Division of Preventative and Behavioral Medicine. University of Massachusetts Medical School, Worcester, MA 01655

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1 USE OF AUXLARY FORMATO A DESG-BASED RADOM PERMUTATO MODEL Wenjun Li Division of Preventative Behavioral Medicine Universit of Massachusetts Medical School, Worcester, MA 0655 Edward J. Stanek Department of Biostatistics Epidemiolog Universit of Massachusetts, Amherst, MA 0003 ABSTRACT This paper illustrates that prediction techniques, commonl used in model-based approach, can be applied to incorporating auiliar information in design-based estimation. Design-based estimators of population parameters (such as mean or total) under simple rom sampling without replacement using auiliar information are developed b etending the designbased prediction method of Stanek, Singer Lencina (004) Stanek Singer (004). The sampling is represented with indicator rom variables, the joint permutation of the response auiliar variables is represented using a set of simultaneous permutation equations. Known auiliar information is incorporated through centering the auiliar variables The content of this article is a part of the first author s dissertation conducted at the Department of Biostatistics Epidemiolog, Universit of Massachusetts, Amherst, Massachusetts. This research was partiall supported b a H grant (H-PHS-R0-HD36848). The authors wish to thank Drs. Julio Singer, John Buonaccorsi, Carol Bigelow for their constructive comments. Corresponding author, address: Wenjun.Li@UMASSMED.EDU Deleted: to Edward J. Stanek. auiiar_simplified_005-ejs.doc 06/9/05

2 on their respective known means. The derived estimators, defined as linear functions of the sample, are unbiased have minimum mean squared error. The have functional forms similar to model-assisted calibration estimators. Unlike these estimators, the are developed using optimalit criteria identical to those used in a model-based prediction approach. Ke words: auiliar variable, design-based inference, prediction, finite sampling rom permutation model, simultaneous permutation Running title: Use of auiliar information auiiar_simplified_005-ejs.doc 06/9/05

3 . TRODUCTO Estimators can be made more precise b accounting for auiliar information such as gender, age, income chronic disease-bearing histor that are partiall or completel known in a population. Methods of improving estimation with auiliar information have been discussed in both model-based (Bolfarine Zacks 99; Valliant et al. 000) design-based (Cassel Deleted: prediction et al. 977; Cochran 977; Särndal Wright 984; Deville Särndal 99; Särndal et al. 99) approaches. Model based approaches include.. Design based approaches include. However, the underling conceptual frameworks differ between the two approaches. The design-based approach is based on a probabilit model that arises from appling a sampling design to an identifiable finite population. Parameters are defined in the population, optimal (unbiased) estimators minimize the epected mean squared error (MSE), with epectation taken over all possible samples (Horvitz Thompson 95; Cassel et al. 977). The model-based approach is based on a probabilit model that is postulated for a set of rom variables in a superpopulation (Rao 997; Brewer 999; Valliant et al. 000). An identifiable population is not defined; instead a realization of a set of superpopulation rom variables sts for the population. Parameters are defined in the superpopulation (not the population), linear combinations of rom variables are the target of prediction, with bias, variance mean squared error of the predictor defined in terms of the epectation over all possible realizations of a model for the superpopulation (Brewer 995). The actual sample design plas no role in the inference. Deleted: probabilit model Deleted: s Deleted: uses the sampling design as the basis for developing estimators. Deleted: E Deleted: prediction- or Deleted: assumes that the target population is a realization of a set of rom variables called superpopulation Deleted:, of which some of the rom variables (a sample) are observed. The Deleted: are Deleted: that connects the variable of interest to a set of auiliar variables The necessit of defining a superpopulation, the lack of an identifiable population is a limitation of the model-based approach. However, this approach has been more readil adapted to comple analsis needs, etending man methods in traditional statistics. We develop a new method for accounting for auiliar variables in a design based framework. The method is based on a rom permutation probabilit model that represents a without replacement simple rom sample (SRS) from a finite population. Parameters are clearl defined in terms of linear auiiar_simplified_005-ejs.doc 06/9/05 3

