---
title: "Nonresponse: weighting classes, propensities and calibration"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Nonresponse: weighting classes, propensities and calibration}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(weightflow)
has_rpart  <- requireNamespace("rpart",  quietly = TRUE)
has_ranger <- requireNamespace("ranger", quietly = TRUE)
```

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`):

```{r wc}
wf <- weighting_spec(sample_survey, base_weights = pw) |>
  step_nonresponse(respondent = responded, method = "weighting_class",
                   by = "region") |>
  prep()
summary(wf)
```

**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:

```{r wc-check}
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)
```

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

```{r prop-logit, warning = FALSE}
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)
```

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.

```{r prop-tree, eval = has_rpart}
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
```

```{r prop-forest, eval = has_ranger}
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
```

### 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:

```{r crossfit, eval = has_ranger}
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
```

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.

```{r weight-model, warning = FALSE}
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)
```

## 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.

```{r cluster}
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
```

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.

```{r nr-calib}
wf <- weighting_spec(sample_survey, base_weights = pw) |>
  step_nonresponse(respondent = responded, method = "calibration",
                   formula = ~ region + sex) |>
  prep()
summary(wf)
```

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

```{r nr-calib-check}
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)
```

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.

```{r nr-calib-integ}
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
```

## 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.
