---
title: "Vignette: Robust Survey Regression"
author: "Beat Hulliger and Tobias Schoch"
output:
    html_document:
        css: "fluent.css"
        highlight: tango
vignette: >
  %\VignetteIndexEntry{Robust Survey Regression Estimator}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
    collapse = TRUE,
    comment = "",
    prompt = TRUE,
    dpi = 36,
    fig.align = "center"
)
```

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```

## Outline

The vignette is organized as follows. In Chapter 1, we guide the reader through
the preparatory steps (loading packages and data). In the chapters that follow,
we discuss regression estimation (with focus on weighted least squares, *M*-
and *GM*-estimators) for 3 different modes of inference.

 - Design-based inference (Chapter 2)
 - Model-based inference (Chapter 3)
 - Compound design- and model-based inference (Chapter 4)

<div class="my-sidebar-blue">
<p style="color: #1f618d;">
**Good to know.**
</p>
<p>
Loosely speaking, the 3 modes of inference reflect how much confidence we place
in the validity of our regression model. In design-based inference (also known
as inference for *descriptive* population parameters), inference is with
respect to the randomization distribution induced by the sampling---the model
is of subordinate importance.
</p>
</div>


## 1 Preparations

First, we load the packages `robsurvey` **and** `survey` ([Lumley](#biblio),
2010, 2021). For regression analysis, the availability of the `survey` package
is **imperative**. It is assumed that the reader is <i>familiar</i> with the
key functions of the <code>survey</code> package, like
<code>svydesign()</code>, etc.

```{r, eval=FALSE}
library("robsurvey", quietly = TRUE)
library("survey")
```

```{r, echo = FALSE}
library(robsurvey, quietly = TRUE)
suppressPackageStartupMessages(library(survey))
```

**Notes.**

* The argument `quietly = TRUE` suppresses the start-up message in the call of
  `library("robsurvey")`.
* Since **version 4.2**, the **survey** package allows the definition of
  pre-calibrated weights (see argument `calibrate.formula` of the function
  `svydesign()`). This vignette uses this functionality (in some places). If
  you have installed an earlier version of the `survey` package, this vignette
  will automatically fall back to calling `svydesign()` without the calibration
  specification. See vignette [Pre-calibrated
  weights](https://CRAN.R-project.org/package=survey/vignettes/precalibrated.pdf)
  of the `survey` package to learn more.

```{r, echo = FALSE, results = "asis"}
survey_version <- packageVersion("survey")
if (survey_version < "4.2") {
cat(paste0('<div class="my-sidebar-orange">\n
<p style="color: #ce5b00;">
**IMPORTANT: PRE-CALIBRATED WEIGHTS ARE NOT SUPPORTED**
</p>
This vignette has been built with version **', survey_version,
'** of the **survey** package. Therefore, `svydesign()` is called without
the `calibrate.formula` argument. As a consequence, some of the variance and
standard error estimates may differ from those with pre-calibrated weights,
i.e., the default specification.<p>
</p>
</div>'))
}
```

1.1 Data

The *counties* dataset contains county-specific information on population,
number of farms, land area, etc. for a sample of n = 100 counties in the U.S.
in the 1990s. The sampling design is simple random sample (without replacement)
from the population of 3&#8239;141 counties. The dataset is tabulated in
[Lohr](#biblio) (1999, Appendix C) and is based on 1994 data of the U.S. Bureau
of the Census. The first 3 rows of the data are

