Basic direct and indirect estimators in sae package

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1 Basic irect an inirect estimators in sae package Isabel Molina *, Yolana Marhuena March, 2015 This ocument presents esign unbiase irect estimators an simple inirect estimators of omain means Ȳ, = 1,..., D. For a general ranom sampling without replacement within each omain U. We enote by π j the inclusion probability of j-th unit from -th omain in the corresponing omain sample s an w j = π 1 j is the corresponing sampling weight. A esignunbiase irect estimator of Ȳ is the Horvitz-Thompson (HT) estimator, given by ˆȲ = N 1 w j Y j. (1) j s Unbiase estimation of the sampling variance of the HT estimator requires availability of the secon orer inclusion probabilities π,jk of each pair of units j an k in s. A simple approximation that avois the use of secon orer inclusion probabilities is obtaine by consiering π,jk π j π k an is given by ˆV π ( ˆȲ ) = 1 N 2 w j (w j 1)Yj. 2 (2) j s Uner Poisson sampling, π,jk = π j π k an in that case the estimator in (2) is exactly unbiase. Uner simple ranom sampling (SRS) without replacement within each area U, = 1,..., D, the HT estimator of the mean Ȳ is the usual sample mean ˆȲ = ȳ = n 1 j s Y j, an the (exactly) unbiase estimator ˆȲ ) = (1 f )S 2/n, for S 2 = j s (Y j of the sampling variance is ˆV π ( ȳ ) 2 /(n 1). When the sampling is with replacement within each omain U, an units are selecte with probabilities P j, j = 1,..., N, proportional to some size measure, if we efine new weights w j = (n P j ) 1, the estimator efine in (1) remains unbiase an the unbiase estimator of the sampling variance is given by ˆV π ( ˆȲ ) = 1 ( f w j Y j ˆȲ ) 2, n j s * Department of Statistics, Universia Carlos III e Mari. Aress: C/Mari 126, Getafe (Mari), Spain, Tf: , Fax: , isabel.molina@uc3m.es 1

2 which becomes S 2/n uner SRS with replacement. The post-stratifie synthetic estimator assumes that ata are istribute into K (large) groups calle post-strata that cut across the omains, an such that the within group mean is constant across omains, that is, if Ȳk enotes the mean in the crossing of post-stratum k an omain an Ȳ+k is the mean of post-stratum k, it hols that Ȳk = Ȳ+k, k = 1,..., K. The groups are assume to have large enough sample sizes to allow irect estimation with high efficiency. Since the mean of omain is given by Ȳ = N 1 K k=1 N kȳk, replacing Ȳk = Ȳ+k by the ratio HT estimator ˆȲ R +k = Ŷ +k / ˆN +k, where Ŷ+k is the irect estimator of the total in post-stratum k an ˆN +k is the irect estimator of the population size N +k in the same post-stratum, we obtain the post-stratifie synthetic estimator ˆȲ SY N = 1 N K N k ˆȲ R +k. Note that this estimator requires the population sizes of the crossings between each post-stratum k an omain, N k for all k an. The irect estimator is inefficient for a omain with small sample size. On the other han, the post-stratifie synthetic estimator is biase when the assumption of constant means across omains within a stratum oes not hol. To balance the bias of a synthetic estimator an the instability of the irect estimator, [1] propose the sample-size epenent (SSD) estimator efine as a composition of the two mentione estimators, that is, ˆȲ SSD = φ ˆȲ k=1 SY N + (1 φ ) ˆȲ, where the composition weight φ epens on the sample size of the omain as { 1, ˆN φ = δn ; ˆN /(δn ), ˆN < δn, for a given constant δ > 0 that controls how much weight is attache to the synthetic estimator, with larger value of δ meaning that more strength is borrowe from other omains. However, if the expecte sample size is small, then ˆN the SSD estimator is not borrowing strength in omains with δn even if they have small sample sizes. Functions irect(), pssynt() an ss() give respectively irect, poststratifie synthetic an sample size epenent estimates. The calls to these functions are: irect(y, om, sweight, omsize, ata, replace = FALSE) pssynt(y, sweight, ps, omsizebyps, ata) ss(om, sweight, omsize, irect, synthetic, elta = 1, ata) Function irect() returns unbiase irect estimates of the area means, where the result epens on the sampling esign specifie through the sampling 2

