This module is part of the. Memobust Handbook. on Methodology of Modern Business Statistics

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1 This moule is part of the Memobust Hanbook on Methoology of Moern Business Statistics 26 March 2014

2 Metho: EBLUP Unit Level for Small Area Estimation Contents General section Summary General escription of the metho Preparatory phase Examples not tool specific Examples tool specific Glossary References... 6 Specific section... 8 Interconnections with other moules Aministrative section... 13

3 General section 1. Summary The aim of Small Area Estimation (SAE) is to compute a set of reliable estimates for each small area for the target variable(s) of interest, whenever the irect estimates (see Weighting an Estimation Main Moule an Weighting an Estimation Generalise Regression Estimator ) cannot be consiere enough reliable, i.e., the corresponent variances (see the moule Quality Aspects Quality of Statistics ) are too high to make those estimates releasable. Small area methos provie a set of techniques to obtain the estimates of interest in the National Statistical Institutes (NSIs) large scale survey, where more etaile information is require, an the sample size is not large enough to guarantee release of irect estimation. SAE methos which increase the reliability of estimates borrowing strength from a larger area. The unit level EBLUP estimator, which is escribe in this moule, is a linear combination of the irect information an a regression synthetic preiction of non-sample units. The fixe part of the moel links the target values to some known auxiliary variables, for each units belonging to the larger area to which the small areas of interest belong to. The area specific ranom effects is instea introuce in orer to take into account the correlation among the units with each small area (between area variation). 2. General escription of the metho The unit level mixe moel can be use when unit-specific auxiliary variables are available in each small area. The area-specific ranom effect terms are consiere in orer to take into account the between area variation through the correlation among units within a small area. The basic unit level linear mixe moel is the neste error regression moel formulate by Battese et al. (1988). It can be expresse as follows: y = x β + u + e (1) i T i i where u e i ~ ii N ~ ii N i = 1, K, N = 1, K, D 2 ( 0, σ u ) 2 ( 0, σ ) e an y i is the variable of interest for the i-th population unit in the -th small area. Assuming non informative sampling esigns, like simple ranom sampling, has been use at the sampling stage, the same moel assume for the population values can be applie for the sample units. Therefore, using a matrix notation, the following moel can be formalise y = x β + z u + e (2) s s s s 3

4 where y s is n-imensional vector of the observe values for the variable y, imensional matrix of the covariate values observe in the sampling units, error vector, z is the ( n D) s an u is the D-imensional vector of area ranom effects. x is the ( p) s n - e s is the n-imensional -imensional incience matrix of the sampling units in the small areas, In orer to obtain the small area estimates base on the above moel, either a preictive or a Bayesian approach can be employe (see Rao, 2003, for more etails). Following the preictive framework, the Best Linear Unbiase Preictor (BLUP) is obtaine by minimising the quaratic loss in the linear unbiase estimator class. The BLUP epens on the variance components an that are usually unknown, so their estimates nee to be compute. Both variance components an fixe effects parameters can be estimate in ifferent ways, for example by means of Maximum Likelihoo (ML) or Restricte Maximum Likelihoo (REML) (Cressie, 1992) methos. Once the parameters of the moel have been estimate, the Empirical Best Linear Unbiase Preictor (EBLUP) base on unit level linear mixe moel is a composite-type estimator. Letting asie the finite population correction factor, it is given by where ˆ θ EBLUP_UNIT = γ y X βˆ x βˆ ( 1 ) X βˆ + + γ (3) T T T γ ˆ σ 2 = u ˆ σ 2 2 u + ˆ σ e n an X is the vector of known population means of the auxiliary variables in the -th area an x is the corresponent vector of sample means. Given the moel, the fixe effects parameter are estimates using the whole available larger area sample information an of course, when the between area variation is small, the EBLUP estimator tens towars the synthetic estimator (being the variance of ranom effects small). More weight is instea given to irect information when the variance of ranom effects is big respect the total variance. There are several extensions of the above escribe basic unit level moel. Since the basic moel oes not take into account for sample ata collecte with a complex sample esign, some methoological evelopment have been irecte to specify more complex moels that take into account the features of the sampling esign. For instance, Stukel an Rao (1999) propose a two-fol neste error regression moel sample ata for ata collecte from a stratifie two-stage sampling. The issue is that, when an informative esign is use the inclusion probabilities of sampling units epen on the values of the target variable the moel which hols for the sample ata is ifferent from the moel assume for the population ata, so that it woul be the cause of severe bias into the preiction. A possible approach with this regar is to explicitly inclue all the esign variables use for the sample selection as covariates or the sampling weights in the specification of the moel. These two options can be untenable when too many esign variables are involve an when the sample weights are not available for non-sample areas or non-sample units. A Pseuo EBLUP estimator was propose by Prasa an Rao (1999) starting from unit linear mixe moel. 4