```{r}
data(counties)
head(counties, 3)
```
where

|   |     |   |     |
| - | --- | - | --- |
| *state* | state | *county* | county |
| *landarea* | land area, 1990 (square miles) | *totpop* | population total, 1992 |
| *unemp* | number of unemployed persons, 1991 | *farmpop* | farm population, 1990 |
| *numfarm* | number of farms, 1987 | *farmacre* | acreage in farms, 1987 |
| *weights* | sampling weight | *fpc* | finite population correction

### 1.2 Goal

The goal is to regress the county-specific size of the farm population in 1990
(variable `farmpop`) on a set of explanatory variables. The simplest
*population* regression model is

$$\begin{equation}
\xi: \quad \mathrm{farmpop}_i = b_0 + b_1 \cdot \mathrm{numfarm}_i + \sqrt{v_i} E_i, \qquad i \in U,
\end{equation}$$

where $U$ is the set of labels of all 3&#8239;141 counties in the U.S.
(population), $b_0, b_1 \in \mathbb{R}$ are unknown regression coefficients,
$v_i$ are known constants (i.e., known up to a constant multiplicative factor,
$\sigma$), and $E_i$ are regression errors (random variables) with mean zero
and unit variance such that $E_i$ and $E_j$ are conditionally independent given
$\mathrm{numfarm}_i$, $i \in U$.

Another candidate model is

$$\begin{equation}
    \xi_{log}: \quad \log(\mathrm{farmpop}_i) = b_0 + b_1 \cdot
        \log(\mathrm{numfarm}_i) + \sqrt{v_i} E_i, \qquad i \in U.
\end{equation}$$

### 1.3 Sampling design

The counties dataset is of size n = 100. In what follows, we restrict attention
to the subset of the 98 counties with $\mathrm{farmpop}_i > 0$. Hence, we
define the sampling design `dn`.

```{r, eval = FALSE}
dn <- svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
                data = counties[counties$farmpop > 0, ],
                calibrate.formula = ~1)
```

```{r, echo = FALSE}
dn <- if (packageVersion("survey") >= "4.2") {
        svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
                  data = counties[counties$farmpop > 0, ],
                  calibrate.formula = ~1)
    } else {
        svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
                  data = counties[counties$farmpop > 0, ])
    }
```

### 1.4 Exploring the data

The figures below show scatterplots of `farmpop` plotted against `numfarm` (in
raw and log scale). The scatterplot in the left panel indicates that the
variance of `farmpop` increases with larger values of `numfarm`
(heteroscedasticity). The same graph but in log-log scale (see right panel)
shows an outlier, which is located far from the bulk of the data.

```{r, echo = FALSE, fig.show = "hold", out.width = "50%", fig.align = "default"}
plot(farmpop ~ numfarm, dn$variables, xlab = "numfarm", ylab = "farmpop",
     cex.axis = 1.2, cex.lab = 1.4)
plot(farmpop ~ numfarm, dn$variables, xlab = "numfarm (log)",
     ylab = "farmpop (log)", log = "xy", cex.axis = 1.2, cex.lab = 1.4)
points(farmpop ~ numfarm, dn$variables[3, ], pch = 19, col = 2)
```

<div class="my-sidebar-blue">
<p style="color: #1f618d;">
**Good to know.**
</p>
<p>
If the sampling weights were not all equal, it is helpful to use a bubble plot
with circles whose area is proportional to the sampling weight; see `svyplot()`
in the `survey` package.
</p>
</div>

## 2 Design-based inference

We consider fitting model $\xi$ (under the assumption of homoscedasticity,
i.e., $v_i \equiv 1$) by *weighted least squares*. The function to do so is
`svyreg()`, and it requires two arguments: a `formula` object (a symbolic
description of the model to be fitted) and survey`design` object (class
`survey::survey.design`).

```{r}
m <- svyreg(farmpop ~ numfarm, dn, na.rm = TRUE)
m
```

The output resembles the one of an `lm()` call. The estimated model can be
summarized using

```{r}
summary(m)
```

The subsequent figure shows the model diagnostic plot of the standardized
residuals vs. fitted values,

```{r, echo = FALSE, out.width = "50%"}
plot(m, which = 1)
```

Other useful methods and utility functions

* `coef()` extracts the estimated coefficients
* `residuals()` extracts the residuals
* `fitted()` extracts the fitted values
* `vcov()` extracts the variance-covariance matrix of the estimated coefficients

<div class="my-sidebar-blue">
<p style="color: #1f618d;">
**Good to know.**
</p>
<p>
Alternatively, linear models can be fitted by weighted least squares with the
<code>svyglm()</code> function in the <code>survey</code> package. The
<code>summary()</code> method for <code>survey::svyglm()</code> computes
<i>p</i>-values under the assumption of the Gaussian distribution, whereas our
method assumes a Student <i>t</i>-distribution.
</p>
</div>

### 2.1 Heteroscedasticity

In the above scatterplot, we observe that the variance of `farmpop` increases
with larger values of `numfarm` (heteroscedasticity). We conjecture that the
$v_i$'s in model $\xi$ are proportional to the square root of the variable
`numfarm`. Thus, we add variable `vi` to the design object `dn` with the
`update()` command

