The vignette covers robust generalized regression (GREG) prediction and robust ratio prediction.

First, we load the namespaces of the packages `robsurvey`

and `survey`

(and attach them to the search path). In order
to use the regression methods, the `survey`

package
**must** be attached to the search path. In addition, we
load the dataset `MU284pps`

.

**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.

The `MU284pps`

dataset is random sample from the MU284
population of Särndal et al. (1992, Appendix B).
The population includes measurements 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 sample is a proportional-to-size sample (PPS) without
replacement of size 50. The sample has been selected by Brewer’s method;
see Tillé (2006, Chap. 7). The sample inclusion
probabilities are proportional to the population size in 1975 (variable
`P75`

). The sampling weight (inclusion probabilities) are
calibrated to the population size and the population total of P75.

The data frame `MU284pps`

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 | weights |
sampling weights |

pi |
finite population correction |

First, we define the sampling design

with the option `pps = "Brewer"`

and the specification
that `fpc`

is equal to the first-order sample inclusion
probabilities, `pi`

.

The variable of interest is revenues from 1985 taxation
(`RMT85`

), and the goal is to estimate the population
revenues total (in million Swedish kronor). From register data, the
population size in 1985 (variable `P85`

) is a know quantity;
it is 8 339 (in thousands). The subsequent graph shows a scatterplot of
`RMT85`

vs. `P85`

(the size of the circles is
proportional to the sampling weight).

We are interested in the ratio estimator of the 1985 revenues total. Consider the following population model

\[ \begin{equation*} \mathrm{RMT85}_i = \mathrm{P85}_i \cdot \theta + \sigma \sqrt{\mathrm{P85}_i} E_i, \qquad i \in U, \end{equation*} \]

where the \(E_i\) are independent and identically distributed random variables with zero mean and unit variance. The parameters \(\theta\) and \(\sigma > 0\) are unknown.

Under the model, the least squares (LS) census estimator of \(\theta\) is \(\theta_N = \sum_{i \in U} y_i / \sum_{i \in U} x_i\). The sample weighted LS estimator is \(\widehat{\theta}_n = \sum_{i \in s}w_i y_i / \sum_{i \in s} w_i x_i\), where \(w_i\) denotes the sampling weight. It is computed by

```
> rat <- svyratio_huber(~RMT85, ~P85, dn, k = Inf)
> rat
Survey ratio M-estimator (Huber psi, k = Inf)
Call:
svyratio_huber(numerator = ~RMT85, denominator = ~P85, design = dn,
k = Inf)
IRWLS converged in 1 iterations
Coefficients:
RMT85/P85
8.379
Scale estimate: 4.891 (weighted MAD)
```

Next, we predict the payroll total based on the estimated regression
parameter \(\widehat{\theta}_n\) (i.e.,
object `rat`

) and the known population size in 1985
(`total = 8339`

).

The estimated mean square error of the ratio predictor is

which is equal to the estimated variance because the ratio predictor
of the total is unbiased for the population total. The estimated total,
standard error and variance covariance can be extracted by the functions
`coef`

, `SE()`

, and `vcov()`

.

A robust ratio predictor (with Huber \(\psi\)-function and robustness tuning constant \(k = 20\)) can be computed by

```
> rat_rob <- svyratio_huber(~RMT85, ~P85, dn, k = 20)
> tot_rob <- svytotal_ratio(rat_rob, total = 8339)
> tot_rob
total SE
RMT85 68529 2550
```

In terms of the estimated standard error, this predictor is considerably more efficient than the (non-robust) ratio predictor. We come to the same conclusion if we consider the approximate mean square error

In place of the Huber estimator, we can use the ratio
*M*-estimator with Tukey biweight (bisquare) \(\psi\)-function; see
`svyratio_tukey()`

. The ratio estimator of the population of
the mean is computed by `svymean_ratio()`

.

Consider the population regression model \[ \begin{equation*} \mathrm{RMT85}_i = \theta_0 + \mathrm{P85}_i \cdot \theta_1 + \mathrm{SS82}_i \cdot \theta_2 + \sigma E_i, \qquad i \in U, \end{equation*} \]

where `CS82`

is the number of Social-Democrat seats in
municipal council. The \(E_i\) are
independent and identically distributed random variables with zero mean
and unit variance. The parameters \(\theta_0,
\ldots, \theta_2\) and \(\sigma
> 0\) are unknown.

