Empirical likelihood equations for survey designs (design-weighted QLS system)

Usage

el_build_equation_system_survey(
  family,
  missingness_model_matrix,
  auxiliary_matrix,
  respondent_weights,
  N_pop,
  n_resp_weighted,
  mu_x_scaled
)

Details

Returns a function that evaluates the stacked EL system for complex survey designs using design weights. Unknowns are \(\theta = (\beta, z, \lambda_W, \lambda_x)\) with \(z = \operatorname{logit}(W)\). Blocks correspond to:

response-model score equations in \(\beta\),
the response-rate equation in \(W\) based on \(\sum d_i (w_i - W)/D_i = 0\),
auxiliary moment constraints \(\sum d_i (X_i - \mu_x)/D_i = 0\),
and the design-based linkage between \(\lambda_W\) and the nonrespondent total: \(T_0/(1-W) - \lambda_W \sum d_i / D_i = 0\), where \(T_0 = N_{\mathrm{pop}} - \sum d_i\) on the analysis scale.

When all design weights are equal and \(N_{\mathrm{pop}}\) and the respondent count match the simple random sampling setup, this system reduces to the Qin, Leung, and Shao (2002) equations (6)-(10).