In this vignette, we discuss how to use robust estimators of location (and scale). The estimators are organized by i) estimating method, 2) population characteristic and 3) type of implemented function.
Barebone methods are strippeddown versions of the survey methods in terms of functionality and informativeness. These functions may serve users and package developers as building blocks. In particular, barebone functions cannot compute variances. The survey methods are much more capable and depend—for variance estimation—on the R package survey
Lumley (2021, 2010).
The losdata
are a simple random sample without replacement of n = 70 patients from a (fictive) hospital population of N = 2 479 patients in inpatient treatment.^{Note 2} First, we load the package and the data.
> library("robsurvey", quietly = TRUE)
> data("losdata")
> attach(losdata)
The first 3 rows of the data are:
> head(losdata, 3)
los weight fpc1 10 34.91549 2479
2 7 34.91549 2479
3 21 34.91549 2479
where
los
lengthofstay in hospital (days)weight
sampling weightfpc
population size (finite population correction)We consider estimating average lengthofstay in hospital (LOS, days).
For the survey methods (not barebone methods), we must load the survey
package (Lumley, 2010, 2021)
> library("survey")
and specify a survey or sampling design object
> dn < svydesign(ids = ~1, fpc = ~fpc, weights = ~weight, data = losdata)
The distribution of variable los is skewed to the right (see boxplot), and we see a couple of rather heavy outliers. On the logarithmic scale, the distribution is slightly skewed to the right. The outliers need not be errors. Following Chambers (1986), we distinguish representative outliers from nonrepresentative outliers.
The outliers visible in the boxplot refer to a few individuals who stayed for a long time in inpatient care. Moreover, we assume that these outliers represent patients in the population that are similar in value (i.e., representative outliers).
The outliers tend to inflate the variance of the weighted mean. Although the outliers are not some kind of error, it is beneficiary to use estimators other than the weighted mean to estimate the population average of los. We are interested in robust estimators which—although being biased as estimators of the population mean—will often have a smaller mean square error than the weighted mean; thus, are more efficient.
The following estimation methods are available.
weighted_mean_trimmed()

weighted_total_trimmed()

In place of the weighted mean, we consider the 5% onesided trimmed weighted population mean of los. The lower end of the distribution is not trimmed (lower bound: LB = 0
). The 5% largest observations are trimmed (upper bound: UB = 0.95
).
> weighted_mean_trimmed(los, weight, LB = 0, UB = 0.95)
1] 9.323529 [
We obtain an estimate of (roughly) 9.3 days. Note that the return value is a scalar.
med (upper bound: UB = 0.95
).
If a barebone method is called with argument info = TRUE
, the function returns a list, the names of which are shown below.
> m < weighted_mean_trimmed(los, weight, LB = 0, UB = 0.95, info = TRUE)
> names(m)
1] "characteristic" "estimator" "estimate" "variance"
[5] "residuals" "model" "design" "call" [
The survey methods are:
svymean_trimmed()

svytotal_trimmed()

As before, we are interested in computing the 5% onesided trimmed weighted population mean of los. In contrast to weighted_mean_trimmed()
, the method svymean_trimmed()
computes the standard error of the estimate using the functionality of the survey
package.
> m < svymean_trimmed(~los, dn, LB = 0, UB = 0.95)
> m
mean SE9.324 1.064 los
The estimated location, variance, and standard error of the estimator can be extracted from object m
with the following commands.
> coef(m)
los 9.323529
> vcov(m)
Variance1.131988
los > SE(m)
1] 1.063949 [
The summary()
method summarizes the most important facts about the estimate. The summary is particularly useful for Mestimators (see below) but less so for other estimators.
> summary(m)
estimator (0, 0.95) of the population mean
Weighted trimmed
mean SE9.324 1.064
los
:
Sampling design
Independent Sampling designsvydesign(ids = ~1, fpc = ~fpc, weights = ~weight, data = losdata)
Additional utility functions are residuals()
, fitted()
, and robweights()
. These functions are mainly relevant for Mestimators.
The barebone methods are:
weighted_mean_winsorized()

weighted_total_winsorized()

weighted_mean_k_winsorized()

weighted_total_k_winsorized()

We compute the 5% onesided winsorized weighted population mean of los. The lower end of the distribution is not winsorized (lower bound: LB = 0
). The 5% largest observations are winsorized (upper bound: UB = 0.95
).
> weighted_mean_winsorized(los, weight, LB = 0, UB = 0.95)
1] 10.40845 [
The onesided winsorized estimators can also be specified in absolute terms by winsorizing a fixed number \(k=1,2,\ldots\) of observations. This estimator is called the onesided kwinsorized mean (and total) and is computed as follows
> weighted_mean_k_winsorized(los, weight, k = 1)
1] 11.40845 [
The survey methods are:
svymean_winsorized()

svytotal_winsorized()

svymean_k_winsorized()

svytotal_k_winsorized()

