Lineari ation Variance Estimation and Allocation

for Twophase Sampling under Mass Imputation

A. Demnati and J.N.K. Rao 

Statistics Canada Carleton University 

Business Survey Methods Division School of Mathematics and Statistics 

Ottawa, Canada, K1A 0T6 Ottawa, Canada, K1S 5B6 

Abdellatif.Demnati@statcan.gc.ca JRao@math.carleton.ca 

Abstract

We consider two-phase sampling in which values of a variable of interest are observed only in 

the second-phase sub-sample. Values for the first-phase units not sampled in the second-phase 

are mass imputed, using values from an administrative file when available and regression 

imputation otherwise. Such two-phase sampling methods are often used in annual business 

surveys to reduce survey costs and respondent burden, assuming that collecting values from 

administrative sources is much cheaper than obtaining values through questionnaires from 

sampled units. We study both naïve and design-consistent estimators for a total or mean under 

the above set-up. We also obtain associated variance estimators using a unified approach 

proposed by Demnati and Rao (2004). Simulation results on the finite sample performance of the 

estimators and associated variance estimators are also presented, using substitution or ratio mass 

imputation. We also study the case of missing sub-sample values and develop estimators of the 

total and associated variance estimators. Sample allocation issues are also studied. 

KEY WORDS: Regression imputation; Sample sizes determination; Substitution method; Twophase sampling. 

1. Introduction

At Statistics Canada, more than sixty annual business surveys are carried out using a holistic design termed as the 

Unified Enterprise Survey (UES). One of the objectives of the UES is to reduce survey cost and response burden by 

replacing some questionnaires with values from administrative data. We assume that obtaining values from 

administrative data is much cheaper than obtaining values through questionnaires. A random sample of size m units 

is first selected. Then a sub-sample of size n is selected and surveyed. The remaining m − n units as well as nonrespondents are imputed using administrative files. Composite imputation involving two or more different methods 

is also often used; for example, the values from administrative file (e.g., tax file), when available, and regression 

imputation otherwise. 

Demnati and Rao (2008) considered the case of simulated census data generated from a probability sample by 

imputing for the non-sampled units and sample non-respondents using auxiliary variables. They studied the 

estimation of a finite population total and other parameters. In this paper, we extended their work to two-phase 

sampling. For variance estimation, we have used the unified approach proposed by Demnati and Rao (2004). We 

have studied naïve estimators based on the simulated first-phase data as well as design-consistent estimators of a 

population total. In section 2, we consider the case of complete response and report the results of a simulation study. 

The case of missing data is studied in section 3. In section 4, we consider stratified simple random sampling in both 

phases and obtain “optimal” first-phase and second-phase strata sample sizes, mh and nh , that minimize the cost 

subject to constraints on the variances of estimators for one or more characteristics of interest. 

2. Complete Response

 
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 
 


2.1 Imputed Estimators

We first consider the case where complete responses are obtained from the second-phase subsample. Imputation is 

performed on all the first-phase units not sampled in the second-phase. Here, the imputed values yˆ k

* are given by 

* * * * * 

yˆ = (1− I ) yˆ + I yˆ with yˆ = t and yˆ based on an imputation model, where t is the value from an k k 1k k 2k 1k k 2k k 

administrative file and the constant I

k is the missing tk indicator. 

A naïve estimator of the finite population total Y = ∑ yk is given by 

(N ) (1) (2|1) (2|1) * Y

ˆ = ∑d {a y + (1− a )yˆ } , (2.1) k k k k k 

(1) (1) (1) (1) (1) (1) where d = a /π , a is the first-phase sample membership indicator variable, π = E (a ) , E denotes k k k k k p k p 

(2|1) design expectation, and ak is the conditional second-phase sample membership indicator variable. 

Suppose the imputation model on the responses yk is specified by a generalized linear model with mean 

E ( y ) =  (β) = h(x

T

β), where x is a p 1 vector of explanatory variables for the model mean, h() is a “link” m k k k k 

function and Em denotes model expectation. For example, the choice h(a) = a gives a linear regression model, and 

a a * h(a) = e /(1+ e ) gives a logistic regression model for binary responses y . The imputed values yˆ are given by k 2k 

* (N ) ˆ(N ) 

yˆ =  (β

ˆ ) , where the naïve estimator β is obtained as the solution to the estimating equations 2k k 

ˆ(N ) l (β) = ∑a I Φ (β)( y −  (β)) = 0 , (2.2) β k k k k k 

(1) (2|1) -1 

a = a a and Φ (β) = [∂ (β) / ∂β][Var ( y )] . For example, Φ (β) = x for linear and logistic regression k k k k k m k k k 

models. 

