Chapter 5 Regression models
Regression models are one of the most widely used tools for analyzing survey data, since they make it possible to study the relationship between a variable of interest and a set of explanatory variables. Through these models, it is possible to assess how certain population characteristics vary according to demographic, social, or economic factors observed in the sample. However, the validity of the results obtained depends on appropriate model specification and on correctly accounting for the characteristics of the survey sampling design.
In general terms, regression models seek to describe and quantify the association between a response (dependent) variable and one or more explanatory (independent) variables, providing elements for statistical interpretation and for drawing inferences about the study population. Nevertheless, the validity of the results obtained depends, to a large extent, on appropriate model specification and on correctly accounting for the characteristics of the survey sampling design (Heeringa et al., 2017).
For example, household income can be analyzed as a response variable as a function of characteristics such as educational attainment and the employment status of household members, considered as explanatory variables. Using information from household surveys, this type of model makes it possible to identify patterns of association, quantify the effect of different socioeconomic factors on income, and generate empirical evidence useful for the design, monitoring, and evaluation of public policies.
However, because household surveys are based on complex sampling designs, classical regression methods, developed under simple random sampling assumptions, may be inappropriate. Ignoring the characteristics of the sampling design can generate biased estimates of regression coefficients, as well as underestimation of their standard errors and variances, compromising the validity of statistical inferences.
Consequently, the analysis of survey data requires close attention to the sampling design. Incorporating survey weights and the adjustments corresponding to stratification and clustering makes it possible to obtain valid and precise inferences. In addition, simplified alternatives have been proposed in some cases, such as the use of normalized weights or approximate weighting approaches, which seek to balance methodological complexity with the practical feasibility of the analysis.
The study of regression under complex sampling designs has a well-documented history. Kish & Frankel (1974) were among the first to discuss the impact of these designs on inferences derived from regression models. Subsequently, Fuller (1975) developed a variance estimator based on linearization techniques for multiple linear regression models with unequal weighting, and introduced specific methods for stratified and two-stage designs.
Later, Shah et al. (1977) addressed the problem of violations of classical assumptions when working with survey data, proposing robust inference alternatives for the parameters. In parallel, Binder (1983) focused on the sampling distributions of regression estimators in finite populations, establishing procedures for estimating variances under complex schemes.
In the following years, Skinner et al. (1989) expanded these contributions through variance estimators for regression coefficients that incorporated stratification and clustering, explicitly recommending the use of linearization methods or alternative techniques for variance estimation. Later, Fuller (2002) provided a compendium of estimation methods applicable to regression models in complex surveys, while Pfeffermann (2011) discussed more recent developments, such as q-weighted weighting methods, presenting empirical evidence of their usefulness.