4. Estimation of the parameters of interest

Andrés Gutiérrez, Leonardo Trujillo and Pedro Luis do Nascimento Silva

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Let $N_{ij}$ be the total number of respondents for the population of interest having a classification $i$ at time $t-1$ and $j$ at time $t.$ Let $R_i$ be the total number of individuals in the population not responding at time $t$ but responding at time $t-1$ with classification $i.$ Let $C_j$ denote the total number of individuals in the population not responding at time $t-1$ but responding at time $t$ with classification $j$ and finally let $M$ be the total number of individuals at the population not responding at any of the two periods of observation. It follows that the total size of the population, $N$, must satisfy:

$$N={\displaystyle \sum _i}{\displaystyle \sum _j}{N}_{ij}+{\displaystyle \sum _j}{C}_j+{\displaystyle \sum _i}{R}_i+M.$$

Defining the following characteristics of interest, it is possible to define the parameters of interest:

$$\begin{array}{cc}{y}_{1ik}& =\{\begin{array}{ll}\mathrm{1,}\hfill & \text{if}\text{the}k-\text{th}\text{individual}\text{responds}\text{at}t-1\text{with}\text{classification}i;\hfill \\ \mathrm{0,}\hfill & \text{otherwise}\text{.}\hfill \end{array}\hfill \\ \\ y_{2jk}& =\{\begin{array}{ll}\mathrm{1,}\hfill & \text{if}\text{the}k-\text{th}\text{individual}\text{responds}\text{at}t\text{with}\text{classification}j;\hfill \\ \mathrm{0,}\hfill & \text{otherwise}\mathrm{.}\hfill \end{array}\hfill \end{array}$$

Then, the product of these quantities, defined as $y_{1ik}{y}_{2jk}$, corresponds to a new characteristic of interest taking the value one if the individual has responded at both times and is classified in the cell $ij$, or zero otherwise. Also,

$$N_{ij}={\displaystyle \sum _{k\in U}}{y}_{1ik}{y}_{2jk}.$$

Define the following dichotomic characteristics:

$$\begin{array}{l}{z}_{1k}=\{\begin{array}{ll}\mathrm{1,}\hfill & \text{ if the }k-\text{th individual responds at }t-1;\hfill \\ \mathrm{0,}\hfill & \text{ otherwise}\text{.}\hfill \end{array}\\ \\ z_{2k}=\{\begin{array}{ll}\mathrm{1,}\hfill & \text{ if the }k-\text{th individual responds at }t;\hfill \\ \mathrm{0,}\hfill & \text{ otherwise}\text{.}\hfill \end{array}\end{array}$$

It follows that

$$\begin{array}{lll}{R}_i\hfill & =\hfill & {\displaystyle \sum _{k\in U}}{y}_{1ik}\left(1-z_{2k}\right)\hfill \\ C_j\hfill & =\hfill & {\displaystyle \sum _{k\in U}}{y}_{2jk}\left(1-z_{1k}\right)\hfill \\ M\hfill & =\hfill & {\displaystyle \sum _{k\in U}}\left(1-z_{1k}\right)\left(1-z_{2k}\right).\hfill \end{array}$$

Let $w_k$ denote the weight for the $k$ -th individual corresponding to a specific sampling strategy (sampling design and estimator) in both waves. Then the following expressions represent the estimators of the parameters of interest:

$$\begin{array}{cc}{\widehat{N}}_{ij}& ={\displaystyle \sum _{k\in S}}{w}_k{y}_{1ik}{y}_{2jk}\hfill \\ {\widehat{R}}_i& ={\displaystyle \sum _{k\in S}}{w}_k{y}_{1ik}\left(1-z_{2k}\right)\hfill \\ {\widehat{C}}_j& ={\displaystyle \sum _{k\in S}}{w}_k{y}_{2jk}\left(1-z_{1k}\right)\hfill \\ \widehat{M}& ={\displaystyle \sum _{k\in S}}{w}_k\left(1-z_{1k}\right)\left(1-z_{2k}\right)\hfill \end{array}$$

for $N_{ij}$, $R_i$, $C_j$ and $M,$ respectively. Note that an unbiased estimation for the population size is given by

