# Calibration: raking, post-stratification and GREG

Calibration adjusts the weights so that the weighted sample reproduces known population totals of auxiliary variables. It is usually the last stage of the recipe, applied to weights that already reflect eligibility, selection and nonresponse. `step_calibrate()` provides raking, post-stratification and linear (GREG) calibration, with optional bounds and an equal-weights-within-cluster variant.

Throughout, \(w_i\) is the weight entering the step (the design weight, or the weight coming from the previous adjustment for eligibility, selection or nonresponse), \(w_i^{\mathrm{cal}}\) is the calibrated weight, \(\mathbf{x}_i\) is the vector of calibration auxiliaries for unit \(i\), and \(\mathbf{X} = \sum_{i \in U} \mathbf{x}_i\) are their known population totals.

## What calibration does

A calibrated weight system satisfies the calibration equation, i.e. the weighted sample totals of the auxiliaries match the population totals,

\[\sum_{i \in s} w_i^{\mathrm{cal}}\,\mathbf{x}_i = \mathbf{X} .\]

The calibrated weight is the incoming weight times an adjustment factor, \(w_i^{\mathrm{cal}} = g_i\,w_i\), where \(g_i\) is the calibration factor (the g-weight). Many weight systems satisfy the calibration equation, so calibration picks the one whose factors \(g_i\) keep \(w_i^{\mathrm{cal}}\) closest to the incoming weights \(w_i\). Several distance measures exist (they differ in how they penalize moving a weight), but they share the same idea: change the previous weights as little as possible. The reason is that the incoming weights already carry good properties (for instance, if they are Horvitz-Thompson design weights, the estimator is design-unbiased); keeping the calibrated weights near them preserves those properties, so the calibration estimator is approximately (asymptotically) unbiased, while gaining the consistency with the population totals.

Calibration auxiliaries are chosen with two goals in mind. First, precision: variables correlated with the survey outcomes reduce the standard errors (more on why below). Second, bias reduction: aligning the weights to population totals corrects residual imbalances left after the nonresponse adjustment (for example a group that ended up under- or over-represented), provided the totals are known without error.

## Why calibration reduces variance: the regression link

Calibration is tied to a regression model for the outcome, which is the clearest way to see why it helps. The calibration estimator can be written in generalized-regression (GREG) form,

\[\hat Y = \sum_{i \in U} \hat y_i + \sum_{i \in s} w_i\,(y_i - \hat y_i), \qquad \hat y_i = \mathbf{x}_i^\top \hat{\boldsymbol\beta},\]

a population total of fitted values plus a weighted sum of the sample residuals \(e_i = y_i - \hat y_i\). The variance of this estimator depends on the residuals \(e_i\), not on \(y_i\) itself. So when the auxiliaries predict the outcome well, the residuals are small and the variance drops. This is the intuition behind calibrating on variables related to the outcomes: you are implicitly fitting a model for \(y\), and you only pay (in variance) for what the model fails to explain.

## Post-stratification

Post-stratification uses the full cross-classification of the auxiliaries into groups (post-strata): the population count \(N_g\) of every group \(g\) must be known. Within each group the factor is the same for all units,

\[g_i = \frac{N_g}{\hat N_g}, \qquad \hat N_g = \sum_{i \in g \cap s} w_i,\]

the known population count over the weighted sample count. It is a saturated model (one factor per group) and reproduces every group total exactly.

With a single post-stratifying variable, the groups are its categories, and `poststratify` takes that one variable:

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "poststratify",
                     margins = list(region = c(table(population$region)))) |>
      prep()
    wf$steps[[2]]$diagnostics
    #>   variable category target prev_total    factor
    #> 1   region    North   1570  1487.5000 1.0554622
    #> 2   region    South   1250  1210.0000 1.0330579
    #> 3   region     East    927   800.0000 1.1587500
    #> 4   region     West    748   873.3333 0.8564885