4 functions of population units. To estimate a parameter, we representing it as a linear combination of sample remainder rom variables, we predict the linear combination of the remainder rom variables borrowing methods from the model based approach. The results provide a solid theoretical bridge between the two frameworks. Surve statisticians have made efforts to bridge the prediction- design-based approaches (are there pure design based model based approaches?), to find estimators that have good properties are interpretable within both approaches (Särndal Wright 984; Brewer et al. 988; Deville Särndal 99; Särndal et al. 99; Brewer 995; Brewer 999; 999; Rao 999). There are two subclasses of such methods for incorporating auiliar information, namel, model-assisted (Cassel et al. 976; Särndal et al. 99) calibration approaches (Deville Särndal 99). n a model-assisted approach, efficient estimators (are these model-based?) with good design-based properties are developed for a superpopulation model, with efficienc determined from a design based? probabilit sampling distribution (Deville Särndal 99). The generalized regression estimator (GREG) is an eample of this tpe (Cassel et al. 977; Särndal et al. 99). n the calibration approach, a class of calibrated estimators is derived b appling a new set of weights that has minimum Deleted: arises from appling a sampling design to an identifiable finite population Deleted: Then, r Deleted: target parameter as a nserted: arises from appling a sampling design to an identifiable finite population. nserted: Then, representing Deleted: a prediction based approach is used to predict the linear combination of the remainder rom variables. nserted: a prediction based approach is used to predict the linear combination of the remainder rom variables. The results provide a solid theoretical bridge between the two frameworks. The population parameters are clearl defined. Estimators are based on traditional statistical methods of prediction of functions of rom variables representing the remainder of the population using a probabilit model arising from the sample design. Deleted: The population parameters are clearl defined. Estimators are based on traditional statistical methods of prediction of functions of rom variables representing the remainder of the population using a probabilit model arising from the sample design. Deleted: Deleted: the distance to the benchmark weights (such as the inverse of inclusion probabilities) while calibrated to known population quantities on some set of auiliar variables. B forcing the new weights be close to initial weights, one wishes to preserve (or approimate) the unbiasedness design-consistence properties of the Horvitz-Thompson estimators (Deville Särndal 99). The GREG calibration approaches combine ideas from model based design based approaches, but lack an integrated theoretical framework. An integrated design based approach that takes advantage of prediction methods was developed b Stanek, Singer Lencina (004). Using a rom permutation model representing SRS, the showed that the estimation of population parameters (such as the mean) can be accomplished b developing predictors of linear functions of non-sampled rom Deleted: Along this line of research, an intriguing question is how to appl Deleted: commonl used in modelbased approaches in a design-based framework. The design-based prediction method Deleted: provides a formal platform for such attempts. auiiar_simplified_005-ejs.doc 06/9/05 4

5 variables using prediction techniques. Furthermore, Stanek Singer (004) etended this approach to prediction of rom effects in balanced two-stage cluster sampling with response error. n this paper, we etend their method to incorporate auiliar information. Beginning with a vector of responses for each subject in a finite population, SRS of subjects is represented b a rom permutation probabilit model that permutes subject response vectors. Auiliar information is incorporated through a simple linear transformation. This paper is organized as follows. We first present definitions notation, introduce the rom permutation model in the simplest scenario with one response one auiliar variable. We then define the population parameter of interest derive the best linear unbiased estimator (BLUE). Subsequentl, results are etended to scenarios with multiple auiliar variables. We conclude b illustrating the method with an eample, include results from a small scale simulation evaluating empirical estimators (Wenjun, d like to follow the notation of the JASA paper, not the JSP paper).. DEFTOS AD OTATO Let the population consist of subjects labeled s =,,,. We assume the labels are non-informative serve onl to identif the subjects. Associated with subject s is a nonstochastic potentiall observable response vector, ( ) interest, s = (( ks) ) s s, where s denotes the response of is a p vector of auiliar variables. We represent the population mean b the vector of means, ( µ µ ), where = ( µ ) + + matri variance is defined b the ( p ) ( p ) = ( ) Σ () p = k k µ is a p vector. The population k Σ, where Σ = Σ, is a p p matri. ndividual terms are Deleted: More specificall, we present a design-based approach that uses prediction methods to estimate population parameters (such as the total or the mean) with auiliar information under simple rom sampling without replacement (SRS). The method frames the problem as simultaneous permutations of multiple variables with an underling rom permutation model. Deleted: o additional model assumptions are required. Deleted: We illustrate this method in the simplest scenario with one response one auiliar variable. Deleted: first, followed b the basics of the Deleted: underling SRS. Deleted: The notations mostl follow Stanek, Singer Lencina (004). We introduce the simultaneous rom permutation models the method for representing sampling with a permutation model net. Deleted: Later Deleted:, we Deleted: the results Deleted: presenting Deleted:, some discussion. Deleted: i nserted: in, d like to follow the notation of the JASA paper, not the JSP paper). Deleted: We consider a case of estimating the prevalence of cigarette smoking in a communit under simple rom sampling without replacement, where the size of the population is finite known. Deleted: b Deleted: each s Deleted: (e.g., j ) there Deleted: vector auiiar_simplified_005-ejs.doc 06/9/05 5

6 defined in the usual manner, such that µ = s, µ k s= = = µ k = k s µ s µ k k ks s=. 3. SAMPLG AD THE RADOM PERMUTATO MODEL s= =, = ( µ ) s ks, s s= We define rom variables representing a simple rom sample of size n from a population as the first n variates in a rom permutation of subjects. We assume that all subjects are roml permuted with equal probabilit assigned to each possible permutation. The rom variables representing a permutation define the rom permutation model. Following Stanek, Singer Lencina (004), we eplicitl represent these rom variables in terms of a set of indicator rom variables U, i =,,,, that have a value of one if subject is s is in position i in a permutation, zero otherwise. Using this notation, response for the subject in position i in a permutation is represented b the rom variable Y = U. Using vectors, Y i i = U where = i i i i selects a subject in position i = (( s )) Defining U = ( U U U ) i is s s= U U U U represents a vector of rom variables that represents the population response vector., the rom variables representing a permutation of response are given b Y= U. Similar vectors can be defined to represent a permutation of auiliar variables, Xk = U k, k =,..., p. The vector matri notation is valuable to simplif representation of the problem the development of estimators. We use properties of the indicator rom variables to evaluate the epected value variance of the rom variables. Taking epectation over all possible permutations, EU is = for all i,..., ; s,..., = =. As a result, E Y = µ for all i=,...,. n a similar i Deleted: Deleted: For simplicit, suppose that the response variable is an indicator of nserted: For simplicit, suppose that the response variable is an indicator of Deleted: is t Deleted: the cigarette smoking status, that there is a single auiliar variable (i.e. p = ) corresponding to nserted: t nserted: that there is a single auiliar variable (i.e. p = ) corresponding to Deleted: s Deleted: the subject s age (which we represent b s ). We assume that age is known for all subjects in the population. nserted: the subject s age (which we represent b s ). nserted: We assume that age is known for all subjects in the population. Deleted: is an auiliar variable (e.g., age) values of j are known for all subjects in the population. Formatted: Bullets umbering Deleted: We represent the... [] Deleted: S Deleted: Suppose Deleted: individuals Deleted: of the population Deleted: the values of the permuted... [] Deleted: formalize this b defining Deleted: Deleted: the Deleted: the th i Deleted: 0 Deleted: Let Deleted: where... [3] auiiar_simplified_005-ejs.doc 06/9/05 6

