In this vignette, we discuss the robust Horvitz-Thompson (RHT) estimator of Hulliger (1995, 1999). The vignette is organized as follows.
First, we load the package.
> library("robsurvey", quietly = TRUE)
workplace sample consists of payroll data on n = 142 workplaces or business establishments (with paid employees) in the retail sector of a Canadian province.
workplacedata are not those collected by Statistics Canada but have been generated by Fuller (2009, Example 3.1.1, Table 6.3).
The original weights of WES were about 2200 for the stratum of small workplaces, about 750 for medium-sized, and about 35 for large workspaces. Several strata containing very large workplaces were sampled exhaustively; see Patak et al. (1998).
> attach(workplace) : The following objects are masked from losdata fpc, weight
The variable of interest is
payroll, and the goal is to estimate the population payroll total in the retail sector (in Canadian dollars).
> head(workplace, 3) ID weight employment payroll strat fpc1 1 786 17 260000 2 10718 2 2 32 661 6873000 1 3432 3 3 36 3 366000 1 3432
For the survey methods (not bare-bone methods), we must load the
survey package (Lumley, 2010, 2021)
and specify a survey or sampling design object
> dn <- svydesign(ids = ~ID, strata = ~strat, fpc = ~fpc, weights = ~weight, data = workplace)
which appears (on print) with the following output in the console
Stratified Independent Sampling design svydesign(ids = ~ID, strata = ~strat, fpc = ~fpc, weights = ~weight, data = workplace)
To get a first impression of the distribution of
payroll, we examine two (design-weighted) boxplots of
payroll (on raw and logarithmic scale). The boxplots are obtained using function
From the boxplot with
payroll on raw scale, we recognise that the sample distribution of
payroll is skewed to the right; the boxplot on logarithmic scale demonstrates that log-transform is not sufficient to turn the skewed distribution into a symmetric distribution. The outliers need not be errors. Following Chambers (1986), we distinguish representative outliers from non-representative outliers (\(\rightarrow\) see vignette “Basic Robust Estimators” for an introduction to the notion of non-/ representative outliers).
The outliers visible in the boxplot refer to a few large workplaces. Moreover, we assume that these outliers represent other workplaces in the population that are similar in value (i.e., representative outliers).
The following bare-bone estimating methods are available:
The functions with postfix
_tukey are M-estimators with the Tukey biweight \(\psi\)-function. The Huber RHT M-estimator of the payroll total can be computed with
> weighted_total_huber(payroll, weight, k = 8, type = "rht") 1] 15587090084[
Note that we must specify
type = "rht" for the RHT [the case
type = "rhj" is discussed in the vignette “Basic Robust Estimators”]. Here, we have chosen the robustness tuning constant \(k = 8\).
The following survey method are available;
The survey method of the RHT (and its standard error) is
> m <- svytotal_huber(~payroll, dn, k = 8, type = "rht") > m total SE1.559e+10 1.198e+09payroll
summary() method summarizes the most important facts about the estimate.
> summary(m) -estimator (type = rht) of the population total Huber M total SE1.559e+10 1.198e+09 payroll : Robustness-function: with k = 8 Psi: 0.9917 mean of robustness weights : Algorithm performancein 4 iterations converged scale (weighted MAD): 89474 with residual : Sampling design Stratified Independent Sampling designsvydesign(ids = ~ID, strata = ~strat, fpc = ~fpc, weights = ~weight, data = workplace)
The estimated location, variance, and standard error can be extracted from object
m with the following commands.
> coef(m) payroll 15587090084 > vcov(m) Variance1.434857e+18 payroll > SE(m) 1] 1197855270[
For M-estimators, the estimated scale (weighted MAD) can be extracted with the
> scale(m) 1] 89474.01[
Additional utility functions are:
residuals()to extract the residuals
fitted()to extract the fitted values under the model in use
robweights()to extract the robustness weights
In the following figure, the robustness weights of object
m are plotted against the residuals. The Huber RHT M-estimator downweights observations at both ends of the residuals’ distribution.
> plot(residuals(m), robweights(m))
An adaptive M-estimator of the total (or mean) is defined by letting the data chose the tuning constant \(k\). This approach is available for the RHT estimator \(\rightarrow\) see vignette “Basic Robust Estimators”, Chap. 5.3 on M-estimators.
CHAMBERS, R. (1986). Outlier Robust Finite Population Estimation. Journal of the American Statistical Association 81, 1063–1069, DOI: 10.1080/01621459.1986.10478374.
FULLER, W. A. (2009). Sampling Statistics, Hoboken (NJ): John Wiley & Sons, DOI: 10.1002/9780470523551.
HULLIGER, B. (1995). Outlier Robust Horvitz–Thompson Estimators. Survey Methodology 21, 79–87.
HULLIGER, B. (1999). Simple and robust estimators for sampling, in: Proceedings of the Survey Research Methods Section, American Statistical Association, pp. 54–63.
HULLIGER, B. (2006). Horvitz–Thompson Estimators, Robustified. In: Encyclopedia of Statistical Sciences ed. by Kotz, S. Volume 5, Hoboken (NJ): John Wiley and Sons, 2nd edition, DOI: 10.1002/0471667196.ess1066.pub2.
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.
PATAK, Z., HIDIROGLOU, M. and LAVALLEE, P. (1998). The methodology of the Workplace and Employee Survey. Proceedings of the Survey Research Methods Section, American Statistical Association, 83–91.