# Machine learning, cross-fitting and robust calibration

The Get started article builds a weighting recipe with classical tools: weighting classes for nonresponse, raking for calibration, a fixed-fence trim. This article covers the methods that go one step further: flexible learners for the response propensity, cross-fitting to keep them honest, penalized calibration, and data-driven trimming. Each is opt-in: the defaults are the classical methods, and a single argument switches the richer one on.

We use the bundled `sample_one` design (a multistage select-one survey) and resolve eligibility, household nonresponse and within-household selection first, so that the examples below start from a clean staged sample.

Throughout, \(s\) denotes the sample and \(U\) the population (universe); \(w_i\) is the weight of unit \(i\); \(X\) are auxiliary totals known for \(U\); and \(\hat\phi_i\), \(\hat m(x_i)\) are estimated response propensities and outcome predictions.

    base <- weighting_spec(sample_one, 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)

## Flexible propensities

Weighting classes assume response is ignorable within cells. A response-propensity model relaxes that: it estimates `P(respond)` from the auxiliaries and adjusts by the inverse propensity (optionally grouped into classes). weightflow can fit that model with logistic regression, a regression tree, a random forest, or gradient boosting, all through the same `engine` argument.

    fit_boost <- base |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "boost",
                       num_classes = 5) |>
      prep()

Gradient boosting captures nonlinearities and interactions among the predictors without you specifying them. That flexibility is useful when nonresponse depends on the covariates in complicated ways, but it comes with a risk.

## The overfitting problem

A flexible learner that predicts the same units it was trained on fits them too well. The estimated propensities are pulled toward the observed outcomes, so some respondents get artificially low fitted propensities. Since the nonresponse adjustment multiplies the weight by the inverse propensity,

\[w_i^{\text{nr}} = \frac{w_i^{\text{prev}}}{\hat\phi_i},\]

a propensity \(\hat\phi_i\) that is biased toward zero produces an enormous weight. The result is a handful of extreme weights that inflate the variance, exactly what weighting is meant to avoid.

The design effect makes this visible. With in-sample boosting, a single propensity class can carry a factor an order of magnitude larger than the rest, and the Kish design effect \(\text{deff} = 1 + \text{CV}^2(w)\) climbs accordingly.

## Cross-fitting

Cross-fitting breaks the circularity. The sample is split into K folds; the propensity of each unit is predicted by a model trained on the other folds, so no unit informs its own prediction. This is the survey-weighting counterpart of the cross-fitting used in double/debiased machine learning (Chernozhukov et al., 2018). In weightflow it is a single argument, `crossfit = K`; when the step has a `cluster`, the folds are formed by cluster, so members of the same household never split across folds and there is no leakage.

    fit_cf <- base |>
      step_nonresponse(respondent = responded, method = "propensity",
                       formula = ~ region + sex + age, engine = "boost",
                       num_classes = 5, crossfit = 5, crossfit_seed = 1) |>
      prep()

The Kish design effect makes the difference concrete. Compare the in-sample boosting with the five-fold cross-fitted model on the same data:

    c(in_sample = design_effect(collect_weights(fit_boost)$.weight)$deff,
      crossfit  = design_effect(collect_weights(fit_cf)$.weight)$deff)
    #> in_sample  crossfit
    #>  2.393923  1.491576

In-sample boosting overfits and inflates the design effect; cross-fitting removes that in-sample optimism and brings it back down. The out-of-sample propensities are also smoother, so the quantile-based classes come out balanced. The estimates barely change; the stability of the weights does.

Cross-fitting is available wherever a model is fitted. In `step_model_calibration()` it plays the same role: a working model \(\hat m(x)\) is fitted for each outcome \(y\), and the weights are calibrated so that the weighted sample total of the prediction matches its population total,

\[\sum_{i \in s} w_i\,\hat m(x_i) = \sum_{i \in U} \hat m(x_i).\]

With `crossfit = K`, the sample predictions \(\hat m(x_i)\) on the left are made out-of-fold (each unit predicted by a model that did not see it), while the population total on the right uses the full model. The example below fits the outcome model for `income` with gradient boosting and cross-fits it:

    fit_mc <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_model_calibration(
        x_formula  = ~ region + sex,
        models     = list(income = y_model(income ~ age + sex + region,
                                           engine = "boost")),
        population = population, crossfit = 5, crossfit_seed = 1) |>
      prep()

    fit_mc$steps[[2]]$diagnostics
    #>              constraint            type   target achieved
    #> (Intercept) (Intercept) X (consistency)     4495     4495
    #> regionSouth regionSouth X (consistency)     1250     1250
    #> regionEast   regionEast X (consistency)      927      927
    #> regionWest   regionWest X (consistency)      748      748
    #> sexM               sexM X (consistency)     2184     2184
    #> income           income       y (model) 89966496 89966496

The calibration still solves its system exactly, so the achieved totals match the targets; cross-fitting only changes how the predictions that enter the system are formed, removing the in-sample optimism of the flexible learner.