7 manner, EUU is i s = ( ) when i i s s, EU ( is ) = when i = i s = s, EUU = otherwise. Using these epressions, we can show that ( Y ) = is i s 0 for all i,..., =, that cov Y, Y = when i i. i i var i We combine these results to epress the epected value variance of a rom permutation using vector notation. As a result, vector; J = E Y = µ where is an column var Y = P, where P = J, is an identit matri,. Similar epressions appl to auiliar variates. 4. SMULTAEOUS RADOM PERMUTATO MODEL We net represent rom variables arising from the rom permutation of response auiliar variates simultaneousl. For simplicit, (in Section 4-7) suppose that the response variable is an indicator of a subject s cigarette smoking status, that there is a single auiliar variable (i.e. p = ) corresponding to the subject s age (which we represent b s ). We assume that age is known for all subjects in the population. We define the simultaneous rom permutation model b concatenating the permutation vectors for response the auiliar variates. The resulting model is similar to the seemingl unrelated regression model (Zellner 963) such that where ( ) var Z= Y X, = Z = Σ P. Y 0 µ = + E X 0 µ. () Z= Gµ + E G, = ( ) µ µ µ. n this model, ( Z) E = Gµ Deleted: dom variables. Then the matri of rom variables representing a joint permutation of two variables is given b U ( ) = ( Y X ), where Y X are the rom vectors corresponding to the response auiliar variables, respectivel. Epressions for the corresponding mean variance can be developed using the properties of U. Since all permutation are equall likel, E ( U) = J, where J is an matri of ones, cov( vec( U) ) = P P, P = J is an identit matri. Formatted: Font: (Default) Arial, pt, Lowered b 3 pt E X = µ Deleted: ( X) = P. Using var Y U 0 = X 0 U show that E Y = Gµ X 0 G =, 0 = µ µ ( ) µ, that Deleted: a Deleted: with a set-up Deleted: as, we can, where Formatted: Left, ndent: Left: 0" Deleted: Deleted: For clarit, let ( )... [4] Z= Y X, the above model can be rewritten in a compact form,... [5] Deleted: () auiiar_simplified_005-ejs.doc 06/9/05 7

8 The mean age, µ, is known since age is assumed to be known for all subjects. This implies that the rom variables arising from permuting the auiliar variable are constrained b X i i = µ =. We eliminate 0 µ from model () b subtracting from µ 0 both sides (which is equivalent to multipling each term in the model b R = 0 P ). Representing the elements of where Z = RZ X b G = RG. Since X = X µ, the transformed model is i i Y = µ + E X 0 Z = G µ + E () P is idempotent, E ( ) Z = G µ, cov Z = Σ P. [The design matri structure is similar to the design matrices described for growth curve models b Rao (967). Put in discussion] 5. SAMPLG AD PARTTOG THE RADOM MATRCES We represent rom variables for a SRS of size n b the first n elements of the rom permutation vectors. The sampled ( Z ) remaining ( ) n 0 n ( n) obtained b pre-multiplication b K =, i.e., KZ ( ( n) n n) 0 non-stochastic, it follows that E ( ) ( µ Z = n 0 ), E ( ) = ( µ n ) Z portions of of Z 0, Z V V, cov =, where V = Σ P n,, V = Σ P Z V, V ( n), Z can be Z =. Since K is Z Deleted: We assume that i =,,, thus known. Formatted: Centered i, µ are Deleted: To incorporate this constraint in the model, we transform X into X b subtracting µ from each element of X, i.e., X j = X j µ or equivalentl X = P X. Since P is idempotent, the column rank of X is the same as that of X cov X = cov X. Let 0 R =, define 0 P Z = RZ, RG = G 0 where ( ) ( ) G = 0, E = RE. The reparameterized model that incorporates the constraint is given b Y = µ + E X 0 or in a compact form,... [6] Deleted: is model has a Deleted: of Deleted: Suppose a sample of size n is selected via SRS from a finite population of size. Deleted: the sample as Deleted: units in a rom permutation of population units. Deleted: SRS occurs when each permutation is equall likel. Deleted: elements Deleted: partitioned into the sampled ( Z ) remaining ( Z ) Deleted: a properl specified permutation matri... [7] auiiar_simplified_005-ejs.doc 06/9/05 8

