Type:	Package
Title:	Robust Small Area Estimation
Version:	0.3
Description:	Empirical best linear unbiased prediction (EBLUP) and robust prediction of the area-level means under the basic unit-level model. The model can be fitted by maximum likelihood or a (robust) M-estimator. Mean square prediction error is computed by a parametric bootstrap.
License:	GPL-3
URL:	https://github.com/tobiasschoch/rsae
BugReports:	https://github.com/tobiasschoch/rsae/issues
Encoding:	UTF-8
NeedsCompilation:	yes
LazyData:	true
Depends:	R (≥ 3.5.0)
Imports:	stats, graphics
Suggests:	knitr, rmarkdown, robustbase
VignetteBuilder:	knitr, rmarkdown
Packaged:	2024-02-06 18:46:57 UTC; tobias
Author:	Tobias Schoch [aut, cre], Burkardt John [cph] (Fortran 90 subroutine zero_rc)
Maintainer:	Tobias Schoch <tobias.schoch@fhnw.ch>
Repository:	CRAN
Date/Publication:	2024-02-06 19:00:02 UTC

Robust Small Area Estimation

Description

The package implements methods to fit the basic unit-level model (also known as model type "B" in Rao, 2003, or nested-error regression model in Battese et al., 1988), to predict area-specific means by the empirical best linear unbiased predictor (EBLUP) or a robust prediction method (Copt and Victoria-Feser, 2009; Heritier et al., 2011), and to compute the mean square prediction error of the predicted area-level means by a parametric bootstrap (Sinha and Rao, 2009; see also Hall and Maiti, 2006; Lahiri, 2003).

The methods are discussed in Schoch (2012).

Details

Implemented methods

• maximum likelihood estimator

• Huber-type M-estimator (RML II of Richardson and Welsh, 1995, not the method proposed in Sinha and Rao, 2009); see Schoch (2012) for details

How to

Data analysis involves the following steps:

1. prepare the data/ specify the model for estimation; see saemodel()

2. fit the model by various (robust) methods; see fitsaemodel()

3. (robustly) predict the random effects and the area means; see robpredict()

References

Battese, G. E., Harter, R. M., and W.A. Fuller (1988). An error component model for prediction of county crop areas using. Journal of the American Statistical Association 83, 28–36. doi:10.2307/2288915

Copt, S. and M.-P. Victoria-Feser (2009). Robust Predictions in Mixed Linear Models, Research Report, University of Geneva.

Lahiri, P. (2003). On the impact of bootstrap in survey sampling and small area estimation. Statistical Science 18, 199–210. doi:10.1214/ss/1063994975

Hall, P. and T. Maiti (2006). On parametric bootstrap methods for small area prediction. Journal of the Royal Statistical Society. Series B 68, 221–238. doi:10.1111/j.1467-9868.2006.00541.x

Heritier, S., Cantoni, E., Copt, S., and M.-P. Victoria-Feser (2009). Robust methods in Biostatistics, New York: John Wiley and Sons.

Rao, J.N.K. (2003). Small Area Estimation, New York: John Wiley and Sons.

Richardson, A.M. and A.H. Welsh (1995). Robust restricted maximum likelihood in mixed linear model. Biometrics 51, 1429–1439. doi:10.2307/2533273

Schoch, T. (2012). Robust Unit-Level Small Area Estimation: A Fast Algorithm for Large Datasets. Austrian Journal of Statistics 41, 243–265. doi:10.17713/ajs.v41i4.1548

Sinha, S.K. and J.N.K. Rao (2009). Robust small area estimation. Canadian Journal of Statistics 37, 381–399. doi:10.1002/cjs.10029

Fitting SAE Models

Description

fitsaemodel fits SAE models that have been specified by saemodel() (or synthetic data generated by makedata()) for various (robust) estimation methods.

Usage

fitsaemodel(method, model, ...)
convergence(object)

## S3 method for class 'fit_model_b'
print(x, digits = max(3L, getOption("digits") - 3L),
    ...)
## S3 method for class 'summary_fit_model_b'
print(x, digits = max(3L, getOption("digits")
    - 3L), ...)
## S3 method for class 'fit_model_b'
summary(object, ...)
## S3 method for class 'fit_model_b'
coef(object, type = "both", ...)

Arguments

Details

Function fitsaemodel() computes the following estimators:

• maximum likelihood (ML): method = "ml",

• Huber-type M-estimation: method = "huberm"; this method is called RML II by Richardson and Welsh (1995); see Schoch (2012)

Maximum likelihood

The ML is not robust against outliers.

