# Nonresponse: weighting classes, propensities and calibration

Nonresponse adjustment inflates the weights of respondents so they also represent the nonrespondents. `step_nonresponse()` offers two routes: weighting classes and response propensity models. This vignette explains both, when each is preferable, and how they are estimated.

Throughout, only active units (weight > 0) take part, so cases already dropped earlier in the recipe (unknown eligibility, ineligible) are excluded automatically. We write \(w_i\) for the weight entering the step, \(r\) for the set of respondents, \(\mathbf{x}_i\) for the auxiliaries known for unit \(i\), and \(w_i^{\mathrm{nr}}\) for the weight after the adjustment.

Both routes rest on the same assumption: response is ignorable given the auxiliaries (missing at random). That is, conditional on \(\mathbf{x}_i\), responding is independent of the survey outcome \(y_i\),

\[P(\text{respond} \mid \mathbf{x}_i, y_i) = P(\text{respond} \mid \mathbf{x}_i) = \phi_i .\]

Under this assumption the respondents, reweighted by the inverse of their response propensity \(\phi_i\), represent the nonrespondents without bias. Choosing auxiliaries that are related both to responding and to the outcomes is therefore what makes the adjustment work.

## Weighting classes

Units are partitioned into cells (the weighting classes) according to one or more categorical auxiliaries, and within each cell the respondents absorb the weight of the nonrespondents. The method rests on a homogeneity assumption: every unit in a cell is taken to have the same response probability, so that within the cell the respondents are a random subsample of the active units (response is MCAR within the cell, MAR across cells). Equivalently, it is a model in which the expected outcome is the same for respondents and nonrespondents of the same cell; the adjustment removes bias to the extent that this within-cell equality holds. Cells should therefore be chosen so that response rates differ between cells while the units inside a cell are homogeneous (i.e., similar in their propensity to respond and, ideally, in the survey outcomes).

This adjustment is the natural choice when nothing is known about the nonrespondents beyond what the sampling frame already carries (e.g., strata, primary sampling units, region, and other design variables available for sampled respondents and nonrespondents alike). When the auxiliaries are known for the whole population rather than only the sample, the same arithmetic becomes post-stratification.

The adjustment factor in a cell \(c\) is the total weight of the active units over the weight of the respondents in that cell,

\[f_c = \frac{\sum_{i \in c} w_i}{\sum_{i \in c \cap r} w_i} .\]

Each respondent’s weight is multiplied by \(f_c\) and nonrespondents go to zero, so \(w_i^{\mathrm{nr}} = f_c\,w_i\) for \(i \in c \cap r\). This is the special case of a propensity model in which \(\phi_i\) is estimated by the (weighted) response rate within the cell: a single estimated propensity shared by every unit of the cell.

In `step_nonresponse()` the cells are specified through the `by` argument, which names the categorical variables that define them (here, `region`):

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      prep()
    summary(wf)
    #>
    #> == Weighting specification (weightflow) ==
    #> Data    : 467 cases
    #> Base wts: pw
    #> Steps   :
    #>   1. nonresponse (weighting class)
    #> Status  : estimated (prep)
    #>
    #> Stage summary:
    #>                     stage n_active sum_wts cv_wts deff_kish n_eff
    #>                      base      467    4371  0.236     1.056   442
    #>  stage_1_step_nonresponse      270    4371  0.144     1.021   265
    #>
    #> deff_kish = 1 + CV^2 (Kish design effect from unequal weighting);
    #> n_eff = n_active / deff_kish. Both worsen with each adjustment and
    #> improve with trimming.
    #>
    #> --- Step 1: nonresponse (weighting class) ---
    #>   cell n_respondents n_nonresponse   factor
    #>   East            52            44 1.846154
    #>  North            78            41 1.525641
    #>  South            72            49 1.680556
    #>   West            68            63 1.926471
    #> Kish deff: 1.056 -> 1.021   |   n_eff: 442 -> 265
    #>
    #> R-indicator (representativity of response): 0.892  (on region)

Validation. By construction the total weight is preserved within each cell (the nonrespondents’ weight is moved to the respondents, not lost). So the weighted total per region after the step equals the base-weight total before it:

    before <- tapply(sample_survey$pw,  sample_survey$region, sum)
    after  <- tapply(wf$final_weight,   sample_survey$region, sum)
    round(cbind(before, after, diff = after - before), 6)
    #>          before     after diff
    #> North 1487.5000 1487.5000    0
    #> South 1210.0000 1210.0000    0
    #> East   800.0000  800.0000    0
    #> West   873.3333  873.3333    0

The differences are zero: weighting classes redistribute, they do not create or destroy weight.

