{*{10}c} -\pi _{(ij1)1}^{{\ast}}\pi _{(ij1)c}^{{\ast}}\\ \vdots \\ \pi _{(ij1)c}^{{\ast}}[1 -\pi _{(ij1)c}^{{\ast}}]\\ \vdots \\ -\pi _{(ij1)(C-1)}^{{\ast}}\pi _{(ij1)c}^{{\ast}}\\ \end{array} \right ) \otimes x_{ij1}: (J - 1)p \times 1 \\ & =& \left [\pi _{(ij1)c}^{{\ast}}(\delta _{ (ij1)c} -\pi _{(ij1)}^{{\ast}})\right ] \otimes x_{ ij1}, {}\end{array}$$

(66)

where ⊗ stands for the Kronecker product, and the formulas for δ _(ij1)c and $\pi _{(ij1)}^{{\ast}}$ vectors are as in (32). Similarly, to compute $\frac{\partial \eta _{ijt\vert t-1}^{{\ast}(c)}(g)} {\partial \beta }$ for (31), we first obtain the local derivatives of the conditional probability function $\eta _{ijt\vert t-1}^{{\ast}(c)}(g)$ given in (10) for t = 2, …, T, as

$$\displaystyle\begin{array}{rcl} \frac{\partial \eta _{ijt\vert t-1}^{{\ast}(c)}(g)} {\partial \beta _{c}^{{\ast}}} & =& \eta _{ijt\vert t-1}^{{\ast}(c)}(g)[1 -\eta _{ ijt\vert t-1}^{{\ast}(c)}(g)]x_{ ijt} \\ \frac{\partial \eta _{ijt\vert t-1}^{{\ast}(c)}(g)} {\partial \beta _{k}^{{\ast}}} & =& -[\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\eta _{ ijt\vert t-1}^{{\ast}(k)}(g)]x_{ ijt},{}\end{array}$$

(67)

yielding the derivative with respect to the global parameter vector β as

$$\displaystyle\begin{array}{rcl} \frac{\partial \eta _{ijt\vert t-1}^{{\ast}(c)}(g)} {\partial \beta } & =& \left (\begin{array}{*{10}c} -\eta _{ijt\vert t-1}^{{\ast}(1)}(g)\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\\ \vdots \\ \eta _{ijt\vert t-1}^{{\ast}(c)}(g)[1 -\eta _{ijt\vert t-1}^{{\ast}(c)}(g)]\\ \vdots \\ -\eta _{ijt\vert t-1}^{{\ast}(C-1)}(g)\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\\ \end{array} \right ) \otimes x_{ijt}: (J - 1)p \times 1 \\ & =& \left [\eta _{ijt\vert t-1}^{{\ast}(c)}(g)(\delta _{ (ij,t-1)c} -\eta _{ijt\vert t-1}^{{\ast}}(g))\right ] \otimes x_{ ijt}, {}\end{array}$$

(68)

where $\eta _{ijt\vert t-1}^{{\ast}}(g)$ is the (C − 1) × 1 vector of transitional probabilities as in (32).

1.2 Algebras for the Likelihood Estimating Equation (39) for γ ≡ γ _M

To compute the derivative in (37) through (38), it is convenient to find the derivative with respect to the local dynamic dependence parameter vector γ _c for $c = 1,\ldots,C - 1$. Thus, we write

$$\displaystyle\begin{array}{rcl} \frac{\partial \eta _{ijt\vert t-1}^{{\ast}(h)}(g)} {\partial \gamma _{c}} & =& \left \{\begin{array}{ll} \delta _{(ij,t-1)g}\eta _{ijt\vert t-1}^{{\ast}(c)}(g)[1 -\eta _{ijt\vert t-1}^{{\ast}(c)}(g)] & \mbox{ for}\;h = c; h,c = 1,\ldots,C - 1 \\ -\delta _{(ij,t-1)g}\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\eta _{ijt\vert t-1}^{{\ast}(h)}(g) & \mbox{ for}\;h\neq c; h,c = 1,\ldots,C - 1 \\ nonumber\ -\delta _{(ij,t-1)g}\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\eta _{ijt\vert t-1}^{{\ast}(C)}(g) & \mbox{ for}\;h = C;\; c = 1,\ldots,C - 1, \end{array} \right.{}\end{array}$$