Huber-type M-estimation

The call for the Huber-type M-estimator (with Huber psi-function) is: fitsaemodel(method = "huberm", model, k), where k is the robustness tuning constant of the Huber psi-function, k \in (0, \infty].

By default, the computation of the M-estimator is initialized by a robust estimate that derives from a fixed-effects model (centered by the median instead of the mean); see Schoch (2012) for the details.

If the data are supposed to be heavily contaminated (or if the default algorithm did not converge), one may try to initialize the algorithm underlying fitsaemodel() by a high breakdown-point estimate. The package offers two initialization methods: NOTE: the robustbase package (Maechler et al., 2022) must be installed to use this functionality.

• init = "lts": least trimmed squares (LTS) regression from robustbase; see ltsReg() and Rousseeuw and Van Driessen (2006),

• init = "s": regression S-estimator from robustbase; see lmrob() and Maronna et al. (2019).

For small and medium size datasets, both methods are equivalent in terms of computation time. For large data, the S-estimator is considerably faster.

Implementation

The methods are computed by (nested) iteratively re-weighted least squares and a derivative of Richard Brent's zeroin algorithm; see Brent (2013, Chapter 4). The functions depend on the subroutines in BLAS (Blackford et al., 2002) and LAPACK (Anderson et al., 2000); see Schoch (2012).

Value

An instance of the class "fitmodel"

References

Anderson, E., Bai, Z., Bischof, C., Blackford, L. S., Demmel, J., Dongarra, J., et al. (2000). LAPACK users' guide (3rd ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM).

Blackford, L.S., Petitet, A., Pozo, R., Remington, K., Whaley, R.C., Demmel, J., et al. (2002). An updated set of basic linear algebra subprograms (BLAS). ACM Transactions on Mathematical Software 28, 135–151. doi:10.1145/567806.567807

Brent, R.P. (2013). Algorithms for minimization without derivatives. Mineola (NY): Dover Publications Inc. (This publication is an unabridged republication of the work originally published by Prentice-Hall Inc., Englewood Cliffs, NJ, in 1973).

Maechler, M., Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M., Conceicao, E.L.T. and M. Anna di Palma (2022). robustbase: Basic Robust Statistics R package version 0.95-0. https://CRAN.R-project.org/package=robustbase

Maronna, R.A., Martin, D., V.J. Yohai and M. Salibian-Barrera (2019): Robust statistics: Theory and methods. Chichester: John Wiley and Sons, 2nd ed.

Richardson, A.M. and A.H. Welsh (1995). Robust restricted maximum likelihood in mixed linear model. Biometrics 51, 1429–1439. doi:10.2307/2533273

Rousseeuw, P. J. and K. Van Driessen (2006). Computing LTS regression for large data sets. Data Mining and Knowledge Discovery 12, 29–45. doi:10.1007/s10618-005-0024-4

Schoch, T. (2012). Robust Unit-Level Small Area Estimation: A Fast Algorithm for Large Datasets. Austrian Journal of Statistics 41, 243–265. doi:10.17713/ajs.v41i4.1548

See Also

fitsaemodel.control()

Examples

# use the landsat data
head(landsat)

# define the saemodel using the landsat data
model <- saemodel(formula = HACorn ~ PixelsCorn + PixelsSoybeans,
    area = ~CountyName,
    data = subset(landsat, subset = (outlier == FALSE)))

# summary of the model
summary(model)

# maximum likelihood estimates
fitsaemodel("ml", model)

# Huber M-estimate with robustness tuning constant k = 2
m <- fitsaemodel("huberm", model, k = 2)
m

# summary of the fitted model/ estimates
summary(m)

# obtain more information about convergence
convergence(m)

# extract the fixed effects
coef(m, "fixef")

# extract the random effects
coef(m, "ranef")

# extract both
coef(m)

Tuning Parameters of fitsaemodel

Description

This function is used to define global settings and parameters that are used by fitsaemodel().

Usage

fitsaemodel.control(niter = 40, iter = c(200, 200), acc = 1e-05,
    dec = 0, decorr = 0, init = "default", k_Inf = 20000, ...)

Arguments

Details

Changing the default values of the parameters may result in failure of convergence or loss of convergence speed.