$$\widehat{N}={\displaystyle \sum _i}{\displaystyle \sum _j}{\widehat{N}}_{ij}+{\displaystyle \sum _j}{\widehat{C}}_j+{\displaystyle \sum _i}{\widehat{R}}_i+\widehat{M}={\displaystyle \sum _s}{w}_k{v}_k$$

where

$$v_k={\displaystyle \sum _i}{y}_{1ik}{\displaystyle \sum _j}{y}_{2jk}+{\displaystyle \sum _j}{y}_{2jk}\left(1-z_{1k}\right)+{\displaystyle \sum _i}{y}_{1ik}\left(1-z_{2k}\right)+\left(1-z_{1k}\right)\left(1-z_{2k}\right).$$

Taking into account the functional form of all the parameters of interest, and noticing that the likelihood function of the model is proportional to (3.1), we arrive at the following result.

Result 4.1 The log-likelihood for the observed data at the population can be rewritten as

$$l_U={\displaystyle \sum _{k\in U}}{f}_k\left(\psi \mathrm{,}{\rho }_{RR}\mathrm{,}{\rho }_{MM}\mathrm{,}\eta \mathrm{,}p\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{z}_1\mathrm{,}{z}_2\right)(4.1)$$

where

$$\begin{array}{l}{f}_k\left(\psi \mathrm{,}{\rho }_{RR}\mathrm{,}{\rho }_{MM}\mathrm{,}\eta \mathrm{,}p\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{z}_1\mathrm{,}{z}_2\right)\\ ={\displaystyle \sum _i}{\displaystyle \sum _j}{y}_{1ik}{y}_{2jk}\ln \left(\psi {\rho }_{RR}{\eta }_i{p}_{ij}\right)\\ +{\displaystyle \sum _i}{y}_{1ik}\left(1-z_{2k}\right)\ln \left({\displaystyle \sum _j}\psi \left(1-{\rho }_{RR}\right){\eta }_i{p}_{ij}\right)\\ +{\displaystyle \sum _j}{y}_{2jk}\left(1-z_{1k}\right)\ln \left({\displaystyle \sum _i}\left(1-\psi \right)\left(1-{\rho }_{MM}\right){\eta }_i{p}_{ij}\right)\\ +\left(1-z_{1k}\right)\left(1-z_{2k}\right)\ln \left({\displaystyle \sum _i}{\displaystyle \sum _j}\left(1-\psi \right){\rho }_{MM}{\eta }_i{p}_{ij}\right)\end{array}$$

where $y_1$ is a vector containing the characteristics $y_{1ik}$, $y_2$ is a vector containing the characteristics $y_{2jk},$ $z_1$ is a vector containing the characteristics $z_{1k}$, and $z_2$ is a vector containing the characteristics $z_{2k}$ (for every $k=1,\dots \mathrm{,}N$ and $i\mathrm{,}j=1,\dots \mathrm{,}G$ ).

Now, in order to obtain estimators of the parameters, it is necessary to maximize this last function. Using standard techniques of maximum likelihood, the corresponding likelihood equations are given by

$${\displaystyle \sum _{k\in U}}{u}_k\left(\theta \right)=0$$

where the vector $u_k$, commonly known as scores, is defined by

$$u_k\left(\theta \right)=\frac{\partial f_k\left(\theta \right)}{\partial \theta }.$$

Also, as it is not usual to survey the whole population, a probability sample is selected and the expression ${\displaystyle {\sum }_{k\in U}}{u}_k\left(\theta \right)$ is considered as a population parameter. In this way, considering $w_k=1/{\pi }_k$ as the corresponding sampling weights, an unbiased estimator for this sum of scores is defined as ${\displaystyle {\sum }_{k\in S}}{w}_k{u}_k\left(\theta \right).$ The next expression is known as the pseudo-likelihood equation and it is an effective way to find estimators for the model parameters taking into account the sampling weights:

$${\displaystyle \sum _{k\in S}}{w}_k{u}_k\left(\theta \right)=0.$$

It is assumed that for the model in this paper, the initial probability of an individual responding at time $t-1$ is the same for all the possible classifications in the survey. Also, the transition probabilities between respondents and nonrespondents do not depend on the classification of the individual in the survey, ${\rho }_{MM}$ and ${\rho }_{RR}$. Considering these assumptions, the following results will let the estimation of the Markov model probabilities take into account the sampling weights.