5 Moreover, multivariate neste error regression moel has been propose in orer to estimate more than one small area parameters of interest simultaneously. This type of moel, applie in Datta et al. (1999), allows to take into account the correlation among the characteristics uner stuy observe in the sample units. Finally, the linear unit level mixe moels shoul be applicable only for continuous observations, then some enhancement moels has been consiere in orer to eal with categorical epenent variables. In that case, Generalise Linear Mixe Moels (GLMM) can be consiere. Within this logistic regression moels with mixe effects are commonly use for estimating small-area proportions (Malec et al., 1997). 3. Preparatory phase Moel selection is crucial preparatory phase since the objective is to lessen the chances of introucing esign-bias into the small area estimates ue to poor moel specification. Moel selection for each target variable was carrie out consiering iagnostic criteria such as maximum likelihoo, AIC, BIC, Conitional AIC (caic), an Cross Valiation (CV) such as in Vaia an Blanchar (2005), Boonstra et al. (2008), an Boonstra, Buelens an Smeets (2009). Once one or several moels has been selecte, it is necessary to assess the fitting quality of the moel(s). The stuy of moel resiuals by graphical representations, like Histograms, Q-Q plots, box-plots an mapping the resiual, allows to check if the moel assumptions are fulfille. 4. Examples not tool specific We refer to Battese, Harter, an Fuller (1988) for an example of ata for application of EBLUP Unit level moel. These ata are taken from a sample survey that have been esigne to estimate crop areas for large regions. The preictions of the crop area for small areas such as counties has generally not been one for the lacking of available ata irecte collecte from these areas. In orer to apply the metho, satellite ata in association with farm-level survey observations has been use. They consiere the estimation of mean hectares of corn an soybeans per segment an the auxiliary variables are the number of pixels classifie as corn an soybeans in each county. In the example were consiere ata for 12 Iowa countries an ata obtaine from lan observatory satellites. Their example relates to application of SAS macros for computing the preictors uner the moel. The same ata is use as an example in for explaining the use of R function mixe.unit.sae.r. 5. Examples tool specific 6. Glossary For efinitions of terms use in this moule, please refer to the separate Glossary provie as part of the hanbook. 5

6 7. References Battese, G. E., Harter, R. M., an Fuller, W. A. (1988), An error-components moel for preiction of county crop areas using survey an satellite ata. Journal of the American Statistical Association 80, BIAS project website: Boonstra, H. J., Buelens, B., an Smeets, M. (2009), Estimation of unemployment for Dutch municipalities. Small Area Estimation 2009 Conference, Elche, Spain, June 29-July 1. Boonstra, H. J., van en Brakel, J., Buelens, B., Krieg, S., an Smeets, M. (2008), Towars small area estimation at Statistics Netherlans. Metron LIV, Brown, J., Chambers, R., Heay, P., an Heasman, D. (2003), Evaluation of small area estimation methos: an application to the unemployment estimates from the UK LFS. Proceeings of Statistics Canaa Symposium Chanra, H. an Chambers, R. (2007), Small area estimation for skewe ata. Small Area Estimation Conference, Pisa, Italy. Cressie, N. (1992), REML Estimation in Empirical Bayes Smoothing of Census Unercount. Survey Methoology 18, D Alò, M., Di Consiglio, L., Falorsi, S., an Solari, F. (2008), The Use of Sample Design Features in Small Area Estimation. ISI 2009 Conference, Durban (South Africa), August. Datta, G. S., Day, B., an Basawa, I. (1999), Empirical best linear unbiase an empirical Bayes preiction in multivariate small area estimation. Journal of Statistical Planning an Inference 75, Datta, G. S., Ghosh, M., Steorts, R., an Maples, J. (2009), Bayesian Benchmarking with Applications to Small Area Estimation property. Small Area Estimation Conference, Elche, Spain. Dick, J. P. (1995), Moelling net unercoverage in the 1991 Canaian Census. Survey Methoology 21, EURAREA Consortium (2004), PROJECT REFERENCE VOLUME, Vol Ghosh, M. an Rao, J. N. K. (1994), Small area estimation: an appraisal. Statistical Science 9, Malec, D., Seransk, J., Moriarity, C. L., an LeClere, F. B. (1997), Small area inference for binary variables in National Health Interview Survey. Journal of the American Statistical Association 92, Molina, I., Saei, A., an Lombaria, M. J. (2007), Small area estimates of labour force participation uner a multinomial logit mixe moel. Journal of the Royal Statistical Society, Series A 170, Montanari, G. E., Ranalli, M. G., an Vicarelli, C. (2009), Estimation of small area counts with the benchmarking property. Small Area Estimation Conference, Elche, Spain. 6

