Variance estimation

weightflow computes weights and also estimates their variances. This vignette shows a few ways to obtain standard errors from a weightflow recipe, and how they relate: a recipe-aware bootstrap, a survey-package linearization, and a recipe-aware jackknife.

Throughout, \(U\) is the population and \(s\) the sample; \(w_i\) is the final weight of unit \(i\); and a population total is written \(Y = \sum_{i \in U} y_i\), estimated by \(\hat Y = \sum_{i \in s} w_i\,y_i\). The sample is drawn in clusters: primary sampling units (PSUs) nested in strata.

Why the adjustments matter for variance

A weighting recipe rarely stops at the design weight. It redistributes unknown eligibility, drops out-of-scope units, adjusts for nonresponse and calibrates to known totals. Each of those stages is estimated from the sample, so each one adds (or, for calibration, often removes) variability.

A linearization that takes the final weights as fixed and applies the ultimate-cluster formula ignores that the nonresponse and calibration steps were themselves estimated. The cleanest way to account for them is to re-run the whole recipe on each replicate, so the replicate weights carry the variability of every stage.

Method 1: a PSU bootstrap that re-applies the recipe

bootstrap_weights() resamples primary sampling units (PSUs) with replacement within strata and re-runs the recipe on each replicate. Pass the inert recipe (do not call prep() first): the bootstrap preps it once per replicate.

dat <- sample_one
dat$age_grp <- cut(dat$age, c(0, 30, 45, 60, Inf),
                   labels = c("18-30", "31-45", "46-60", "60+"))

spec <- weighting_spec(dat, base_weights = pw) |>
  step_unknown_eligibility(unknown = unknown_elig, by = "region",
                           cluster = "household_id") |>
  step_drop_ineligible(ineligible = ineligible) |>
  step_nonresponse(respondent = hh_responded, method = "weighting_class",
                   by = "region", cluster = "household_id") |>
  step_select_within(prob = p_within) |>
  step_nonresponse(respondent = responded, method = "weighting_class",
                   by = c("region", "sex", "age_grp")) |>
  step_calibrate(method = "raking",
                 margins = list(region = c(table(population$region)),
                                sex    = c(table(population$sex))))

boot <- bootstrap_weights(spec, replicates = 200, strata = "region",
                          psu = "psu", seed = 2024, progress = FALSE)
boot
#> <weightflow bootstrap>
#>   replicates : 200
#>   units      : 417 (active: 209)
#>   strata     : region
#>   psu        : psu

The multiplier is the Rao-Wu rescaling bootstrap. Consider a stratum \(h\) with \(n_h\) PSUs, from which \(m_h\) are drawn with replacement (by default \(m_h = n_h - 1\)). Let \(t_{hi}^{*}\) be the number of times PSU \(i\) is selected in a replicate. Every unit in that PSU has its weight rescaled by

\[\lambda_{hi} = 1 - \sqrt{\tfrac{m_h}{n_h - 1}} + \sqrt{\tfrac{m_h}{n_h - 1}}\;\frac{n_h}{m_h}\,t_{hi}^{*},\]

so the replicate weight is \(w_i^{*} = \lambda_{hi}\,w_i\). The factor has expectation one over the resampling, \(\mathbb{E}(\lambda_{hi}) = 1\), which keeps each replicate design-unbiased, and the construction never turns it negative, so the recipe can be re-prepped on every replicate without invalid weights. Whole PSUs are kept together (every unit in a drawn PSU is retained), as the design’s clustering requires.

Estimates with bootstrap standard errors

Writing \(\hat\theta\) for the point estimate and \(\hat\theta_b\) for its value on replicate \(b\) (each computed from the re-prepped replicate weights), the bootstrap variance is the average squared deviation across the \(B\) replicates,

\[\widehat{\operatorname{Var}}(\hat\theta) = \frac{1}{B} \sum_{b=1}^{B} \big(\hat\theta_b - \hat\theta\big)^2 .\]

boot_mean(boot,  "income")     # mean income
#>   estimate       se ci_lower ci_upper
#> 1 21615.21 872.7788 19904.59 23325.82
boot_total(boot, "employed")   # total employed
#>   estimate       se ci_lower ci_upper
#> 1 1927.219 140.9421 1650.978 2203.461
boot_mean(boot,  "employed")   # employment rate
#>    estimate         se  ci_lower  ci_upper
#> 1 0.4287473 0.03102821 0.3679331 0.4895615