Value

A list with entries

• niter

• iter

• acc

• k_Inf

• init

• dec

• decorr

• add

See Also

fitsaemodel()

Examples

# use the landsat data
head(landsat)

# define the saemodel using the landsat data
model <- saemodel(formula = HACorn ~ PixelsCorn + PixelsSoybeans,
    area = ~CountyName,
    data = subset(landsat, subset = (outlier == FALSE)))

# summary of the model
summary(model)

# obtain the maximum likelihood estimates with, for instance, 'niter = 50'
# number of outer-loop iterations (by default: niter = 40). Here, we use
# 'niter = 50' for the sake of demonstration, not because it is needed.
fitsaemodel("ml", model, niter = 50)

LANDSAT Data: Prediction of County Crop Areas Using Survey and Satellite Data

Description

The landsat data is a compilation of survey and satellite data from Battese et al. (1988). It consists of data on segments (primary sampling unit; 1 segement approx. 250 hectares) under corn and soybeans for 12 counties in north-central Iowa.

Usage

data(landsat)

Format

A data frame with 37 observations on the following 10 variables.

SegmentsInCounty

number of segments per county.

SegementID

sample segment identifier (per county).

HACorn

hectares of corn for each sample segment (as reported in the June 1978 Enumerative Survey).

HASoybeans

hectares of soybeans for each sample segment (as reported in the June 1978 Enumerative Survey).

PixelsCorn

number of pixels classified as corn for each sample segment (LANDSAT readings).

PixelsSoybeans

number of pixels classified as soybeans for each sample segment (LANDSAT readings).

MeanPixelsCorn

county-specific mean number of pixels classified as corn.

MeanPixelsSoybeans

county-specific mean number of pixels classified as soybeans.

outlier

outlier indicator; observation number 33 is flagged as outlier.

CountyName

county names (factor variable): Cerro Gordo, Hamilton, Worth, Humboldt, Franklin, Pocahontas, Winnebago, Wright, Webster, Hancock, Kossuth, Hardin.

Details

The landsat data in Battese et al. (1988) is a compilation of the LANDSAT satellite data from the U.S. Department of Agriculture (USDA) and the 1978 June Enumerative Survey.

Survey data

The survey data on the areas under corn and soybeans (reported in hectares) in the 37 segments of the 12 counties (north-central Iowa) have been determined by USDA Statistical Reporting Service staff, who interviewed farm operators. A segment is about 250 hectares.

Satellite data

For the LANDSAT satellite data, information is recorded as "pixels". A pixel is about 0.45 hectares. The USDA has been engaged in research toward transforming satellite information into good estimates of crop areas at the individual pixel and segments level. The satellite (LANDSAT) readings were obtained during August and September 1978.

Data for more than one sample segment are available for several counties (i.e., unbalanced data).

Observations No. 33 has been flaged as outlier; see Battese et al. (1988, p. 28).

Source

The landsat data is from Table 1 of Battese et al. (1988, p. 29).

References

Battese, G. E., Harter, R. M., and W.A. Fuller (1988). An error component model for prediction of county crop areas using. Journal of the American Statistical Association 83, 28–36. doi:10.2307/2288915

See Also

landsat_means

Examples

head(landsat)

Means of the LANDSAT Data for Corn and Soybeans

Description

The landsat data is a compilation of survey and satellite data from Battese et al. (1988). The county-specific population means of pixels of the segments under corn and soybeans, respectively, are available in the data.frame landsat_means.

Usage

data(landsat_means)

Format

A data frame with 12 observations (counties) on the following 3 variables.

(intercept)

all ones.

MeanPixelsCorn

county-specific mean of number of pixels classified as corn (LANDSAT readings).

MeanPixelsSoybeans

county-specific number of pixels classified as soybeans (LANDSAT readings).

Details

The data.frame landsat_means is an aggregation of the data.frame landsat.

References

Battese, G. E., Harter, R. M., and W.A. Fuller (1988). An error component model for prediction of county crop areas using. Journal of the American Statistical Association 83, 28–36. doi:10.2307/2288915

Examples

head(landsat_means)

Synthetic Data Generation for the Basic Unit-Level SAE Model

Description

This function generates synthetic data (possibly contaminated by outliers) for the basic unit-level SAE model.