## Ridge (penalized) calibration

Calibration forces the weighted sample to reproduce known population totals. With many control variables, forcing every constraint exactly can again push the weights to extremes. Ridge calibration relaxes the constraints in a controlled way: each target is allowed a small, penalized deviation. In the linear case the calibration system \(A\lambda = (X - \hat X)\) gains a penalty on its diagonal,

\[\big(A + \operatorname{diag}(s / c_j)\big)\,\lambda = X - \hat X,\]

where \(c_j\) is the cost of constraint \(j\) and \(s\) scales the penalty to the system, making it unit-free. A single, scale-free `penalty` governs the trade-off: large values stay (almost) exact, small values relax more and tighten the weights.

    pop_totals <- c("(Intercept)" = nrow(population),
                    regionSouth = sum(population$region == "South"),
                    regionEast  = sum(population$region == "East"),
                    regionWest  = sum(population$region == "West"),
                    sexM        = sum(population$sex == "M"))

    fit_ridge <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "linear", formula = ~ region + sex,
                     totals = pop_totals, penalty = 1) |>
      prep()

    fit_ridge$steps[[2]]$diagnostics
    #>                variable target achieved deviation
    #> (Intercept) (Intercept)   4495  4438.23    -56.77
    #> regionSouth regionSouth   1250  1237.24    -12.76
    #> regionEast   regionEast    927   848.83    -78.17
    #> regionWest   regionWest    748   841.12     93.12
    #> sexM               sexM   2184  2236.07     52.07

Under ridge the achieved totals no longer match the targets exactly; the `deviation` column reports the (small) gap, which is the price paid for steadier weights. As `penalty` decreases, the deviations grow and the calibration factors concentrate around one.

## Potter (MSE-optimal) trimming

Trimming caps extreme weights, but the cap is usually picked by hand. Potter’s method chooses it from the data. Writing \(\tau\) for the candidate cutoff (weights above \(\tau\) are capped), the method evaluates a grid of values of \(\tau\) and minimizes an estimate of the mean squared error of the weighted total,

\[\text{MSE}(\tau) = \text{bias}(\tau)^2 + \text{var}(\tau),\]

balancing the bias introduced by trimming at \(\tau\) (related to the weight removed above the cutoff) against the variance contributed by the weights that remain. The chosen cutoff is the \(\tau\) with the smallest estimated MSE.

    trimmed_tukey <- base |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = c("region", "sex")) |>
      step_trim_weights(method = "tukey") |>
      prep()

    trimmed_potter <- base |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = c("region", "sex")) |>
      step_trim_weights(method = "potter") |>
      prep()

    trimmed_tukey$steps[[6]]$diagnostics[, c("method", "upper", "n_capped")]
    #>   method  upper n_capped
    #> 1  tukey 91.367        1
    trimmed_potter$steps[[6]]$diagnostics[, c("method", "upper", "n_capped")]
    #>   method  upper n_capped
    #> 1 potter 60.127        4

The two rules give different cutoffs: the Tukey far-out fence is a fixed, conservative rule, while Potter searches for the cutoff that minimizes estimated MSE, often more aggressive, capping a few more weights when that lowers the overall error.

## Putting it together

These methods compose like any other step, and the recipe-aware bootstrap covers them automatically: because it re-applies the whole recipe on each replicate (re-fitting the propensity model, re-running the cross-fitting, re-solving the calibration), the standard errors reflect the variability these methods introduce, not just the final weights. See the Variance estimation article for the bootstrap, and Get started for the staged logic these methods plug into.

## References

- Chernozhukov, V., et al. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1), C1–C68.
- Breidt, F. J., & Opsomer, J. D. (2017). Model-assisted survey estimation with modern prediction techniques. Statistical Science, 32(2), 190–205.
- Bardsley, P., & Chambers, R. L. (1984). Multipurpose estimation from unbalanced samples. Applied Statistics, 33(3), 290–299.
- Potter, F. J. (1990). A study of procedures to identify and trim extreme sample weights. Proc. ASA Survey Research Methods Section, 225–230.