Advanced Topics in Survey Sampling

Transcription

1 Advanced Topics in Survey Sampling Jae-Kwang Kim Wayne A Fuller Pushpal Mukhopadhyay Department of Statistics Iowa State University World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Advanced Topics in Survey Sampling 7/23-24/ / 318 Outline 1 Probability sampling from a finite population 2 Use of auxiliary information in estimation 3 Use of auxiliary information in design 4 Replication variance estimation 5 Models used in conduction with sampling 6 Analytic studies Kim & Fuller & Mukhopadhyay (ISU & SAS) Advanced Topics in Survey Sampling 7/23-24/ / 318

2 Chapter 1 Probability sampling from a finite universe Jae-Kwang Kim Wayne A Fuller Pushpal Mukhopadhyay Department of Statistics Iowa State University World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Probability Sampling U = {1, 2,, N} : list for finite population, sampling frame F = {y 1, y 2,, y N } : finite population/finite universe A ( U) : index set of sample A : set of possible samples Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

3 Sampling Design Definition p( ) is a sampling design p(a) is a function from A to [0, 1] such that 1 p(a) [0, 1], a A, 2 a A p(a) = 1. i.e. p(a) is a probability mass function defined on A. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Notation for Sampling I i = { 1 if i A 0 otherwise d = (I 1, I 2,, I N ) n = N i=1 I i: (realized) sample size π i = E [I i ] : first order inclusion probability π ij = E [I i I j ] : second order inclusion probability Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

4 Design-estimator Characteristics Definition ˆθ = ˆθ(y i ; i A) is design unbiased for θ N = θ (y 1, y 2,, y N ) E{ˆθ F} = θ N, (y 1, y 2,, y N ), where E{ˆθ F} = a A ˆθ(a)p(a) Definition ˆθ is a design linear estimator ˆθ = i A w iy i, where w i are fixed with respect to the sampling design Note: If w i = π 1 i, ˆθ = i A w iy i is the Horvitz-Thompson estimator. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Theorem Theorem If ˆθ = i A w iy i is a design linear estimator, then E{ˆθ n F} = N i=1 w iπ i y i V {ˆθ n F} = N i=1 N j=1 (π ij π i π j ) w i y i w j y j. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

5 Proof of Theorem Because E{I i } = π i and because w i y i, i = 1, 2,..., N, are fixed, N N E{ˆθ F} = E{I i F}w i y i = π i w i y i i=1 i=1 Using E(I i I k ) = π ik and Cov(I i, I k ) = E(I i I k ) E(I i )E(I k ), { N } V {ˆθ F} = V I i w i y i F N = i=1 i=1 k=1 i=1 k=1 N Cov(I i, I k )w i y i w k y k N N = (π ik π i π k )w i y i w k y k Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Horvitz-Thompson Estimator of Total Corollary Let π i > 0 i U. Then, ˆTy = i A ( ) (i) E ˆT y F = T y (ii) V ( ˆTy F) = N i=1 N j=1 π 1 i y i satisfies (π ij π i π j ) y i π i y j π j Proof of (ii): Substitute π 1 i for w i of Theorem Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

6 Unbiased Variance Estimation Theorem Let π ij > 0, i, j U and let ˆθ = i A w iy i be design linear. Then, ˆV = 1 (π ij π i π j ) w i y i w j y j satisfies π ij i A j A ( ) ) E ˆV F = V (ˆθ F Proof : Let g(y i, y j ) = (π ij π i π j )w i y i w j y j. By Theorem N E π 1 ij g(y i, y j ) F = N g(y i, y j ) i,j A i=1 j=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Simple Random Sampling (SRS) Choose n units from N units without replacement with equal probability. 1 Each subset of n distinct units is equally likely to be selected. 2 There are ( N n) samples of size n from N. 3 Give equal probability of selection to each subset with n units. Definition Sampling design for SRS: P(A) = / (N ) 1 n if A = n 0 otherwise. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

7 Lemma Under SRS, the inclusion probabilities are π i = π ij = [( )] N 1 ( )( ) 1 N 1 = n n 1 n 1 N [( )] N 1 ( )( )( ) 1 1 N 2 = n 1 1 n 2 n (n 1) N (N 1) for i j. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Simple Random Samples Let ȳ n = n 1 i A y i, ˆV = n 1 ( 1 N 1 n ) s 2 n, s 2 n = (n 1) 1 i A (y i ȳ n ) 2, Ȳ N = N 1 N i=1 y i, and S 2 N = (N 1) 1 N i=1 (y i ȳ N ) 2. Then, (i) E (ȳ n F) = ȳ N (ii) V (ȳ n F) = n 1 ( 1 N 1 n ) SN 2 ( ) (iii) E ˆV F = V (ȳ n F) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

8 Poisson Sampling Definition: Estimation (of T y = N i=1 y i) I i inde Bernoulli (π i ), i = 1, 2,, N. ˆT y = N I i y i /π i, E{ ˆT y F} = i=1 N π i y i /π i i=1 Variance ( ) Var ˆT y F = N (π i πi 2 )yi 2 /πi 2 = i=1 N i=1 ( ) 1 1 yi 2 π i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Poisson Sampling ( ) Optimal design: minimize Var ˆT y F subject to N i=1 π i = n min N i=1 π 1 i y 2 i + λ( π 2 i yi 2 N π i n) i=1 = λ π i y i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

9 A Design Result Definition Superpopulation model (ξ) : the model for y = (y 1, y 2,..., y N ). Definition Anticipated Variance : the expected value of the design variance of the planned estimator calculated under the superpopulation model. Theorem (1.2.3) Let y = {y 1, y 2,, y N } be iid (µ, σ 2 ), let d = (I 1, I 2,, I N ) be independent of y, and define ˆT y = i A π 1 i y i. Then V { ˆT y T y } is minimized at π i = n/n and π ij = n(n 1)/N(N 1), i j. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Proof of Theorem V { ˆT y T y } = V {E( ˆT y T y F)} + E{V ( ˆT y T y F)} N N = E (π ij π i π j ) y i y j π i π j (i) min N (ii) N i=1 i=1 π 1 i i=1 = µ 2 N i=1 j=1 N (π ij π i π j ) σ 2 π i π j j=1 s.t. N i=1 π i = n π i = N 1 n N j=1 (π ij π i π j )π 1 i π 1 j = V { N i=1 I iπ 1 V { N i=1 I iπ 1 i } = 0 for π i = N 1 n and π ij = N i=1(π i π 2 i ) 1 π 2 i i } 0 n(n 1) N(N 1). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

10 Discussion of Theorem (i) Finding π i and π j that minimize V {ˆθ F} is not possible because V {ˆθ F} is a function of N unknown values Godambe (1955), Godambe & Joshi (1965), Basu (1971) (ii) If y 1, y 2,, y N ξ for some model ξ (superpopulation model), then we sometimes can find an optimal strategy (design and estimator). Under iid & HT estimation, the optimal design is SRS. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Example: Stratified Sampling Definition 1 The finite population is stratified into H subpopulations. U = U 1 U H 2 Within each population (or stratum), samples are drawn independently in the strata. Pr (i A h, j A g ) = Pr (i A h ) Pr (j A g ), for h g where A h is the index set of the sample in stratum h, h = 1, 2,, H. Example: Stratified SRS 1 Stratify the population. Let N h be the population size of U h. 2 Sample size allocation: Determine n h. 3 Perform SRS independently (select n h sample elements from N h ) in each stratum. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

11 Estimation under Stratified SRS 1 HT estimator: where ȳ h = n 1 2 Variance h i A h y hi. ( ) Var ˆTy = ˆT y = H h=1 H N h ȳ h h=1 N 2 h n h where S 2 h = (N h 1) 1 N h i=1 ( yhi Ȳ h ) 2. 3 Variance estimation ˆV ( ) ˆT y = H h=1 N 2 h n h where s 2 h = (n h 1) 1 i A h (y hi ȳ h ) 2. ( 1 n ) h Sh 2 N h ( 1 n ) h sh 2 N h Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Optimal Strategy under Stratified Sampling Theorem (1.2.6) Let F be a stratified finite population in which the elements in stratum h are realizations of iid(µ h, σh 2 ) random variables. Let C be the total cost for sample observation and assume that it costs c h to observe an element in stratum h. Then a sampling and estimation strategy for T y that minimizes the anticipated variance in the class of linear unbiased estimators and probability design is: Select independent simple random samples in each stratum, selecting nh in stratum h, where n h N hσ h ch with C = j n h c h, subject to n h N h, and use the HT estimator. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

12 Comments on Theorem Anticipated variance AV {ˆθ θ N } = E{E[(ˆθ θ N ) 2 F]} [E{E(ˆθ θ N F)}] 2 For HT estimation, E(ˆθ θ N F) = 0 and the anticipated variance becomes AV { ˆT H y T y } = Nh 2 ( n 1 h N 1 ) h σ 2 h Minimizing H optimal allocation h=1 n 1 h h=1 N2 h σ2 h subject to H h=1 n hc h = C leads to the n h N hσ h ch. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Sample Selection Using SAS R PROC SURVEYSELECT Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

13 PROC SURVEYSELECT Probability-based random sampling equal probability selection PPS selection Stratification and clustering Sample size allocation Sampling weights inclusion probabilities joint inclusion probabilities Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

14 Sampling Methods Simple random with and without replacement Systematic Sequential PPS Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 PPS Sampling Methods With and without replacement Systematic Sequential with minimum replacement Two units per stratum: Brewer s, Murthy s Sampford s method Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

15 Digitech Cable Company Digital TV, high-speed Internet, digital phone 13,471 customers in four states: AL, FL, GA, and SC Customer satisfaction survey (high-speed Internet service) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Sampling with Stratification Can afford to call only 300 customers The sampling frame contains the list of customer identifications, addresses, and types Need adequate sampling units in every stratum (state and type) Select simple random sample without replacement within strata Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

16 Sampling Frame CustomerID State Type AL Platinum GA Gold GA Gold GA Platinum FL Platinum SC Platinum FL Gold AL Gold Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Sort Sampling Frame by Strata before Selection proc s o r t data=customers ; by S t a t e Type ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

17 Select Stratified Sample proc s u r v e y s e l e c t data=customers method=s r s n=300 seed = out=samplestrata ; s t r a t a S t a t e Type / a l l o c=prop ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 The SURVEYSELECT Procedure Selection Method Strata Variables Simple Random Sampling State Type Allocation Input Data Set Proportional CUSTOMERS Random Number Seed Number of Strata 8 Total Sample Size 300 Output Data Set SAMPLESTRATA Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

18 Strata Sizes State Type SampleSize PopSize AL Gold AL Platinum FL Gold FL Platinum GA Gold GA Platinum SC Gold SC Platinum Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Data Collection Important practical considerations that the computer cannot decide for you: Telephone interview Rating, age, household size,... Auxiliary variables: data usage, average annual income, home ownership rate,... Callbacks, edits, and imputations Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

19 Survey Objective: Digitech Cable Rate customer satisfaction Are customers willing to recommend Digitech? Is satisfaction related to household size? race? Is usage time related to data usage? Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Digitech Cable Data Collection Survey variables Rating Recommend Usage time Household size Race Auxiliary variables Data usage Neighborhood income Competitors Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

20 Large Sample Results for Survey Samples Complex designs : Weights Few distributional assumptions Heavy reliance on large sample theory Central Limit Theorem Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Review of Large Sample Results Mann and Wald notation for order in probability Sequence: X 1, X 2,, X n, (g n > 0) X n = o p (g n ) X n /g n 0 in probability lim P[ X n/g n > ɛ] = 0 n X n = O p (g n ) X n /g n bounded in probability P[ X n /g n > M ɛ ] < ɛ for some M ɛ, ɛ > 0, n Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

21 Examples of Order in Probability Let x n N(0, n 1 ). Then the following statements hold P{ x n > 2n 1/2 } < 0.05 n P{ x n > Φ 1 (1 0.5ɛ)n 1/2 } < ɛ n therefore x n = O p (n 1/2 ). If x n N(0, n 1 σ 2 ), then x n = O p (?) If x n N(µ, n 1 σ 2 ), then x n = O p (?) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Example of o p Again, let x n N(0, n 1 ). lim P{ x n > k} = 0 k > 0 x n = o p (1) n lim P{ n1/4 x n > k} = 0 k > 0 x n = o p (n 1/4 ) n Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

22 Properties of Order in Probability f n > 0, g n > 0 X n = O p (f n ), Y n = O p (g n ), then X n Y n = O p (f n g n ) X n s = O p (fn s ), s 0 X n + Y n = O p (max{f n, g n }). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Chebychev s Inequality For given r > 0 with E{ X r } <, P[ X A ɛ] E{ X A r } ɛ r. Corollary E{Xn 2 } = O(an) 2 X n = O p (a n ) By Chebyshev s inequality, [ ] Xn P > M ɛ a n E{X 2 n } a 2 nm 2 ɛ < ɛ choose M ɛ = K/ɛ, where K is the upper bound of a 2 n E{X 2 n }. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

23 Central Limit Theorems Lindeberg: X 1, X 2, : independent (µ i, σi 2), B2 n = n i=1 σ2 i 1 n E{(X i µ i ) 2 (Xi µ i ) L I ( X i µ i > ɛb n )} 0 N(0, 1) B n B 2 n i=1 Liapounov: X 1, X 2, : independent (µ i, σi 2), B2 n = n i=1 σ2 i n i=1 E { X i µ i 2+δ} = o(bn 2+δ ), for some δ > 0 n i=1 (X i µ i ) L N(0, 1) B n Note: Liapounov condition Lindeberg condition Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Slutsky s Theorem {X n }, {Y n } are sequences of random variables satisfying X n L X, p lim Yn = c X n + Y n L X + c Y n X n L cx Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