) The finite population parameter θ N induced by the estimator Y

ˆ(N is given by 

(N ) (2|1) (2|1) * θ = E (Y

ˆ ) ≈ ∑π y + ∑(1−π ) y , (2.3) N p k k k k 

* * ˆ(N ) where y = E ( yˆ ). It is clear from (2.3) that θ depend on the selection probabilities. For example, the use of Y k p k N 

under simple random sampling at both stages induces a mixture of two totals as finite population parameter: 

(2|1) (2|1) * (2|1) θ ≈ f ∑ y + (1− f )∑ y where f is the conditional sampling fraction for the second-phase sample. N k k 

ˆ(N ) The sampling bias induced by the estimator Y in estimating the finite population total Y is given by 

π

(2|1) * θ −Y ≈ −∑(1− )( y − y ) ≡ B . (2.4) N k k k 

In order to remove the sampling bias, one may first estimate the bias (2.4) by 

(2|1) Bˆ = − ∑d (1−π )( y − yˆ*

) , (2.5) k k k k 

ˆ(N ) and then adjust Y to get the bias-adjusted estimator 

(1) * Y

ˆ = ∑d y + ∑(d − d ) yˆ , (2.6) k k k k k 

* with yˆ =  (β

ˆ

) , the estimator β

ˆ is obtained as solution to the estimating equations 2k k 

l

ˆ (β) = ∑d I Φ (β)(y −  (β)) = 0 , (2.7) β k k k k k 

where d = a /π and π = E (a ) . The design-based estimator Yˆ

is approximately unbiased for Y : E (Y

ˆ

) ≈ Y . k k k k p k p 

We may also write Y ˆ as 

(1) (2|1) (2|1) * Y

ˆ = ∑d {d y + (1− d ) yˆ } . k k k k k 

2.2 Variance Estimation
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We suppose first that the parameter of interest is (N ) T (1) θ = E (Y

ˆ ) . Let d = (d ,d ) , where d = d , and d = d . N p k 1k 2k 1k k 2k k 

The Demnati–Rao (DR) variance estimator (Demnati and Rao, 2004) is simply given by 

(N ) θ DR (Y

ˆ ) =θ(z) , (2.8) 

where θ(u) , in operation notation, is the variance estimator of the linear combination Uˆ = ∑u

T

d , u = (u ,u )

T

a k k k 1k 2k 

N ) th vector of constants, z = ∂f ( ) / ∂b | , f ( ) = Y

ˆ(

, is a 2 N matrix with k column d , and is a k b k = d d k b 

th (1) T 

b d 

2 N matrix of arbitrary real numbers with k column b = (b ,b ) . The design based variance estimator θ(u) k k k 

of the total Uˆ is given by 

θ(u) = ∑ ∑ u

T

cov (d ,d )u , (2.9) k t k p k t t 

with 

(1) (1) 1−ω 1−ω  kt kt cov (d ,d ) = diag(d )  diag(d ) , (2.10) p k t k   t 1−ω

(1)   kt kt 1−ω

(1) (1) (1) (1) (1) where ω = π π /π , ω = π π /π , and (π ,π ) denote respectively the first and two phase joint inclusion kt k t kt kt k t kt kt kt 

probabilities. Substituting (2.10) into (2.9) we get 

θ(u) =θ

(1) (u ,u ) + 2c (u ,u ) +θ (u ,u ) , (2.11) 

s 1 1 s 1 2 s 2 2 

with 

(1) (1) (1) (1) θ (x, y) = ∑∑d d (1−ω )x y , (2.12) s k t kt k t 

(1) (1) 

c (x, y) = ∑ ∑d d (1−ω )x y , (2.13) s k t kt k t 

and 

θ (x, y) = ∑∑d d (1−ω )x y . (2.14) s k t kt k t 

We may write cs(x, y) as 

(1) (1) (1) (1) 

c (x, y) = ∑ ∑d d (1−ω )x v =θ (x,v), (2.15) s k t kt k t s 

(2|1) (2|1) ( 2|1) ( 2|1) where v = d y and d = a /π is the conditional second-phase weight. It is seen from (2.15) that c (x, y) k k k k k k s 

is computed from the first-phase sampling variance estimator. 