Usage

makedata(seed = 1024, intercept = 1, beta = 1, n = 4, g = 20, areaID = NULL,
         ve = 1, ve.contam = 41, ve.epsilon = 0, vu = 1, vu.contam = 41,
         vu.epsilon = 0)

Arguments

Details

Let y_i denote an area-specific n_i-vector of the response variable for the areas i = 1,..., g. Define a (n_i \times p)-matrix X_i of realizations from the std. normal distribution, N(0,1), and let \beta denote a p-vector of regression coefficients. Now, the y_i are drawn using the law y_i \sim N(X_i\beta, v_e I_i + v_u J_i) with v_e and v_u the variances of the model error and random-effect variance, respectively, and I_i and J_i denoting the identity matrix and matrix of ones, respectively.

In addition, we allow the distribution of the model/residual and area-level random effect to be contaminated (cf. Stahel and Welsh, 1997). Notably, the laws of e_{i,j} and u_i are replaced by the Tukey-Huber contamination mixture:

• e_{i,j} \sim (1-\epsilon^{ve})N(0,v_e) + \epsilon^{ve}N(0, v_e^{\epsilon})

• u_{i} \sim (1-\epsilon^{vu})N(0,v_u) + \epsilon^{vu}N(0, v_u^{\epsilon})

where \epsilon^{ve} and \epsilon^{vu} regulate the degree of contamination; v_e^{\epsilon} and v_u^{\epsilon} define the variance of the contamination part of the mixture distribution.

Four different contamination setups are possible:

• no contamination (i.e., ve.epsilon = vu.epsilon = 0),

• contaminated model error (i.e., ve.epsilon != 0 and vu.epsilon = 0),

• contaminated random effect (i.e., ve.epsilon = 0 and vu.epsilon != 0),

• both are conaminated (i.e., ve.epsilon != 0 and vu.epsilon != 0).

Value

An instance of the class saemodel.

References

Schoch, T. (2012). Robust Unit-Level Small Area Estimation: A Fast Algorithm for Large Datasets. Austrian Journal of Statistics 41, 243–265. doi:10.17713/ajs.v41i4.1548

Stahel, W. A. and A. Welsh (1997). Approaches to robust estimation in the simplest variance components model. Journal of Statistical Planning and Inference 57, 295–319. doi:10.1016/S0378-3758(96)00050-X

See Also

saemodel(), fitsaemodel()

Examples

# generate a model with synthetic data
model <- makedata()
model

# summary of the model
summary(model)

Robust Prediction of Random Effects, Fixed Effects, and Area-Specific Means

Description

Function robpredict() predicts the area-level means by (1) the empirical best linear unbiased predictor (EBLUP) or (2) a robust prediction method which is due to Copt and Victoria-Feser (2009). In addition, the function computes the mean square prediction error (MSPE) of the predicted area-level means by a parametric bootstrap method.

Usage

robpredict(fit, areameans = NULL, k = NULL, reps = NULL, seed = 1024,
    progress_bar = TRUE)

## S3 method for class 'pred_model_b'
print(x, digits = max(3L, getOption("digits") - 3L),
    ...)
## S3 method for class 'pred_model_b'
plot(x, type = "e", sort = NULL, ...)
## S3 method for class 'pred_model_b'
residuals(object, ...)
## S3 method for class 'pred_model_b'
as.matrix(x, ...)
## S3 method for class 'pred_model_b'
head(x, n = 6L, ...)
## S3 method for class 'pred_model_b'
tail(x, n = 6L, keepnums = TRUE, addrownums, ...)

Arguments

Details

Function robpredict() computes predictions of the area-level means and—if required—an estimate of the area-specific mean square prediction error (MSPE).

Prediction of area-level means

• Case 1: If areameans is a matrix with area-level means of the explanatory variables, then the computation of the fixed effects effects are based on areameans.

• Case 2: If areameans = NULL, then the predictions are based on the sample data that have been used to fit the model.

Mean square prediction error

• If reps = NULL, the number of bootstrap replicates is not specified; hence, MSPE is not computed.

• If reps is a positive integer and areameans is not NULL (see Case 1 above), then a (robust) parametric bootstrap estimate of MSPE is computed as proposed by Sinha and Rao (2009); see also Lahiri (2003) and Hall.

Robustness

• The EBLUP obtains if k = NULL, i.e., if the robustness tuning constant k is unspecified.

• Robust predictions of the area-level means are computed if k is a nonnegative real number. Small values of k imply that outliers are heavily downweighted; formally, the EBLUP corresponds to choosing the tuning constant k equal to infinity. The value of the tuning constant k specified in robpredict() can be different from the tuning constant k used in fitting the model. The robust prediction method is due to Copt and Victoria-Feser (2009); see also Heritier et al. (2009, 113-114) and differs from the method in Sinha and Rao (2009).