9 V, = V, = Σ J n ( n), Pab, = a J a J = n n n n. Consequentl, the b partitioned model that reflects SRS sampling can be represented as Z G = µ + E, () Z G Deleted: is an n ( n) matri of ones. Deleted: 3 where = ( ) ( ) G = 0. G n 0 n n ( n) 6. PARAMETER OF TEREST This is problematic. The coefficients c i depend on position, so can t think of a rational for an coefficient other than ci = k for all i =,...,. f c i can be some other values, then cy i= i i ma be a rom variable. f it is a rom variable, then we would predict it, not Formatted: Font: (Default) Arial, pt, Lowered b 8 pt estimate it. think we should focus on estimating the population mean (not the total), in which case we don t have to introduce new notation. This would also impl that ci = k. What do ou think? We assume that the parameter of interest is a linear function of the permuted rom n i= i i i= i i i= n+ i i, or in matri format, ( don t like the variables, namel, T = cy = cy + cy notation T, since this represents a rom variable) T = CZ = C Z + C Z, () Deleted: 4 where C= ( c 0 ), = ( ) ( ) C c 0 n ( n) c = ( ) = ( c c c ) c c c n n+ n+ C = c 0, c = ( c c c ), c denoting vectors of known constants. f c =, i =,,,, then T is the population total of the response variate. After sampling, onl i the second term in the right h side of () will be unknown; thus, estimatingt, is equivalent to Deleted: (4) auiiar_simplified_005-ejs.doc 06/9/05 9

10 find a predictor of cy i n i i C Z = = +. We develop the best linear unbiased predictor (BLUP) of C Z, refer to the estimator of µ as the BLUE of µ. 7. BLUE OF A LEAR FUCTO We require the estimator of cy i i to be a linear function of the sample, n n wy i i wx i= i= i i i= n+ wz = +, where w = ( w ) w is a n vector of coefficients, ( ) w = w w w w = ( w w w ), to be unbiased, i.e., E n ( ) ( = E ) n wz C Z, have minimum MSE. The estimator of T is given b n n n ( ) i= i i ( i= i i i= i i ) P = C + w Z = cy + w Y + w X. (3) Deleted: 5 The unbiased constraint requires that wg C G = 0. The variance of P is, var P = wvw wv C + C V C. With such setup, the prediction theorem of Roall (976) can be applied. The minimum variance unbiased estimator is obtained b minimizing the function, { } Φ w = wv w w V C + wg C G, (4) where λ is a Lagrangian multiplier, resulting in After simplification, (5) reduces to, where f = n, λ ( ) ( ) ˆ w = V V, + G GV G G GV V, C. (5) wˆ i f = c i= n+ i, i =,, n. (6) n f β β =. Consequentl, the estimator its variance are Deleted: without difficult. Deleted: amel, an Deleted: with minimum variance Deleted: 6 Deleted:. The unique solution for w is in a format similar to that presented in Roall (976), Deleted: 7 Deleted: (7) Deleted: 8 auiiar_simplified_005-ejs.doc 06/9/05 0

11 n ˆ P= cy + c Y X i i n n β µ = = + i i i ( ) (7) Formatted: Font: (Default) Arial, pt, Lowered b 5 pt Deleted: 9 ( ˆ ) n n n i i i i i= i= i= i= n+ var P = c c + n f c c n n n + f c c c n n n n ( ρ ) ( f ) i= i i= n+ i i= n+ i c i n i ( ρ ) = + +, n n (8) Deleted: 0 where ρ = is the correlation coefficient of Y on X. The estimator of a population total µ ma be obtained as an application of (7) (8) b setting c i =, i =,,,. t follows that f wˆ i = f β, i =,,, n, that, Tˆ ny ( n) Y ( X ) n β µ = +, (9) ( Tˆ ) ( ρ ) var f n, (0) = Deleted: (9) Deleted: (0) Deleted: l Deleted: Deleted: where Y is the sample mean of the response variable X is the sample mean of the auiliar variable. This epression is similar to the epressions commonl seen in linear regression estimators, but includes a finite population correction factor. ( This needs re-writing if this is the onl parameter considered.) variance The estimators derived in the previous section depend on the covariance. While ma be known,, on the is often unknown in practice. A simple commonl Deleted: between the response variable the auiliar variable Deleted: of the auiliar variable. used approach is to replace, with their sample moments, respectivel, ˆ n n ( Y Y ), ˆ i n = Y i i Y Xi X ˆ n n = = X i i X n = = = i, auiiar_simplified_005-ejs.doc 06/9/05

12 where Y n = Yi n n X i = n i = given b Pˆ z var ( ˆ α P) = X. An approimate normal-based interval estimator for ˆP is i ±, where ˆP is defined b replacing variance components in (8) or (0) Deleted: (0) Deleted: () b their estimates. [ Are ou going to discuss what happens when the auiliar means are not known?] 8. EXTESO TO CASES WTH MULTPLE COVARATES (With the previous changes, this section can be shortened. haven t done so.)etension of the above results to scenarios with multiple auiliar variables is straightforward. Suppose p auiliar variables are used in estimation. The matri of rom variables representing a joint permutation of a response variable the p covariates is given b ( p) = ( p) U Y X X, where X k is the rom vector corresponding to the th k covariate, k =,,, p. A simultaneous rom permutation model can be defined similar to with Z = ( Y X Xp ), G = ( p+ ) = ( p) E ( Z) = Gµ, that cov X Z = Σ P, where Σ = X ΣX covariance between Y X = ( X X Xp ) µ µ µ µ. t follows that, X is a column vector of, Σ X is the variance-covariance matri of X. When auiliar means µ, k =,, p, are known, similar transformation can be applied k to Z, such that Z 0 = Z. A reparameterized model incorporating known auiliar 0 p P information has a form identical to Model, but here = ( p ) G 0. With these, linear unbiased minimum variance estimation can be derived following the same steps in Section 7. auiiar_simplified_005-ejs.doc 06/9/05