(2|1) (1) (2|1) * It remains to evaluate z . We have f ( ) = ∑b π y + ∑(b − b π ) yˆ (b) with k b k k k k k k k 

* (

ˆ(N ) ˆ( N ) ( 2|1) 

yˆ (b) = (1− I )t + I  β (b)) and l (β(b)) = ∑b π I Φ (β(b))(y −  (β(b))) = 0 , where b is the second k k k k k β k k k k k k 

column of the matrix b. Hence, 

yˆ

* 

z = ∂f ( ) / ∂b | =

k (2.16) k b k b = d  (N ) * 

g ( y − yˆ ),  k k k 

with 

(N ) (2|1) (N ) (N ) -1 

g = π −π J

ˆ (β)[Jˆ (β

ˆ

)] Φ (β

ˆ )I , (2.17) k k k θ β k a k 

ˆ ˆ ˆ (N ) (N ) (N ) ( ˆ N ) where J (β) = −∂Y (β)/ ∂β , and J (β) = −∂l (β) / ∂β . θ β β

Under the linear model 

 = x

T

β and Var ( y ) = σ

2

/ c , (2.18) 

ˆ

k k m k k 

(N ) ˆ −1 for some specified constants c , Φ (β) = x c , β is estimated by β = Q ∑a I c x y with k k k k (N ) k k k k k 

Qˆ T ˆ ˆ (N ) (1) ( 2|1) T (N ) ˆ = ∑a I c x x . We have J (β) = ∑d (1− a )I x , J (β) = Q , and (N ) k k k k k θ k k k k θ (N ) 
 

 

 

 

 

 


 
 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(N ) (2|1) (1) (2|1) T −1 

g = π + π ∑ d (1− a )I x Qˆ x c I . (2.19) k k k t t t t t (N ) k k k 

A variance estimator of the bias-adjusted estimator Yˆ is given by (2.9) with uk replaced by 

yˆ k

* 

z = ∂f ( )/ ∂b | = , (2.20) k b k b d =  * 

g ( y − yˆ ),  k k k 

where 

-1 

g =1− J

ˆ (β)[Jˆ (β

ˆ

)] Φ (β

ˆ

)I , k θ β k k 

ˆ ˆ / * T ˆ f ( ) = Y , Jˆ (β) = −∂Y ∂β , and Jˆ (β) = −∂l

ˆ (β)/ ∂β . Under the linear model, we have yˆ = x β , d θ β β 2k k 

ˆ ˆ −1 ˆ β = Q ∑d I c x y with Q = ∑d I c x x

T

, and k k k k k k k k k k 

(1) T −1 

g =1+ ∑ (d − d )I x Qˆ x c I . (2.21) k t t t t t k k k 

2.3 Simulation Study

We conducted a small simulation study to examine the pe ˆ(N ) rformances of the estimators Y and Yˆ and associated 

variance estimators, when substitution imputation or ratio imputation is used. The more general case of values from 

administrative file when available and regression imputation otherwise is not studied in this section. We first 

generated a finite population {y ,..., y ,..., y } , with y = ( y , y , y )

T

, of size N = 393 from the following 1 k N k 1k 2k 3k 

1/ 2ε

1/ 2ε1/ 2 models: y = x + x , y =1.2x + x , and y = 5 +1.2x + x ε , where ε are independent observations 1k k k k 2k k k k 3k k k k k 

generated from N(0,1) , and the fixed xk are the “number of beds” for the Hospitals population studied in Valliant et 

al. (2000, p.424-427). We stratified the population into two strata with 272 units k having xk ≤ 350 in stratum 1 

and 121 units k with xk > 350 in stratum 2. We selected R = 5,000 stratified two-phase simple random samples of 

sizes m = m = 50 and n = n =15 . Our vector parameter of interest is the finite population total θ = (Y ,Y ,Y )

T 

. 1 2 1 2 N 1 2 3 

Values for the first-phase units not sampled in the second phase are mass imputed. For each variable l , l =1,2,3 , 

four estimators are considered. Two estimators used the substitution imputation method: the naïve estimator 

(N ) (1) (1) θ

ˆ = ∑ d {a y + (1− a )x } , and the corresponding design-based estimator θˆ = ∑ d y + ∑(d − d )x , lS k k lk k k lS k lk k k k 

l = 1,2,3 . The other two estimators used the ratio imputation method: the naïve estimator 

ˆ ˆ ˆ ˆ (N ) (1) (N ) (1) θ = ∑ d {a y + (1− a )x β } , and the associated design-based estimator θ = ∑ d y + ∑(d − d )x β , lR k k lk k k l lR k lk k k k l 

N ) where β

ˆ( = ∑ a y / ∑ a x and βˆ = ∑ d y / ∑ d x . The estimator θˆ reduces to the two-phase ratio estimator: l k lk k k l k lk k k lR 

(1) Y

ˆ

R = Y

ˆ

(Xˆ / Xˆ) . 

