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)

First, we load the packages `robsurvey`

**and** `survey`

(Lumley, 2010, 2021). For regression analysis, the availability of the `survey`

package is **imperative**.

```
> library("robsurvey", quietly = TRUE)
> library("survey")
```

*Note.* The argument `quietly = TRUE`

suppresses the start-up message in the call of `library("robsurvey")`

.

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 3 141 counties. The dataset is tabulated in Lohr (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

```
> data(counties)
> head(counties, 3)
state county landarea totpop unemp farmpop numfarm farmacre weights1 AL Escambia 948 36023 1339 531 414 90646 31.41
2 AL Marshall 567 73524 3189 1592 1582 136599 31.41
3 AK Prince of Wales 7325 6408 383 71 2 214 31.41
fpc1 3141
2 3141
3 3141
```

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 |

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 3 141 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

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

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`

.

```
> dn <- svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
+ data = counties[counties$farmpop > 0, ])
```

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.

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`

).

```
> m <- svyreg(farmpop ~ numfarm, dn, na.rm = TRUE)
> m
Weighted least squares
:
Callsvyreg(formula = farmpop ~ numfarm, design = dn, na.rm = TRUE)
:
Coefficients
(Intercept) numfarm -121.310 1.921
: 563.4 Scale estimate
```

The output resembles the one of an `lm()`

call. The estimated model can be summarized using

```
> summary(m)
:
Callsvyreg(formula = farmpop ~ numfarm, design = dn, na.rm = TRUE)
:
Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max. -1381.71 -309.50 -13.04 0.00 181.30 2637.47
:
CoefficientsPr(>|t|)
Estimate Std. Error t value -121.3105 93.7991 -1.293 0.196
(Intercept) 1.9215 0.1934 9.936 <2e-16 ***
numfarm ---
: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Signif. codes
: 563.4 on 3076.18 degrees of freedom Residual standard error
```

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

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

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

`> 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"`

).

```
> svyreg(farmpop ~ -1 + numfarm, dn, var = ~vi, na.rm = TRUE)
Weighted least squares
:
Callsvyreg(formula = farmpop ~ -1 + numfarm, design = dn, var = ~vi,
na.rm = TRUE)
:
Coefficients
numfarm 1.775
: 99.65 Scale estimate
```

Note that we have dropped the regression intercept.

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)

```
> m <- svyreg_huberM(farmpop ~ -1 + numfarm, dn, var = ~vi, k = 1.3, na.rm = TRUE)
> m
-estimator (Huber psi, k = 1.3)
Survey regression M
:
Callsvyreg_huberM(formula = farmpop ~ -1 + numfarm, design = dn,
k = 1.3, var = ~vi, na.rm = TRUE)
in 6 iterations
IRWLS converged
:
Coefficients
numfarm 1.669
: 83.35 (weighted MAD) Scale estimate
```

and the summary obtains by

```
> summary(m)
:
Callsvyreg_huberM(formula = farmpop ~ -1 + numfarm, design = dn,
k = 1.3, var = ~vi, na.rm = TRUE)
:
Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max. -165.951 -64.126 -10.095 6.848 47.253 458.062
:
CoefficientsPr(>|t|)
Estimate Std. Error t value 1.66859 0.09758 17.1 <2e-16 ***
numfarm ---
: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Signif. codes
: 83.35 on 3077.18 degrees of freedom
Residual standard error
:
Robustness weights
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.2369 1.0000 1.0000 0.9294 1.0000 1.0000
```

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.

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.

Variable `farmacre`

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

object `dn`

.

`> dn_exclude <- na.exclude(dn)`

The formula object of our model is

`> 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`

.

`> 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.

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, 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.)

```
> 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.

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

`tukeyWgt()`

`huberWgt()`

`simpsonWgt()`

, see Simpson et al. (1992)

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

is

```
> m <- svyreg_tukeyGM(f, dn_exclude, k = 4.6, xwgt = tukeyWgt(dis))
> summary(m)
:
Callsvyreg_tukeyGM(formula = f, design = dn_exclude, k = 4.6, xwgt = tukeyWgt(dis))
:
Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max. -1.004303 -0.352241 -0.052866 0.004878 0.360099 3.389738
:
CoefficientsPr(>|t|)
Estimate Std. Error t value 1.62725 0.56957 2.857 0.00431 **
(Intercept) log(numfarm) 1.30079 0.07044 18.466 < 2e-16 ***
log(totpop) -0.06550 0.03176 -2.062 0.03929 *
log(farmacre) -0.20161 0.04617 -4.366 1.31e-05 ***
---
: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Signif. codes
: 0.5549 on 2979.95 degrees of freedom
Residual standard error
:
Robustness weights
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0000 0.6270 0.7850 0.7114 0.8818 0.9959
```

Model-based inferential statistics are computed using a *dummy* `survey.design`

object under the assumption of a simple random sample without replacement. We define

```
> dn0 <- svydesign(ids = ~1, weights = ~1,
+ data = counties[counties$farmpop > 0, ])
```

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`

)

`> 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

```
> summary(m, mode = "model")
:
Callsvyreg_huberM(formula = log(farmpop) ~ log(numfarm), design = dn0,
k = 1.3, na.rm = TRUE)
:
Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max. -1.203816 -0.314133 0.007801 0.003245 0.335280 3.647648
:
CoefficientsPr(>|t|)
Estimate Std. Error t value -0.13991 0.29129 -0.48 0.632
(Intercept) log(numfarm) 1.08916 0.04663 23.36 <2e-16 ***
---
: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Signif. codes
: 0.4934 on 96 degrees of freedom
Residual standard error
:
Robustness weights
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.1758 1.0000 1.0000 0.9557 1.0000 1.0000
```

Likewise, we can call `vcov(m, mode = "model")`

to extract the estimated *model-based* covariance matrix of the regression estimator.

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); see also motivating example.

The MU284 population of Särndal et al. (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 (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.

```
> data(MU284strat)
> dn <- svydesign(ids = ~1, strata = ~ Stratum, fpc = ~fpc, weights = ~weights,
+ data = MU284strat)
```

The Huber *M*-estimator of regression is

`> 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.

```
> summary(m, mode = "compound")
:
Callsvyreg_huberM(formula = RMT85 ~ REV84 + P85 + S82 + CS82, design = dn,
k = 2)
:
Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max. -77.877 -18.469 5.044 66.250 17.635 2690.755
:
CoefficientsPr(>|t|)
Estimate Std. Error t value 130.973688 22.260303 5.884 1.15e-08 ***
(Intercept) 0.005729 0.003304 1.734 0.0840 .
REV84 9.499801 0.285577 33.265 < 2e-16 ***
P85 -3.845513 0.542752 -7.085 1.14e-11 ***
S82 -1.965082 1.010420 -1.945 0.0528 .
CS82 ---
: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Signif. codes
: 23.96 on 279 degrees of freedom
Residual standard error
:
Robustness weights
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.01781 1.00000 1.00000 0.95076 1.00000 1.00000
```

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.

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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.

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

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