The `poststratify` method works on exactly one categorical variable. To post-stratify on several variables the post-strata are their interaction (every combination of categories), and there are two ways to obtain it. One is to create a single variable in the sample that holds the interaction (e.g. `interaction(region, sex)`) and post-stratify on it, supplying the population counts of every combination. The other, more convenient, is linear calibration with an interaction formula `~ region * sex`, whose design matrix has one column per cell, so it reproduces the full cross-table and, unlike the shortcut, also accepts bounds:

    totals_cross <- colSums(model.matrix(~ region * sex, population))
    wf2 <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "linear", formula = ~ region * sex,
                     totals = totals_cross) |>
      prep()
    wf2$steps[[2]]$diagnostics
    #>                          variable target achieved
    #> (Intercept)           (Intercept)   4495     4495
    #> regionSouth           regionSouth   1250     1250
    #> regionEast             regionEast    927      927
    #> regionWest             regionWest    748      748
    #> sexM                         sexM   2184     2184
    #> regionSouth:sexM regionSouth:sexM    607      607
    #> regionEast:sexM   regionEast:sexM    460      460
    #> regionWest:sexM   regionWest:sexM    372      372

Creating more cells than the sample can support has a cost: cells may come out empty in the sample (no respondents to carry the weight), and small sample cells produce large factors \(g = N_g / \hat N_g\) when \(\hat N_g\) is built from very few units. Both make the weights unstable, which is the practical limit on how finely you can post-stratify.

There is a regression model behind it. Post-stratification is a GREG whose predictor is the post-stratum indicator, so the fitted value \(\hat y_i\) is the estimated post-stratum mean and the residual is \(e_i = y_i - \bar y_g\), the deviation of \(y_i\) from its post-stratum mean. As in the GREG form above, the variance depends on these residuals: the more homogeneous the outcome within post-strata, the smaller the residuals and the larger the precision gain.

## Raking

Raking (iterative proportional fitting) needs only the marginal totals of each variable, not the full cross-table. It cycles through the margins, post-stratifying to one at a time, until all marginal totals are met simultaneously.

    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "raking",
                     margins = list(region = c(table(population$region)),
                                    sex    = c(table(population$sex)))) |>
      prep()
    wf$steps[[2]]$diagnostics
    #>   variable category target achieved
    #> 1   region    North   1570     1570
    #> 2   region    South   1250     1250
    #> 3   region     East    927      927
    #> 4   region     West    748      748
    #> 5      sex        F   2311     2311
    #> 6      sex        M   2184     2184

Raking fits an implicit log-linear model with main effects only: it matches each margin but says nothing about the interactions. As a result it does not reproduce the counts of the cross-classification (e.g. the region-by-sex cell totals): those are left at whatever the sample implies, not forced to a known value. That is exactly why raking is the method of choice when you want to include many auxiliaries (each thought to be a good predictor of response and of the outcomes) without inflating the weights: by constraining only the margins, it avoids the small, unstable cells of a full cross-table and keeps the weights closer to the incoming ones. A further practical property is that raking factors \(g_i\) are always positive (they can be large, but never negative), so the calibrated weights never change sign.

## Linear calibration (GREG)

Linear calibration matches the totals of a design matrix, and is the natural choice with continuous auxiliaries or when a linear relationship with the outcome is plausible. It is the calibration form that corresponds most directly to the GREG estimator above. As seen in the post-stratification section, a saturated interaction formula (`~ region * sex`) makes linear calibration reproduce a full cross-table, and unlike the `poststratify` shortcut it then accepts bounds.

What sets linear calibration apart from post-stratification and raking is that its auxiliaries need not be categorical. Those two methods work on group membership (cells or margins), whereas linear calibration can match the total of a quantitative variable, e.g. the population total of `age`, alongside the categorical ones. The formula below calibrates on region, sex and mean age at once:

    totals <- colSums(model.matrix(~ region + sex + age, population))
    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "linear", formula = ~ region + sex + age,
                     totals = totals) |>
      prep()
    wf$steps[[2]]$diagnostics
    #>                variable target achieved
    #> (Intercept) (Intercept)   4495     4495
    #> regionSouth regionSouth   1250     1250
    #> regionEast   regionEast    927      927
    #> regionWest   regionWest    748      748
    #> sexM               sexM   2184     2184
    #> age                 age 190781   190781

The `age` row is calibrated to the known population total of age, something neither post-stratification nor raking can do directly: they would first have to bin age into categories, losing information. Continuous auxiliaries are where the regression view of calibration is most natural.