24 Theorem 1.3.1: Samples of Samples Theorem y 1,, y N are iid with d.f F (y) and c.f. ϕ(t) = E{e ity }, i = 1 d = (I 1,, I N ) : random vector independent of y (y k ; k A) d are iid with c.f. ϕ(t) Proof: In book. A SRS of a SRS is a SRS. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Application of Theorem y 1,, y N iid N(µ y, σ 2 y ) SRS of size n N from F N (y k, k A) d are iid N(µ y, σ 2 y ) and (y k, k U A c ) d are iid N(µ y, σ 2 y ) ȳ n N(µ y, σ 2 y /n N ) and ȳ N n N(µ y, σ 2 y /(N n N )) indep of ȳ n ȳ n ȲN N [ 0, n 1 N (1 f N )σ 2 y ] and [ n 1 N (1 f N )s 2 n] 1/2 (ȳn Ȳ N ) t n 1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

25 Finite Population Sampling Motivation Is x n x N = o p (1)? Does x n x N ˆV { x n x N F N } L N(0, 1)? We ll be able to answer these shortly. n N isn t very interesting Need n and N n Need a sequence of samples from a sequence of finite populations Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Sequence of Samples from a Sequence of Populations Approach 1 {F N } is a sequence of fixed vectors Approach 2 {y 1, y 2,, y N } is a realization from a superpopulation model. Notation U N = {1, 2,, N} : N-th finite population F N = {y 1N,, y NN } y in : observation associated with i-th element in the N-th population A N : sample index set selected from U N with size n N = A N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

26 Design Consistency Definition ˆθ is design consistent for θ N if for every ɛ > 0, lim P{ ˆθ θ N > ɛ F N } = 0 N almost surely, where P( F N ) denotes the probability induced by the sampling mechanism. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Design Consistency for ȳ n in SRS Approach 1 (fixed sequence) Assume a sequence of finite populations {F N } s.t. lim N N 1 N (y i, yi 2 ) = (θ 1, θ 2 ), and θ 2 θ1 2 > 0. i=1 By Chebyshev s inequality, where f N = n N N 1. P { ȳ n Ȳ N ɛ F N } n 1 N (1 f N)S 2 N ɛ 2 Approach 2 y 1,, y N iid(µ, σ 2 y ) lim N S 2 N = σ2 y a.s. V [ȳ n Ȳ N F N ] = O p (n 1 N ) ( ) ȳ n ȲN FN = O p (n 1/2 N ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

27 Central Limit Theorem (1.3.2) Part 1 Theorem (Part 1) {y 1N,, y NN } iid(µ, σ 2 ) and 2 + δ moments (δ > 0) SRS, ȳ n = n 1 N i=1 I iy in, Ȳ N = N 1 N i=1 y in [V (ȳ n Ȳ N )] 1/2 (ȳ n Ȳ N ) d L N(0, 1). Proof : Write ȳ n Ȳ N = N 1 N i=1 c in y in, where c in = ( n 1 NI i 1 ). B 2 n = N 2 V (ȳ n Ȳ N ) = N 2 V (ȳ n Ȳ N d) = N i=1 c2 in σ2 = (N n)n/nσ 2 Applying Lindberg CLT Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Theorem Part 2 Theorem (Part 2) Furthermore, if {y in } has bounded fourth moments, then [ ˆV (ȳ n ȲN)] 1/2 (ȳ n ȲN) L N(0, 1). Proof : Want to apply Slutsky s theorem: ȳ n Ȳ N ˆV (ȳ n Ȳ N ) = ȳ n Ȳ N V (ȳn Ȳ N ) V (ȳn Ȳ N ) ˆV (ȳ n Ȳ N ) L N(0, 1). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

28 Proof of Part 2, Continued Then it is enough to show that V (ȳ n Ȳ N ) ˆV (ȳ n Ȳ N ) = (n 1 N 1 )σ 2 y (n 1 N 1 )s 2 n p 1. To show s 2 n p σ 2 y, note that sn 2 σy 2 = 1 σ 2 y = 1 σ 2 y = 1 σ 2 y p 1 n 1 1 n 1 1 n n (y i ȳ n ) 2 i=1 n i µ) i=1(y 2 1 σy 2 n (y i µ) 2 + O p (n 1 ) i=1 1 if E(y i µ) 4 M 4 < n n 1 (ȳ n µ) 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Comment on Theorem The CLT in Theorem is derived with Approach 2 (using superpopulation model) 2 The result can be extended to stratified random sampling (textbook) {y hin } iid (µ h, σ 2 h ) H N H N ˆθ n = N 1 N h ȳ hn, Ȳ N = N 1 N h Ȳ hn h=1 h=1 [ ˆV (ˆθ n ȲN)] 1/2 (ˆθ n ȲN) L N(0, 1) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

29 Poisson Sampling Population: y 1, y 2,..., y N Probabilities: π 1, π 2,..., π N The sampling process is Bernoulli trial for each i (Independent trials) Sample is those i for which the trial is a success N independent random variables Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 CLT under Approach 1 (Fixed Sequences) y 1, y 2, : sequence of real vectors π 1, π 2, : sequence of selection probabilities g i = (1, y i, α N π 1 i, α N π 1 i y i ) where α N = E(n N )/N = n BN /N x i = g i I i, I i Bernoulli(π i ) (i.e. Poisson sampling) Let ˆµ x = n 1 BN N i=1 x i = n 1 BN N i=1 g ii i and µ xn = n 1 BN N i=1 g iπ i E ( ˆµ x F N ) = µ xn ˆT y,ht = N i=1 I i y i π 1 i = α 1 N = α 1 N N i=1 α N π 1 i y i I i N x i4. i=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

30 Theorem 1.3.3, Part 1 Theorem (Part 1) Assume Poisson sampling (i) lim N n 1 BN N i=1 g iπ i = µ x (ii) lim N n 1 BN N i=1 π i(1 π i )g i g i = Σ xx = Σ 11, Σ 22 : positive definite [ Σ11 Σ 12 Σ 12 Σ 22 ], (iii) lim N max 1 k N (γ g k ) 2 N i=1 (γ g i ) 2 π i (1 π i ) = 0 γ s.t. γ Σ xx γ > 0. Then, n BN ( ˆµ x µ xn ) F N L N(0, Σxx ). Proof: textbook Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Theorem 1.3.3, Part 2 Theorem (Part 2) Under conditions (i)-(iii), if (iv) lim N n 1 BN N i=1 π i g i 4 = M 4, then [ ˆV ( ˆT y )] 1/2 ( ˆT y T y ) F N N(0, I ), where ˆV ( ˆT y ) = ( ) ( ) π i (1 π i ) yi yi. π i π i π i i A Proof: textbook Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

31 Theorem CLT for SRS Theorem {y i } : sequence of real numbers with bounded 4th moments SRS without replacement ˆV 1/2 n (ȳ n ȳ N ) F N L N(0, 1) The result is obtained by showing that there is a SRS mean that differs from Poisson mean by o p (n 1/2 ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Function of Means (Theorem 1.3.7) Theorem (i) n( x n µ x ) L N(0, Σ xx ) (ii) g(x) : continuous and differentiable at x = µ x (iii) h x (x) = g(x) : continuous at x = µ x x n[g( x n ) g(µ x )] L N(0, h x(µ x )Σ xx h x (µ x )). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

32 Proof of Theorem [Step 1] By a Taylor expansion g ( x n ) = g (µ x ) + ( x n µ x ) h x (µ ) where µ x is on the line segment joining x n and µ x. [Step 2] Show µ x µ x = o p (1) [Step 3] Using the assumption that h x (x) is continuous at x = µ x, show that h x (µ x) h x (µ x ) = o p (1). [Step 4] Because n [g ( xn ) g (µ x )] = n ( x n µ x ) [ h x (µ x) ] we apply Slutsky s theorem to get the result. = n ( x n µ x ) [ h x (µ x ) + o p (1) ], Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Example on Curvature Approximation: x 2 n x n Let µ = 2 and V ( x n ) = Then N(µ, V ( x n )) N(µ 2, (2µ) 2 V ( x n )) E{ x 2 n} = = µ 2. V { x 2 n} = 2(0.01) 2 + 4(2 2 )(0.01). = 4µ 2 V ( x n ). Let µ = 2 and V ( x n ) = 3. Then E{ x 2 n} = µ 2. V { x 2 n} = 2(2) 2 + 4(2 2 )(3) 4µ 2 V ( x n ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

33 Large Sample Bias n 1/2 (ˆθ θ) L N(0, 1) does not imply E{ˆθ} θ. For example, if then E } {ȳn x n (ȳ n, x n ) N((µ y, µ x ), Σ) and µ x 0, is not defined. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Ratio Estimation (Theorem 1.3.8, Part 1) Theorem x i = (x 1i, x 2i ), X 1N 0, ˆR = x 2,HT / x 1,HT nn 1 (ˆT x T xn ) L N(0, M xx ) n( ˆR R N ) N(0, h N M xx h N ) where ˆT x = i A π 1 i x i, T xn = N i=1 x i, R N = X 2N, and X 1N h N = x 1 1N ( R N, 1) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

34 Proof of Theorem Part 1 ˆR = x 2,HT = X 2N + 1 ( x 2,HT x 1,HT X 1N X X 2N ) X 2N 1N X 2 ( x 1,HT X 1N ) + Remainder 1N Method 1. Mean value theorem & continuity of the first order partial derivatives Remainder= o p (n 1/2 ) Method 2. Second order Taylor expansion + continuity of the second order partial derivatives Remainder= O p (n 1 ) ( R ) ˆR = R + ( x x x HT X N ) ( ) R = R + ( x HT X N ) + 1 ( x 2 ( x HT X 2 R ) N ) x x x ( x HT X N ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Theorem Part 2 Theorem In addition, if {V (ˆT x F N )} 1 ˆV HT (ˆT x ) I = o p (n 1/2 ), then [ ˆV ( ˆR)] 1/2 ( ˆR R N ) L N(0, 1) where ˆV ( ˆR) = i A j A π 1 ij (π ij π i π j )π 1 i ˆd i π 1 j ˆd j, ˆd i = ˆT x1 1 (x 2i ˆRx 1i ), ˆT x = i A π 1 i x i, T xn = N i=1 x i, R N = x 2N x 1N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

35 Remarks on Ratios 1 Variance estimation : ˆV ( ˆR) = ˆV ( i A π 1 i ˆd i ) ˆd i = ˆT 1 x1 (x 2i ˆRx 1i ) 2 If x 1i = 1 and x 2i = y i, then Hájek estimator V (ȳ π ȳ N F) ȳ π = i A π 1 i y i i A π 1 i. = V [N 1 i A V [N 1 ˆT y F] = V [N 1 i A π 1 i (y i ȳ N ) F] π 1 i y i F]. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Approximations for Complex Estimators ˆθ is defined through an estimating equation w i g(x i, ˆθ) = 0. i A Let Ĝ(θ) = w i g(x i ; θ) i A G(θ) = N N 1 g(x i ; θ) i=1 Ĥ(θ) = Ĝ(θ)/ θ H(θ) = G(θ)/ θ Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

36 Theorem Theorem Under suitable conditions, n( ˆθ θ N ) L N(0, V) where V = n[h(θ N )] 1 V {Ĝ(θ N F N )}[H(θ N )] 1 Also, V ( ˆθ) can be estimated by ˆV = n[ĥ( ˆθ)] 1 ˆV {Ĝ( ˆθ F 1 N )}[Ĥ( ˆθ)] Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Comments It is difficult to show CLT for general HT estimator Exception: Large number of strata CLT requires: Large samples (that is, effectively large) Moments: no extreme observations No extreme weights Functions: curvature small relative to s.e Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

37 Basic Estimators Using SAS PROC SURVEYMEANS PROC SURVEYFREQ Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 PROC SURVEYMEANS Univariate analysis: population totals, means, ratios, and quantiles Variances and confidence limits Domain analysis Poststratified analysis Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

38 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 PROC SURVEYFREQ One-way to n-way frequency and crosstabulation tables Totals and proportions Tests of association between variables Estimates of risk differences, odds ratios, and relative risks Standard errors and confidence limits Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

39 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Digitech Cable Describe: Satisfaction ratings Usage time Satisfaction ratings based on household sizes Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

40 PROC SURVEYMEANS ods g r a p h i c s on ; proc surveymeans data=responsedata mean t o t a l=t o t ; s t r a t a S t a t e Type ; weight SamplingWeight ; c l a s s Rating ; v a r Rating UsageTime ; run ; ods g r a p h i c s o f f ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 The SURVEYMEANS Procedure Data Summary Number of Strata 8 Number of Observations 300 Sum of Weights Statistics Variable Level Label Mean Std Error of Mean UsageTime Computer Usage Time Rating Extremely Unsatisfied Customer Satisfaction Unsatisfied Customer Satisfaction Neutral Customer Satisfaction Satisfied Customer Satisfaction Extremely Satisfied Customer Satisfaction Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

41 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 PROC SURVEYFREQ proc s u r v e y f r e q data=responsedata ; s t r a t a S t a t e Type ; weight SamplingWeight ; t a b l e s Rating / c h i s q t e s t p =( ) ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

42 The SURVEYFREQ Procedure Customer Satisfaction Rating Frequency Weighted Frequency Std Err of Wgt Freq Percent Test Percent Std Err of Percent Extremely Unsatisfied Unsatisfied Neutral Satisfied Extremely Satisfied Total E Frequency Missing = 16 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Rao-Scott Chi-Square Test Pearson Chi-Square Design Correction Rao-Scott Chi-Square DF 4 Pr > ChiSq F Value Num DF 4 Den DF 1104 Pr > F Sample Size = 284 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

43 Domain Estimation Domains are subsets of the entire population Domain sample size is not fixed Variance estimation should account for random sample sizes in domains Degrees of freedom measured using the entire sample Use the DOMAIN statement Do NOT use the BY statement for domain analysis Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Describe Usage Based on Household Sizes proc surveymeans data=responsedata mean t o t a l=t o t ; s t r a t a S t a t e Type ; weight SamplingWeight ; v a r UsageTime ; domain H o u s e h o l d S i z e ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