```{r}
dn <- update(dn, vi = sqrt(numfarm))
```

The argument `var` of `svyreg()` is now used to specify the heteroscedastic
variance (it can be defined as a `formula`, i.e. `~vi`, or the variable name in
quotation marks, `"vi"`).

```{r}
svyreg(farmpop ~ -1 + numfarm, dn, var = ~vi, na.rm = TRUE)
```

Note that we have dropped the regression intercept.

### 2.2 Regression *M*-estimator

We continue to assume the heteroscedastic model $\xi$, but now we compute a
weighed regression *M*-estimator. Two types of estimators are available:

* `svyreg_huberM()` *M*-estimator with Huber or asymmetric Huber
  $\psi$-function (the latter obtains by specifying argument `asym = TRUE`);
* `svyreg_tukeyM()` *M*-estimator with Tukey biweight (bisquare)
  $\psi$-function.

The tuning constant of both $\psi$-functions is called `k`. The Huber
*M*-estimator of regression with `k = 3` is (note that we use `na.rm = TRUE` to
remove observations with missing values)

```{r}
m <- svyreg_huberM(farmpop ~ -1 + numfarm, dn, var = ~vi, k = 1.3,
                   na.rm = TRUE)
m

```
and the summary obtains by

```{r}
summary(m)
```

The output of the summary method is almost identical with the one of an `lm()`
call. The paragraph on *robustness weights* is a numerical summary of the
*M*-estimator's robustness weights, which can be extracted from the estimated
model with `robweights()`.

The subsequent figure shows the graph for `plot(residuals(m), robweights(m))`.
From the graph, we can see by how much the residuals have been downweighted.

```{r, echo = FALSE, out.width = "50%"}
plot(residuals(m), robweights(m), cex.axis = 1.2, cex.lab = 1.4)
```

<div class="my-sidebar-orange">
<p style="color: #ce5b00;">
**IMPORTANT (failure of convergence)**
</p>
<p>
In the above example, the algorithm converged in 6 IRLWS (iteratively
reweighted least squares) iterations. In case the algorithm does not converge,
the default settings of the IRWLS algorithm can be modified. The following
settings can be specified/ changed (as function arguments) in the call of
<code>svyreg_huberM()</code> or <code>svyreg_tukeyM()</code>:

* `maxit = 50`: maximum number of iterations to use
* `tol = 1e-5`: numerical tolerance criterion to stop the iterations
* `init = NULL`: initialization of the regression *M*-estimator; if `init =
  NULL` the regression estimator is initialized by weighted least squares;
  otherwise, `init` can be specified as a starting value, i.e., a $p$-vector of
  regression coefficients.

See also `svyreg_control()`.
</p>
</div>

### 2.3 Regression *GM*-estimator

We want to fit the population regression model

$$
\begin{equation*}
    \log(\mathrm{farmpop}_i) = b_0 +  b_1 \log(\mathrm{numfarm}_i) + b_2
        \log(\mathrm{totpop}_i) + b_3 \log(\mathrm{farmacre}_i) + \sqrt{v_i}
        E_i, \qquad i \in U,
\end{equation*}
$$

with sample data by the weighted generalized *M*-estimator (GM) of regression.
*GM*-estimators are robust against high leverage observations (i.e., outliers
in the design space of the model) while *M*-estimators of regression are not.
Two types of *GM*-estimators are available:

* `svyreg_huberGM()` with Huber or asymmetric Huber $\psi$-function (the latter
  obtains by specifying argument `asym = TRUE`);
* `svyreg_tukeyGM()` with Tukey biweight (bisquare) $\psi$-function.

#### Preparation

Variable `farmacre` contains 3 missing values. For ease of analysis, we exclude
the missing values from the `survey.design` object `dn`.

<!-- 2023-12-31: Design without calibrate.formula argument because survey pkg
has a bug in na.excluce and na.omit for is.calibrated designs -->

```{r}
dn <- svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
                data = counties[counties$farmpop > 0, ])
dn_exclude <- na.exclude(dn)
```

The formula object of our model is

```{r}
f <- log(farmpop) ~ log(numfarm) + log(totpop) + log(farmacre)
```

With the help of the formula object `f`, we form a matrix of the
**explanatory** variables (excluding the regression intercept) of our model.
This matrix is called `xmat`.