Value

An instance of the S3 class pred_model_b

NOTE

Users of Rgui.exe on Windows are recommended to call robpredict() with argument progress_bar = FALSE because Rgui.exe does not handle calls to txtProgressBar() well (the execution time of the same job increases and it tends to stall the execution of R). Users of R-Studio and Rterm.exe are not affected.

References

Copt, S. and M.-P. Victoria-Feser (2009). Robust Predictions in Mixed Linear Models, Research Report, University of Geneva.

Lahiri, P. (2003). On the impact of bootstrap in survey sampling and small area estimation. Statistical Science 18, 199–210. doi:10.1214/ss/1063994975

Hall, P. and T. Maiti (2006). On parametric bootstrap methods for small area prediction. Journal of the Royal Statistical Society. Series B 68, 221–238. doi:10.1111/j.1467-9868.2006.00541.x

Heritier, S., Cantoni, E., Copt, S., and M.-P. Victoria-Feser (2009). Robust methods in biostatistics. New York: John Wiley and Sons.

Schoch, T. (2012). Robust Unit-Level Small Area Estimation: A Fast Algorithm for Large Datasets. Austrian Journal of Statistics 41, 243–265. doi:10.17713/ajs.v41i4.1548

Sinha, S.K. and J.N.K. Rao (2009). Robust small area estimation. Canadian Journal of Statistics 37, 381–399. doi:10.1002/cjs.10029

See Also

saemodel(), makedata(), fitsaemodel()

Examples

# use the landsat data
head(landsat)

# set up the model
model <- saemodel(formula = HACorn ~ PixelsCorn + PixelsSoybeans,
    area = ~CountyName,
    data = subset(landsat, subset = (outlier == FALSE)))

# summary of the model
summary(model)

# Huber M-estimate with robustness tuning constant k = 2
m <- fitsaemodel("huberm", model, k = 2)
m

# summary of the fitted model/ estimates
summary(m)

# robust prediction of the random effects and the area-level means (robust
# EBLUP) using the counts-specific means (landsat_means)
head(landsat_means)

# for robust prediction, we use the robustness tuning constant 'k = 1.8'
m_predicted <- robpredict(m, landsat_means, k = 1.8)
head(m_predicted)

# extract prediction as matrix
as.matrix(m_predicted)

# extract residuals from the predictions
head(residuals(m_predicted))

# prediction incl. MSPE; parametric bootstrap with only 'reps = 10'
# replications (for demonstration purposes; in practice, 'reps' should be
# considerably larger)
m_predicted_mspe <- robpredict(m, landsat_means, k = 1.8, reps = 10,
                               progress_bar = FALSE)
head(m_predicted_mspe)

Setting Up a SAE Model

Description

Function saemodel() is used to specify a model. Once a model has been specified, it can be fitted using fitsaemodel() by different estimation methods.

Usage

saemodel(formula, area, data, type = "b", na.omit = FALSE)

## S3 method for class 'saemodel'
print(x, ...)
## S3 method for class 'saemodel'
summary(object, ...)
## S3 method for class 'saemodel'
as.matrix(x, ...)

Arguments

Details

Function saemodel() is used to specify a model.

• model is a symbolic description (formula of the fixed-effects model to be fitted.

  A typical model has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response (explanatory variables); see formula.

  A formula has an implied intercept term. To remove this use either y ~ x - 1 or y ~ 0 + x; see formula for more details of allowed formulae.

• area is a symbolic description (formula) of the random effects (nested error structure). It must be right-hand side only formula consisting of one term, e.g., ~ areaDefinition.

The data must no contain missing values.

The design matrix (i.e., matrix of the explanatory variables defined the right-hand side of model) must have full column rank; otherwise execution is terminated by an error.

Once a model has been specified, it can be fitted by fitsaemodel().

Value

An instance of the S3 class "saemodel"

References

Rao, J.N.K. (2003). Small Area Estimation, New York: John Wiley and Sons.

See Also

makedata(), fitsaemodel()

Examples

# use the landsat data
head(landsat)

# set up the model
model <- saemodel(formula = HACorn ~ PixelsCorn + PixelsSoybeans,
    area = ~CountyName,
    data = subset(landsat, subset = (outlier == FALSE)))

# summar of the model
summary(model)