3 weight vector sweight an the argument replace for with or without replacement sampling. We must provie the area population sizes in the ata frame omsize, whose first column must contain the area coes. In pssynt(), we must specify our selecte post-stratifying variable in argument ps. The population sizes of each crossing between omain an post-strata must be specifie in the ata frame omsizebyps, whose first column must be again the area coes. Function ss() gives SSD estimators obtaine by composition of irect an synthetic estimators. We nee to introuce the irect estimators (irect) an the synthetic estimators (synthetic) to compose, together with the constant δ (elta) involve in the SSD estimator. Domain coes (om) an omain population sizes (omsize) are also require arguments. The vector of sampling weights (sweight) must be inclue in the three functions. The variables specifie in y, om, sweight an ps can be selecte from the ata set specifie in argument ata. Example. Poverty mapping In this example, we calculate several simple estimates of poverty inciences in Spanish provinces, namely irect estimates, post-stratifie synthetic estimates with eucation levels as post-strata an SSD estimates obtaine from the composition of irect an post-stratifie synthetic estimates. The poverty incience for a province is the province mean of a binary variable taking value 1 when person s income is below a given poverty line an 0 otherwise. Direct estimates can be obtaine easily applying the usual theory for means to this binary variable. First, we loa the ata set incomeata containing the input ata for each iniviual an the ata sets sizeprov an sizeproveu containing the population sizes an the population sizes by eucation level, respectively. > library("sae") > ata("incomeata") > ata("sizeprov") > ata("sizeproveu") Next, we efine the poverty line z, calculate the binary variable poor, with value 1 if the corresponing income value is below the poverty line an 0 otherwise, an calculate province poverty inciences as province means of this variable. > z < > poor <- as.integer(incomeata$income < z) We use the province name provlab as the omain coe (om) an calculate irect estimates. 3

4 > Popn <- sizeprov[, c("provlab", "N")] > <- irect(y = poor, om = incomeata$provlab, + sweight = incomeata$weight, omsize = Popn) Next, we calculate post-stratifie synthetic estimates with eucation levels as post-strata. For the function pssynt(), we construct the ata frame omsizebyps, containing the omain coes provlab in the first column an, in the remaining columns, the province sizes by eucation level. The names of the columns (except for the first one) in this ata frame must be the eucation levels, namely 0 (age<16), 1 (primary eucation), 2 (seconary eucation) an 3 (post-seconary eucation): > Popn.euc <- sizeproveu[,-2] > colnames(popn.euc) <- c("provlab","0", "1", "2", "3") > PSYN.euc <- pssynt(y = poor, sweight = incomeata$weight, + ps = incomeata$euc, + omsizebyps = Popn.euc) We calculate SSD estimates by composition of the previous irect an poststratifie estimates, an taking the efault value elta=1 in function ss(). Again, the first columns of omsize, irect an synthetic must be the province names. > SSD <- ss(om = provlab, sweight = weight, omsize = Popn, + irect = [, c("domain", "Direct")], + synthetic = PSYN.euc, ata = incomeata) We collect the province names, sample sizes an the three sets of percent poverty incience estimates in the ata frame results: > results <- ata.frame(province = $Domain, + SampleSize = $SampSize, + = $Direct * 100, + PSYN.euc = PSYN.euc$PsSynthetic * 100, + SSD = SSD$ss * 100) > print(results, row.names = FALSE) Province SampleSize PSYN.euc SSD Alava Albacete Alicante Almeria Avila Baajoz Baleares Barcelona Burgos

5 Caceres Caiz Castellon Ceuta CiuaReal Coroba CorunaLa Cuenca Gerona Granaa Guaalajara Guipuzcoa Huelva Huesca Jaen Leon Leria Lugo Mari Malaga Melilla Murcia Navarra Orense Ovieo Palencia PalmasLas Pontevera RiojaLa Salamanca Santaner Segovia Sevilla Soria Tarragona Tenerife Teruel Toleo Valencia Vallaoli Vizcaya Zamora Zaragoza These estimates are plotte in the Figure for each province (area), with provinces sorte by ecreasing sample size. This Figure shows that irect esti- 5

6 mates an SSD estimates are very similar, with irect estimates slightly more unstable. However, the post-stratifie synthetic estimates appear to be too stable, giving practically the same values for all provinces. This estimator is base on the unrealistic assumption of constant poverty incience for all the population with the same eucation level an therefore might be seriously biase. > # Sorte results by ecreasing sample size > results <- results[orer(results$samplesize, + ecreasing = TRUE), ] > plot(results$, type = "n", + xlab = "area (sorte by ecreasing sample size)", + ylab = "Estimate", cex.axis = 1.5, cex.lab = 1.5) > points(results$, type = "b", col = 3, lw = 2, pch = 1) > points(results$psyn.euc, type= "b", col = 5, lw = 2, pch = 2) > points(results$ssd, type = "b", col = 2, lw = 2, pch = 5) > legen("bottom", legen = c("direct", "Post-strat euc", "SSD"), + ncol = 1, col = c(3, 5, 2), lw = rep(2, 3), + pch = c(1, 2, 5), cex = 1.3) Estimate Direct Post strat euc SSD area (sorte by ecreasing sample size) Comparing irect estimates with the EB estimates of poverty inciences obtaine in the ata frame results.eb of Example 5 in [2], we can see that estimates iffer significantly for the 5 selecte provinces an the CVs show great gains in efficiency of EB estimates as compare with irect estimates. 6

7 > [c("42","5","34","44","40"), -4] Domain SampSize Direct CV 42 Soria Avila Palencia Teruel Segovia References [1] Drew, D., Singh, M.P. & Chouhry, G.H. (1982). Evaluation of small area estimation techniques for the Canaian Labour Force Survey. Survey Methoology 8, [2] Molina, I. & Marhuena, Y. (1982). sae: An R package for Small Area Estimation. R Journal, Uner revision. 7