5.2 Use of sampling weights
When working with data from surveys based on complex sampling designs, the classical assumptions of regression models are rarely strictly satisfied, since observations do not come from simple and independent random samples, but from selection schemes that incorporate elements such as stratification, clustering, and unequal probabilities of selection. For example, the presence of clustering may violate the assumption of independence of errors, since individuals belonging to the same household, segment, or geographic area tend to share similar characteristics. Similarly, unequal probabilities of selection and weighting adjustments can generate heterogeneity in the variance of observations, violating the assumption of homoscedasticity.
In this context, a fundamental question arises: how should sampling weights be incorporated into regression models? The answer is not straightforward, since survey weights reflect not only probabilities of selection, but also various adjustments associated with nonresponse, calibration, and coverage corrections. Although their use makes it possible to produce estimates that are representative of the population, the direct incorporation of weights into models can also reduce statistical efficiency and increase estimator variability.
In general terms, the literature distinguishes two main approaches to this problem (Heeringa et al., 2017). The first corresponds to the design-based approach, whose objective is to obtain valid inferences for the target population while respecting the characteristics of the sample selection process. From this perspective, sampling weights are essential for correcting unequal probabilities of inclusion and producing unbiased estimates of the regression coefficients. However, this approach does not protect against possible model specification errors; that is, even when the estimates are valid from a design perspective, the model may not adequately represent the relationships existing in the population.
The second corresponds to the model-oriented approach, according to which weights are not necessarily required as long as the model is correctly specified and the sampling mechanism is non-informative. Under this assumption, the relationships between the variables observed in the sample coincide with those in the population, so the sampling design does not introduce relevant biases in parameter estimation. From this perspective, incorporating weights could even be counterproductive, by unnecessarily increasing the variance of the estimators and, consequently, the standard errors.
The discussion about whether to use weights in regression models has been extensively developed in the specialized literature, particularly in works such as Skinner et al. (1989) and Pfeffermann (2011). In practice, a frequent methodological recommendation is to estimate models both with and without weights and then compare the results obtained. If including the weights produces important changes in the estimated coefficients or modifies the substantive conclusions of the analysis, this suggests that the sampling design is informative or that the model has specification problems, making the use of weights advisable (United Nations Statistics Division, 2026). Conversely, if the coefficients remain relatively stable and the weights only increase the standard errors, it may be considered that the model adequately captures the structure of the data and that the use of weights is not strictly necessary.
In applied terms, this decision usually depends on the analytical purpose of the study. When the objective is to carry out descriptive inference and produce estimates that are representative of the population, the use of weights is indispensable. By contrast, in contexts of analytical inference oriented toward the study of associations, causal relationships, or hypothesis tests, both weighted and unweighted models may be used, especially when the model incorporates variables related to the sampling design, such as strata or clusters. Nevertheless, the use of unweighted models must be carefully justified, since it implies assuming more restrictive conditions about the sample selection mechanism and the correct specification of the statistical model.
In summary, when weighting is chosen, weights correct possible biases from over- or underrepresentation of certain groups and help obtain more accurate variance estimates. Within the design-based approach, this allows the results to approximate unbiased values comparable to those that would be obtained in a complete census, even when the model is not optimally formulated. However, when weights are highly dispersed, they can increase the variance of the estimated parameters and make the estimates unstable, which is why in explanatory or analytical contexts unweighted models may, at times, yield more consistent and efficient results.
In any case, if the model is misspecified, ignoring sampling weights can lead to biased, uninformative estimates, or even estimates lacking inferential validity. For this reason, the appropriate selection of the variables included in the model is a fundamental aspect of the analysis. To address these issues, various adjustment strategies have been proposed to achieve a balance between both objectives. The most commonly used procedures include the following:
Senate-type weights: this procedure adjusts the weights so that their sum matches the sample size rather than the population size. The objective is to maintain the relative representativeness of the units while reducing the dispersion of the original weights, which is particularly advantageous in surveys where there is high variability among expansion factors. Thus, the new weights are \(w_k^{Senate} = w_k \times \frac{n}{\sum w_k}\).
Normalized weights: in this approach, the original weights are rescaled so that their sum is equal to one, which avoids an unnecessary increase in variance in the models. This technique is especially useful when working with different data subsets (for example, models estimated in subpopulations) or when seeking to minimize variance inflation. Thus, the new weights are \(w_k^{Normalized} = \frac{w_k}{\sum w_k}\).
It is important to note that, in both methods, the adjusted weights are obtained through direct multiplicative transformations of the original sampling weights. Therefore, they should not be used to calculate population sizes or totals. Likewise, these procedures do not alter estimates of ratios, such as means or proportions, since in those cases the weights cancel out in the quotient. In practice, the use of these adjustments can be considered a pragmatic solution in contexts where specialized survey software is not available. However, when tools such as the R packages survey (Lumley, 2024) and srvyr (Freedman Ellis & Schneider, 2024), described in previous chapters, are available, rescaling is no longer necessary, since these packages apply the appropriate treatment to preserve both representativeness and the inferential properties of the model.