The weighted LS estimate is computed by

```
> wls <- svyreg(RMT85 ~ P85 + SS82, dn)
> wls
Weighted least squares
Call:
svyreg(formula = RMT85 ~ P85 + SS82, design = dn)
Coefficients:
(Intercept) P85 SS82
61.756 11.590 -7.344
Scale estimate: 161.2
```

The summary method shows that variable `CS82`

contributes
significantly to the explanation of the response variable.

```
> summary(wls)
Call:
svyreg(formula = RMT85 ~ P85 + SS82, design = dn)
Residuals:
Min. 1st Qu. Median Mean 3rd Qu. Max.
-1066.08 -84.13 -25.84 -4.62 22.43 2001.02
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 61.756 36.108 1.710 0.0883 .
P85 11.590 1.211 9.572 <2e-16 ***
SS82 -7.344 2.946 -2.493 0.0132 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 161.2 on 281 degrees of freedom
```

The diagnostic plots (below) indicate several issues. In particular, the fitted values are smaller than the responses (see “Response vs. Fitted values”). As a result, the fit tends to underestimate.

Letting the issues aside, we want to predict the 1985 revenues total.
The population totals of `P85`

and `SS82`

are
known quantities (from registers) and are passed to the function
`svytotal_reg()`

via the `totals`

argument.
Because the model includes a regression intercept, we must also specify
the population size `N`

. The total is predicted using

```
> tot <- svytotal_reg(wls, totals = c(P85 = 8339, SS82 = 6301), N = 284,
+ type = "ADU")
> tot
total SE
RMT85 67914 2588
```

where `type = "ADU"`

defines the “standard” GREG
predictor, which is an asymptotically unbiased (ADU) estimator/
predictor, hence the name. By default, the argument
`check.names`

is set to `TRUE`

in the call of
`svytotal_reg()`

. Thus, the names of arguments of
`totals`

are checked against the names of the estimated
regression coefficients. If the names of `totals`

are not
specified, we call the function with `check.names = FALSE`

.
The estimated total, standard error and variance covariance can be
extracted by the functions `coef`

, `SE()`

, and
`vcov()`

.

The (estimated) approximate mean square error (MSE; which coincides with the estimated variance of the predictor) is

The estimated total, standard error and variance covariance can be
extracted by the functions `coef`

, `SE()`

, and
`vcov()`

.

Consider the regression model from the last paragraph. Now, we
compute a robust GREG predictor of the 1985 revenues total. The
regression *M*-estimator with Tukey biweight (bisquare) \(\psi\)-function and the robustness tuning
constant \(k=15\) is

```
> rob <- svyreg_tukeyM(RMT85 ~ P85 + SS82, dn, k = 15)
> rob
Survey regression M-estimator (Tukey psi, k = 15)
Call:
svyreg_tukeyM(formula = RMT85 ~ P85 + SS82, design = dn, k = 15)
IRWLS converged in 6 iterations
Coefficients:
(Intercept) P85 SS82
-13.6992 8.3275 -0.2038
Scale estimate: 16.46 (weighted MAD)
```

The diagnostic plots look better than for the weighted LS estimator.

The robust GREG predictor of the 1985 revenues total is

```
> tot <- svytotal_reg(rob, totals = c(P85 = 8339, SS82 = 6301), N = 284,
+ type = "huber", k = 50)
> tot
total SE
RMT85 66641 1300
```

where the prediction is based on the Huber \(\psi\)-function with tuning constant \(k = 50\). The tuning constant for robust prediction should in general be larger than the one used for regression estimation. Observe that we have “mixed” the \(\psi\)-functions: Regression estimation based on the Tukey biweight \(\psi\)-function and prediction with the Huber \(\psi\)-function.

The (estimated) approximate MSE of the robust GREG predictor is

which is considerably smaller than the MSE of the “standard” GREG.
The estimated total, standard error and variance covariance can be
extracted by the functions `coef`

, `SE()`

, and
`vcov()`

.

See the help file of `svymean_reg()`

or
`svytotal_reg()`

to learn more.

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.

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

TILLE, Y. (2006). *Sampling Algorithms*. New York:
Springer-Verlag.