44 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Describe Rating Based on Household Sizes proc s u r v e y f r e q data=responsedata ; s t r a t a S t a t e Type ; weight SamplingWeight ; t a b l e s H o u s e h o l d S i z e Rating / p l o t=a l l ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318

45 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 1 7/23-24/ / 318 Chapter 2 Use of Auxiliary Information in Estimation World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

46 Ratio Estimation Population : Observe x N = N 1 N i=1 x i Sample : Observe ( x HT, ȳ HT ) = N 1 i A π 1 i (x i, y i ) Ratio estimator ȳ rat = x N ȳ HT x HT Let R N = x 1 N ȳn be the population ratio, where ( x N, ȳ N ) = N 1 N i=1 (x i, y i ). Assume that ( x HT, ȳ HT ) ( x N, ȳ N ) = O p (n 1/2 ). Assume that x N 0. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Asymptotic Properties of Ratio Estimator (1) Linear approximation: Proof ȳ rat ȳ N = ȳ HT R N x HT + O p (n 1 ) ȳ rat ȳ N = x 1 HT x N (ȳ HT R N x HT ) { } = 1 + O p (n 1/2 ) (ȳ HT R N x HT ) = ȳ HT R N x HT + O p (n 1 ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

47 Asymptotic Properties of Ratio Estimator (2) Bias approximation: Uses second order Taylor expansion x 1 HT ȳ HT = R N + x 1 N (ȳ HT R N x HT ) + x 2 N { R N ( x HT x N ) 2 ( x HT x N ) (ȳ HT ȳ N ) +O p (n 3/2 ). Under moment conditions for x 1 HT ȳ HT and x N 0, Bias(ȳ rat ) = E(ȳ rat ȳ N ) = x 1 N [R N V ( x HT ) Cov( x HT, ȳ HT )] + O(n 2 ) = O(n 1 ). Thus, bias of ˆθ = ȳ rat is negligible because } R.B.(ˆθ) = Bias(ˆθ) Var(ˆθ) 0 as n. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Asymptotic Properties of Ratio Estimator (3) Given the conditions of Theorem ˆθ θ Var(ˆθ) N(0, 1), and ˆV (ˆθ) = ˆV (ȳ HT ˆR x HT ) = ( xht x N ) 2 ˆV (ȳ HT ˆR x HT) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

48 Other Properties of Ratio Estimator 1 Ratio estimator is the best linear unbiased estimator under y i = x i β + e i, e i (0, x i σ 2 ) 2 Scale invariant, not location invariant 3 Linear but not design linear 4 Ratio estimator in stratified sample, ȳ st,s = ȳ st,c = H h=1 ( H h=1 W h ȳ h ( xhn x h ) W h x hn ) H h=1 W hȳ h H h=1 W h x h : separate ratio estimator : combined ratio estimator where W h = N h /N. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / Regression estimation Sample : Observe (x i, y i ). Population : Observe x i = (1, x 1i ) i = 1, 2,..., N or only x N. Interested in estimation of ȳ N = N 1 Regression model N y i i=1 y i = x i β + e i e i independent of x j for all i and j, e i ind (0, σe). 2 Under Normal model, regression gives the best predictor Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

49 Regression Model: Estimation Linear Estimator: ȳ w = i A w iy i. y i = x i β + e i, e i ind (0, σ 2 e) To find the best linear unbiased estimator of x N β = E{ȳ N }, ( ) { } min V w i y i X, x N s.t. E w i y i ȳ N X, x N = 0 i A min s.t. i A X = (x 1, x 2,..., x n) w 2 i w i x i = x N i A i A Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Best Linear Unbiased Estimator Lagrange multiplier method Q = 1 2 i A w 2 i Q w i = w i + λ x i + λ ( i A i A w ix i = x N λ = x N ( i A x i x i) 1 w i x i x N ) w i = x i ( i A x i x i) 1 x N = x N (X X ) 1 x i Regression estimator is the solution that minimizes the variance in the class of linear estimators that are unbiased under the model. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

50 Properties of Regression Estimator 1 Linear in y. Location-Scale Invariant. 2 Alternative expression: For x i = (1, x 1i ), ȳ reg = ȳ n + ( x 1N x 1n ) ˆβ 1 = i A w i y i ˆβ 1 = [ ] 1 (x 1i x 1n ) (x 1i x 1n ) (x 1i x 1n ) y i i A i A [ ] 1 w i = 1 n + ( x 1N x 1n ) (x 1i x 1n ) (x 1i x 1n ) (x 1i x 1n) i A 3 Writing ȳ reg = ȳ n + ( x 1N x 1n ) ˆβ 1 = ˆβ 0 + x 1N ˆβ1, the regression estimator can be viewed as the predicted value of Y = β 0 + x 1 β 1 + e at x 1 = x 1N under the regression model. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Example: Artificial population Population Plot y ( 3, 4 ) x Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

51 Example (Cont d): SRS of size n = 20 Sample Plot (n=20) y ( 2.61, 3.42 ) ( 3, 4 ) ( 3, 3.85 ) pop. mean = 4 sam. mean = 3.42 reg. est. = x Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Properties of Regression Estimator (2) 4 For the mean adjusted regression model the OLS estimator of γ 0 is y i = γ 0 + (x 1i x 1N )γ 1 + e i, ˆγ 0 = ȳ n ( x 1n x 1N ) ˆγ 1 where ˆγ 1 = ˆβ 1. That is, ˆγ 0 = ȳ reg. 5 Under the linear regression model: a ȳ reg unbiased (by construction) { N } E(ȳ reg ȳ N X N ) = E x i ˆβ N (x i β + e i ) X N = i=1 N x i β i=1 i=1 N x i β = 0 i=1 ( E( ˆβ X N ) = β & E(e i X N ) = 0) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

52 Properties of Regression Estimator (3) b Variance V (ȳ reg ȳ N X, x N ) = n 1 (1 f )σ 2 e +( x 1N x 1n )V ( ˆβ 1 X )( x 1N x 1n ) V ( ˆβ 1 X) = [ i A (x 1i x 1n ) (x 1i x 1n ) ] 1 σ 2 e ȳ reg = ȳ n + ( x 1N x 1n ) ˆβ 1 = β 0 + x 1n β 1 + ē n + ( x 1N x 1n )β 1 + ( x 1N x 1n )( ˆβ 1 β 1 ) = β 0 + x 1N β 1 + ē n + ( x 1N x 1n )( ˆβ 1 β 1 ) ȳ N = β 0 + x 1N β 1 + ē N ȳ reg ȳ N = ē n ē N + ( x 1N x 1n )( ˆβ 1 β 1 ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Properties of Regression Estimator (4) If x 1i Normal, where k = dim(x i ) V {ȳ reg ȳ N } = 1 n (1 f ) [1 + ] k σe 2 n k 2 V {ȳ n ȳ N } = n 1 (1 f )σ 2 y If R 2 adj = 1 σ2 e σ 2 y k n 2, then V {ȳ n ȳ N } V {ȳ reg ȳ N }. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

53 Best Linear Predictor Let y i for i A be observations from the model y i = x i β + e i, with e i ind(0, σ 2 e). Predict ȳ N n = x N n β + ē N n. The BLUP of ē N n is 0, and the BLUP (estimator) of x N n β is x N n ˆβ. Thus, the BLUP of ȳ N N 1 [nȳ n + (N n) x N n ˆβ] = N 1 [n x n ˆβ + (N n) x N n ˆβ] because ȳ n x n ˆβ = 0. = x N ˆβ is Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 General Population: SRS Regression Since ȳ n x n ˆβ = 0, ȳ reg ȳ N = x N ˆβ ȳn = ȳ n + ( x N x n ) ˆβ ȳ N = ȳ n ȳ N + ( x N x n )β N + ( x N x n )( ˆβ β N ) = ā n ā N + ( x N x n )( ˆβ β N ) a i = y i x i β N, ā N = 0 ( ) 1 ( β N = x ix i i U i U x iy i ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

54 Bias of Regression Estimator Design bias is negligible (assume moments) } E[ȳ reg F N ] = ȳ N + E{ā n ā N F N } + E {( x N x n )( ˆβ β N ) F N = ȳ N + E {( x N x n )( ˆβ } β N ) F N. Bias(ȳ reg F N ) = E[( x n x N )( ˆβ β N ) F N ] } = tr {Cov( x n, ˆβ ) F N k [Bias(ȳ reg F N )] 2 V ( x j,n F N )][V ( ˆβ j F N )] = O(n 1 )O(n 1 ). j=1 Bias(ȳ reg F N ) = O(n 1 ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Variance of Approximate Distribution ȳ reg ȳ N = ā n ā N + ( x N x n )( ˆβ β N ) = ā n ā N + O p (n 1 ) V (ā n F N ) = (1 f )n 1 S 2 a By Theorem 1.3.4, [V (ā n F N )] 1/2 (ā n ā N ) L N(0, 1). Recall ā N = 0. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

55 Estimated Variance ˆV {ȳ reg } = (1 f )n 1 (n k) 1 i A â 2 i, where â i = y i x i ˆβ and k = dimension of x i. âi 2 = i A i A = i A = i A [ ] 2 a i x i ( ˆβ β N ) a 2 i 2( ˆβ β N ) i A a 2 i + O p (1) x ia i + ( ˆβ β N ) ( i A x ix i ) ( ˆβ β N ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Limiting Distribution of Coefficients Theorem (SRS) Assume (y i, x i ) iid, existence of eighth moments. ( ) 1 ˆβ = ( ˆM xx ) 1 ˆM xy = x ix i n 1 x iy i i A n 1 i A ( β N = (M xx,n ) 1 M xy,n = N 1 i U x ix i ) 1 N 1 i U x iy i V { ˆβ β N F N } 1/2 ( ˆβ β N ) F N L N(0, I) V { ˆβ β N F N } = n 1 (1 f N )M 1 xx,nv bb,n M 1 xx,n V bb,n = N 1 i U x ia 2 i x i, a i = y i x i β N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

56 Proof of Theorem ˆβ β N = ˆM 1 xx ˆM xa =: ˆM 1 xx ( n 1 i A b i ) where ˆM xa = n 1 i A x i a i. Given moment conditions, n { ˆM xa M xa,n } L N [0, (1 fn )V bb,n ] (Theorem 1.3.4) where M xa,n = N 1 i U x ia i = 0. Now ˆM xx p Mxx,N and, by Slutsky s theorem, the result follows. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Regression for General Designs (Theorem 2.2.1) F N = {z 1, z 2,, z N } where z i = (y i, x i ), Z n = (z 1, z 2,, z n). Define ( ) ˆM ˆM zφz = n 1 Z nφ 1 n Z n = xφx ˆM xφy ˆM yφx ˆM yφy and ˆβ = ˆM 1 xφx ˆM xφy, where Φ n :n n matrix (positive definite) e.g. Φ n = (N/n)diag{π 1,, π n }. Assume (i) V { z HT z N F N } = O p (n 1 ) a.s. where z HT (ii) = N 1 i A π 1 i z i ˆMzφz M zφz,n = O p (n 1/2 ) a.s. and ˆM zφz nonsingular. (iii) K 1 < Nn 1 π i < K 2 (iv) [V { z HT z N F N }] 1/2 ( z HT z N ) L N(0, I) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

57 Theorem Theorem Given moments, design consistent HT estimators, (i) ˆβ β N = M 1 xφx,n b HT + O p (n 1 ) a.s. where β N = (M xx,n ) 1 M xy,n = ( i U x i x ) 1 i i U x i y i. (ii) [ ˆV ( ˆβ F N )] 1/2 ( ˆβ β N ) L N(0, I) where b i = n 1 Nπ i ξ i a i, a i = y i x i β N, ξ i is the i-th column of X nφ 1 n, b HT = N 1 i A π 1 b i and ˆV ( ˆβ F N ) = ˆM 1 xφx ˆV ( b HT) ˆM 1 xφx i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Proof of Theorem ˆβ β N = ˆM 1 xφx (n 1 X nφ 1 n a n ) n 1 X nφ 1 n a n = O p (n 1/2 ) ˆM 1 xφx = M 1 xφx,n + O p(n 1/2 ) ˆβ β N = M 1 xφx,n (n 1 X nφ 1 n a n ) + O p (n 1 ) n 1 X nφ 1 n a n = n 1 ξ i a i = N 1 i A i A π 1 i b i = b HT ξ i is the i-th column of X nφ 1 n, b i = n 1 Nπ i ξ i a i. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

58 Remarks on Theorem The choice of Φ n is arbitrary. (i.e. The result in Theorem holds for given Φ n ) A simple case is Φ n = (N/n)diag{π 1,, π n } 2 Variance estimation: ˆV ( ˆβ) = ˆM 1 xφx ˆV bb ˆM 1 xφx where ˆV bb is the estimated sampling variance of b HT ˆb i = n 1 Nπ i ξ i â i and â i = y i x i ˆβ. calculated with 3 Result holds for a general regression estimator. That is, the asymptotic normality of x N ˆβ follows from the asymptotic normality of ˆβ. 4 Theorem states the consistency of ˆβ for β N. But we are also interested in the consistency of the estimator of ȳ N. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Theorem 2.2.3: Design Consistency of ȳ reg for ȳ N Theorem { } Let p lim β β N F N = 0. Then, p lim {ȳ reg ȳ N F N } = 0 p lim N 1 N N a i = 0 i=1 where a i = y i x i β N and ȳ reg = x N β. Proof : Because β is design consistent for β N, { } p lim {ȳ N x N β FN = p lim N 1 = p lim { N i=1 N N 1 i=1 (y i x i β) FN } (y i x i β N ) F N }. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