```{r}
xmat <- model.matrix(f, dn_exclude$variables)[, -1]
```

Below we show the pairwise scatterplots of `pairs(xmat)`. The pairwise
distributions are mostly elliptically contoured and show some outliers.

```{r, echo = FALSE, out.width = "50%"}
pairs(xmat)
```

#### Robust Mahalanobis distances

The intermediate goal is to compute the robust Mahalanobis distances of the
observations in `xmat`. Observations with large distances are considered
outliers and will be downweighted in the subsequent regression analysis. In
order to obtain the robust Mahalanobis distances, we compute the robust
multivariate location and scatter matrix of `xmat` using the (weighted) BACON
algorithm in package `wbacon` ([Schoch](#biblio), 2021), and extract the robust
Mahalanobis distances `dis`. (If package `wbacon` is not available, the robust
center and scatter matrix are given, such that the Mahalanobis distances can be
computed.)

```{r}
if (requireNamespace("wbacon", quietly = TRUE)) {
    # package wbacon is available
    mv <- wbacon::wBACON(xmat, weights = weights(dn_exclude))
    # distances
    dis <- wbacon::distance(mv)
} else {
    # package wbacon is not available
    center <- c(6.285968, 10.195002, 12.047715)
    scatter <- matrix(0, 3, 3)
    scatter[lower.tri(scatter, TRUE)] <- c(0.678646, 0.441020, 0.415634,
                                           2.191174, -0.302097, 1.040932)
    scatter <- scatter + t(scatter) - diag(3) * scatter
    # distances
    dis <- sqrt(mahalanobis(xmat, center, scatter))
}
```

The boxplot of `dis` shows a couple of observations with large Mahalanobis
distance. These observations are considered as potential outliers.

```{r, echo = FALSE, out.width = "50%", fig.asp = 0.5}
boxplot(dis, horizontal = TRUE, xlab = "dis")
```

Three functions are available to downweight excessively large distances (see
also figure)

* `tukeyWgt()`
* `huberWgt()`
* `simpsonWgt()`, see [Simpson et al.](#biblio) (1992)

```{r, echo = FALSE, fig.show = "hold", out.width = "33%", fig.align = "default"}
x <- seq(0, 6, length = 500)
cex_axis <- 1.5; cex_lab <- 1.5; cex_main <- 1.8
plot(x, huberWgt(x, k = 2), type = "l", xlab = "x", ylab = "",
     cex.axis = cex_axis, cex.lab = cex_lab, main = "huberWgt(x, k = 2)",
     cex.main = cex_main, ylim = c(0, 1))
plot(x, tukeyWgt(x, k = 4), type = "l", xlab = "x", ylab = "",
     cex.axis = cex_axis, cex.lab = cex_lab, main = "tukeyWgt(x, k = 4)",
     cex.main = cex_main, ylim = c(0, 1))
plot(x, simpsonWgt(x, Inf, 4), type = "l", xlab = "x", ylab = "",
     cex.axis = cex_axis, cex.lab = cex_lab,
     main = "simpsonWgt(x, a = Inf, b = 4)", cex.main = cex_main,
     ylim = c(0, 1))
```

#### Regression estimation and inference

The *GM*-estimator of regression with Tukey biweight function and downweighting
of high leverage observations using `tukeyWgt(dis)` is

```{r}
m <- svyreg_tukeyGM(f, dn_exclude, k = 4.6, xwgt = tukeyWgt(dis))
summary(m)
```

## 3 Model-based inference

Model-based inferential statistics are computed using a *dummy* `survey.design`
object under the assumption of a simple random sample without replacement. We
define

