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.
First, we load the packages robsurvey
and survey
(Lumley,
2010, 2021). For regression analysis, the availability of the
survey
package is imperative. It is
assumed that the reader is familiar with the key functions of the
survey
package, like svydesign()
, etc.
Notes.
quietly = TRUE
suppresses the start-up
message in the call of library("robsurvey")
.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 of the survey
package to learn more.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 3 141 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 weights
1 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
fpc
1 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
.
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 surveydesign
object (class survey::survey.design
).
> m <- svyreg(farmpop ~ numfarm, dn, na.rm = TRUE)
> m
Weighted least squares
Call:
svyreg(formula = farmpop ~ numfarm, design = dn, na.rm = TRUE)
Coefficients:
(Intercept) numfarm
-121.310 1.921
Scale estimate: 563.4
The output resembles the one of an lm()
call. The
estimated model can be summarized using
> summary(m)
Call:
svyreg(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
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -121.3105 93.7991 -1.293 0.196
numfarm 1.9215 0.1934 9.936 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 563.4 on 3076.18 degrees of freedom
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 coefficientsresiduals()
extracts the residualsfitted()
extracts the fitted valuesvcov()
extracts the variance-covariance matrix of the
estimated coefficientsIn 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
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
Call:
svyreg(formula = farmpop ~ -1 + numfarm, design = dn, var = ~vi,
na.rm = TRUE)
Coefficients:
numfarm
1.775
Scale estimate: 99.65
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
Survey regression M-estimator (Huber psi, k = 1.3)
Call:
svyreg_huberM(formula = farmpop ~ -1 + numfarm, design = dn,
k = 1.3, var = ~vi, na.rm = TRUE)
IRWLS converged in 6 iterations
Coefficients:
numfarm
1.669
Scale estimate: 83.35 (weighted MAD)
and the summary obtains by
> summary(m)
Call:
svyreg_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
Coefficients:
Estimate Std. Error t value Pr(>|t|)
numfarm 1.66859 0.09758 17.1 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 83.35 on 3077.18 degrees of freedom
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 <- svydesign(ids = ~1, fpc = ~fpc, weights = ~weights,
+ data = counties[counties$farmpop > 0, ])
> dn_exclude <- na.exclude(dn)
The formula object of our model is
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
.
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)
Call:
svyreg_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
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.62725 0.56957 2.857 0.00431 **
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 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5549 on 2979.95 degrees of freedom
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, ],
+ 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);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
)
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")
Call:
svyreg_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
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.13991 0.29129 -0.48 0.632
log(numfarm) 1.08916 0.04663 23.36 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4934 on 96 degrees of freedom
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 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
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.
The Huber M-estimator of regression is
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")
Call:
svyreg_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
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 130.973688 22.260303 5.884 1.15e-08 ***
REV84 0.005729 0.003304 1.734 0.0840 .
P85 9.499801 0.285577 33.265 < 2e-16 ***
S82 -3.845513 0.542752 -7.085 1.14e-11 ***
CS82 -1.965082 1.010420 -1.945 0.0528 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 23.96 on 279 degrees of freedom
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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