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7/31/2019 Gen Linear

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General linear models in small areaestimation: an assessment in

agricultural surveys

Carlo Russo, Massimo Sabbatini, and Renato SalvatoreUniversity of Cassino, Italy

The MEXSAI Conference


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Some small area estimation

references Ghosh M., Rao J. N. K. (1994), Small area estimation: an

appraisal, Statistical Science, Vol. 9, No. 1, pp. 55-93

He Z., Sun D. (2000), Hierarchical Bayes estimation of

hunting success rates with spatial correlations,Biometrics, 56, 102-109

Malec D., Sedransk J., Moriarity C. L., LeClere F. B.(1997), Small area inference for binary variables in theNational Health Intgerview Survey, Journal of the

American Statistical Association, Vol. 92, 439, 815-826 Rao J. N. K. (2002), Small area estimation with

applications to Agriculture, Proceedings of theConference on agricultural and environmental statisticalappications in Rome, Vol. III, 555-564

Rao J. N. K. (2003), Small area estimation, Wiley, London


GENERAL LINEAR MODELS IN SMALL AREA ESTIMATION: AN ASSESSMENT IN AGRICULTURAL SURVEYS


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Small area estimation: a simple

outlineThe term small area usually denote a small

geographical area, such as a county, a province, anadministrative area or a census division

From a statistical point of view the small area is a smalldomain, that is a small subpopulation constituted byspecific demographic and socioeconomic group ofpeople, within a larger geographical areas

Sample survey data provide effective reliable estimatorsof totals and means for large areas and domains. But itis recognized that the usual direct survey estimatorsperforming statistics for a small area, have

unacceptably large standard errors, due to thecircumstance of small sam le size in the area




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outline In fact, sample sizes in small areas are reduced, due to

the circumstance that the overall sample size in asurvey is usually determined to provide specific

accuracy at a macro area level of aggregation, that isnational territories, regions ad so on (Ghosh and Rao,1994)

Small area statistics are important tools for planning

agricultural policies in specific regional andadministrative areas

But important is also the information demand from othersectors, such as private, especially for questions related

to local social and economics conditions, in local areamarketin research, and so on




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outline

The small area statistics are based on a collection ofstatistical methods that borrow strength form related

or similar small areas through statistics models thatconnect variables of interest in small areas with vectorsof supplementary data, such as demographic,behavioral, economic notices, coming fromadministratvive, census and specific sample surveys

records

Small area efficient statistics provide, in addition of this,excellent statistics for local estimation of population,farms, and other characteristics of interest in post-

censual years




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outlineThe most commonly used tecniques for small area

estimation are the empirical Bayes (EB) procedures, thehierarchical Bayes (HB) and the empirical best linearunbiased prediction (EBLUP) procedures (Rao, 2003)

Some utilization of this tecniques in agrigulturalstatistics are related to the implementation of satellitedata, and, in general, of differently-oriented sumpleysurveys in model-based frameworks

There are two types of small area models that includerandom area-specific effects: in the first type, the basicarea level model, connection through response and areaspecific auxiliary variables is established, because the

limited availability at such type of data at unit level




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outlineThe second type are the unit level area models, in which

element-specific auxiliary data are available for thepopulation elements (Ghosh and Rao, 1994; Rao, 2002)

The simplest way to perform small area statistics is,however, to derive synthetic estimates from large areadata assumptions on related local areas: sintheticestimators are generally used because of theirapplicability to general sampling designs and of their

improving efficiency in relation to exploiting informationfrom similar small areas

The problem is that such type of estimators arepotentially design-biased. Following the composite

estimate approach to small area analyis, the way ofbalance the bias of synthetic estimator against the




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outline4)

4) is the direct area estimator with sampling errors

Combining 2) and 4):

5)

5) this model involves design random variables and, at the

same time, the model-based random variables. It is anexample of general linear mixed model (GLMM) withdiagonal covariance structure

mieiii ,..., 1=+=

mieuz iiiT

ii ,..., 1=++= x




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outlineThe BLUP (best linear unbiased prediction) estimator is a

weighted average of the design-based estimator and theregression-synthetic estimator

The MSE of the BLUP estimator depends on the varianceparameter of the random area effects

In practical applications this parameter is unknown, and itis replaced by an estimator

Then, we have a two-stage estimator, called empiricalBLUP (EBLUP)

Since the MSE of the EBLUP estimator is insensitive to thechoiche of the random area effect varaince estimator, itis larger than the BLUP estimator

Assuming normality of random effects, the relatedvariance area parameters can be estimated either by

maximum likelihood (ML) or restricted maximumlikelihood (REML) methods




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outline Instead of EBLUP and EB, if we follow the HB

(hierarchical Bayes) approach, first a prior distributionon the model parameters is specified, and then theposterior distribution of the parameters of interest is

obtained

The usually estimation small area problem are solvedexploiting the posterior distribution framework. Theevaluation of parameters of interest is obtained by its

posterior mean-based estimate, and the precision of theestimate in terms of its MSE is measured with theposterior variance