```{r}
dn0 <- svydesign(ids = ~1, weights = ~1,
                 data = counties[counties$farmpop > 0, ],
                 calibrate.formula = ~1)
```

which differs from our original design `dn` in the following aspects:

* `weights = ~1` (i.e., the sampling weights are set to 1.0);
* the argument `fpc` (finite sampling correction) is not specified.

We consider fitting model $\xi_{log}$ by the Huber regression *M*-estimator
(based on the design object `dn0`)

```{r}
m <- svyreg_huberM(log(farmpop) ~ log(numfarm), dn0, k = 1.3, na.rm = TRUE)
```

In the call of `summary()`, we specify the argument `mode = "model"` for
model-based inference (default is `mode = "design"`) to obtain

```{r}
summary(m, mode = "model")
```

Likewise, we can call `vcov(m, mode = "model")` to extract the estimated
*model-based* covariance matrix of the regression estimator.

<div class="my-sidebar-blue">
<p style="color: #1f618d;">
**Good to know** (comparison with function <code>rlm</code> in package
<code>MASS</code>).
</p>
<p>
For the purpose of comparison, we have fitted the above model with the function
<code>rlm()</code> in package <code>MASS</code> ([Venables and
Ripley](#biblio), 2002) with tuning constant <code>k = 1.3</code>.

```{r, echo=FALSE}
if (requireNamespace("MASS", quietly = TRUE)) {
    summary(MASS::rlm(log(farmpop) ~ log(numfarm),
                      data = counties[counties$farmpop > 0, ],
                      k = 1.3, na.action = na.omit))
} else {
    cat("Package MASS is not available\n")
}
```
</p>
</div>

## 4 Compound design- and model-based inference

Suppose we wish to estimate the regression coefficients that refer to the
superpopulation (i.e., a more general population than the finite population).
Furthermore, our sample data represent a large fraction of the finite
population (i.e., $n/N$ is not small). In statistical inference about the
coefficients, we need to account for the sampling design and the model because
the quantity of interest is the superpopulation parameter and the sampling
fraction is not negligible ([Rubin-Bleuer and Schiopu-Kratina, 2005; Binder and
Roberts, 2009](#biblio)); see also motivating example.

<div class="my-sidebar-blue">
<p style="color: #1f618d;">
**Motivating example**
</p>
<p>
Let $\bar{y}$ be the mean of a sample of size $n$. For ease of discussion, we
shall assume a simple random sample (without replacement) from a finite
population of size $N$. The mean of the finite population is denoted by
$\bar{Y}$. Here, we wish to refer to <i>a more general population</i> than the
finite population. Therefore, $\bar{y}$ is regarded as an estimator of the
super-population mean $\mu$ with the following decomposition


$$
\sqrt{n} (\bar{y} - \mu) = \sqrt{n}(\overline{y} - \overline{Y}) +
    \sqrt{\frac{n}{N}} \cdot \sqrt{N}(\overline{Y} - \mu).
$$

If the sampling fraction, $n/N$, is small, the second term on the right-hand
side is negligible regarding the limiting distribution of $\sqrt{n}(\bar{y} -
\mu)$---therefore design-based inference is only concerned with
$\sqrt{n}(\bar{y} - \bar{Y})$. If, however, the sampling fraction is not small,
statistical inference about $\mu$ should take the limiting distribution of
$\sqrt{N}(\bar{Y} - \mu)$ into account.
</p>
</div>

### 4.1 The MU284 population

The MU284 population of [Särndal et al.](#biblio) (1992, Appendix B) is a
dataset with observations on the 284 municipalities in Sweden in the late 1970s
and early 1980s. It is available in the `sampling` package; see [Tillé and
Matei](#biblio) (2021). The population is divided into two parts based on 1975
population size (`P75`):

* the MU281 population, which consists of the 281 smallest municipalities;
* the MU3 population of the three biggest municipalities/ cities in Sweden
  (Stockholm, Göteborg, and Malmö).

The three biggest cities take exceedingly large values (representative
outliers) on almost all of the variables. To account for this (at least to some
extent), a stratified sample has been drawn from the MU284 population using a
take-all stratum. The sample data, `MU284strat`, is of size $n=60$ and consists
of

* a stratified simple random sample (without replacement) from the MU281
  population, where stratification is by geographic region (REG) with
  proportional sample size allocation;
* a take-all stratum that includes the three biggest cities/ municipalities
  (population M3).

| | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| *Stratum* | $S_1$ | $S_2$ | $S_3$ | $S_4$ | $S_5$ | $S_6$ | $S_7$ | $S_8$ | *take all* |
| *Population stratum size* | 24 | 48 | 32 | 37 | 55 | 41 | 15 | 29 | 3 |
| *Sample stratum size* | 5 | 10 | 6 | 8 | 11 | 8 | 3 | 6 | 3 |
| | | | | | | | | | |

The overall sampling fraction is $60/284 \approx 21\%$ (and thus not
negligible). The data frame `MU284strat` includes the following variables.

| | | |
| - | ---- | - | ---- |
*LABEL* | identifier variable | *P85* | 1985 population size (in $10^3$) |
*P75* | 1975 population size (in $10^3$) | *RMT85* | revenues from the 1985 municipal taxation (in $10^6$ kronor) |
*CS82* | number of Conservative seats in municipal council | *SS82* | number of Social-Democrat seats in municipal council (1982) |
*S82* | total number of seats in municipal council in 1982 | *ME84* | number of municipal employees in 1984 |
*REV84* | real estate values in 1984 (in $10^6$ kronor) | *CL* | cluster indicator |
*REG* | geographic region indicator | *Stratum* | stratum indicator |
*weights* | sampling weights | *fpc* | finite population correction |

We load the data and generate the sampling design.