13 Detailed derivation is given in the Appendi. The unique solution for the vector of coefficients w is where = X X = ( β β βp ) f wˆ = ci n n, i= n+ f β β Σ with β =, k =,,, p. don t underst this. The diagonal element of Σ is k k k k. The diagonal element of Σ is not k unless Σ is diagonal.?? = X X = ( β β βp ) β Σ Accordingl, the estimator its variance are p ˆ P = cy + c Y X i= i= n+ n k= n i i i β k( k µ k) () Deleted: 3 var ( Pˆ ) = c c + n( f ) c c n n n i i i i i n = i= n i= n i= n+ Deleted: 4 + f c c c n n n n ( ρ ) ( f ) i i i X i= i= n+ i= n+ ci ( X ) n ρ n i= n+ +, () where Y is the sample mean of the response variable X k is the sample mean of the k -th auiliar variable, ρ = Σ is the multiple correlation coefficient of Y on X X X X otice that the onl difference between () (8) is the multiple correlation coefficient instead of ρ. X. A straightforward application of () () is to obtain estimator of population total T ρ X Deleted: (4) Deleted: (0) Deleted: (3) Deleted: (4) b letting c i =, i =,,,, that is, auiiar_simplified_005-ejs.doc 06/9/05 3

14 p ˆ T = ny + ( n) Y β k ( Xk µ k) n, (3) k = ( Tˆ ) ( ρ ) f var = X n. (4) This epression is similar to the epressions commonl seen in multiple linear regression models (Grabill 976), but includes a finite population correction factor. Results (3) (4) are also Deleted: 5 Deleted: 6 Deleted: (5) Deleted: (6) the same as difference estimators with optimal coefficients (Montanari 987). [Somewhere there should be the relationship with the GREG calibration approaches.] 9. APPLCATO EXAMPLES n this section, we appl the BLUE to an eample related to the estimation of the total number of smokers in a finite population of size. The smoking status is coded as for smoker 0 otherwise. The age of the subjects is denoted b, could be continuous or dichotomous (or categorical), e.g., 0 for ounger (<65) or for older ( 65). When age is continuous, µ is the population mean age; when age is dichotomous, µ is its proportion of persons aged 65. n either case, we assume µ is known. The estimator ˆP its variance constitute simple eamples of (9) (0). Furthermore, β = is the population Deleted: () Deleted: () regression coefficient of smoking status on age or age categories, Y is the proportion of smokers, X is the mean age (or proportion aged 65) in the sample, squared population correlation coefficient between smoking status age. ρ = is the When the auiliar variable is dichotomous, i.e., the proportion of subjects 65 is known, the total numbers of persons aged <65 65 in the population ( a b, respectivel) are also known, ( ) =. When a b are unknown, we replace auiiar_simplified_005-ejs.doc 06/9/05 4

15 with their sample estimates, m n ˆ = n n ( m) ( ) ˆ nn a b ma m b =, where n a n n n n ( ) a b n b are numbers of the ounger older in the sample, m ma mb = +, m a numbers of the ounger the older smokers in the sample, respectivel. As a result, ˆ nn a b ma m b β =. Consequentl, n( n ) ab na nb m b are the m nanb nb n a ma m b ˆ µ = + n n n b a na nb its variance is V ( ) ( m) ( ) ˆ µ f mn = n n n ρ f we replace, ( ˆ ) nn a b nn a b ma m b, where ˆ ρ =. n( n ) a b m( n m) na nb with their sample estimates ˆ, ˆ then ˆ β = ma mb n a n, ˆ m m µ a a b b = +, its estimated variance is b na nb V ( ˆ µ ) nn a ˆ = n n b ( ) f ma m b = m +. This estimator is a post-stratified estimator of prevalence n( n ) na n b rate, which has been considered in calibration (Deville Särndal 99) model-assisted approaches (Särndal et al. 99). Etension to scenarios where age is classified in multiple categories is straightforward. When population covariances of the auiliar variables are used, the estimator is p m nk n nk ˆ µ = mk m n n( n ), its variance is k = k n var ( ˆ µ ) ( m) ( ) n ( n ) p f mn ( nm mn ) =. When sample variance of the k k n n n k = k, auiiar_simplified_005-ejs.doc 06/9/05 5