The HB approach is computationally intensive, involvingin much cases high dimensional integration




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outline Some tools, such as Gibbs sampling and importance

sampling, the latter jointly employed with Monte Carlonumerical integration methods, are commonly used in

order to overcome some computational problems

In the recent years, comparative studies concerning theEBLUP, EB, and HB approaches lead in general to closevalues of predictors. All of the three in certain particular

situations can work better than others




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Qualitative data

Qualitative data are becoming relevant in theagricultural economics field for two major reasons:firstly theoretical development stress the relevance of

discrete and intrinsically qualitative phenomena,secondly the increasing sophistication of the statisticsapproach in the field allows economist to drawquantitative conclusions from discrete data

The qualitative data about households, including therole of women, services availability, presence/absenceof infrastructures are considered as relevant factor inthe analysis that require close consideration in anyeconomic model in the field




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Qualitative data

Agricultural economists are also interested to the socialanalysis of the rural territory. The segmentation of theuniverse, based on qualitative variables (such asgender, age, education) becomes relevant to define the

dynamics of specific groups and to analyze issues ofinterest

The shift of the policy focus from producers support torural development in high income countries is one of themajor factor determining the new interest in theanalysis of the qualitative aspects of agriculture

In the contest of qualitative data analyses, bothcontinuos and binary or nonnumeric data are availableby the large data sets exploration of some arrays, suchas in agricultural census data. The complete exploitationof that large number of informations about farms is

often feasible only with some explorative data analyses,in particular homogeneity and correspondence analysis




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Qualitative data

On the other hand, it is recognized that some complexaspects of farms structure are correctly pointed out ifwe implement in economic models, at the same time, all

possible information

Small area statistics are powerful methods in estimatingsmall area farms characteristics, but some agriculturalpolicies need further information, especially thoserelated with particular classes of farms

The apparteinance of farms in well-recognized classes,jointly used with other area information, is then a basicpolicy-makers tool

From this standpoint is very useful try to achieve smallarea random effects models that combine continuous

and categorical predictors and use binary responsevariables. The goal is to estimate proportions of farms




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The general logistic-linear mixed model

small area analysisThe extension of the GLM models to binary response

variables small area analysis is given in Malec et al.,1997 and 1999. The related unit level model combine,

in the paper application example, small area-specificcovariates with unit level demographic andsocioeconomic data. Then estimates was stated relatingindividuals and classes, using a HB approach

In that GLM model, it is assumed that each individual inthe population is assigned to one of mutually exclusiveand exhaustive classes, based on the individualsdemographic and socioeconomic status

Given a vector of random effects the estimation ofparameters of GLMM model for binary responsesrequests computation of high dimensional integrals,with dimension equal to the number of levels of the




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The general logistic-linear mixedmodel

small area analysis One approach in literature was done in the contest of

HB framework. He and Sun, 2000, given an example ofhierarchical Bayes estimation procedure of a logistic-

linear mixed model in hunting success rates at the sub-area level for post-season harvest surveysThe model implements fixed week effects and random

geographic effects, in the contest of autoregressive (AR)

and conditional autoregressive (CAR) approach to theanalysis of spatial correlations between neighboringsub-areas. The process of estimation needs, as in thecase of the GLM represented by the logistic-linear modelabove, Gibbs sampling procedures




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small area analysis We introduce in the paper a Monte Carlo Newton-

Raphson ML procedure (McCulloch, 1997) in estimatingparameters in the following general logistic-linear mixedmodel

The estimation problem in closed form likelihoodintegral expressions is proposed to solve numerically viaMonte Carlo approach.

Another problem is how to generate starting values ofthe parameters in likelihood expressions if, previously,we dont specify the vector of random effects. A naturalway to solve the problem is to adopt the Metropolis

algorithm, that is a simple Markov Chain Monte Carlo(MCMC) algorithm

iiTkikikik uppp +== )logit())/(log( X1




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small area analysisThe basic characteristic of a MCMC is that the sequence

of generated points takes a kind of random walk inparameter space, instead of each point being

generated, one independently from another

Moreover, the probability of jumping from one point toan other depends only on the last point and not on theentire previous history (this is the peculiar property of a

Markov chain)

The paper shows the Monte Carlo approach to theNewton-Raphson procedure of estimating logistic linearparameters estimation via an iterative procedure that

leads to convergent MLE estimates, under assumptionof normalit




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Conclusions

In this paper, a Monte Carlo Newton-Raphson algorithmhas been outlined, assuming normality of random areaeffects, in order to approach the MLE estimation issues

related to the logistic-linear mixed model, in the contextof qualitative small area estimation

As generally recognized, the focus of the recenteconomic theory on qualitative data can be summarizein two major points: the increasing interest in theanalysis of discrete phenomena, and the explanatorypower of qualitative variable in describing the currenttrend in the agricultural sector

Statistical methods able to convey the qualitativeinformation in the estimation models are able toincrease efficiency




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Thank you

Please find much more methodological details inthe paper available on the conference website

e-mail to: [email protected]

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