59 Condition for Design Consistency Corollary ( ) Assume design consistency for z HT Let ȳ reg = x N ˆβ, ˆβ = (X nφ 1 n and for sample moments. X n ) 1 X nφ 1 n y n If γ n such that X n γ n = Φ n D 1 π J n (1) where D π = diag(π 1, π 2,, π n ) and J n is a column vector of 1 s, then ȳ reg is design-consistent for ȳ N. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Proof of Corollary By Theorem 2.2.3, we have only to show that N 1 i U (y i x i β N ) = 0, where β N = p lim ˆβ. Now, (y X n ˆβ) Φ 1 n X n = 0 (y X n ˆβ) Φ 1 n X n γ n = 0 (y X n ˆβ) D 1 π J n = N(ȳ HT x HT ˆβ) = 0 p lim{ȳ HT x HT β N } = p lim{ȳ HT x HT ˆβ} = 0 & p lim{ȳ HT x HT β N } = N 1 i U (y i x i β N ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

60 Remarks on Corollary Condition Φ n Dπ 1 J n C(X n ) is a crucial condition for the design consistency of the regression estimator of the form ȳ reg = x N ˆβ with ˆβ = (X nφ 1 n X n ) 1 X nφ 1 n y n. 2 If condition Φ n Dπ 1 J n C(X n ) does not hold, one can expand the X n matrix by including z 0 = Φ n Dπ 1 J n and use Z n = [z 0, X n ] to construct the regression estimator: ȳ reg = z N ˆγ = [ z ( z 0N, x N ) 0 Φ 1 n z 0 z 0 Φ 1 n X nφ 1 n z 0 X nφ 1 n X n X n ] 1 ( z 0 Φ 1 n y X nφ 1 n y ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Examples for Φ n D 1 π J n C(X n ) 1 Φ n = D π and x i = (1, x 1i ) ȳ reg = ȳ π + ( x 1,N x 1,π ) ˆβ 1 where (ȳ π, x 1,π ) = ( ) 1 i A π 1 i i A π 1 i (y i, x 1i ) ˆβ 1 = { i A π 1 i (x 1i x 1,π ) (x 1i x 1,π ) } 1 i A π 1 i (x 1i x 1,π ) y i. 2 Φ n = I n, x i = ( π 1 ) i, x 1i = (wi, x 1i ), w N = N 1 N i=1 w i, (ȳ ω, x 1,ω ) = ( i A π 1 i ( ˆβ 0,ols, ˆβ 1,ols ) = ( i A x i x i ȳ reg,ω = w N ȳ ω + ( x 1,N w N x 1,ω ) ˆβ 1,ols w i ) 1 i A π 1 i (y i, x 1i ) ) 1 i A x i y i. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

61 Design Optimal Regression Estimator (Theorem 2.2.4) Theorem Sequence of populations and designs giving consistent estimators of moments. Consider ȳ reg ( ˆβ) = ȳ π + ( x 1N x 1π ) ˆβ for some ˆβ (i) ȳ reg is design consistent for ȳ N for any reasonable ˆβ. (ii) ˆβ = [ ˆV ( x 1π )] 1 Ĉ( x 1π, ȳ 1π ) minimizes the estimated variance of ȳ reg ( ˆβ) (iii) CLT for ȳ reg ( ˆβ ) can be established Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Remarks on Theorem Optimal estimator can be viewed as Rao-Blackwellization based on ( ) (( ) ( )) x π x N V( x N, π ) C( x π, ȳ π ) ȳ N C( x π, ȳ π ) V (ȳ π ) ȳ π 2 GLS interpretation ( x π x N ȳ π ) = ( 0 1 ) ȳ N + ( e1 e 2 ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

62 2.3 Linear Model Prediction Population model : y N = X N β + e N, e N (0, Σ eenn ) e N independent of X N. Assume Σ eenn known Sample model : y A = X A β + e A, e A (0, Σ eeaa ) Best linear predictor (BLUP) ˆθ = N 1 y i + {ŷ i + Σ eeāa Σ 1 eeaa(y A X A ˆβ)} i A i Ā where ŷ i = x i ˆβ and ˆβ = (X AΣ 1 eeaax A ) 1 X A Σ 1 eeaay A Q : When is the model-based predictor ˆθ design consistent? Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Model and Design Consistency Theorem (2.3.1) If Σ eeaa (D 1 π J n J n ) Σ eeaā J N n C(X A ), then ˆθ = ȳ HT + ( x N x HT ) ˆβ and ˆθ ȳ N = ā HT ā N + O p (n 1 ) where a i = y i x i β N Analogous to Corollary for prediction. If the model satisfies the conditions for design consistency, then the model is called full model. Otherwise, it is called reduced model (or restricted model). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

63 Model and Design Consistency General strategy (for general purpose survey) a Pick important y b Find a model y = X β + e c Use ȳ reg = x N ˆβ (Full model) or use ȳreg,π = ȳ π + ( x N x π ) ˆβ where ˆβ = (X nσ 1 ee X n ) 1 (X nσ 1 ee y n ) If the design consistency condition does not hold, we can expand the X A matrix by including z 0 such as Σ eeaa Dπ 1 J n, Z = [z 0, X ]. If z 0N is not known, use ȳ reg,π of (c). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / Nonlinear Models (All x Values Known) Superpopulation model y i = α(x i ; θ) + e i, E(e i ) = 0, e i indep. of x j, for all i and j. 1 ȳ c,reg = ȳ HT + N 1 i U α(x i; ˆθ) N 1 i A π 1 i α(x i ; ˆθ) [ 2 ȳ m,reg = N 1 i A y i + ] i Ā α(x i; ˆθ) Remark: ȳ c,reg = ȳ m,reg if i A (π 1 i Bennett, 1988). 1)(y i ŷ i ) = 0. (Firth and Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

64 Consistency of Nonlinear Regression (Theorem 2.3.2) Theorem (i) There exist θ N such that ˆθ θ N = O p (n 1/2 ), a.s.. (ii) α(x, θ) is a continuous differentiable function of θ with derivative uniformly continuous on B, a closed set containing θ N. (iii) The partial derivative h(x i ; θ) = α(x i ; θ)/ θ satisfies sup N 1 π 1 i h(x i ; θ) h(x i ; θ) θ B = O p(n 1/2 ) a.s. i A i U ȳ c,reg ȳ N = ā HT ā N + O p (n 1 ) where a i = y i α(x i ; θ N ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Calibration Minimize ω V ω s.t. ω X = x N (ω Vω)(aX V 1 Xa ) (ω Xa ) 2 with equality iff ω V 1/2 ax V 1/2 ω ax V 1 ω = kax V 1, k : constant ω X = kax V 1 X & x N (X V 1 X ) 1 = ka ω = x N (X V 1 X ) 1 X V 1 ω V ω x N (X V 1 X ) 1 x N Note Minimize V (ω y X, d) s.t. E(ω y ȳ N X, d) = 0. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

65 Alternative Minimization Lemma α : given n-dimensional vector Let ω a = arg min ω ω Vω s.t ω X = x N Let ω b = arg min ω (ω α) V(ω α) s.t ω X = x N If V α C(X), then ω a = ω b. Proof : (ω α) V(ω α) = ω Vω α Vω ω Vα + α Vα = ω Vω λ X ω ω Xλ + α Vα where V α = Xλ = ω Vω 2λ x N + α Vα ω X = x N If α = Dπ 1 J n, then V α C(X ) is the condition for design consistency in Corollary Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 General Objective Function min i A G(ω i, α i ) s.t. ω i x i = x N i A Lagrange multiplier method g(ω i, α i ) λ x i = 0 where g(ω i, α i ) = G ω i ω i = g 1 (λ x i, α i ) where λ is from i A g 1 (λ x i, α i )x i = x N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

66 GREG Estimator min Q(ω, d) = i A d 1 i (ω i d i )q i + λ x i = 0 ω i = d i + λ d i x i/q i i A ω i x i = i A ( ) 2 ωi d i 1 q i s.t. d i d i x i + λ i A d i x ix i /q i ω i x i = x N. i A λ = ( x N x HT )( i A d i x ix i /q i ) 1 w i = d i + ( x N x HT )( i A d i x ix i /q i ) 1 d i x i/q i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Other Objective Functions Pseudo empirical likelihood Q(ω, d) = d i log Kullback-Leibler distance: Q(ω, d) = ω i log where d i = 1/π i. ( ωi d i ( ωi d i ), ω i = d i /(1 + x i λ) ), ω i = d i exp(x i λ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

67 Theorem Deville and Särndal (1992) Theorem Let G(ω, α) be a continuous convex function with a first derivative that is zero for ω = α. Under some regularity conditions, the solution ω i that minimizes G(ω i, α i ) s.t. ω i x i = x N i A i A satisfies i A ω i y i = i A α i y i + ( x N x α ) ˆβ + O p (n 1 ) where ˆβ = ( i A x i x ) 1 i/φ ii i A x i y i/φ ii and φ ii = 2 G(ω i, α i )/ ωi 2 ωi. =α i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Weight Bounds ω i = d i + d i λ x i /c i can take negative values (or take very large values) Add L 1 ω i L 2 to ω i x i = x N. Approaches 1 Huang and Fuller (1978) 2 Husain (1969) Q(w i, d i ) = d i Ψ ( wi d i ), Ψ : Huber function min ω ω + γ(ω X x N ) Σ 1 x x (ω X x N ) for some γ 3 Other methods, quadratic programming. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

68 Comments Regression estimation is large sample superior to mean and ratio estimation for k << n. Applications require restrictions on regression weights ( w i > 1/N ) Model estimator is design consistent if X γ = Σ ee D 1 π J. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Regression Estimators Using SAS PROC SURVEYREG Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

69 Regression Estimators Response variable is correlated to a list of auxiliary variables Population totals for the auxiliary variables are known Efficient estimators can be constructed by using a linear contrast from a regression model Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Digitech Cable Can you improve the estimate of the average usage time by taking data usage into account? Average data usage (MB) for the population is available Data usage for every unit in the sample is available Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

70 ESTIMATE Statement proc s u r v e y r e g data=responsedata p l o t= f i t ( weight=heatmap shape=hex n b i n s =20); s t r a t a S t a t e Type ; weight SamplingWeight ; model UsageTime = DataUsage ; e s t i m a t e R e g r e s s i o n Estimator i n t e r c e p t 1 DataUsage ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 The SURVEYREG Procedure Regression Analysis for Dependent Variable UsageTime Fit Statistics R-Square Root MSE Denominator DF 292 Estimate Label Estimate Standard Error DF t Value Pr > t Regression Estimator <.0001 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

71 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Poststratification Using SAS PROC SURVEYMEANS Strata identification is unknown, but strata totals or percentages are known Stratification after the sample is observed Use poststratification to produce efficient estimators adjust for nonresponse bias perform direct standardization Variance estimators must be adjusted Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

72 Digitech Cable Known distribution of race in the four states Adjust the distribution of race in the sample to match the population Estimate the average usage time after adjusting for race Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 POSTSTRATA Statement proc surveymeans data=responsedata mean s t d e r r ; s t r a t a S t a t e Type ; weight SamplingWeight ; v a r UsageTime ; p o s t s t r a t a Race / p o s t p c t=racepercent ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318

73 Poststratified Estimator The SURVEYMEANS Procedure Statistics Variable Label Mean Std Error of Mean UsageTime Computer Usage Time A set of poststratified-adjusted weights is created The variance estimator uses the poststratification information Store the poststratified-adjusted replicate weights from PROC SURVEYMEANS and use the adjusted replicate weights in other survey procedures Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 2 7/23-24/ / 318 Chapter 3 Use of Auxiliary Information in Design World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

74 Design Strategy Find the best strategy (design, estimator) for ȳ N under the model y i = x i β + e i, e i ind (0, γ ii σ 2 ) x N, γ ii known, β, σ 2 unknown Estimator class : ˆθ = i A w iy i : linear in y and E{(ˆθ ȳ N ) d, X N } = 0, so ˆθ is model-unbiased & design consistent. Criterion : Anticipated variance AV{ˆθ ȳ N } = E{V (ˆθ ȳ N F)} Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Candidate Estimator ˆθ = N 1 ( i A y i + i A c x i ˆβ ˆβ = (X D 1 γ X ) 1 X D 1 γ y = ( i A x i x ) 1 ( i/γ ii i A x i y ) i/γ ii D γ = diag(γ 11, γ 22,..., γ nn ) If the vector of γ ii is in the column space of X, i A (y i x i ˆβ) = 0 and ˆθ = ȳ reg = x N ˆβ. If π i γ 1/2 ii and the vector of γ 1/2 ii in in the column space of X, ȳ reg = ȳ HT + ( x N x HT ) ˆβ. ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

75 Theorem (Isaki and Fuller, 1982) Under moment conditions, if π i γ 1/2 ii, γ 1/2 ii = x i τ 1, and γ ii = x i τ 2 for some τ 1 and τ 2, then ( ) 2 lim nav{ȳ reg ȳ N } = lim 1 N γ 1/2 N N ii n N N N 2 γ ii σ 2 and i=1 lim n [AV{ȳ reg ȳ N } AV{Ψ l ȳ N }] 0 N for all Ψ l D l and all p P c, where i=1 D l = {Ψ l ; Ψ l = i A α i y i and E{(Ψ l ȳ N ) d, X } = 0} and P c is the class of fixed-sample-size nonreplacement designs with fixed probabilities admitting design-consistent estimators of ȳ N. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Proof of Theorem For Ψ l = α y E{Ψ l ȳ N d, X N } = 0 α X = x N V {Ψ l ȳ N d, X N } = { α D γ α 2N 1 α D γ J n + N 2 J ND γn J N } σ 2 α D γ J n = α X τ 2 = x N τ 2 = N 1 J NX N τ 2 = N 1 J ND γn J N V {Ψ l ȳ N d, X N } = { α D γ α N 2 J ND γn J N } σ 2 Enough to find α that minimizes α D γ α s.t. α X = x N α = x N (X D 1 γ α D γ α x N (X Dγ 1 (See Section 2.7) X ) 1 X Dγ 1 X ) 1 x N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