The utility functions coef()
, vcov()
, SE()
, summary()
, residuals()
, fitted()
, and robweights()
are available.
Winsorization and trimming act directly on the values of an estimator. Other estimation methods reduce the sampling weight of potential outliers instead. A hybrid method of winsorization and weight reduction to treat influential observations has been proposed by Dalén (1987). An observation \(y_i\) is called influential if its expanded value, \(w_iy_i\), is exceedingly large. Let \(c>0\) denote a winsorization or censoring cutoff value. Dalén’s estimator "Z2"
and "Z3"
of the population \(y\)total are given by \(\sum_{i \in s} [w_i y_i]_{\circ}^c\), where \(\circ\) is a placeholder for "Z2"
or "Z3"
and \[
\begin{align*}
[w_i y_i]_{Z2}^c =
\begin{cases}
w_i y_i & \text{if} \quad w_i y_i \leq c, \\
c & \text{otherwise},
\end{cases}
&\qquad \text{and} \qquad
[w_i y_i]_{Z3}^c =
\begin{cases}
w_i y_i & \text{if} \quad w_i y_ \leq c, \\
c + (y_i  c/w_i) & \text{otherwise}.
\end{cases}
\end{align*}
\]
Estimator "Z2"
censors the terms \(w_iy_i\) at \(c\). In estimator "Z3"
, observations \(y_i\) such that \(w_iy_i > c\) contribute to the estimated total only with \(c\) plus the excess over the cutoff, \((w_iy_i  c)\). Note that the excess over the threshold has a weight of 1.0 (Lee, 1995). An estimator of the population \(y\)mean obtains by dividing the estimator of the estimated \(y\)total by the (estimated) population size.
From a practical point of view, the choice of constant \(c\) in Dalén’s estimators is rather tricky because we cannot only derive \(c\) from a large order statistic, say \(y_{(k)}\), \(k < n\) (like for trimming). Instead, the corresponding weight \(w_{(k)}\) needs to be taken into account.
The barebone methods are:
weighted_mean_dalen()

weighted_total_dalen()

The estimators "Z2"
and "Z3"
can be specified by the argument type
; by default, type = "Z2"
. The censoring threshold \(c\) is implemented as argument censoring
.
> weighted_mean_dalen(los, weight, censoring = 1500)
2 of 71 observations censored
1] 10.73129 [
The survey methods are:
svymean_dalen()

svytotal_dalen()

The utility functions coef()
, vcov()
, SE()
, summary()
, residuals()
, fitted()
, and robweights()
are available.
The barebone methods are:
weighted_mean_huber()

weighted_total_huber()

weighted_mean_tukey()

weighted_total_tukey()

The estimators with postfix _huber
and _tukey
are based on, respectively, the Huber and Tukey (biweight) \(\psi\)function.
As losdata is a simple random sample, Mestimators of type = "rht"
is the method of choice. Here, we compute the Hubertype robust weighted Mestimator of the mean with robustness tuning constant \(k=8\).
> weighted_mean_huber(los, weight, type = "rwm", k = 8)
1] 11.17228 [
The function huber2()
is an implementation of the weighted Huber proposal 2 estimator. It is only available as barebone method.^{Note 3}
> huber2(los, weight, k = 8)
1] 13.02817 [
The survey methods are:
svymean_huber()

svytotal_huber()

svymean_tukey()

svytotal_tukey()