(69)

for all g = 1, …, C, with

$$\displaystyle{\delta _{(ijt)g} = \left \{\begin{array}{ll} [01'_{g-1},1,01'_{C-1-g}]'&\mbox{ for}\;g = 1,\ldots,C - 1 \\ 01_{C-1} & \mbox{ for}\;g = C, \end{array} \right.}$$

as in (32), leading to

$$\displaystyle\begin{array}{rcl} \frac{\partial log\;L_{i}(\tau _{i})} {\partial \gamma _{c}} & \equiv & \frac{\partial log\;L_{i}(\beta,\gamma _{M},\sigma _{\tau })} {\partial \gamma _{c}} \\ & =& \sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ h=1}^{C}\sum _{ g=1}^{C}\left \{ \frac{y_{ijth}} {\eta _{ijt\vert t-1}^{{\ast}(h)}(g)} \frac{\partial \eta _{ijt\vert t-1}^{{\ast}(h)}(g)} {\partial \gamma } \right \} \\ & =& \sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ g=1}^{C}y_{ ijtc}\delta _{(ij,t-1)g}[1 -\eta _{ijt\vert t-1}^{{\ast}(c)}(g)] \\ & & -\sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ g=1}^{C}\sum _{ h\neq c}^{C} \frac{y_{ijth}} {\eta _{ijt\vert t-1}^{{\ast}(h)}(g)}\delta _{(ij,t-1)g}\left (\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\eta _{ ijt\vert t-1}^{{\ast}(h)}(g)\right ) \\ & =& \sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ g=1}^{C}y_{ ijtc}\delta _{(ij,t-1)g} \\ & & -\sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ g=1}^{C}\sum _{ h=1}^{C}y_{ ijth}\delta _{(ij,t-1)g}\left (\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\right ) \\ & =& \sum _{j=1}^{n_{i} }\sum _{t=2}^{T}\sum _{ g=1}^{C}\delta _{ (ij,t-1)g}\left [y_{ijtc} -\left (\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\right )\right ], {}\end{array}$$

(70)

for $c = 1,\ldots,C - 1$.

1.3 Algebras for the Likelihood Estimating Equation (47) for σ _τ

The derivative $\frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \sigma _{\tau }}$ in (46) follows from (9) and is given by

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi _{(ij1)c}^{{\ast}}} {\partial \sigma _{\tau }} & =& \frac{\partial } {\partial \sigma _{\tau }}\left \{\begin{array}{ll} \frac{\exp (x'_{ijt}\beta _{c}^{{\ast}}+\sigma _{\tau }\tau _{ i})} {1+\sum _{g=1}^{C-1}\exp (x'_{ijt}\beta _{g}^{{\ast}}+\sigma _{\tau }\tau _{i})} & \mbox{ for}\;c = 1,\ldots,C - 1 \\ \frac{1} {1+\sum _{g=1}^{C-1}\exp (x'_{ijt}\beta _{g}^{{\ast}}+\sigma _{\tau }\tau _{i})} & \mbox{ for}\;c = C,\end{array} \right. \\ & =& \left \{\begin{array}{ll} \tau _{i}\pi _{(ij1)c}^{{\ast}}\pi _{(ij1)C}^{{\ast}} &\mbox{ for}\;c = 1,\ldots,C - 1 \\ -\tau _{i}\pi _{(ij1)C}^{{\ast}}[1 -\pi _{(ij1)C}^{{\ast}}]&\mbox{ for}\;h = C; \end{array} \right.{}\end{array}$$

(71)

and similarly, the derivative $\frac{\partial \eta _{ijt\vert t-1}^{{\ast}(c)}(g)} {\partial \sigma _{\tau }}$ in (46) follows from (10), and it has the formula given by

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial \sigma _{\tau }}[\eta _{ijt\vert t-1}^{{\ast}(c)}(g)]& =& \frac{\partial } {\partial \sigma _{\tau }}\left \{\begin{array}{ll} \frac{\exp \,\left [x'_{ijt}\beta _{c}^{{\ast}}+\gamma '_{ c}y_{ij,t-1}^{(g)}+\sigma _{\tau }\tau _{ i}\right ]} {1\,+\,\sum _{v=1}^{C-1}\exp \,\left [x'_{ijt}\beta _{v}^{{\ast}}+\gamma '_{v}y_{ij,t-1}^{(g)}+\sigma _{\tau }\tau _{i}\right ]},&\mbox{ for}\;c = 1,\ldots,C - 1 \\ \frac{1} {1\,+\,\sum _{v=1}^{C-1}\exp \,\left [x'_{ijt}\beta _{v}^{{\ast}}+\gamma '_{v}y_{ij,t-1}^{(g)}+\sigma _{\tau }\tau _{i}\right ]},&\mbox{ for}\;c = C,\\ \end{array} \right. \\ & =& \left \{\begin{array}{ll} \tau _{i}\eta _{ijt\vert t-1}^{{\ast}(c)}(g)\eta _{ijt\vert t-1}^{{\ast}(C)}(g) &\mbox{ for}\;c = 1,\ldots,C - 1 \\ -\tau _{i}\eta _{ijt\vert t-1}^{{\ast}(C)}(g)[1 -\eta _{ijt\vert t-1}^{{\ast}(C)}(g)]&\mbox{ for}\;c = C. \end{array} \right.{}\end{array}$$

(72)