16 p k mk auiliar variables is used, the BLUE of µ is ˆ µ =, its variance is n var ( ˆ µ ) f m = n n. p k m k = nk ( ) k = k (This section needs some work, rewriting with new simulation results) mpact of choices of variance components on performance of the estimators To underst the impact of estimating variance components on the performance of the estimators, we conducted a Monte Carlo simulation. A series of hpothetical populations of sizes 50, 00, 00, 400, 800,600 were generated; each with an auiliar variable representing 50% men. n all populations, the prevalence of smokers is 60% for men 0% for women. We then used () (3) to estimate the population prevalence. The simulation stud was carried out as follows. From each population of size = 50,00, 00,400,800,600, a series of 5,000 samples of size n, where n = 5,50,00,00,400,800 n<, were drawn via SRS. For each sample, we estimated the proportion of smokers, its variance using the following methods: ) Simple Epansion Estimator without auiliar information (SEE), ) Design-based Estimator using Sample (DES) estimate of population variance of gender ( ˆ ) estimated covariance ( ˆ ), 3) Design-based Estimator using known Population variance (DEP) of gender ) sample estimates of ( ˆ ) (, 4) Design-based estimator using known Population variance covariance (POP). The 95% confidence intervals (Cs) were computed using the variance estimators based on normal approimations. We then compared the overall performance of these estimation methods in terms of mean squared error (MSE), coverage rates auiiar_simplified_005-ejs.doc 06/9/05 6

17 average length of the confidence intervals. Simulation results are summarized in Tables. These simulations confirm that all these estimators are asmptoticall unbiased, or at least that the biases are negligible; especiall when the sample size is not small. There was a %-6% reduction in the MSE of the estimators relative to SEE for small large sample sizes (Table ). There is evidence that using known population auiliar variances (DEP) is superior to using its sample estimates (DES) in terms of MSE reduction, especiall when sample sizes are relativel small (n<00). The coverage rate of the confidence intervals constructed using the variance estimators of the four methods are all ver close to their nominal level (95%), but the coverage rates of DEP DES confidence intervals are lower than SEE POP. The width of the DEP DES confidence intervals tend to be the narrower than those of SEE POP estimators. When sample sizes are moderate or large, there are little differences in coverage rates between the methods, or in the difference between DEP POP methods. 0. DSCUSSO We proposed an alternative method of developing estimators in the presence of auiliar information using a rom permutation model under SRS. This framework does not require parametric distributions. The rom variables are attributable onl to rom sampling (permutation). We represented the joint permutation of response auiliar variables with a setup similar to seemingl unrelated regression. Best linear unbiased estimators are derived under design-based framework mimicking estimation techniques that are commonl applied in prediction-based approaches (Roall 976; Valliant et al. 000). With this proposed method, we obtain eplicit epressions for the joint permutation of response multiple auiliar variables, formulae for the population mean vector their variance-covariance matri. These results are similar to those commonl seen in literature regarding seemingl unrelated regression (Zellner 96; Zellner 963; Revankar 974; Revankar auiiar_simplified_005-ejs.doc 06/9/05 7

18 976; Srivastava Giles 987; Gao Huang 000; Liu 000), but our results incorporate the finiteness of the population do not rel on parametric or superpopulation model assumptions. The estimators presented in this article are identical to the estimators obtained under a design-based approach (Cochran 977) assuming a linear regression model or the difference estimator (See Section 6.3 in (Särndal et al. 99)) GREG estimators derived under modelassisted approach (Särndal et al. 99). n fact, an appeal of the approach is providing a formal foundation for these estimators. n addition, the estimators their variances are the same as for asmptoticall design-unbiased estimators their asmptotic design-variance derived from a model-calibration approach based on a superpopulation framework (Wu Sitter 00). However, our results depend neither on the required assumptions b superpopulation models model-assisted approaches nor on arbitrar distance measures between adjusted initial weights that are critical for the conventional calibration approach. n addition, the estimators proposed here their variances are eact not approimated. These estimators require the knowledge of both X Σ X, their variances are alwas smaller than the variances of simple epansion estimators. The gain in the precision of prediction is proportional to the squared (multiple) correlation coefficient of the response variable auiliar variable(s). When both X Σ X are known, the derived estimator is the same as difference estimator with optimal coefficients (Montanari 987). When both X Σ X are unknown, we appl these results b substituting their sample estimates (method of moments), the resulting estimators will be the same as usual calibration estimators. The approach ields unbiased linear estimators whose weights are not sampledependent (when variance components are known). n contrast, usual calibration technique of incorporating auiliar information leads to weights that are conditioned on the observed sample, the corresponding estimators are not necessaril unbiased. The design-based rom auiiar_simplified_005-ejs.doc 06/9/05 8

19 permutation model estimators are optimized in terms of its mean squared error or its variance, a direct natural measure of precision; while calibration estimators are optimized in terms of the distance between adjusted naïve weights, with no assurance about minimum MSE or variance. Another advantage of the estimators is that the are alwas unique when the population variance (or variance-covariance matri) of the auiliar variable(s) is known nonsingular. Finall, we acknowledge the limitations of this research. We considered onl linear unbiased minimum variance estimators under SRS use of auiliar variables at the estimation stage. t remains to underst the theor applications of this design-based approach in other estimation/prediction problems, such as small-domain estimation, unit nonresponse adjustment, estimation of non-linear functions. t is interesting to eamine whether relaing the unbiasedness constraints or using other optimal criteria such as log-determinants will result in improvement or loss of efficienc in estimation. Another challenge is development of estimation/prediction methods for more comple sampling design, such as stratified or cluster rom sampling longitudinal surve sampling. We are currentl making efforts toward these directions, building on the results of these methods Stanek Singer (004). auiiar_simplified_005-ejs.doc 06/9/05 9