Result 4.2 Under the assumptions of the model, the resulting maximum pseudo-likelihood estimators for $\psi ,$ ${\rho }_{RR}$ and ${\rho }_{MM}$ are given by

$$\begin{array}{ll}{\widehat{\psi }}_{mpv}& =\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}+{\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}}\hfill \\ {\widehat{\rho }}_{RR\mathrm{,}mpv}& =\frac{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}}\hfill \\ {\widehat{\rho }}_{MM\mathrm{,}mpv}& =\frac{\widehat{M}}{{\displaystyle {\sum }_j{\widehat{C}}_j}+\widehat{M}}\hfill \end{array}$$

respectively.

Result 4.3 Under the assumptions of the model, the resulting maximum pseudo-likelihood estimators for ${\eta }_i$ and $p_{ij}$ are obtained through iteration until convergence of the next expressions

$$\begin{array}{lll}{\widehat{\eta }}_{i\mathrm{,}mpv}^{(v+1)}\hfill & =\hfill & \frac{{\displaystyle {\sum }_j{\widehat{N}}_{ij}}+{\widehat{R}}_i+{\displaystyle {\sum }_j\left({\widehat{C}}_j{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}+{\displaystyle {\sum }_i{\widehat{R}}_i}+{\displaystyle {\sum }_j{\widehat{C}}_j}}\hfill \\ {\widehat{p}}_{ij\mathrm{,}mpv}^{(v+1)}\hfill & =\hfill & \frac{{\widehat{N}}_{ij}+\left({\widehat{C}}_j{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}{{\displaystyle {\sum }_j{\widehat{N}}_{ij}}+{\displaystyle {\sum }_j\left({\widehat{C}}_j{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}/{\displaystyle {\sum }_i{\widehat{\eta }}_i^{(v)}{\widehat{p}}_{ij}^{(v)}}\right)}}\hfill \end{array}$$

respectively. The superindex $\left(v\right)$ denotes the value of the estimation for the parameters of interest at the $v-th$ iteration.

The results before provide an exhaustive frame for the implementation of the two-stage Markovian model in order to take into account the sampling weights in longitudinal surveys. Another question of interest is how to choose the initial values $\left\{{\widehat{\eta }}_i^{(0)}\right\}$ and $\left\{{\widehat{p}}_{ij}^{(0)}\right\}$. In general, any set of values is valid if they follow the initial restrictions. These are

$$\begin{array}{cc}{\displaystyle \sum _i}{\widehat{\eta }}_i^{(0)}& =1\hfill \\ {\displaystyle \sum _j}{\widehat{p}}_{ij}^{(0)}& =1.\hfill \end{array}$$

However, following the guidelines at Chen and Fienberg (1974) and considering the hypothetical case where all of the individuals responded in both periods, then $M=0\text{, }{R}_i=0$ (for every $i=1,\dots \mathrm{,}G$ ) and $C_j=0$ (for every $j=\mathrm{1,}\dots \mathrm{,}G$ ) and their sampling estimations are also null. Given this, and considering the expressions of the resulting estimators, a sensible choice is given by

$$\begin{array}{cc}{\widehat{\eta }}_i^{(0)}& =\frac{{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}{{\displaystyle {\sum }_i{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}}\hfill \\ {\widehat{p}}_{ij}^{(0)}& =\frac{{\widehat{N}}_{ij}}{{\displaystyle {\sum }_j{\widehat{N}}_{ij}}}.\hfill \end{array}$$

Lastly, this iterative approach is commonly implemented for estimation problems by maximum likelihood in contingency tables. However, some approaches for the fit of log-linear models in contingency tables for complex survey designs can be found at Clogg and Eliason (1987), Rao and Thomas (1988), Skinner and Vallet (2010), among others. The next result provides an approach to gross flow estimation considering the sampling weights at both periods of interest.

Result 4.4 Under the assumptions of the model, a sampling estimator of ${\mu }_{ij}$ is

$${\widehat{\mu }}_{ij\mathrm{,}mpv}=\widehat{N}{\widehat{\eta }}_{i\mathrm{,}mpv}{\widehat{p}}_{ij\mathrm{,}mpv}.$$

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