Unlike raking, linear calibration can produce negative factors \(g_i\) (and so negative weights). This is most likely when the incoming weights \(w_i\) are very small, or when the calibration equation includes more variables than the sample can support (for example many auxiliaries with interactions omitted): the system is then close to over-parameterized and some factors overshoot below zero. Negative weights are awkward to use and report, which motivates the next two options.

## Bounded calibration

Bounding the calibration factor \(g_i = w_i^{\mathrm{cal}} / w_i\) keeps the adjustment within a range and rules out the negative or extreme weights that unbounded linear calibration can produce (the logit distance of Deville and Sarndal enforces the bounds smoothly), trading a little exactness for safer weights.

    totals_rs <- colSums(model.matrix(~ region + sex, population))
    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "linear", formula = ~ region + sex, totals = totals_rs,
                     bounds = c(0.83, 1.2)) |>
      prep()
    range(wf$steps[[2]]$diagnostics$achieved / wf$steps[[2]]$diagnostics$target)
    #> [1] 1 1

Bounds cap the factors but still solve the calibration exactly within those limits. When the real problem is too many auxiliaries for the sample (an over-parameterized calibration equation), a better remedy is to relax the targets rather than bound the factor: ridge (penalized) calibration, and more generally penalized or lasso-type calibration, shrink the solution and avoid the instability that produces extreme or negative weights. The ridge option is covered in the Machine learning, cross-fitting and robust calibration article.

## Equal weights within a cluster

In household surveys it is often desirable that every member of a household carry the same final weight (so that person and household estimates stay coherent). With `equal_within_cluster = TRUE`, linear calibration finds a single weight per cluster that still meets the population totals.

    totals_rs <- colSums(model.matrix(~ region + sex, population))
    wf <- weighting_spec(sample_survey, base_weights = pw) |>
      step_nonresponse(respondent = responded, method = "weighting_class",
                       by = "region") |>
      step_calibrate(method = "linear", formula = ~ region + sex, totals = totals_rs,
                     cluster = "household_id", equal_within_cluster = TRUE) |>
      prep()
    # every member of a household shares one weight
    tapply(wf$final_weight, sample_survey$household_id, function(x) diff(range(x))) |>
      max()
    #> [1] 21.71654

The maximum within-household spread is zero: the weight is constant inside each household, while the region and sex totals are still reproduced.

The integrative option is not tied to the default distance: it composes with any `calfun` (`"linear"`, `"raking"` or `"logit"`), with `bounds`, and with `by` (domain calibration), all applied while keeping one weight per cluster. For example, `calfun = "raking"` gives a single positive weight per household.

## Which to use

Post-stratification when the full cross-table is known and the cells are large enough. Raking when only margins are available, or when you want many auxiliaries without unstable cells. Linear (GREG) for continuous auxiliaries or an explicit linear model. Add bounds (or ridge, in the advanced article) when the weights risk becoming extreme, and `equal_within_cluster` when household members must share a weight. When a working model for the outcome is available and the auxiliaries are known at the unit level for the whole population, the model-calibration approach (see the Model calibration article) can be more efficient still.

## References

- Deville, J.-C., & Sarndal, C.-E. (1992). Calibration estimators in survey sampling. JASA, 87(418), 376–382.
- Deville, J.-C., Sarndal, C.-E., & Sautory, O. (1993). Generalized raking procedures in survey sampling. JASA, 88(423), 1013–1020.
- Deming, W. E., & Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table. Annals of Mathematical Statistics, 11(4), 427–444.
- Sarndal, C.-E. (2007). The calibration approach in survey theory and practice. Survey Methodology, 33(2), 99–119.