Survey Calibration

Introduction

Calibration is a widely used technique in survey sampling. Suppose \(m\) sampling units in a survey have been assigned initial weights \(d_i\) for \(i = 1,\ldots,m\), and furthermore, there are \(n\) auxiliary variables whose values in the sample are known. Calibration seeks to improve the initial weights \(d_i\) by finding new weights \(w_i\) that incorporate this auxiliary information while perturbing the initial weights as little as possible, , the ratio \(g_i = w_i/d_i\) must be close to one. Such reweighting improves precision of estimates (Chapter 7, T. S. Lumley (2010)).

Let \(X \in {\mathbf R}^{m \times n}\) be the matrix of survey samples, with each column corresponding to an auxiliary variable. Reweighting can be expressed as the optimization problem (see Davies, Gillard, and Zhigljavsky (2016)):

\[ \begin{array}{ll} \mbox{minimize} & \sum_{i=1}^m d_i\phi(g_i) \\ \mbox{subject to} & A^Tg = r \end{array} \]

with respect to \(g \in {\mathbf R}^m\), where \(\phi:{\mathbf R} \rightarrow {\mathbf R}\) is a strictly convex function with \(\phi(1) = 0\), \(r \in {\mathbf R}^n\) are the known population totals of the auxiliary variables, and \(A \in {\mathbf R}^{m \times n}\) is related to \(X\) by \(A_{ij} = d_iX_{ij}\) for \(i = 1,\ldots,m\) and \(j = 1,\ldots,n\).

Raking

A common calibration technique is raking, which uses the penalty function \(\phi(g_i) = g_i\log(g_i) - g_i + 1\) as the calibration metric.

We illustrate with the California Academic Performance Index data in the survey package (T. Lumley (2018)) which also supplies facilities for calibration via the function calibrate. Both the population dataset (apipop) and a simple random sample of \(m = 200\) (apisrs) are provided. Suppose that we wish to reweight the observations in the sample using known totals for two variables from the population: stype, the school type (elementary, middle or high) and sch.wide, whether the school met the yearly target or not. This reweighting would make the sample more representative of the general population.

The code below estimates the weights using survey::calibrate.

data(api)
design_api <- svydesign(id = ~dnum, weights = ~pw, data = apisrs)
formula <- ~stype + sch.wide
T <- apply(model.matrix(object = formula, data = apipop),
           2,
           sum)

cal_api <- calibrate(design_api, formula, population = T, calfun = cal.raking)
w_survey <- weights(cal_api)

The CVXR formulation follows.

di <- apisrs$pw
X <- model.matrix(object = formula, data = apisrs)
A <- di * X
n <- nrow(apisrs)
g <- Variable(n)
constraints <- list(t(A) %*% g == T)

## Raking
Phi_R <- Minimize(sum(di * (-entr(g) - g + 1)))
p <- Problem(Phi_R, constraints)
res <- solve(p)
w_cvxr <- di * res$getValue(g)

We compare the results below in a table which show them to be identical.

## Using functions in the *un echoed* preamble of this document...
build_table(d1 = build_df(apisrs, "Survey", w_survey),
            d2 = build_df(apisrs, "CVXR", w_cvxr),
            title = "Calibration weights from Raking")

Other Calibration Metrics

Two other penalty functions commonly used are:

• Quadratic \[ \phi^{Q}(g) = \frac{1}{2}(g-1)^2; \]
• Logit \[ \phi^{L}(g; l, u) = \frac{1}{C}\biggl[ (g-l)\log\left(\frac{g-l}{1-l}\right) + (u-g)\log\left(\frac{u-g}{u-1}\right) \biggr] \mbox{ for } C = \frac{u-l}{(1-l)(u-1)}. \]

It is again easy to incorporate these in our example and compare to survey results.