20 Reference Bolfarine, H. S. Zacks (99). Prediction Theor for Finite Populations. ew York, Springer- Verlag. Brewer, K. R. W. (995). Combining design-based model-based inference (Chapter 30). Business Surve Methods. B. B. Co, DA; Chinnappa, B; Christianson, A; Colledge, MJ; Kott, PS. ew York, John Wile & Sons, nc.: Brewer, K. R. W. (999). "Cosmetic calibration with unequal probabilit sampling." Surve Methodolog 5: 05-. Brewer, K. R. W. (999). "Design-based or prediction-based inference? Stratified rom vs stratified balanced sampling." nternational Statistical Review 67(): Brewer, K. R. W., M. Hanif, et al. (988). "How earl Can Model-based Prediction Designbased Estimation Be Reconciled?" Journal of the American Statistical Association 83: 8-3. Cassel, C. M., C.-E. Särndal, et al. (977). Foundations of nference in Surve Sampling. ew York, Y, Wile. Cassel, C. M., C. E. Särndal, et al. (976). "Some Results on Generalized Difference Estimation Generalized Regression Estimation for Finite Populations." Biometrika 63: Cochran, W. G. (977). Sampling Techniques. ew York, John Wile Sons. Deville, J. C. C. E. Särndal (99). "Calibration Estimators in Surve Sampling." Journal of the American Statistical Association 87: Gao, D.-d. R.-B. Huang (000). "Some finite sample properties of Zellner estimator in the contet of m seemingl unrelated regression equations." Journal of Statistical Planning nference 88: Grabill, F. A. (976). Theor application of the linear model. Belmont, CA, Wadsworth Publishing Compan, nc. Horvitz, D. G. D. J. Thompson (95). "A generalization of sampling without replacement from a finite universe." Journal of the American Statistical Association 47(60): Liu, J. S. (000). "MSEM dominance of estimators in two seemingl unrelated regressions." Journal of Statistical Planning nference 88(): Montanari, G. E. (987). "Post-sampling Efficient QR-prediction in Large-sample Surves." nternational Statistical Review 55: 9-0. Rao, C. R. (967). Least square theor using an estimated dispersion matri its applications to measurement of signals. Proceedings of teh 5th Berkele Smposium on Mathematical Statistics Probabilit, Berkele, CA. Rao, J.. K. (997). "Developments in sample surve theor: an appraisal." Canadian Journal of Statistics 5(): -. Rao, J.. K. (999). "Some Current Trends in Sample Surve Theor Methods." Sankha: The ndian Journal of Statistics: Series B 6(): -57. Revankar,. S. (974). "Some Finite Sample Results in the Contet of Two Seemingl Unrelated Regression Equations." Journal of the American Statistical Association 69(345): Revankar,. S. (976). "Use of Restricted Residuals in SUR Sstems: Some Finite Results." Journal of the American Statistical Association 7(353): Roall, R. M. (976). "The linear least-squares prediction approach to two-stage sampling." Journal of the American Statistical Association 7: Särndal, C. E., B. Swensson, et al. (99). Model assisted surve sampling. ew York, Springer- Verlag. auiiar_simplified_005-ejs.doc 06/9/05 0

21 Särndal, C. E. R. L. Wright (984). "Cosmetic Form of Estimators in Surve Sampling." Scinavian Journal of Statistics : Srivastava, V. K. D. E. A. Giles (987). Seemingl unrelated regression equations models: estimation inference. ew York, Dekker. Stanek, E. J., J. d. M. Singer, et al. (004). "A unified approach to estimation prediction under simple rom sampling." Journal of Statistical Planning nference : Stanek, E. J. J. M. Singer (004). "Predicting Rom Effects from Finite population clustered samples with response error." Journal of the American Statistical Association 99(468): Valliant, R., A. H. Dorfman, et al. (000). Finite Population Sampling nference, a Prediction Approach. ew York, John Wile & Sons. Wu, C. B. R. R. Sitter (00). "A model-calibration approach to using complete auiliar information from surve data." Journal of the American Statistical Association 96(453): Zellner, A. (96). "An efficient method of estimating seemingl unrelated regression tests for aggregation bias." Journal of the American Statistical Association 57(98): Zellner, A. (963). "Estimators of Seemingl Unrelated Regression Equations: Some Eact Finite Sample Results." Journal of the American Statistical Association 58: auiiar_simplified_005-ejs.doc 06/9/05

22 Table Mean squared error ( 0,000) of four estimators n=5 n=50 n=00 SEE DES DEP POP SEE DES DEP POP SEE DES DEP POP n=00 n=400 n= SEE Simple Epansion Estimator DES Design-based Estimator using the Sample Estimate of the Variance DEP Design-based Estimator using the partiall known Population Variance POP Design-based Estimator using known Population Variance auiiar_simplified_005-ejs.doc 06/9/05

23 Table Coverage rates width of 95% confidence intervals n=5 n=50 n=00 SEE DES DEP POP SEE DES DEP POP SEE DES DEP POP n=00 n=400 n=800 SEE DES DEP POP SEE DES DEP POP SEE DES DEP POP SEE Simple Epansion Estimator DES Design-based Estimator using the Sample Estimate of the Variance DEP Design-based Estimator using the partiall known Population Variance POP Design-based Estimator using known Population Variance auiiar_simplified_005-ejs.doc 06/9/05