For any other statistic, pass a function of the weights and the data to bootstrap_estimate():

bootstrap_estimate(boot, function(w, d) {
  ok <- !is.na(d$income) & w > 0
  stats::median(rep(d$income[ok], times = round(w[ok])))   # weighted median (approx.)
})
#>   estimate     se ci_lower ci_upper
#> 1    18136 930.15 16312.94 19959.06

Method 2: hand the weights to the survey package

as_svydesign() builds an ultimate-cluster linearization design from a prepped recipe. It is fast, but treats the calibration as fixed.

fitted <- prep(spec)
des <- as_svydesign(fitted, ids = "psu", strata = "region")
survey::svymean(~income, des, na.rm = TRUE)
#>         mean     SE
#> income 21615 989.34

To keep the recipe’s adjustments in the variance while still using survey, feed it the bootstrap replicate weights from method 1:

rep_des <- as_svrepdesign(boot)
survey::svymean(~income, rep_des, na.rm = TRUE)
#>         mean     SE
#> income 21615 872.78

This matches boot_mean(boot, "income") exactly, because as_svrepdesign() sets scale = 1 / B, rscales = 1 and mse = TRUE.

Replicate weights for a tidyverse workflow

collect_replicate_weights() attaches the point weight (.weight) and the replicate weights (rep_1 … rep_B) to the active respondents, ready for srvyr.

df <- collect_replicate_weights(boot)
d_rep <- srvyr::as_survey_rep(df, weights = .weight,
                              repweights = dplyr::starts_with("rep_"),
                              type = "bootstrap", combined.weights = TRUE,
                              scale = 1 / attr(df, "R"), rscales = 1, mse = TRUE)
srvyr::summarise(d_rep, mean_income = srvyr::survey_mean(income, na.rm = TRUE))
#> # A tibble: 1 × 2
#>   mean_income mean_income_se
#>         <dbl>          <dbl>
#> 1      21615.           873.

Method 3: a delete-a-PSU jackknife that re-applies the recipe

The jackknife is the natural sibling of the bootstrap: instead of resampling PSUs, it deletes one PSU at a time and re-runs the whole recipe, so the replicate weights again carry the variability of every stage. jackknife_weights() builds the stratified delete-a-PSU jackknife (JKn) with strata/psu; the unstratified JK1 follows from strata = NULL.

jk <- jackknife_weights(spec, strata = "region", psu = "psu", progress = FALSE)
jk
#> <weightflow jackknife>
#>   replicates : 48 (delete-a-PSU)
#>   units      : 417 (active: 209)
#>   strata     : region
#>   psu        : psu

jack_mean(jk,  "income")     # mean income, with the JKn variance
#>   estimate       se ci_lower ci_upper
#> 1 21615.21 939.1042  19774.6 23455.82
jack_total(jk, "employed")   # total employed
#>   estimate       se ci_lower ci_upper
#> 1 1927.219 155.2414 1622.952 2231.487

For a total it matches survey’s replicate jackknife exactly. As with the bootstrap, the replicate weights bridge to survey/srvyr through as_svrepdesign(jk), so any estimand or domain can be estimated downstream with the recipe’s uncertainty built in.

Which one to use

Use the recipe-aware bootstrap (method 1, in any of its three forms) when the nonresponse and calibration steps are a meaningful part of the design and you want their uncertainty reflected; it is the more honest variance. Use the linearization (method 2) for a quick, well-understood standard error when the adjustments are minor or you only need the design-and-clustering part. The jackknife (method 3) is the recipe-aware alternative to the bootstrap when a deterministic, replicate-based variance is preferred; it matches survey’s replicate jackknife for totals.

A few practical notes. More replicates give a more stable bootstrap SE; 200 is fine for exploration, 500-1000 for final figures. Each stratum needs at least two PSUs to be resampled (single-PSU strata are left untouched, with a warning). If a replicate leaves a calibration or weighting-class cell empty it is dropped with a warning; coarser by cells make the bootstrap more robust.