1. Introduction

Andrés Gutiérrez, Leonardo Trujillo and Pedro Luis do Nascimento Silva

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Survey techniques are commonly used in order to estimate some parameters of interest in a finite population. The inference for these parameters is based on the probability distribution induced by the sampling design used to get the sample of individuals. In most of the cases for official statistics, the sample design under consideration is complex in the sense of not providing a simple random sample of the population.

After getting a probabilistic sample, sometimes it is necessary to consider the classification of the individuals in the sample through different categories in one or more nominal variables. This classification can be incorporated in a contingency table in order to summarize two variables or the temporal variations in a single variable at two different periods of time. However, in order to get accurate estimates, it is not advisable to ignore the sampling design in the inference for the parameters of interest.

Another common problem in this type of survey is nonresponse for some sample units, which can rarely be considered random or ignorable. Therefore it is necessary to consider some approach that can compensate for the potentially nonignorable nonresponse. Chen and Fienberg (1974), Stasny (1987) and recently Lu and Lohr (2010) have considered two-stage models in order to classify the individuals in a sample for two different times with nonignorable nonresponse. However, this approach ignored the sampling design that is complex and also informative for most surveys conducted for producing official statistics.

This article considers a common scenario for longitudinal surveys where the main aim is to estimate the number of population individuals belonging to several cells in a contingency table according to the categories of a variable measured at two different points in time. We also consider the modeling of the nonresponse that can affect the estimates if it is ignored. The inferential processes are tied to the complex survey design used to collect the information in the sample.

For instance, in labour force surveys, it is possible to find complex classifications depending on the labour force status of the respondents at two consecutive periods of observation and measurement. The aim is to estimate the number of people that in a past period were working and are still working in the current period of observation. Another possible objective is to estimate the number of people who were unemployed in the last period of observation and are still unemployed in the current period of the survey or the number of people that in the last period of observation were employed and in the current period are unemployed or vice versa. For this example, all the entries on Table 1.1 are considered as parameters of interest. Note that even under a census, the counts in Table 1.1 may not be observable due to nonresponse.

Table 1.1
Parameters of interest in a contingency table corresponding to a labour force survey at two consecutive periods of observation.
Table summary
This table displays the results of Parameters of interest in a contingency table corresponding to a labour force survey at two consecutive periods of observation.. The information is grouped by Period 1 (appearing as row headers), Period 2 (appearing as column headers).

Period 1	Period 2
	Employed	Unemployed	Inactive	Total
Employed	$X_{11}$	$X_{12}$	$X_{13}$	$X_{1+}$
Unemployed	$X_{21}$	$X_{22}$	$X_{23}$	$X_{2+}$
Inactive	$X_{31}$	$X_{32}$	$X_{33}$	$X_{3+}$
Total	$X_{+1}$	$X_{+2}$	$X_{+3}$	$X_{++}$

Kalton (2009) stated that, in terms of the marginal totals, it is possible to estimate the net flows through a direct comparison between the two periods of observation. Then, it is possible to determine if the unemployment rate increased or decreased and also in what magnitude. For example, comparing that on period 1 there were $X_{1+}={\displaystyle {\sum }_j}{X}_{1j}$ people employed, whereas on period 2 there were $X_{+1}={\displaystyle {\sum }_i}{X}_{i1}$ people employed. Nevertheless, a more detailed analysis can be obtained analyzing the gross flows as a decomposition of the net flows. In this way, if the unemployment rate increased one percentage point, it is possible to conclude if this increase was due to the fact that one percentage point of the employed people lost their job or because ten percentage points of the employed people lost their job and nine percentage points of the unemployed people found a new job. This is possible comparing the values $X_{ij}$.

Also, given that in a complex survey it is possible to have unequal sampling weights and clustering and stratification effects, the likelihood function of the sampling data is difficult to find in an analytical way. Then, using classical methods of maximum likelihood would no longer be convenient for survey data from complex surveys. Then, the standard analysis must be modified to take into account the sampling weights and the sampling effects of a complex survey such as weighted estimation of proportions, variance estimation based on the sampling design and generalized corrections for the design effects (Pessoa and Silva 1998).

Section 2 surveys the basic statistical concepts used in this paper, such as survey estimators, nonresponse and categorical data inference. Section 3 proposes a superpopulation model describing the probabilistic behavior of the assignment of the individuals according to the categories of the variable considered in the survey. This corresponds to a two-stage Markov chain model. Some basic concepts of pseudo-likelihood estimation are also reviewed in Section 3. Then, in Section 4, we propose some estimators for the model parameters and the counts in the gross flows contingency table. These estimators are design-unbiased and the mathematical expressions to estimate their variance are shown in Section 5. Section 6 considers both an empirical application and a Monte Carlo simulation in order to test the proposed methodology when the data in the survey is obtained under a simple and a complex survey design. Our simulation shows that other methodological approaches lead to biased estimation. Section 7 considers a practical application for estimating gross flows for the Pesquisa Mensal de Emprego (PME survey) in Brazil. In Section 8, we highlight the strengths and shortcomings of the proposed method. All the mathematical proofs are presented in the Appendix.

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