76 Remarks on Theorem (1) Under the model, y i = x i β + e i, ȳ reg ȳ N = ē HT ē N + O p (n 1 ). AV(ȳ reg ȳ N ) = AV(ē HT ē N ) N = N 2 [(1 π i )π 1 i ]γ ii σ 2 i=1 min AV(ȳ reg ȳ N ) s.t. N i=1 π i = n, π i = n ( N ) 2 N AV(ȳ reg ȳ N ) N 2 n 1 (Godambe-Joshi lower bound) i=1 γ 1/2 ii ( N j=1 γ1/2 jj i=1 γ ii σ 2 ) γ 1/2 ii Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Remarks on Theorem (2) For model: y i = x i β + e i, e i (0, γ ii σ 2 ), best strategy is ȳ reg = ȳ HT + ( x N x HT ) ˆβ with π i γ 1/2 ii To achieve a sampling design with π i γ 1/2 ii 1 Use Poisson sampling : n is not fixed (Not covered by Theorem). 2 Use systematic sampling 3 Use approximation by stratified random sampling Choose U h s.t. i U h γ 1/2 ii γ 11 γ 22 γ NN. = H 1 i U γ1/2 ii, Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

77 Stratification, Example y i = β 0 + x i β 1 + e i, e i (0, σ 2 e) and x i = i, i = 1, 2, N = 1, 600 stratified sampling with N h = N/H(= M), n h = n/h, n = 64 ȳ st = H h=1 V (ȳ st ) = 1 H 2 N h N ȳh = 1 H H h=1 H h=1 ȳ h V (ȳ h ) = 1 H 2 H h=1 1 n h σ 2 yh = 1 n σ2 w where σ 2 w = H 1 H h=1 σ2 yh, σ2 yh = E [ (y hi ȳ h ) 2]. = M β2 1 + σ2 e (M 2 1)/12 = variance of {1, 2,, M} Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Example continued Number of V (ȳ st ) V { ˆV (ȳ st }) Strata ρ 2 = 0.25 ρ 2 = 0.9 ρ 2 = 0.25 ρ 2 = ρ 2 = 1 (σ 2 y ) 1 σ 2 e Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

78 Remarks on Stratification 1 For efficient point estimation, increase H 2 V { ˆV (ȳ st )} depends on ρ: V { ˆV (ȳ st )} can decrease in H and then increase in H Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 One per Stratum A common procedure is to select one unit per stratum and to combine or collapse two adjacent strata to form a variance estimation stratum E ˆV cal {ȳ col } = 0.25(y 1 y 2 ) 2 [ ˆVcal {ȳ col }] = 0.25(µ 1 µ 2 ) (σ1 2 + σ2) 2 Two-per-stratum design V {ȳ 2,st } = 0.125(µ 1 µ 2 ) (σ σ 2 2) Controlled two-per-stratum design ( 3.1.4) can be used to reduce the variance of the two-per-stratum design Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

79 Cluster Sampling Population of cluster of elements May have different cluster sizes Cluster size can be either known or unknown at the design stage Clusters are sampling units Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Model for Cluster Sampling y ij = µ y + b i + e ij, i = 1, 2,, N, j = 1, 2,, M i b i (0, σ 2 b ), e ij (0, σ 2 e), M i (µ M, σ 2 M) and b i, e ij, and M i are independent. M i y i = y ij (M i µ y, γ ii ) Mi j=1 γ ii = V (y i M i ) = M 2 i σ 2 b + M iσ 2 e Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

80 Strategies for Mean per Element 1 ˆθ n,1 = M n 1 ȳ n : SRS 2 ˆθ n,2 = M 1 HT ȳ HT : with π i M i 3 ˆθ n,3 = M 1 HT ȳ HT : with π i γ 1/2 ii Then V (ˆθ n,3 ) V (ˆθ n,2 ) V (ˆθ n,1 ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Two Stage Sampling Population of N clusters (PSUs) Select a sample of n 1 clusters Select a sample of m i elements from M i elements in cluster i Sampling within clusters is independent A model: y ij = µ y + b i + e ij b i (0, σ 2 b ) ind of e ij (0, σ 2 e) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

81 Estimation of Mean per Element where ȳ i = m 1 i m i j=1 y ij. ( N ) 1 θ N = M i ˆθ SRC = i=1 ( n N i=1 i=1 M i) 1 n i=1 M i j=1 y ij M i ȳ i With equal M i, equal m i, and SRS at both stages, V(ˆθ SRC θ N ) = 1 ( 1 n ) 1 σb 2 n 1 N + 1 [ 1 n ] 1m n 1 m NM. = 1 [σb 2 n + 1 ] 1 m σ2 e σ 2 e Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Optimal Allocation Cost function: C = c 1 n 1 + c 2 n 1 m Minimize V{ˆθ θ N } s.t. C = c 1 n 1 + c 2 n 1 m: m = [ σ 2 e σ 2 b ] 1/2 c 1 c 2 Proof: C is fixed. Minimize C V {ˆθ θ n } = n 1 1 ( σ 2 b + m 1 σe 2 ) (c1 n 1 + c 2 n 1 m) = σ 2 b c 1 + σ 2 ec 2 + ( σ 2 b c 2m + c 1 σ 2 em 1). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

82 Two-Phase Sampling 1 Phase-one : Select A 1 from U. Observe x i 2 Phase-two : Select A 2 from A 1. Observe (x i, y i ) π 1i = Pr[i A 1 ] π 2i 1i = Pr[i A 2 i A 1 ] π 2i = π 1i π 2i 1i Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Two-Phase Sampling for Stratification x i = (x i1,, x ig ) { 1 if i group g x ig = 0 otherwise G ˆT 2pr,st = ˆN 1g ȳ 2πg : reweighted expansion estimator g=1 ˆN 1g = i A1 π 1 1i x ig ȳ 2πg = i A 2 π 1 1i π 1 2i 1i x ig y i i A 2 π 1 1i π 1 2i 1i x ig ( If π 2i 1i = f 2g = n ) 2g, i group g, then ȳ 2πg = n 1g i A 2 π 1 1i x ig y i i A 2 π 1. 1i x ig Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

83 Theorem Let the second phase sample be stratified random sampling with π 2i 1i fixed and constant within groups. Under moment conditions, [V {ȳ 2p,st F N }] 1/2 (ȳ 2p,st ȳ N ) F N L N(0, 1) where V (ȳ 2p,st F N ) = V (ȳ 1π F N ) + E ȳ 1π = i A 1 w 1i y i, w 1i = S 2 1ug = (n 1g 1) 1 ū 1g = n 1 1g i A 1g w 1i y i G g=1 n 2 1g j A1 π 1 1j i A 1g (u i ū 1g ) 2 ( 1 1 ) n 2g n 1g 1 π 1 1i S 2 1ug F N Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Proof of Theorem ȳ 2p,st ȳ N = (ȳ 2p,st ȳ 1π ) + (ȳ 1π ȳ N ) (i) {V [ȳ 2p,st ȳ 1π F N ]} 1/2 (ȳ 2p,st ȳ 1π ) (A 1, F N ) L N(0, 1) ȳ 2p,st ȳ 1π = G n 1g (ū 2g ū 1g ) g=1 ū 2g = 1 n 2g V {ȳ 2p,st ȳ 1π A 1, F N } = E{ȳ 2p,st ȳ 1π A 1, F N } = 0 G g=1 w i y i, ū 1g = 1 w i y i n 1g i A 2g i A 1g n 2 1g (ii) V [ȳ 1π ȳ N F N ] 1/2 (ȳ 1π ȳ N ) F N L N(0, 1) By Theorem (p 54-55), the result follows ( n 1 2g n 1 1g ) S 2 1ug Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

84 Separate Samples with Common Characteristics Two independent surveys A 1 : observe x A 2 : observe (x, y) Interested in estimating θ = ( x N, ȳ N ) x 1 x 2 ȳ 2 = V = V e 1 e 2 e 3 ( x N ȳ N = ) + u = Zθ + e ˆθ GLS = (Z V 1 Z) 1 Z V 1 u e 1 e 2 e 3 V V 22 V 23 0 V 32 V 33 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Composite Estimation Example(Two Time Periods: O = observed) Sample t = 1 t = 2 A O O core panel part (detecting change) B O C O supplemental panel survey (cross sectional) Sample A : ȳ 1A, ȳ 2A, Sample B : ȳ 1B, Sample C : ȳ 2C Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

85 Two Time Periods GLS V ȳ 1B ȳ 1A ȳ 2A ȳ 2C e 1 e 2 e 3 e 4 = = n 1 ( ȳ1,n ȳ 2,N ) + e 1 e 2 e 3 e 4 B n 1 A n 1 A ρ 0 0 n 1 A ρ n 1 A n 1 C Composite estimator ˆθ = (Z V 1 Z) 1 Z V 1 y Design is complex because more than one item. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Rejective Sampling y i = x i β + e i, x i, i = 1, 2,..., N known V xx = V { x p } under initial design P d, with p i initial selection probability. Procedure: Select sample using P d If ( x p x N )V 1 xx ( x p x N ) < K d keep sample. Otherwise reject and select a new sample Continue until sample is kept Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

86 Result for Rejective Sampling If ȳ reg = z N ˆβ design consistent under Pd x i c = (1 p i ) 1 p i, for some c, then ȳ reg = z N ˆβ design consistent for Rejection ˆβ = 1 z i φ i p 2 i z i z i φ i p 2 i y i i A rej i A rej z i = (x i, z 2i ), z 2i design var, eg. stratum indicators φ i = (1 p i ) for Poisson φ i = (N h 1) 1 (N h n h ) stratification ˆV {ȳ reg } is design consistent for V rej {ȳ reg } Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Sample Design (Fairy Tale) Client: Desire estimate of average daily consumption of Cherrios by females who made any purchase at store x between January 1, 2015 and July 1, 2015 (here is list) with a CV of 2% Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

87 Design Discussion Objectives? What is population of interest? What data are needed? How are data to be collected? What has been done before? What information (auxiliary data) available? How soon must it be done? Budget Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 BLM Sage Grouse Study Bureau of Land Management Rangeland Health Emphasis on sage grouse habitat Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

88 Sage Grouse Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Chapter 3 7/23-24/ / 318 Sage Grouse Kim & Fuller & Mukhopadhyay (ISU & SAS)

89 Sample Frame, Sample Units Public Land Survey System (PLSS) Central and western US Grid system of square miles (sections) (1.7km) Quarter section 0.5mi on a slide (segment) Much ownership based on PLSS ISU has used PLSS as a frame Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Low Sage Grouse Density Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

90 Sample Selection Stratify area (Thiessen polygons) Select random point locations (two per stratum) Segment containing point is sample segment Select observation points in segment Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Stratification Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

91 Sample Points Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Selection Probabilities Segments vary in size Segment rate less than 1/300 Segment π i treated as proportional to size Joint probability treated as π i π j Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

92 Variables x hi = Elevation of selected point y hi = created variable at point (Range health) C hi = acre size of segment h: stratum, i: segment Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Weights Segment probability = T 1 Ch (n hc hi ) T Ch = acre size of stratum Point weight (acres) = π 1 hi = [T 1 Ch (n hc hi )] 1 C hi m 1 hi m hi = number of points observed in segment hi Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

93 PPS - Fixed Take Let One point = one acre C hi = acre size of segment i in stratum h m = fixed number of points per segment Probability of selection (point j in segment i): P(acre ij ) = P(seg i )P(acre ij seg i ) = T 1 ch C hi ( mc 1 ) hi = T 1 ch m Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 One Point Per Segment ȳ st = (136) 1 68 h=1 2 y hi = i=1 [ ˆV (ȳ st )] 0.5 = = [ 68 ] 0.5 [ h=1 N 2 N 2 h ˆV (ȳ h ) 68 (136) 1 h=1 i=1 ] (y hi ȳ h ) 2 = Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

94 Regression Estimator y hi = x hi β 1 + δ j,hi = H δ j,hi β 1+j + e hi j=1 { 1 if j = h 0 otherwise. x hi = (x hi, δ 1,hi, δ 2,hi,, δ H,hi ) ȳ reg = ( x N x st ) ˆβ = ȳ st + ( x N x st ) ˆβ 1. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Regression Estimator x N = mi. known, x st = ȳ reg = ȳ st + ( x N x st ) ˆβ 1 = (0.0003)(0.9788) = (0.0165) ˆβ 1 = { 68 h=1 2 i=1(x hi x h ) 2} 1 68 h=1 2 (x hi x h )(y hi ȳ h ) i=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

95 Regression Weights 68 h=1 ȳ reg = 68 h=1 i=1 2 w hi y hi w hi = n 1 + ( x N x st ) ˆV {ȳ reg } = whi 2 h=1 i=1 { 68 2 h=1 i=1 i=1 w 2 hi = n 1 + ( x N x st ) 2 { 68 h=1 (x hi x h ) 2 } 1 {y hi ȳ h (x hi x h ) ˆβ 1 } 2 } 1 2 (x hi x h ) 2 i=1 (x hi x h ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Model Variance y hj = β 0 + β 1 (x hi x hn ) + e hi, e hi (0, σ 2 e) V {ȳ reg } = [ n 1 + ( x N x st ) 2 { 68 h=1 ] 2 (x hi x h ) 2 } 1 σe 2 i=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

96 Sample Number Two x N = x st = ȳ reg = (0.0289)(1.0203) = [ V {ȳ reg } = n 1 + ( x N x st ) 2 { 68 h=1 i=1 ˆV {ȳ reg } = { }10 2 = (0.0172) 2 h=1 i=1 ] 2 (x hi x h ) 2 } 1 σe ˆσ e 2 = (67) 1 {y hi ȳ h (x hi x h ) ˆβ 1 } 2 = Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318 Comments on Design Stratification Model for design Error model selection probabilities Avoid Over design Variance estimation Simple - explanation for users Rejective - avoids bad samples Data will be used for more than designed for Be prepared (budget cut, use sampling again). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 3 7/23-24/ / 318