Let θ

ˆ denote an estimator of a population total θ N and θ(θ

ˆ

) be the associated variance estimator. We calculated the 

simulated relative bias of θ

ˆ and θ(θ

ˆ

) as RB(θ

ˆ

) = (θ

ˆ −θ ) /θ and RB{θ(θˆ

)} = {θ(θ

ˆ

) − MSE(θ

ˆ

)}/ MSE(θ

ˆ

) , where 

. N N 

ˆ −1 R ˆ θˆ θ = R ∑ θ is the mean of the estimates from the simulated samples r = 1,...,R and 

. r =1 r r 

−1 R 2 MSE(θ

ˆ

) = R ∑ (θ

ˆ −θ ) is the simulated mean squared error (MSE). We calculated RB(θˆ

) , RB{θ(θ

ˆ

)} and 

r =1 r N 

(

ˆ T ˆ T ˆ T

)

T ˆ (

ˆ(N ) ˆ ˆ(N ) ˆ )

T mean squared error (MSE) ratios for each component of the vector θ1 2 3 ,θ ,θ with θl lS lS lR lR = θ ,θ ,θ ,θ

(l =1,2,3 ), and those values are reported in Table 1. 

ˆ(N ) It is clear from Table 1 that the substitution naïve estimator θ S can lead to large relative bias and loss in efficiency 

when the model generating the variable of interest is different from the model generating the imputed values. For 

example, θ

ˆ

S

( performs poorly for the variables 2 and 3. On the other hand, the design based estimator θ

ˆ performs N ) 

ˆ(N ) well regardless of the population model and imputation method. The variance estimator θL for θ S leads to serious 
 

 


 
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ˆ

underestimation of MSE for variables 2 and 3. On the other hand, variance estimator θL for θ

ˆ performs well with 

(N ) ˆ(N ) small RB . The naïve estimator θ R under ratio imputation is more robust to model deviations than θ S

. 

3. Missing Responses

In this section, we consider the case of missing second-phase responses, y . Imputation is performed on all the firstphase missing responses. The resulting first-phase sample is complete. An estimator of the finite population total 

Y = ∑ yk is given by 

(1) * Y

ˆ

(τ ) = ∑ d o τ y + ∑(d − d o τ ) yˆ , (3.1) k k θ ,k k k k k θ ,k k 

for specified τ , where o = 1 if y is observed, and o = 0 otherwise. Here, the imputed value yˆ

* is given by θ ,k k k k 2k 

* 

yˆ =  (β

ˆ

) where the estimator β

ˆ

is obtained as solution to the estimating equations 2k k 

l

ˆ (β) = ∑ d o τ β

I Φ (β)(y −  (β)) = 0 , (3.2) β k k ,k k k k k 

for specified τ β ,k. 

In the case of complete response, the choice of τ =τ =1 gives the design-based estimator θ ,k β ,k 

(1) * ( 2|1) Y

ˆ = ∑d y + ∑(d − d ) yˆ , and the choice of τ =τ = π leads to the naïve estimator k k k k k θ ,k β ,k k 

(N ) (1) (2|1) ( 2|1) * Y

ˆ = ∑d [a y + (1− a ) yˆ ], studied in section 2. In the case of incomplete response, the choice of k k k k k 

−1 (1) τ = τ = ξ gives the design-response based estimator Y

ˆ = ∑d (o /ξ ) y + ∑(d − d (o /ξ )) yˆ

*

, where θ ,k β ,k k ξ k k k k k k k k k 

ξ = E (o ) and E denotes model expectation under response mechanism. k r k r 

We suppose that the parameter of interest is T (1) θ = E E (Y

ˆ

(τ )). Let d = (d ,d ,d ) , where d = d , d = d N p r k 1k 2k 3k 1k k 2k k 

and d = d o . Then the Demnati–Rao (DR) variance estimator (Demnati and Rao, 2007) is simply given by 

θ

3k k k 

DR (Y

ˆ

(τ )) =θ(z) , (3.3) 

T where θ(u) , in operation notation, is the variance estimator of the linear combination Uˆ = ∑uk dk

, 

T th u = (u ,u ,u ) a vector of constants, z = ∂f ( ) / ∂b | , f ( ) = Y

ˆ

(τ ) , is a 3 N matrix with k k 1k 2k 3k k b k = d d b d 

column d , and is a 3 N matrix of arbitrary real numbers with k th column b = (b ,b ,b )