The Huber Mestimator of the mean (and its standard error) can be computed with
> m < svymean_huber(~los, dn, type = "rwm", k = 8)
> m
mean SE11.17 1.328 los
The summary()
method summarizes the most important facts about the Mestimate
> summary(m)
estimator (type = rwm) of the population mean
Huber M
mean SE11.17 1.328
los
:
Robustnessfunction: with k = 8
Psi: 0.9877
mean of robustness weights
:
Algorithm performancein 4 iterations
converged scale (weighted MAD): 5.93
with residual
:
Sampling design
Independent Sampling designsvydesign(ids = ~1, fpc = ~fpc, weights = ~weight, data = losdata)
The estimated scale (weighted MAD) can be extracted with the scale()
function. Additional utility functions are coef()
, vcov()
, SE()
, residuals()
, fitted()
, and robweights()
. The following figure shows a plot of the robustness weights against the residuals. We see that large residuals are downweighted.
> plot(residuals(m), robweights(m))
An adaptive Mestimator of the total (or mean) is defined by letting the data chose the tuning constant \(k\). Let \(\widehat{T}\) denote the weighted total, and let \(\widehat{T}_k\) be the Huber Mestimator of the weighted total with robustness tuning constant \(k\). Under quite general regularity conditions, the estimated mean square error (MSE) of \(\widehat{T}_k\) can be approximated by (see e.g., Gwet and Rivest, 1992; Hulliger, 1995)
\[\widehat{\mathrm{mse}}\big(\widehat{T}_{k}\big) \approx \mathrm{var}\big(\widehat{T}_{k}\big) +\big(\widehat{T}  \widehat{T}_{k}\big)^2.\]
The minimum estimated risk (MER) estimator (Hulliger, 1995) selects \(k\) such that \(\widehat{\mathrm{mse}}\big(\widehat{T}_{k}\big)\) is minimal (among all candidate estimators). Now, suppose that we have been working on the Mestimator with \(k=8\).
> m < svymean_huber(~los, dn, type = "rwm", k = 8)
Next, we compute the MER, starting from the current Mestimate, i.e., object m
.
> mer(m)
: [1, 10]
Search intervalfor k = 10
Minimum found : 35%
Rel. efficiency gain
mean SE11.46 1.478 los
Hence, the MER is 35% more efficient than the classical estimator as an estimator of the population total.
The weighted quantile (and median) can be computed by
> weighted_quantile(los, weight, probs = c(0.1, 0.9))
10% 90%
3 22
> weighted_median(los, weight)
50%
8
When all weights are equal, weighted_quantile()
is equal to base::quantile()
with argument type = 2
.
The normalized weighted median absolute deviations about the weighted median can be computed with
> weighted_mad(los, weight)
1] 5.930408 [
By default, the normalization constant to make the weighted MAD an unbiased estimator of scale at the Gaussian core model is constant = 1.482602
. This constant can be changed if necessary.
The normalized weighted interquartile range can be computed with
> weighted_IQR(los, weight)
1] 6.6717 [
By default, the normalization constant to make the weighted IQR an unbiased estimator of scale at the Gaussian core model is constant = 0.7413
. This constant can be changed if necessary.
^{1} All barebone methods can be called with the argument info = TRUE
to return a list with the following entries: characteristic (e.g., mean), estimator (e.g., trimmed estimator), estimate (numerical value), variance (by default: NA
), robust (list of arguments that specify robustness), residuals (numerical vector), model (list of data used for estimation), design (by default: NA
), call.
^{2} We have constructed the losdata
as a showcase; though, the LOS measurements are real data that we have taken from the \(201\) observations in Ruffieux et al. (2000). Our losdata
are a sample of size \(n = 70\) from the \(201\) original observations.
^{3} The function huber2()
is is similar to MASS::hubers()
(Venables and Ripley, 2002). It differs from the implementation in MASS
in that it allows for weights and is initialized by the (normalized) weighted interquartile range (IQR) not the median absolute deviations (MAD).
The paper of Chambers (1986) is the landmark paper about outliers in finite population sampling. Lee (1995) and Beaumont and Rivest (2009) are a good starting point to learn about robustness in finite population sampling.
Trimming and winsorization are discussed in Lee (1995) and Beaumont and Rivest (2009). The variance estimators of the weighted trimmed and winsorized estimators are straightforward adaptions of the classical estimators; see Huber and Ronchetti (2009, Chap. 3.3) or Serfling (1980, Chap. 8). A rigorous treatment in the context of finite population sampling can be found in Shao (1994).
Rao (1971) was among the first to propose weight reduction. Consider a sample of size \(n\), and suppose that the \(i\)th observation is an outlier. He suggested to reduce the outlier’s sampling weight \(w_i\) to one, and redistribute the weight difference \(w_i−1\) among the remaining observations. As a result, observation \(i\) does not represent other values like it. Dalén’s estimator offers a more general notion of weight reducution; see Dalén (1987) and also Chen et al. (2017).
In the context of finite population sampling, Mestimators were first studied by Chambers (1986). He investigated robust methods in the model or prediction based framework of Royall and Cumberland (1981). Modelassisted estimators were introduced (for ratio estimation) by Gwet and Rivest (1992) and studied by Lee (1995), and Hulliger (1995, 1999, 2005). A recent comprehensive treatment can be found in Beaumont and Rivest (2009).
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