## Response propensities

Instead of cells, the probability of responding is modelled from auxiliaries known for respondents and nonrespondents alike,

\[\phi_i = P(\text{respond} \mid \mathbf{x}_i),\]

and estimated by \(\hat\phi_i\). The model is fitted on the active units, weighted by the current weights, and two routes follow. With `num_classes = NULL`, each respondent is weighted by the inverse propensity, \(w_i^{\mathrm{nr}} = w_i / \hat\phi_i\). With an integer `num_classes`, units are grouped into that many classes formed from quantiles of \(\hat\phi_i\) and a weighting-class adjustment is applied within each, which is more robust to a misspecified model.

### Logistic regression

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "logit",
                       num_classes = 5) |>
      prep()
    summary(wf)
    #>
    #> == Weighting specification (weightflow) ==
    #> Data    : 467 cases
    #> Base wts: pw
    #> Steps   :
    #>   1. nonresponse (propensity: logit, 5 classes)
    #> Status  : estimated (prep)
    #>
    #> Stage summary:
    #>                     stage n_active sum_wts cv_wts deff_kish n_eff
    #>                      base      467    4371  0.236     1.056   442
    #>  stage_1_step_nonresponse      270    4371  0.155     1.024   264
    #>
    #> deff_kish = 1 + CV^2 (Kish design effect from unequal weighting);
    #> n_eff = n_active / deff_kish. Both worsen with each adjustment and
    #> improve with trimming.
    #>
    #> --- Step 1: nonresponse (propensity: logit, 5 classes) ---
    #>  propensity_class   n mean_prop   factor
    #>     [0.499,0.524]  94 0.5122335 2.224138
    #>     (0.524,0.547]  93 0.5341818 1.837719
    #>     (0.547,0.594] 101 0.5721654 1.602273
    #>     (0.594,0.643]  86 0.6162161 1.574468
    #>     (0.643,0.684]  93 0.6600845 1.576271
    #> Kish deff: 1.056 -> 1.024   |   n_eff: 442 -> 264
    #>
    #> R-indicator (representativity of response): 0.889  (on region, sex, age)

Because the model is fitted with survey weights, a logistic fit may print a “non-integer #successes” message: that is expected for a weighted binomial fit and does not affect the estimated propensities.

### Trees, forests and boosting

The same propensity can be estimated with a regression tree (`engine = "tree"`, package `rpart`), a random forest (`engine = "forest"`, package `ranger`), or gradient boosting (`engine = "boost"`, package `xgboost`), which capture nonlinearities and interactions without specifying them. More flexibility is not free, though: a very flexible model can overfit the response and produce more dispersed adjustment factors, which raises the variance of the weights (a higher design effect). Compare the deff after each engine below, the forest and boosting typically yield the largest, the weighting classes the smallest.

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "tree",
                       num_classes = 5) |>
      prep()
    design_effect(wf$final_weight)$deff
    #> [1] 1.055763

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "forest",
                       num_classes = 5) |>
      prep()
    design_effect(wf$final_weight)$deff
    #> [1] 1.149559

### Flexibility, overfitting, and cross-fitting

The reason flexibility is not free deserves a closer look. A very flexible model can fit the noise of the particular sample in addition to the signal (overfitting). When the propensity is then predicted for the very units the model was trained on, the estimates \(\hat\phi_i\) are pulled toward the observed responses: some respondents receive artificially low propensities, and since the adjustment is \(1/\hat\phi_i\), those units get extreme weights that inflate the variance. The model is not bad at prediction; i.e., it predicts too well in-sample and poorly out of it.

The remedy is cross-fitting: estimate each unit’s propensity with a model trained on other units (held-out folds), so the prediction is out-of-sample and free of this optimism. weightflow provides it through the `crossfit` argument:

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "forest",
                       num_classes = 5, crossfit = 5, crossfit_seed = 1) |>
      prep()
    design_effect(wf$final_weight)$deff
    #> [1] 1.050402

The Machine learning, cross-fitting and robust calibration article develops the boosting engine and cross-fitting in full, with a worked comparison of the design effect with and without cross-fitting.