7 Pfeffermann, D. (2002), Small area estimation New evelopments an irections. International Statistical Review 70, Pfeffermann, D. an Tiller, R. (2006), Small-area estimation with state-space moels subject to benchmark constraints. Journal of the American Statistical Association 101, Prasa, N. G. N. an Rao, J. N. K. (1999), On robust small area estimation using a simple ranom effects moel. Survey Methoology 25, Pushpal, K, Mukhopahyay, P. K., an McDowell, A. (2011), Small Area Estimation for Survey Data Analysis Using SAS Software. SAS Institute Inc., Cary. Rao, J. N. K. (2003), Small area estimation. John Wiley & Sons, Hoboken, New Jersey. SAE ESSnet (2012), Deliverables of the project. Saei, A. an Chambers, R. (2003), Small Area Estimation uner Linear an Generalize Linear Mixe Moels with Time an Area Effects. Methoology Working Paper- M03/15, University of Southampton, Unite Kingom. Stukel, D. M. an Rao, J. N. K. (1999), Small area estimation uner two-fol neste errors regression moels. Journal of Statistical Planning an Inference 78, Torabi, M., Datta, G. S., an Rao, J. N. K. (2009), Empirical Bayes estimation of small area means uner a neste error linear regression moel with measurement errors in the covariates. Scaninavian Journal of Statistics 38, Vaia, F. an Blanchar, S. (2005), Conitional Akaike information for mixe-effects moels. Biometrika 92,

8 Specific section 8. Purpose of the metho The metho is use for small area estimation, when irect estimates usually applie for official statistics are too unreliable an unit level auxiliary information are available. 9. Recommene use of the metho 1. The metho can be applie for estimation when auxiliary information/covariates are available for each sample unit. The mean or total population values nee to be known at area level. 2. A linear moel can be use when the ata are continuous an normally istribute. A transformation of the ata may be require before moelling to make the ata normally istribute. 3. The metho is useful to improve irect estimator if a set of covariates with a strong relationship with the target variable is available. 4. If the target variable is not continuous or normally istribute a generalise linear moel might be applie. For instance, the variable of interest at unit level is often binary, so that the logistic or probit moel shoul be more appropriate. 5. Both unit an area level auxiliary information can be consiere. 10. Possible isavantages of the metho 1. If the moel is not correctly specifie the estimator can be affecte from severe bias. 2. The basic metho o not consier the sampling strategy to select the units. 3. When aing up small omains estimates to a larger omain, it is not ensure that irect estimator at larger level is obtaine. A simple way to guarantee this type of consistency is by means of ratio ajustment of the EBLUP unit level estimator. Benchmarking can be also set as a constraint to obtain small area estimates. This woul prouce ifferent methos that will not be reporte in the present hanbook (Wang, Fuller, an Qu, 2008; Pfeffermann an Tiller, 2006; Montanari, Ranalli, an Vicarelli, 2009; Datta et al., 2009). 4. The moel assumes symmetry of the istribution, while in some cases, like in business survey, skewness may be present. If transformation of variables o not suffice to reuce skewness, avance metho may be consiere. For instance by employing M-quantile moels (Chanra an Chambers, 2007). 5. Stanar small area moels generally consier only i.i.. area ranom effects, whereas more realistic an efficient moels might inclue further structure ranom effects, such as time for repeate surveys an spatial autocorrelate ranom effects. 11. Variants of the metho 1. Variants of the metho are given by the ifferent estimation methos for the variance component of moel (3), e.g., Maximum Likelihoo ML or Restricte Maximum Likelihoo (REML) (see Cressie, 1992), or Metho of moments. 8

9 2. On the basis of the assume moel, an estimator which uses only the regression component is given by the unit level synthetic estimator: ˆSynth_unitlevel θ = X T β ˆ This estimator is always applie for no sample omain. 3. For repeate sample surveys, extensions aime to introuce time ranom effects can be also consiere. 4. In orer to consier the spatial autocorrelation among areas a unit level moel with spatially correlate area effect can be consiere. The spatial correlation can be introuce through the variance-covariance matrix of the ranom effects in function of the istance between areas or by moelling irectly the ranom effects by means of SAR-type moel. 5. Multinomial moels are consiere in Molina et al. (2007). 12. Input ata Input ata set can be classifie accoring to the source of information neee to apply the metho. The first ata set contains sample information whereas the secon one contains information provie from auxiliary source at area level. Specific software tools may nee various structure for the input to prouce estimation. We refer to links in section 27 for software tools that make possible the application of EBLUP unit level. 1. Ds-input1 = a sample ata set contains the target variable an auxiliary variables observe for each sampling unit. 2. Ds-input2 = a ata set (macroata) with mean or total values of covariates for each omain, an population size of the omain. 13. Logical preconitions 1. Missing values 1. EBLUP unit level estimator oes not account explicitly for missing values in the sample observations. 2. Erroneous values 1. Stanar small area methos o not take in consieration errors in the target variables an covariates. Possible misspecification of the auxiliary variables or correction in the variables are not taken into account by EBLUP unit level moel (see Torabi et al., 2009). 3. Other quality relate preconitions Other types of preconitions 1. Normality is often assume for the estimation of the MSE. 2. Sampling esign is usually not consiere in the inference. 9

