Chapter 1 Introduction
The analysis of surveys with complex designs has a well-established tradition in the statistical literature, grounded in the design-based inference approach. As noted by Särndal et al. (2003) and Gutiérrez (2016), this approach recognizes the probability distribution induced by the sample selection process as the only probabilistic mechanism governing inference. Under this paradigm, the statistical properties of estimators, such as unbiasedness, precision, and consistency, are evaluated with respect to the distribution of all possible samples that the design can generate, without imposing distributional assumptions on the variable of interest.
Within this framework, variance estimation has played a central role and has evolved around three broad complementary methodological families. The first, the ultimate cluster method (Hansen et al., 1953), simplifies the multilevel structure by concentrating variability in the primary sampling units (PSUs), treating selection as if it were carried out with replacement. The second, Taylor linearization (Binder, 1983), extends this principle to nonlinear parameters, such as means, proportions, and ratios, through a first-order approximation that reduces the variance estimation problem to that of a total. Finally, the third family, replication methods such as jackknife, bootstrap, and balanced repeated replication, estimates variance from the dispersion among estimates obtained through systematic sample replicates.
In parallel, in the field of statistical modeling, foundational contributions such as those by Kish & Frankel (1974), Fuller (1975), and Binder (1983) laid the groundwork for weighted regression and variance estimation for coefficients under stratified and multistage designs. Building on this development, Pfeffermann (1993) formalized the maximum pseudo-likelihood (MPL) method, which incorporates sampling weights into the log-likelihood function to produce design-consistent estimators, thereby extending the generalized linear models of Nelder & Wedderburn (1972) to the context of surveys with complex designs.
More recently, Heeringa et al. (2017) have synthesized these contributions within a unified analytical framework that coherently integrates the sampling design, weighting, and modeling, establishing their work as a standard reference for contemporary practice in complex survey analysis.
This document is intended for readers with training in statistics or data science, or with equivalent professional experience. It is especially designed for those who have knowledge of the foundations of sampling theory, such as inclusion probabilities, estimators, and confidence intervals, of linear and logistic regression models, and of introductory programming in R.
In this context, the document is positioned as an intermediate-to-advanced text aimed at deepening and consolidating this knowledge, facilitating its rigorous application in complex and technically demanding settings. Its purpose is to accompany researchers in the transition from theoretical foundations to their proper implementation in the production of official statistics, academic research in the social sciences, public policy evaluation, and data analysis under international quality standards.