Author: Beat Hulliger and Tobias Schoch

# Vignette: Robust Survey Regression

#### Beat Hulliger and Tobias Schoch

## 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)

## 1 Preparations

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.

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

### 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 3 141 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`.

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

### 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.

## 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`).

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

### 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

    > 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

    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.

### 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)

    > 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.

### 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`.

    > 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

    > 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.

#### 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, 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)

#### Regression estimation and inference

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

## 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

    > 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`)

    > 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")

    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.

## 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); see also motivating example.

### 4.1 The MU284 population

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, calibrate.formula = ~-1 + Stratum)

### 4.2 Model fit and statistical inference

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")

    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.

## References

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

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

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. 21105/joss.03238

SIMPSON, D. G., RUPPERT, D. AND CARROLL, R. J. (1992). On one-step GM estimates and stability of inferences in linear regression. Journal of the American Statistical Association 87, 439–450. DOI: 10.2307/2290275

TILLE, T. and MATEI, A. (2021). sampling: Survey Sampling. R package version 2.9. https://CRAN.R-project.org/package=sampling.

VENABLES, W. N. AND RIPLEY, B. D. (2002). Modern Applied Statistics with S, New York: Springer-Verlag, 4th edition. DOI: 10.1007/978-0-387-21706-2