```{r}
data(MU284strat)
dn <- svydesign(ids = ~1, strata = ~ Stratum, fpc = ~fpc, weights = ~weights,
                data = MU284strat, calibrate.formula = ~-1 + Stratum)
```

### 4.2 Model fit and statistical inference

The Huber *M*-estimator of regression is

```{r}
m <- svyreg_huberM(RMT85 ~ REV84 + P85 + S82 + CS82, dn, k = 2)
```

The compound design- and model-based distribution of the estimated coefficients
obtains by specifying the argument `mode = "compound"` in the call of the
`summary()` method.

```{r}
summary(m, mode = "compound")
```

We may call `vcov(m, mode = "compound")` to extract the estimated covariance
matrix of the regression estimator under the compound design- and model-based
distribution.

---

## References {#biblio}

BINDER, D. A. AND ROBERTS, G. (2009). Design- and Model-Based Inference for
Model Parameters. In: *Sample Surveys: Inference and Analysis* ed. by
Pfeffermann, D. and Rao, C. R. Volume 29B of *Handbook of Statistics*,
Amsterdam: Elsevier, Chap. 24, 33–54. [DOI: 10.1016/
S0169-7161(09)00224-7](https://doi.org/10.1016/S0169-7161(09)00224-7)

LOHR, S. L. (1999). *Sampling: Design and Analysis*, Pacific Grove (CA):
Duxbury Press.

LUMLEY, T. (2010). *Complex Surveys: A Guide to Analysis Using R: A Guide to
Analysis Using R*, Hoboken (NJ): John Wiley & Sons.

LUMLEY, T. (2021). survey: analysis of complex survey samples. R package
version 4.0, URL https://CRAN.R-project.org/package=survey.

RUBIN-BLEUER, S. AND SCHIOPU-KRATINA, I. (2005). On the Two-phase framework for
joint model and design-based inference. *The Annals of Statistics* **33**,
2789–2810. [DOI: 10.1214/
009053605000000651](https://doi.org/10.1214/009053605000000651)

SÄRNDAL, C.-E., SWENSSON, B. AND WRETMAN, J. (1992). *Model Assisted Survey
Sampling*, New York: Springer-Verlag.

SCHOCH, T. (2021). wbacon: Weighted BACON algorithms for multivariate outlier
nomination (detection) and robust linear regression. *Journal of Open Source
Software* **6**, 3238. [DOI: 10.
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