## Quadratic
Phi_Q  <- Minimize(sum_squares(g - 1) / 2)
p <- Problem(Phi_Q, constraints)
res <- solve(p)
w_cvxr_q <- di * res$getValue(g)
w_survey_q <- weights(calibrate(design_api, formula, population = T, calfun = cal.linear))

u <- 1.10; l <- 0.90
w_survey_l <- weights(calibrate(design_api, formula, population = T, calfun = cal.linear,
                                bounds = c(l, u)))
Phi_L <- Minimize(sum(-entr((g - l) / (u - l))  -
                      entr((u - g) / (u - l)))) 
p <- Problem(Phi_L, c(constraints, list(l <= g, g <= u)))
res <- solve(p)
w_cvxr_l <- di * res$getValue(g)

Further Metrics

Following examples in survey::calibrate, we can try a few other metrics.

First, the hellinger distance.

hellinger <- make.calfun(Fm1 = function(u, bounds)  ((1 - u / 2)^-2) - 1,
                         dF= function(u, bounds) (1 -u / 2)^-3 ,
                         name = "Hellinger distance")
w_survey_h <- weights(calibrate(design_api, formula, population = T, calfun = hellinger))
Phi_h <- Minimize(sum((1 - g / 2)^(-2)))
p <- Problem(Phi_h, constraints)
res <- solve(p)
w_cvxr_h <- di * res$getValue(g)

Next, the derivative of the inverse hyperbolic sine.

w_survey_s <- weights(calibrate(design_api, formula, population = T, calfun = cal.sinh,
                                bounds = c(l, u)))
Phi_s <- Minimize(sum( 0.5 * (exp(g) + exp(-g))))
p <- Problem(Phi_s, c(constraints, list(l <= g, g <= u)))
res <- solve(p)
w_cvxr_s <- di * res$getValue(g)

Session Info

sessionInfo()

## R version 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin21.6.0 (64-bit)
## Running under: macOS Ventura 13.0
## 
## Matrix products: default
## BLAS:   /usr/local/Cellar/openblas/0.3.21/lib/libopenblasp-r0.3.21.dylib
## LAPACK: /usr/local/Cellar/r/4.2.1_4/lib/R/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] grid      stats     graphics  grDevices datasets  utils     methods  
## [8] base     
## 
## other attached packages:
## [1] survey_4.1-1     survival_3.4-0   Matrix_1.5-1     dplyr_1.0.10    
## [5] kableExtra_1.3.4 CVXR_1.0-11     
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.9        svglite_2.1.0     gurobi_9.5-2      lattice_0.20-45  
##  [5] rcbc_0.1.0.9001   Rmosek_10.0.25    assertthat_0.2.1  digest_0.6.30    
##  [9] utf8_1.2.2        gmp_0.6-6         slam_0.1-50       R6_2.5.1         
## [13] evaluate_0.17     highr_0.9         httr_1.4.4        blogdown_1.13    
## [17] pillar_1.8.1      rlang_1.0.6       rstudioapi_0.14   jquerylib_0.1.4  
## [21] rmarkdown_2.17    splines_4.2.1     osqp_0.6.0.5      webshot_0.5.4    
## [25] stringr_1.4.1     bit_4.0.4         munsell_0.5.0     compiler_4.2.1   
## [29] xfun_0.34         pkgconfig_2.0.3   systemfonts_1.0.4 Rglpk_0.6-4      
## [33] htmltools_0.5.3   mitools_2.4       Rcplex_0.3-5      tidyselect_1.2.0 
## [37] tibble_3.1.8      bookdown_0.29     codetools_0.2-18  fansi_1.0.3      
## [41] viridisLite_0.4.1 MASS_7.3-58.1     jsonlite_1.8.3    lifecycle_1.0.3  
## [45] DBI_1.1.3         magrittr_2.0.3    scales_1.2.1      cli_3.4.1        
## [49] stringi_1.7.8     cachem_1.0.6      Rmpfr_0.8-9       xml2_1.3.3       
## [53] bslib_0.4.0       generics_0.1.3    vctrs_0.5.0       tools_4.2.1      
## [57] bit64_4.0.5       glue_1.6.2        fastmap_1.1.0     yaml_2.3.6       
## [61] colorspace_2.0-3  cccp_0.2-9        rvest_1.0.3       ECOSolveR_0.5.4  
## [65] knitr_1.40        sass_0.4.2

Source

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