### Weighting the propensity model

By default the propensity model is fitted using the weights that enter the step, that is, the base design weights or the weights already adjusted by earlier steps in the cascade (unknown eligibility, an earlier nonresponse stage). Setting `weight_model = FALSE` fits it unweighted, so those incoming weights enter only the \(1/\hat\phi_i\) adjustment and not the model fit. Fitting unweighted can lower the variance of the propensity estimates when the incoming weights are unrelated to response given the covariates, at the cost of possible bias if they are related (Little and Vartivarian 2003). Only the model fit changes; the adjustment always uses the incoming weights.

    f <- function(wm) weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "logit",
                       num_classes = NULL, weight_model = wm) |>
      prep()
    c(weighted   = design_effect(f(TRUE)$final_weight)$deff,
      unweighted = design_effect(f(FALSE)$final_weight)$deff)
    #>   weighted unweighted
    #>   1.021551   1.022794

## Person or household level

Nonresponse can occur at the person level (within a reached household) or at the household level (the whole household is not reached). The `cluster` argument moves the adjustment to the household: each household counts once with its weight, and the redistribution (or the propensity model) is done over households, then assigned to their members.

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region", cluster = "household_id") |>
      prep()
    design_effect(wf$final_weight)$deff
    #> [1] 1.06383

The level is dictated by what is known about the nonrespondents: household auxiliaries and a whole-household outcome call for `cluster`; person-level auxiliaries within reached households do not. Note that the effective sample size drops more at the household level, since households (not persons) are the independent units being adjusted.

## Nonresponse by calibration

A third route calibrates the respondents’ weights so that, over the auxiliaries, they reproduce the totals the active units (respondents and nonrespondents) carried before the step. With `method = "calibration"` and `totals = NULL` (the default) the targets are those incoming, design-weighted totals, so the adjustment is the two-phase, sample-level calibration estimator (Sarndal and Lundstrom 2005): the calibrated respondent estimates reproduce the pre-nonresponse cascade estimates exactly. Supply `totals` to calibrate to external population totals instead.

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "calibration",
                       formula = ~ region + sex) |>
      prep()
    summary(wf)
    #>
    #> == Weighting specification (weightflow) ==
    #> Data    : 467 cases
    #> Base wts: pw
    #> Steps   :
    #>   1. nonresponse (calibration: linear, sample-level)
    #> Status  : estimated (prep)
    #>
    #> Stage summary:
    #>                     stage n_active sum_wts cv_wts deff_kish n_eff
    #>                      base      467    4371  0.236     1.056   442
    #>  stage_1_step_nonresponse      270    4371  0.146     1.021   264
    #>
    #> deff_kish = 1 + CV^2 (Kish design effect from unequal weighting);
    #> n_eff = n_active / deff_kish. Both worsen with each adjustment and
    #> improve with trimming.
    #>
    #> --- Step 1: nonresponse (calibration: linear, sample-level) ---
    #>     variable    target achieved
    #>  (Intercept) 4370.8333  4370.83
    #>  regionSouth 1210.0000  1210.00
    #>   regionEast  800.0000   800.00
    #>   regionWest  873.3333   873.33
    #>         sexM 2284.1667  2284.17
    #> nonresponse calibration to sample-level totals; g in [1.491, 1.950]
    #> Kish deff: 1.056 -> 1.021   |   n_eff: 442 -> 264
    #>
    #> R-indicator (representativity of response): 0.890  (on region, sex)

Validation. The respondents, reweighted, reproduce the auxiliary totals of all active units before the step:

    X      <- model.matrix(~ region + sex, sample_survey)
    resp   <- sample_survey$responded == 1
    before <- colSums(sample_survey$pw * X)               # respondents + nonrespondents
    after  <- colSums(wf$final_weight[resp] * X[resp, ])  # respondents, calibrated
    round(rbind(before, after, diff = after - before), 4)
    #>        (Intercept) regionSouth regionEast regionWest     sexM
    #> before    4370.833        1210        800   873.3333 2284.167
    #> after     4370.833        1210        800   873.3333 2284.167
    #> diff         0.000           0          0     0.0000    0.000

The distance (`calfun`), `bounds` and ridge `penalty` carry over from `step_calibrate()`. With `equal_within_cluster = TRUE` and a `cluster` the adjustment is integrative: the responding members of a household share one calibration factor, so the weights stay constant within household.

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "calibration",
                       formula = ~ region, cluster = "household_id",
                       equal_within_cluster = TRUE) |>
      prep()
    design_effect(wf$final_weight)$deff
    #> [1] 1.020774

## Which to use

Weighting classes need categorical auxiliaries and enough respondents per cell; they are simple and transparent. Propensity models handle continuous predictors and many auxiliaries at once, and the tree/forest engines relax functional-form assumptions. Using propensity classes (`num_classes`) rather than the direct \(1/\hat\phi_i\) keeps the adjustment stable when the model is imperfect, at the cost of some efficiency. In all cases, model the response on auxiliaries that are both predictive of responding and related to the survey outcomes.