---
title: "Calibration: raking, post-stratification and GREG"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Calibration: raking, post-stratification and GREG}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(weightflow)
```

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:

```{r poststrat}
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
```

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:

```{r poststrat-multi}
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
```

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.

```{r raking}
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
```

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:

```{r linear}
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
```

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.

```{r bounded}
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)
```

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.

```{r equal-cluster}
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()
```

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.