10 14. Tuning parameters 1. Parameters for the convergence of the iterative metho: number of iterations an/or stopping rule, starting value for the variance components of the moels. 15. Recommene use of the iniviual variants of the metho 1. For non-sample area only synthetic type estimates can be compute. 2. For estimation of ranom component of the variance, software tools applies ML or REML. 16. Output ata 1. Ds-output1 = the target values estimates for each omain an corresponing MSE. 17. Properties of the output ata 1. User may check MSE bias iagnostic (see SAE ESSnet site of the resulting estimates. 18. Unit of input ata suitable for the metho Sample unit level information for target variable an covariates to fit the moel an to estimates the moel parameters inclue the area ranom effects. Population area level means or totals for each omain to compute the estimator. 19. User interaction - not tool specific 1. Moel selection, the choice of which auxiliary variables to inclue in the moel, e.g., by means of AIC an BIC, caic 2. Satisfy the moel hypotheses, like symmetry an homogeneity. A transformation of the variable may be neee. 3. Specification of starting value for the variance of the ranom effects an tuning parameters for convergence 4. Choice of metho for variance component estimation 5. The use of the quality metho shoul provie some evience regaring spatial bias/autocorrelation at ifferent level of aggregation of estimates. Finally MSE for assessing reliability of estimates has to monitore (see guielines on Logging inicators 1. Run time of the application 2. Number of iteration to attain convergence in the estimation process 3. Out of the range estimation of the target parameter can be attaine when linear mixe moel is assume, in this case the normal assumption shoul be relaxe. 10

11 4. Unerestimate of MSE can be possible uner normality assumption an preictive approach to inference. Generalise linear mixe moels an Hierarchical Bayes approach to inference may alternatively be applie. 5. Characteristics of the input ata, for instance problems size. 21. Quality inicators of the output ata 1. MSE 2. Moel Bias iagnostic 3. Benchmark 4. Moel selection iagnostic: AIC, BIC, caic 5. Analysis of the resiual 6. Spatial istribution of area level resiual (Maps) 22. Actual use of the metho The metho is applie in: 1. Netherlans, for the prouction of the yearly estimates of unemployment fractions for all Dutch municipalities. 2. Spain, to prouce reliable quarterly estimates of consumption expenitures of househol an for the survey of the information Society-Families. 3. Unite Kingom, to prouce 2007/08 mile layer super output area MSOA-level estimates of the proportion of househols in poverty for Englan an Wales, calculate base on equivalise househol income after housing costs an prouce using the same methoology that was use to prouce mean income estimates. 4. Brazil, to generate estimates of poverty an inequality for 5500 Brazilian municipalities. Interconnections with other moules 23. Themes that refer explicitly to this moule 1. Weighting an Estimation Main Moule 2. Weighting an Estimation Small Area Estimation 3. Quality Aspects Quality of Statistics 24. Relate methos escribe in other moules 1. Weighting an Estimation Generalise Regression Estimator 2. Weighting an Estimation Synthetic Estimators for Small Area Estimation 3. Weighting an Estimation Composite Estimators for Small Area Estimation 4. Weighting an Estimation EBLUP Area Level for Small Area Estimation (Fay-Herriot) 11

12 5. Weighting an Estimation Small Area Estimation Methos for Time Series Data 25. Mathematical techniques use by the metho escribe in this moule 1. ML or REML by means of Newton-Raphson algorithm 26. GSBPM phases where the metho escribe in this moule is use Calculate aggregates 27. Tools that implement the metho escribe in this moule 1. Eurarea SAS macro an function ( 2. R functions prouce by ESSnet SAE ( 3. R package sae2 (BIAS project website: 4. SAMPLE project coes in Process step performe by the metho Estimation of parameters in isaggregate omains 12

13 Aministrative section 29. Moule coe Weighting an Estimation-M-EBLUP Unit Level for SAE 30. Version history Version Date Description of changes Author Institute first version Michele D Alò, Anrea Fasulo secon version Michele D Alò, Anrea Fasulo secon version Michele D Alò, Anrea Fasulo thir version Michele D Alò, Anrea Fasulo preliminary release final version within the Memobust project ISTAT ISTAT ISTAT ISTAT 31. Template version an print ate Template version use 1.0 p Print ate :35 13