97 Chapter 4 Replication Variance Estimation World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Jackknife Variance Estimation Create a new sample by deleting one observation x (k) = n x x k n 1 x (k) x = x k x n 1 n 1 n n ( x (k) x) 2 = k=1 1 n(n 1) n (x k x) 2 = n 1 sx 2 k=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

98 Alternative Jackknife Weights n k=1 x (k) ψ = ψx k + (1 ψ) x (k) x (k) ψ x = ψ(x k x) + (1 ψ)( x (k) x) x (k) 1 ψ ψ x = (ψ n 1 )(x k x) = ( nψ 1 n 1 )(x k x) ψ (nψ x)2 1)2 n = (n 1) 2 (x k x) 2 ( x (k) If (nψ 1) 2 = n 1 n n (1 f ), then k=1 k=1 ( x (k) ψ x)2 = 1 n (1 f )s2 x. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Random Group Jackknife n = mb : m groups of size b, x 1,..., x m x = 1 m x i m i=1 ˆV ( x) = 1 m 1 m 1 x (k) n = n x b x k n b x (k) b x = n x b x k n b ˆV RGJK ( x) m 1 m Unbiased but d.f. = m 1. m k=1 ( x (k) m ( x i x) 2 i=1 = m x x k m 1 x = 1 m 1 ( x k x) b x) 2 = 1 m(m 1) m ( x k x) 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 k=1

99 Theorem Theorem F N = {y 1,..., y N } : sequence of finite population y i iid(µ y, σ 2 y ) with finite 4 + δ moments g( ) : continuous function with continuous first derivative at µ y n 1 n n {g(ȳ (k) ) g(ȳ)} 2 = [g (ȳ)] 2 ˆV (ȳ) + o p (n 1 ) k=1 where ˆV (ȳ) = n 1 s 2 and g (ȳ) = g(ȳ) ȳ. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Proof of Theorem By a Taylor linearization, g(ȳ (k) ) = g(ȳ)+g (ȳ k )(ȳ (k) ȳ) = g(ȳ)+g (ȳ)(ȳ (k) ȳ)+r nk (ȳ (k) ȳ) for some ȳ k B δ k (ȳ), δ k = ȳ (k) ȳ where R nk = g (ȳ k ) g (ȳ). Thus, n k=1 { g(ȳ (k) ) g(ȳ)} 2 = = n { g (ȳk )} 2 (ȳ (k) ȳ) 2 k=1 n { g } 2 (ȳ) + R nk (ȳ (k) ȳ) 2 k=1 (i) max 1 k n ȳk ȳ 0 in probability. (ii) max 1 k n g (ȳk ) g (ȳ) 0 in probability. n 1 n n {g(ȳ (k) ) g(ȳ)} 2 = [g (ȳ)] 2 ˆV (ȳ) + o p (n 1 ) k=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

100 Remainder with Second Derivatives If g( ) has continuous second derivatives at µ y, then n 1 (n 1) n k=1 [g(ȳ (k) ) g(ȳ)] 2 = [g (ȳ)] 2 ˆV (ȳ) + O p (n 2 ). Proof : [g(ȳ (k) ) g(ȳ)] 2 = [g (ȳ k )]2 (ȳ (k) ȳ) 2 g (ȳk )2 = [g (ȳ)] 2 + 2[g (ȳk (ȳk k ȳ) [g (ȳk )]2 = [g (ȳ)] 2 + K 1 ȳk ȳ for some K 1. Thus, since ȳ k ȳ = O p(n 1 ), we have the result. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Jackknife Often Larger than Taylor ˆR = x 1 ȳ ˆR (k) ˆR = [ x (k) ] 1 ȳ (k) x 1 ȳ = [ x (k) ] 1 (y k ˆRx k )(n 1) 1 ˆV JK ( ˆR) = n 1 n ( ˆR (k) ˆR) 2 n k=1 1 n = [ x (k) ] 2 (y k n(n 1) ˆRx k ) 2 k=1 vs. ˆVL ( ˆR) = 1 n(n 1) E[( x (k) ) 2 ] E[( x) 2 ] n ( x) 2 (y k ˆRx k ) 2 k=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

101 Quantiles ξ p = Q(p) = F 1 (p), p (0, 1) where F (y) is cdf ˆξ p = ˆQ(p) = inf { ˆF (y) p}, p = 0.5 for median y ( ) 1 ˆF (y) = w i w i I (y i y) i A i A To reduce the bias, use interpolation: ˆξ p = ˆξ p0 + ˆξ p1 ˆξ p0 ˆF (ˆξ p1 ) ˆF (ˆξ p0 ) {p ˆF (ˆξ p0 )} where ˆξ p1 = inf x1,...,x n {x; ˆF (x) p} & ˆξ p0 = sup x1,...,x n {x; ˆF (x) p} Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Test Inversion for Quantile C.I. Construct acceptance region for H 0 : p = p 0 {p 0 2[ ˆV (ˆp 0 )] 1/2, p 0 + 2[ ˆV (ˆp 0 )] 1/2 } Invert p-interval to give C.I. for ξ p0 { ˆQ(p 0 2[ ˆV (ˆp 0 )] 1/2 ), ˆQ(p 0 + 2[ ˆV (ˆp 0 )] 1/2 )} Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

102 Plots of CDF and Inverse CDF CDF (F) Inverse CDF (Q) p ζ p ζ p p Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Bahadur Representation Let ˆF (x) be an unbiased estimator of F (x), the population CDF of X. For given p (0, 1), we can define ζ p = F 1 (p) to be the p-th population quantile of X. Let ˆζ p = ˆF 1 (p) be the p-th sample quantile of X using ˆF. Also, define ˆp = ˆF (ζ p ). Bahadur (1966): ζ p = ˆF 1 (ˆp) = ˆF 1 (p) + d ˆF 1 (p) (ˆp p) dp = ˆF 1 (p) + df 1 (p) (ˆp p) dp = ˆζ p + 1 (ˆp p). f (ζ p ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

103 Variance Estimation for Quantiles (1) Bahadur Representation ( ) p(1 p) n(ˆξ p ξ p ) N 0, [F (ξ p )] 2 (SRS) ˆQ (p) = [ ˆF (ξ p )] 1 = ˆV (ˆξ p ) = ˆγ 2 ˆV { ˆF (ξp )} V [F (ˆξ b )]. = ( ) F 2 V (ˆξ p ) ξ ˆQ(ˆp + 2 V (ˆp)) ˆQ(ˆp 2 ˆV (ˆp)) (ˆp + 2 ˆV (ˆp)) (ˆp 2 V (ˆp)) =: ˆγ Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Variance Estimation for Quantiles (2) 1 Jackknife variance estimator is not consistent. Median ˆθ = 0.5(x m + x m+1 ) for n=2m even ˆV JK = 0.25(n 1)[x m+1 x m ] 2 2 Bootstrap and BRR are O.K. 3 Smoothed quantile ˆξ p = ˆξ p0 + ˆγ[p ˆF (ˆξ p0 )] ˆξ (k) p = ˆξ p0 + ˆγ[p ˆF (k) (ˆξ p0 )] ˆV JK (ˆξ p ) = k c k (ˆξ (k) p ˆξ p ) 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

104 4.4 Two-Phase Samples ȳ 2p,reg = ȳ 2p,REE = ȳ 2π + ( x 1π x 2π ) ˆβ 2 ˆβ 2 = w 2i (x i x 2π ) (x i x 2π ) i A 2 w 1 2i = π 1i π 2i 1i, w 1i = π 1 1i 1 i A 2 w 2i (x i x 2π ) y i ˆV JK (ȳ 2p,REE ) = L k=1 [ ] c k ȳ (k) 2 2p,REE ȳ 2p,REE Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Two-Phase Samples where w (k) 2i = (w (k) 1i )(π 1 2i 1i ) x (k) 1π = i A1 w (k) 1i 1 i A1 w (k) 1i x i ( x (k) 2π, ȳ (k) 2π ) = i A2 w (k) 2i 1 w (k) 2i (x i, y i ) i A2 ˆβ (k) 2 = w (k) 2i (x i x (k) 2π ) (x i x (k) i A2 2π ) 1 w (k) 2i (x i x (k) 2π ) (y i ȳ (k) 2π ) i A 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

105 Theorem Kim, Navarro, Fuller (2006 JASA), Assumptions (i) Second phase is stratified with π 2i 1i constant within group (ii) K L < Nn 1 π 1i < K U for some K L &K U (iii) V { ˆT 1y F} K M V { ˆT 1y,SRS } where ˆT 1y = i A 1 π 1 1i y i (iv) nv { ˆT 1y F} = N N i=1 j=1 Ω ijy i y j where N i=1 Ω ij = O(N 1 ) ( ) 2 (v) E ˆV 1 ( ˆT 1y ) F V ( ˆT 1y F) 1 = o(1) (vi) E{[c k ( ˆT (k) 1y ˆT 1y ) 2 ] 2 F} < K L L 2 [V ( ˆT 1y )] 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Theorem Kim, Navarro, Fuller (2006 JASA), Result ˆV {ȳ 2p,reg } = V (ȳ 2p,reg F) 1 N 2 where κ 2i = π 2i 1i N i=1 κ 1 2i (1 κ 2i )e 2 i + o p (n 1 ) e i = y i ȳ N (x i x)β N [ N ] 1 N β N = (x i x N ) (x i x N ) (x i x N ) y i. i=1 i=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

106 Variance Estimation Replication is computationally efficient for large surveys, simple for users Jackknife works if Taylor appropriate Grouping for computational efficiency Theoretical improvement for quantiles Problem with rare items Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Variance Estimation Using SAS Always use weights, strata, clusters, and domains Taylor series linearization Replication variance estimation Balanced repeated replication (BRR) Jackknife repeated replication (delete-one jackknife) User-specified replicate weights Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

107 Taylor Series Linearization Variance Estimation Use first-stage sampling units (PSUs) Compute pooled variance from strata Compute stratum variance based on cluster (PSU) totals Use the VARMETHOD=TAYLOR option Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Taylor Series Linearization Variance Estimation ˆV (ˆθ) = h (n h 1) 1 n h (1 f h ) i (e hi+ ē h.. ) 2 e hi+ = j e rc,hi+ = j hij hij w hij w hij 1 1 w hij (y hij ȳ... ) w hij (δ rc,hij ˆP rc ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

108 VARMETHOD=TAYLOR proc s u r v e y f r e q data=responsedata varmethod=t a y l o r t o t a l=t o t ; s t r a t a S t a t e Type ; weight SamplingWeight ; t a b l e s Rating ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Replication Variance Estimation Using SAS Methods include delete-one jackknife, BRR, and user-specified replicate weights The quantity of interest is computed for every replicate subsample, and the deviation from the full sample estimate is measured Design information is not necessary if the replicate weights are supplied Use the VARMETHOD=JACKKNIFE BRR option Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

109 Replication Variance Estimation Using SAS Create R replicate samples based on the replication method specified For any statistic θ, compute ˆθ for the full sample and ˆθ (r) for every replicate sample The replication variance estimator is ˆV (ˆθ) = r α r (ˆθ (r) ˆθ) 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 SAS Statements and Options VARMETHOD= TAYLOR JACKKNIFE BRR OUTWEIGHT= OUTJKCOEFS= REPWEIGHTS statement JKCOEFS= TOTAL RATE = Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

110 Create Replicate Weights proc surveymeans data=responsedata varmethod=j a c k k n i f e ( o u t w e i g h t=resdatajk o u t j k c o e f s=resjkcoef ) ; s t r a t a S t a t e Type ; weight SamplingWeight ; v a r UsageTime ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 The SURVEYMEANS Procedure Variance Estimation Method Jackknife Number of Replicates 300 Statistics Variable Label N Mean Std Error of Mean 95% CL for Mean UsageTime Computer Usage Time Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

111 Use Replicate Weights proc s u r v e y f r e q data=resdatajk ; weight SamplingWeight ; t a b l e s Rating / c h i s q t e s t p =( ) ; t a b l e s Recommend ; r e p w e i g h t s RepWt : / j k c o e f s=resjkcoef ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 The SURVEYFREQ Procedure Variance Estimation Method Replicate Weights Jackknife RESDATAJK Number of Replicates 300 Customer Satisfaction Rating Frequency Weighted Frequency Std Err of Wgt Freq Percent Test Percent Std Err of Percent Extremely Unsatisfied Unsatisfied Neutral Satisfied Extremely Satisfied Total Frequency Missing = 16 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318

112 Recommend Recommend Frequency Weighted Frequency Std Err of Wgt Freq Percent Std Err of Percent Total Frequency Missing = 11 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 4 7/23-24/ / 318 Chapter 5 Models Used in Conjunction with Sampling World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

113 Nonresponse Unit Nonresponse: weight adjustment Item Nonresponse: imputation Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Two-Phase Setup for Item Nonresponse Phase one (A): Observe x i Phase two (A R ) : Observe (x i, y i ) π 1i = Pr[i A] : inclusion probability phase one (known) π 2i 1i = Pr[i A R i A] : inclusion probability phase two (unknown) Response indicator variable: R i = { 1 if i AR 0 if i / A R for i A Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

114 Two-Phase Setup for Unit Nonresponse We are interested in estimating the population mean of Y using weighted mean of the observations: i A ȳ R = R w i y i i A R w i where w i = π 1 1i ˆπ 1 2i 1i Regression weighting approach ȳ reg,1 = x N ˆβ or ȳ reg,2 = x 1 ˆβ where x 1 = ( i A π 1 1i ) 1 ( i A π 1 1i x i ) and ˆβ = ( 1i x ix i ) 1 ( i A R π 1 i A R π 1 1i x iy i ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Theorem Theorem (i) V [N 1 i A π 1 1i (x i, y i ) F] = O(n 1 ) (ii) V [ ˆV (ȲHT) F] = O(n 3 ) (iii) K L < π 2i 1i < K U, π 1 2i 1i = x i α for some α, (iv) x i λ = 1 for all i for some λ, (iv) R i : independent ȳ reg,1 ȳ N = 1 N where π 2i = π 1i π 2i 1i, e i = y i x i β N, and β N = ( i U π 2i 1ix i x i) 1 i U π 2i 1ix i y i. i A R π 1 2i e i + O p (n 1 ), Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