24 Appendi Proof of result (3) (4) When there are p auiliar variables, a simultaneous rom permutation model can be defined similar to with Z = vec ( Y X Xp ) = ( p+ ) G. When auiliar means µ, k =,, p, are known, let k X = P X. The transformation can be represented as Z = RZ, where 0 R = 0 p P, = ( p+ ) G 0, RG G 0 where = ( p ) E = RE. The reparameterized model that incorporates the constraints thus has a form identical to Model. p+ n 0 n ( n) To represent the SRS sampling, let K =, thus the portioned p+ ( ( n) n n) 0 simultaneous permutation model takes the same form of (), Deleted: (3) Z G µ = + E, Z G G 0 G = n 0 ( n). p where = ( n np) A linear function of the rom variables can be defined similar to (), with Deleted: (4) C= ( c 0 p ), = ( np ) C c 0 C = c 0 ( n). The estimator of C p Z can then be defined as a linear function of the sample, ( p ) w w w, wz, where = ( ) w = w w w, is a ( p+ ) n vector of coefficients. With these, a linear unbiased minimum variance estimation can be derived b minimizing the function, { } Φ w = wv w wv C + wg C G. λ Deleted: 6/9/005 nserted: 6/9/005 Deleted: 6/8/005 Li Stanek: Covariance adjusted estimation 6/9/005 A

25 Specificall, differentiating the above function with respect to w λ, setting the derivatives to zeros results in the following estimating equations V G wˆ V, C = ˆ, λ G 0 G C G 0 G = n 0 ( n). The unique solution is p where = ( n np) ( ) ( ) ˆ w = V V, + G G V G G G V V, C. (A.) Since V Σ J, Pn, n= n n V = Σ P n,,, = n ( n) n n n G V G u Σ u with n P n n, n =, we have the following useful identities, ( ) = ( ) ( p ) V V J. Subsequentl, (A.) can be simplified as n u = 0, = p+ n ( n) follows, wˆ = J + ( u Σ u ) ( ) ( ) n n Σ uu P C n ( ) c = u n + uσ u Σ u n. n n p+ n n n, n n Since Σ ( XΣX X ) X ( ΣX X X ) ( ) ( ) X = = X ΣX ΣX X XΣXX ΣX X X, ( ) ( ) ( u Σ u Σ u XΣXX) ( XΣXX) ( ) = = = ΣXX β ΣXX XΣXX where β= Σ. Therefore, we have X X nc wˆ = u n + n n n. β Deleted: 6/9/005 nserted: 6/9/005 Deleted: 6/8/005 Li Stanek: Covariance adjusted estimation 6/9/005 A

26 Consequentl, Pˆ = C + w Z = cy + c Y β X µ n n X n p = cy i i+ ci Y β k( Xk µ k) i= i= n+ n k= where ρ var X X X X 0 ( Pˆ ) = ( C + w) V( C + w) = + ( ) n n n ci ci 0 n f ci ci 0 i n = i= n i= n n i= n+ + f c c c n n n n i i i ( ρ X ) ( f ) i= n i= n+ i= n+ ci ( X ) 0 n ρ n i= n+ +, = Σ is the multiple correlation coefficient of Y on X. 0 Deleted: 6/9/005 nserted: 6/9/005 Deleted: 6/8/005 Li Stanek: Covariance adjusted estimation 6/9/005 A3

27 Page 6: [] Deleted Ed Stanek 6/9/005 8:59 AM We represent the population b an matri ( ), where are column vectors of the response auiliar values, respectivel. The population mean total for the response auiliar variates are given b µ = j j = T =, j= j µ = j j = T = j. We define = j, j= j = = ( ) j = ( j )( j ) j = j =, summarize the variance covariance of the two variates b Σ =. Page 6: [] Deleted Ed Stanek 6/9/005 9:07 AM the values of the permuted units Page 6: [3] Deleted Ed Stanek 6/9/005 9:9 AM where = U U U U, represent a matri of indicator ran i i i i Page 7: [4] Deleted Ed Stanek 6/9/005 :36 PM ( X) = = E µ var X P. Using Y U 0 =, we can show that X 0 U E Y = Gµ, where X 0 G, = 0 ( ) = µ µ µ Y, that cov = Σ P. X Page 7: [5] Deleted Ed Stanek 6/9/005 0:5 AM For clarit, let ( ) Z= Y X, the above model can be rewritten in a compact form, Z= Gµ + E. Page 8: [6] Deleted Ed Stanek 6/9/005 0:5 AM

28 To incorporate this constraint in the model, we transform X into X b subtracting µ from each element of X, i.e., X = X µ or equivalentl j j X = P X. Since P is idempotent, the column rank of X is the same as that of X cov ( X ) = cov X. Let R 0 0 P, define = = Z RZ, RG = ( G 0 ) G 0, where = ( ) the constraint is given b or in a compact form, E = RE. The reparameterized model that incorporates Y = µ + E X 0 = µ + Z G E, () where Page 8: [7] Deleted Ed Stanek 6/9/005 :40 PM partitioned into the sampled ( Z ) remaining ( ) Z portions through