115 Proof of Theorem Since ˆβ = ( i A R π 1 1i x i x i) 1 i A R π 1 1i x i y i, ˆβ β N = O p (n 1/2 ) where β N = ( i U π 2i 1ix i x i) 1 i U π 2i 1ix i y i. ȳ reg,1 ȳ N = x N ˆβ xn β N N (y i x i β N ) = π 1 2i 1i π 2i 1i(y i x i β N ) i=1 = i=1 N (α x i)π 2i 1i (y i x i β N ) = 0 i=1 Use π 1 2i 1i = x i α, transform x i to show x N ( ˆβ β N ) = N 1 π 1 2i e i + O p (n 1 ). i A R Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Variance Estimation for ȳ reg,1 ȳ reg,1 = i A R x N j A R π 1 1j x jx j 1 π 1 1i x i y i =: 1 N i A R 1 π 1i 1 ˆπ 2i 1i y i Small f = n/n, let ˆb j = ˆπ 1 2j 1jêj, ê j = y j x j ˆβ. ˆV = 1 N 2 i A R π 1ij π 1iπ 1j π 1ij j A R ˆb i π 1i ˆbj π 1j Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

116 Justification Variance V w 2i e i F = (π 2ij π 2i π 2j ) w 2i w 2j e i e j i A R i U j U = π 2i 1i π 2j 1j (π 1ij π 1i π 1j )w 2i w 2j e i e j i j;i,j U + i U(π 2i π 2 2i)w 2 2ie 2 i where π 2ij = { π1ij π 2i 1i π 2j 1j for i j π 1i π 2i 1i for i = j. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Justification (Cont d) Expectation of variance estimator E 1ij (π 1ij π 1i π 1j )w 2i e i w 2j e j F π 1 i A R j A R = i U(π 1i π 2 1i)π 2i 1i w 2 2ie 2 i = i U + i j;i,j U π 2i 1i π 2j 1j (π 1ij π 1i π 1j )w 2i e i w 2j e j (π 2ij π 2i π 2j )w 2i e i w 2j e j j U + i U π 2i (π 2i π 1i )w 2 2ie 2 i, where w 2i = N 1 π 1 2i. The second term is the bias of the variance estimator and it is of order O(N 1 ). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

117 Variance Estimation for ȳ reg,2 ȳ reg,2 = x 1 ˆβ ȳ reg,2 ȳ N = ( x 1 x N )β N + x N ( ˆβ β N ) + O p (n 1 ) = ( x 1 x N )β N + N 1 π 1 2i (y i x i β N ) + O p (n 1 ). i A R Variance estimator ˆV 2 = 1 N 2 i A j A π 1ij π 1i π 1j π 1ij ˆb i2 π 1i ˆb j2 π 1j where ˆb j2 = (x j x 1 ) ˆβ + (N x 1 ) ( i AR π 1 1i x i x i) 1 Rj x jêj. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Imputation Fill in missing values with plausible values Provides a complete data file: we can apply standard complete data methods By filling in missing values, analyses by different users will be consistent Good imputation model reduces the nonresponse bias Makes full use of information Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

118 A Hot Deck Imputation Procedure Partition the sample into G groups: A = A 1 A 2 A G. In group g, we have n g elements, r g respondents, and m g = n g r g nonrespondents. For each group A g, select m g imputed values from r g respondents with replacement (or without replacement). Imputation model: y i iid(µ g, σ 2 g ), i A g (respondents and missing) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Example 5.2.1: Hot Deck Imputation Under SRS - y i : study variable. subject to missing - x i : auxiliary variable. always observed (Group indicator) - I i : sampling indicator function for unit i - R i : response indicator function for y i - yi : imputed value A g = A Rg A Mg with A Rg = {i A g ; R i = 1} and A Mg = {i A g ; R i = 0}. Imputation: y j = y i with probability 1/r g for i A Rg and j A Mg. Imputed estimator of ȳ N : ȳ I = n 1 i A {R i y i + (1 R i ) yi } =: n 1 y Ii i A Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

119 Variance of Hot Deck Imputed Mean V (ȳ I ) = V {E I (ȳ I y n )} + E {V I (ȳ I y n )} G G = V n g ȳ Rg + E n 2 n 1 g=1 g=1 ( ) m g 1 r 1 g S 2 Rg where ȳ Rg = rg 1 i A Rg y i and SRg 2 = (r g 1) 1 i A Rg (y i ȳ Rg ) 2, y n = (y 1, y 2,..., y n ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Variance of Hot Deck Imputed Sample (2) Model : y i i A g iid(µ g, σ 2 g ) V {ȳ I } = V {ȳ n } + n 2 G g=1 G = V {ȳ n } + n 2 c g σg 2 g=1 n g m g rg 1 σg 2 + n 2 G g=1 m g (1 rg 1 )σg 2 Reduced sample size: n 2 n 2 g (r 1 g ng 1 )σg 2 Randomness due to stochastic imputation: n 2 m g (1 rg 1 )σg 2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

120 Variance Estimation Naive approach: Treat imputed values as if observed Naive approach underestimates the true variance! Example: Naive: ˆV I = n 1 S 2 I E { S 2 I } } n = E {(n 1) 1 (y Ii ȳ I ) 2 i=1. = (n 1) 1 [ E{(y Ii µ) 2 } V {ȳ I } ]. = E(Sy,n) 2 Bias corrected estimator ˆV = ˆV G I + c g SRg 2 g=1 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Other Approaches for Variance Estimation Multiple imputation: Rubin (1987) Adjusted jackknife: Rao and Shao (1992) Fractional imputation: Kim and Fuller (2004), Fuller and Kim (2005) Linearization: Shao and Steel (1999), Kim and Rao (2009) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

121 Fractional Imputation Basic Idea Split the record with missing item into M imputed values Assign fractional weights Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Fractional imputation Features Split the record with missing item into m(> 1) imputed values Assign fractional weights The final product is a single data file with size nm. For variance estimation, the fractional weights are replicated. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

122 Fractional imputation Example (n = 10) ID Weight y 1 y 2 1 w 1 y 1,1 y 1,2 2 w 2 y 2,1? 3 w 3? y 3,2 4 w 4 y 4,1 y 4,2 5 w 5 y 5,1 y 5,2 6 w 6 y 6,1 y 6,2 7 w 7? y 7,2 8 w 8?? 9 w 9 y 9,1 y 9,2 10 w 10 y 10,2 y 10,2?: Missing Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Fractional imputation (categorical case) Fully Efficient Fractional Imputation (FEFI) If both y 1 and y 2 are categorical, then fractional imputation is easy to apply. We have only finite number of possible values. Imputed values = possible values The fractional weights are the conditional probabilities of the possible values given the observations. Can use EM by weighting method of Ibrahim (1990) to compute the fractional weights. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

123 FEFI Example (y 1, y 2 : dichotomous, taking 0 or 1) ID Weight y 1 y 2 1 w 1 y 1,1 y 1,2 2 w 2 w2,1 y 2,1 0 w 2 w2,2 y 2,1 1 3 w 3 w3,1 0 y 3,2 w 3 w3,2 1 y 3,2 4 w 4 y 4,1 y 4,2 5 w 5 y 5,1 y 5,2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 FEFI Example (y 1, y 2 : dichotomous, taking 0 or 1) ID Weight y 1 y 2 6 w 6 y 6,1 y 6,2 7 w 7 w7,1 0 y 7,2 w 7 w7,2 1 y 7,2 8 w 8 w8,1 0 0 w 8 w8,2 0 1 w 8 w8,3 1 0 w 8 w8, w 9 y 9,1 y 9,2 10 w 10 y 10,1 y 10,2 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

124 FEFI Example (Cont d) E-step: Fractional weights are the conditional probabilities of the imputed values given the observations. w ij = ˆP(y (j) i,mis y i,obs) = ˆπ(y i,obs, y (j) i,mis ) Mi l=1 ˆπ(y i,obs, y (l) i,mis ) where (y i,obs, y i,mis ) is the (observed, missing) part of y i = (y i1,, y i,p ). M-step: Update the joint probability using the fractional weights. ˆπ ab = 1ˆN n i=1 M i j=1 w i w ij I (y (j) i,1 = a, y (j) i,2 = b) with ˆN = n i=1 w i. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 FEFI Example (Cont d) Variance estimation Recompute the fractional weights for each replication Apply the same EM algorithm using the replicated weights. E-step: Fractional weights are the conditional probabilities of the imputed values given the observations. w (k) ij = ˆπ (k) (y i,obs, y (j) i,mis ) Mi l=1 ˆπ(k) (y i,obs, y (l) i,mis ) M-step: Update the joint probability using the fractional weights. ˆπ (k) ab = 1 ˆN (k) n M i i=1 j=1 w (k) i w (k) ij I (y (j) i,1 = a, y (j) i,2 = b) where ˆN (k) = n i=1 w (k) i. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

125 FEFI Example (Cont d) Final Product Replication Weights Weight x y 1 y 2 Rep 1 Rep 2 Rep L w 1 x 1 y 1,1 y 1,2 w (1) 1 w (2) 1 w (L) 1 w 2 w2,1 x 2 y 2,2 0 w (1) 2 w (1) 2,1 w (2) 2 w (2) 2,1 w (L) 2 w (L) 2,1 w 2 w2,2 x 2 y 2,2 1 w (1) 2 w (1) 2,2 w (2) 2 w (2) 2,1 w (L) 2 w (L) 2,2 w 3 w3,1 x 3 0 y 3,2 w (1) 3 w (1) 3,1 w (2) 3 w (2) 3,1 w (L) 3 w (L) 3,1 w 3 w3,2 x 3 1 y 3,2 w (1) 3 w (1) 3,2 w (2) 3 w (2) 3,2 w (L) 3 w (L) 3,2 w 4 x 4 y 4,1 y 4,2 w (1) 4 w (2) 4 w (L) 4 w 5 x 5 y 5,1 y 5,2 w (1) 5 w (2) 5 w (L) 5 w 6 x 6 y 6,1 y 6,2 w (1) 6 w (2) 6 w (L) 6 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 FEFI Example (Cont d) Final Product Replication Weights Weight x y 1 y 2 Rep 1 Rep 2 Rep L w 7 w7,1 x 7 0 y 7,2 w (1) 7 w (1) 7,1 w (2) 7 w (2) 7,1 w (L) 7 w (L) 7,1 w 7 w7,2 x 7 1 y 7,2 w (1) 7 w (1) 7,2 w (2) 7 w (2) 7,2 w (L) 7 w (L) 7,2 w 8 w8,1 x w (1) 8 w (1) 8,1 w (2) 8 w (2) 8,1 w (L) 8 w (L) 8,1 w 8 w8,2 x w (1) 8 w (1) 8,2 w (2) 8 w (2) 8,2 w (L) 8 w (L) 8,2 w 8 w8,3 x w (1) 8 w (1) 8,3 w (2) 8 w (2) 8,3 w (L) 8 w (L) 8,3 w 8 w8,4 x w (1) 8 w (1) 8,4 w (2) 8 w (2) 8,4 w (L) 8 w (L) 8,4 w 9 x 9 y 9,1 y 9,2 w (1) 9 w (2) 9 w (L) 9 w 10 x 10 y 10,1 y 10,2 w (1) 10 w (2) 10 w (L) 10 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

126 Fractional Hot-Deck Imputation Using SAS PROC SURVEYIMPUTE Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Missing Values Exclude observations with missing weights Analyze missing levels as a separate level Delete observations with missing values (equivalent to imputing missing values with the estimated values from the analysis model [SAS default]) Analyze observations without missing values in a separate domain (equivalent to imputing missing values by 0 [NOMCAR]) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

127 Nonresponse Follow-up interviews Weight adjustment, poststratification Hot-deck and fractional hot-deck imputation Multiple imputation (MI and MIANALYZE procedures in SAS) Other techniques Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 PROC SURVEYIMPUTE Hot-deck imputation Simple random samples Proportional to weights Approximate Bayesian bootstrap Fully efficient fractional imputation (FEFI) Imputation-adjusted replicate weights Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

128 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 FEFI Detail Use all possible levels that a missing item can take, given the levels of the observed items Assign fractional weights proportional to the weighted frequencies of the imputed levels in the observed data Unit X Y Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

129 FEFI: Initialization Fill in missing values Use the complete cases to compute fractional weights Recipient ImpWt Unit X Y Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 M Step: Compute Proportions The FREQ Procedure Frequency Percent Row Pct Col Pct X Table of X by Y Y 0 1 Total Total Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

130 E Step: Adjust Fractional Weights Recipient ImpWt Unit X Y Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Repeat E-M Steps in Replicate Samples Recipient ImpWt FracWgt ImpRepWt_1 ImpRepWt_2 ImpRepWt_9 Unit X Y Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

131 Digitech Cable Item nonresponse: Rating, Recommend,... Impute missing items from the observed data Use imputation cells Create an imputed data set and a set of replicate weights for future use Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 PROC SURVEYIMPUTE proc s u r v e y i m p u t e data=responsedata method= f e f i ; s t r a t a S t a t e Type ; weight SamplingWeight ; c l a s s Rating Recommend H o u s e h o l d S i z e Race ; v a r Rating Recommend H o u s e h o l d S i z e Race ; c e l l s I m p u t a t i o n C e l l s ; output out=imputeddata o u t j k c o e f=j K C o e f f i c i e n t s ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

132 The SURVEYIMPUTE Procedure Missing Data Patterns Group Rating Recommend HouseholdSize Race Freq Sum of Weights Unweighted Percent 1 X X X X X X X. X X X X X X X Missing Data Patterns Group Weighted Percent Imputation Summary Observation Status Number of Observations Sum of Weights Nonmissing Missing Missing, Imputed Missing, Not Imputed 0 0 Missing, Partially Imputed 0 0 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 NumberOfDonors NumberOfUnits NumberOfRows Output data set contains the imputed data New variables: Replicate Weights, Recipient Index, Unit ID Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

133 UnitID Recipient ImpWt SamplingWeight ImputationCells Rating Recommend HouseholdSize Race Neutral 0 Medium Other Satisfied 0 Medium Other Unsatisfied 0 Medium Other Extremely Unsatisfied 0 Large NA Extremely Unsatisfied 0 Large White Extremely Unsatisfied 0 Medium Black Extremely Unsatisfied 0 Medium Hispanic Extremely Unsatisfied 0 Medium White Extremely Unsatisfied 0 Small Black Extremely Unsatisfied 0 Small Hispanic Extremely Unsatisfied 0 Small NA Extremely Unsatisfied 0 Small Other Extremely Unsatisfied 0 Small White Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Analyses of FEFI Data The imputed data set can be used for any analyses Use the imputation-adjusted weights Use the imputation-adjusted replicate weights The number of rows in the imputed data set is NOT the same as the number of observation units Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

134 Use the Imputed Data to Estimate Usage proc surveymeans data=imputeddata mean varmethod=j a c k k n i f e ; weight ImpWt ; v a r UsageTime ; r e p w e i g h t s ImpRepWt : / j k c o e f s=j K C o e f f i c i e n t s ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 The SURVEYMEANS Procedure Data Summary Number of Observations 450 Sum of Weights Variance Estimation Method Replicate Weights Jackknife IMPUTEDDATA Number of Replicates 300 Statistics Variable Label Mean Std Error of Mean UsageTime Computer Usage Time Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

135 Use the Imputed Data to Estimate Rating proc s u r v e y f r e q data=imputeddata varmethod=j a c k k n i f e ; weight ImpWt ; t a b l e s Rating Recommend ; r e p w e i g h t s ImpRepWt : / j k c o e f s=j K C o e f f i c i e n t s ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 The SURVEYFREQ Procedure Customer Satisfaction Rating Frequency Weighted Frequency Std Err of Wgt Freq Percent Std Err of Percent Extremely Unsatisfied Unsatisfied Neutral Satisfied Extremely Satisfied Total E Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

136 Recommend Recommend Frequency Weighted Frequency Std Err of Wgt Freq Percent Std Err of Percent Total E Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / Small area estimation Basic Setup Original sample A is decomposed into G domains such that A = A 1 A G and n = n n G n is large but n g can be very small. Direct estimator of Y g = i U g y i Ŷ d,g = i A g 1 π i y i Unbiased May have high variance. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

137 If there is some auxiliary information available, then we can do something: Synthetic estimator of Y g Ŷs,g = Xg ˆβ where X g = i U g x i is the known total of x i in U g and ˆβ is an estimated regression coefficient. Low variance (if x i does not contain the domain indicator). Could be biased (unless i U g (y i x ib) = 0) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Composite estimation: consider Ŷ c,g = α g Ŷ d,g + (1 α g ) Ŷs,g for some α g (0, 1). We are interested in finding αg that minimizes the MSE of Ŷ c. The optimal choice is α g = MSE MSE (Ŷd,g ) For the direct estimation part, MSE (Ŷs,g ) + MSE (Ŷd,g ) (Ŷs,g ) = V (Ŷd,g ) can be estimated. ) } For the synthetic estimation part, MSE (Ŷs,g = E {(Ŷ s,g Y g ) 2 cannot be computed directly without assuming some error model. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

138 Area level estimation Basic Setup Parameter of interest: Ȳ g = Ng 1 Model and u g ( 0, σ 2 u). Also, we have with V g = V ( ˆȲ d,g ). i U g y i Ȳ g = X g β + u g ˆȲ d,g ( Ȳ g, V g ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Area level estimation (Cont d) Two model can be written ˆȲ d,g = Ȳ g + e g X g β = Ȳ g u g where e g and u g are independent error terms with mean zeros and variance V g and σ 2 u, respectively. Thus, the best linear unbiased predictor (BLUP) can be written as where α g = σ 2 u/(v g + σ 2 u). ˆȲ g = αg ˆȲ d,g + ( 1 αg ) X g β Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

139 Area level estimation (Cont d) MSE: If β, V g, and σu 2 are known, then ( ) ( ) MSE ˆȲ g = V ˆȲ g Ȳ g { ( ) = V αg ˆȲ d,g Ȳg + ( 1 αg ) ) ( X } g β Ȳg = ( α g ) 2 Vg + ( 1 α g ) 2 σ 2 u = α g V g = ( 1 α g ) σ 2 u. Note that, since 0 < α g < 1, MSE ( ˆȲ g ) < V g and MSE ( ˆȲ g ) < σ 2 u. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Area level estimation (Cont d) If β and σ 2 u are unknown: 1 Find a consistent estimator of β and σu. 2 2 Use ˆȲ g (ˆα g, ˆβ) = ˆα g ˆȲ d,g + ( 1 ˆα g ) X g ˆβ. where ˆα g = ˆσ 2 u/( ˆV g + ˆσ 2 u) Estimation of σu: 2 Method of moment ˆσ u 2 = { G ( k g ˆȲ d,g X ˆβ) } 2 g ˆV d,g, G p g { where k g ˆσ u 2 + ˆV } 1 g and G g=1 k g = 1. If ˆσ u 2 is negative, then we set it to zero. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

140 Area level estimation (Cont d) MSE { } MSE ˆȲ g (ˆα g, ˆβ) = V = V { ˆȲ g (ˆα g, ˆβ) Ȳ g } { ˆα g ( ) ˆȲ d,g Ȳ g + ( 1 ˆα g ) ( X ˆβ )} g Ȳ g ) 2 {σu 2 + X g V ( ˆβ) X } g = ( αg ) 2 Vg + ( 1 αg +V (ˆα g ) { V g + σu 2 } = αg V g + ( 1 αg )2 X g V ( ˆβ) X g +V (ˆα g ) { V g + σu 2 } MSE estimation (Prasad and Rao, 1990): { } MSE ˆ ˆȲ g (ˆα g, ˆβ) = ˆα g ˆV g + ( 1 ˆα g )2 X ˆV g ( ˆβ) X g { } +2 ˆV (ˆα g ) ˆV g + ˆσ u 2. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318 Unit level estimation Unit level estimation: Battese, Harter, and Fuller (1988). Use a unit level modeling y gi = x giβ + u g + e gi and It can be shown that Ŷ g = i U g {x ˆβ } gi + û g. ˆȲ g = ˆα g Ȳreg,g + ( 1 ˆα g ) Ȳs,g where and Ȳ reg,g = ˆȲ d,g + ( X g ˆ X d,g ) ˆβ Ȳ s,g = X g ˆβ. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 5 7/23-24/ / 318

141 Chapter 6 Analytic Studies World Statistics Congress Short Course July 23-24, 2015 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Parameters Two types of parameters Descriptive parameter: How many people in the United States were unemployed on March 10, 2015? Analytic parameter: If personal income (in the United States) increases 2%, how much will the consumption of beef increase? Basic approach to estimating analytic parameters 1 Specify a model that describes the relationship among the variables (often called superpopulation model). 2 Estimate the parameters in the model using the realized sample. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

142 Parameter Estimation for Model Parameter 1 θ N : finite population characteristic for θ satisfying E(θ N ) = θ + O(N 1 ) and V (θ N θ) = O(N 1 ), where the distribution is with respect to the model. } 2 ˆθ: estimator of θ N satisfying E {ˆθ θ N F N = O p (n 1 ) and } V {ˆθ F N = O p (n 1 ) almost surely. E(ˆθ) = θ + O(n 1 ) { )} V (ˆθ θ) = E V (ˆθ FN + V (θ N ) ˆV (ˆθ θ) = ˆV (ˆθ F N ) + ˆV (ˆθ N θ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 A Regression Model The finite population is a realization from a model y i = x i β + e i (2) where the e i are independent (0, σ 2 ) random variables independent of x j for all i and j. We are interested in estimating β from the sample. First order inclusion probabilities π i are available. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

143 Estimation of Regression Coefficients OLS estimator ( ) 1 ˆβ ols = x ix i x iy i Probability weighted estimator i A i A ˆβ π = ( i A π 1 i x ix i ) 1 i A π 1 i x iy i If superpopulation model also holds for the sample, then OLS estimator is optimal. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Informative Sampling Non-informative sampling design (with respect to the superpopulation model) satisfies P (y i B x i, i A) = P (y i B x i ) (3) for any measurable set B. The left side is the sample model and the right side is the population model. Informative sampling design: Equality (3) does not hold. Non-informative sampling for regression implies E { x ie i i A } = 0. (4) If condition (4) is satisfied and moment conditions hold, ˆβ ols is consistent Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

144 Hypothesis Testing Thus, one may want to test (4), or test directly { } { } H 0 : E ˆβols = E ˆβπ (5) 1 From the sample, fit a regression of y i on (x i, z i ) where z i = π 1 2 Perform a test for γ = 0 under the expanded model y = Xβ + Zγ + a where a is the error term satisfying E(a X, Z) = 0. Justification : Testing (5) is equivalent to testing E { Z (I P X ) y } = 0 (6) where P X = X (X X ) 1 X. Since ˆγ = {Z (I P X ) Z} 1 Z (I P X ) y, testing for γ = 0 is equivalent to testing for (6). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 i x i Remarks on Testing When performing the hypothesis testing, design consistent variance estimator is preferable. Rejecting the null hypothesis means that we cannot directly use the OLS estimator under the current model. Include more x s until the sampling design is non-informative under the expanded model. (Example 6.3.1) Use the probability weighted estimator or use other consistent estimators (Section 6.3.2) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

145 Example 6.3.1: Birthweight and Age Data y: gestational age x: birthweight stratified sample of babies OLS result ŷ = x (0.370) (0.012) Weighted regression ŷ = x (0.535) (0.016) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Example (Cont d) DuMouchel & Duncan test: Fit the OLS regression of y on (1, x, w, wx), where w i = π 1 i, to obtain ( ˆβ 0, ˆβ 1, ˆγ 0, ˆγ 1 ) = (22.088, 0.583, 8.287, 0.326) (0.532) (0.033) (0.861) (0.332) The hypothesis is rejected. (F (2, 86) = ) Thus, we cannot use OLS method in this data. Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

146 Example (Cont d) However, if we include a quadratic term into the model then the sampling design becomes noninformative. OLS result where x 2 = 0.01(x 30) 2. Weighted regression ŷ = x x 2 (0.343) (0.010) (0.082) ŷ = x x 2 (0.386) (0.011) (0.108) DuMouchel & Duncan test: F (3, 84) = Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Estimators Under Informative Sampling Pfeffermann and Sverchkov (1999) estimator: Minimize Q(β) = i A w i w i (y i x i β) 2 where w i = π 1 i, w i = E (w i x i, i A). w i can be a function of x i. (Estimated) GLS estimator: Minimize where v 2 i = E Q(β) = w i (y i x i β) 2 /vi 2 (7) i A } {w i (y i x i β) 2 x i. 1 Obtain ˆβ π and compute ê i = y i x i ˆβ π. 2 Fit a (nonlinear) regression model a 2 i = w i ê 2 i on x i, a 2 i = q a (x i ; γ a ) + r ai to get ˆv 2 i = q a (x i ; ˆγ a ) and insert ˆv 2 i in (7). Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

147 Comments Fitting models to complex survey data Always test for informative design If the hypothesis of noninformative design is rejected: Examine model Use HT estimator or more complex design consistent estimator Variance estimation for clusters and two-stage designs must recognize clusters Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Linear and Logistic Regression Using SAS PROC SURVEYREG PROC SURVEYLOGISTIC Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

148 PROC SURVEYREG Linear regression Regression coefficients Significance tests Estimates and contrasts Regression estimator Comparisons of domain means Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

149 PROC SURVEYLOGISTIC Categorical response Logit, probit, complementary log-log, and generalized logit regressions Regression coefficients Estimates and contrasts Odds ratios Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

150 Pseudo-likelihood Estimation Finite population parameter is defined by the population likelihood, l N (θ N ) = U N log {L(θ N, x i )} A sample-based estimate of the likelihood is used to estimate the parameter, l π (ˆθ) = { } A π 1 i log L(ˆθ, x i ) Variance estimators assume fixed population values, V (ˆθ F N ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Taylor Series Linearization Variance Estimation Sandwich variance estimator that accounts for strata, cluster, and weights: ˆV (ˆθ) = I 1 GI 1 G = (n p) 1 (n 1) h (n h 1) 1 n h (1 f h ) i (e hi+ ē h.. ) (e hi+ ē h.. ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

151 Replication Variance Estimation Estimate θ in the full sample and in every replicate sample: ˆV (ˆθ) = r α r (ˆθ (r) ˆθ)(ˆθ (r) ˆθ) Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Digitech Cable Customer satisfaction survey Describe usage time based on data usage after adjusting for race Describe usage time based on race Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

152 Linear Regression proc s u r v e y r e g data=imputeddata ; weight ImpWt ; c l a s s Race ; model UsageTime = DataUsage Race / s o l u t i o n ; r e p w e i g h t s ImpRepWt : / j k c o e f s=j K C o e f f i c i e n t s ; lsmeans Race / d i f f ; output out=regout p r e d i c t e d=f i t t e d V a l u e s r e s i d u a l=r e s i d u a l s ; run ; Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318

153 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318 Kim & Fuller & Mukhopadhyay (ISU & SAS) Chapter 6 7/23-24/ / 318