T

. We have, k b k 1k 2k 3k 

yˆ k

* 



z = ∂f ( )/ ∂b | = 0 (3.4) k b k b d = 

 * 

g (τ )( y − yˆ ),  k k k 

with 

-1 

g (τ ) =τ −τ J

ˆ (β

ˆ

)[J

ˆ (β

ˆ

)] Φ (β

ˆ

)I (3.5) k θ ,k β ,k θ β k k 

where J

ˆ (β) = −∂Yˆ

(τ ) / ∂β , and J

ˆ (β) = −∂l

ˆ (β) / ∂β . Under linear model with τ =τ =τ , we have θ β β θ ,k β ,k k 

ˆ ˆ ˆ β = Q

-1 ∑d τ o I c x y , Q = ∑d τ o I c x xT

, and k k k k k k k k k k k k k k 

(1) T −1 

g (τ ) =τ {1+ ∑(d − d o τ )I x Qˆ x c I }. (3.6) k k t t t t t t k k k 

It remains to evaluate θ(u) . We have 

θ(u) = θ (u ) +θ (u ) , (3.7) 

s s r o 
 

 

 

 

 

 

 

 


 
 
 
 



 
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where u = u and u = (u ,u + o u )

T

. k ;o 3k k ;s 1k 2k k k ;o 

The design based variance estimator θ (u ) is given by (2.9) with u replaced by u . 

s s k k ;s 

The response estimated variance θr(u) , assuming that the order of expectation can be interchange so that 

E E = E E , is given by 

p r r p 

θ (u) = ∑ ∑ d d ω o o (ξ

ˆ −ξˆ ξˆ

)ξ

ˆ −1

u u , (3.8) r k t kt k t kt k t kt k t 

where ˆ ˆ ˆ ξ = Eˆ (o ) , ξ = Eˆ(o o ) , and E denotes the estimator of model expectation under a model on o . A k r k kt r k t r k 

variance estimator of Y

ˆ

(τ ) is then given by (3.7) with u replaced by z given by (3.4). Note that θ = Y if Y

ˆ

(τ ) k k N 

is unbiased for Y under the assumed response mechanism and the sampling design. 

4. Determination of “Optimal” Sample Si es

4.1 Complete Response

Let y be the value obtained using a questionnaire, and t or x be the value obtained from administrative sources. k k k 

We assume that obtaining values from administrative sources is much cheaper than obtaining values through 

questionnaire. We consider the case where a stratified first-phase sample of size m = (m ,...,m ,...,m ) of t or x is 1 h H k k 

obtained, and then from each first-phase sample s

(1) , a sub-sample s of size n is interviewed and the response y h h h k 

is recorded. Our interest is to find the “optimal” sample sizes m = (m ,...,m ,...,m ) and n = (n ,...,n ,...,n ) for 1 h H 1 h H 

estimating the finite population parameter θ N that minimize the cost subject to constraints on the variances and the 

sample sizes. 

Consider first the general form of an estimator in the complete response case: 

(1) T Uˆ = ∑ d u + ∑ d u = ∑ u d (4.1) k 1k k 2k k k 

T (1) T where u = (u ,u ) and d = (d ,d ) . If the conditional second-phase inclusion probabilities are constant, then k 1k 2k k k k 

(1) (1) (1) (1) (1−ω ) /ω (1−ω ) /ω  kt kt kt kt Cov (d ,d ) =   . (4.2) p k t  (1) (1) 

  (1−ωkt kt kt kt ) /ω (1−ω ) /ω

Even when the conditional second-phase inclusion probabilities are random the right side of equation (4.2) is often 

used as an approximation to Cov (d ,d ) . Using (4.2), the sampling variance of the estimated total Uˆ

is given by p k t 

Var (u) = ∑ ∑u

TCov (d ,d )u . (4.3) p k p k t t 

Substituting (4.2) into (4.3), we get 

(1) (1) Var (u) = ∑ ∑[(1−ω ) /ω ]u u p kt kt 1k 1t 

+ ∑ ∑[(1−ω ) /ω ]u u (4.4) kt kt 2k 2t 

(1) (1) + 2∑ ∑[(1−ω ) /ω ]u u . kt kt 1k 2t 

Under stratified simple random sampling (STSRS) at both stages: sh

(1) is a simple random sample of size mh from 

(1) (1) stratum h and s is simple random sample of size n from s , we have π = m / N , π = n / N , and for h h h hk h h hk h h 

k ≠ t , π

(1) 

= m (m −1) /[N (N −1)], and π = n (n −1)/[N (N −1)]. We get from (4.4), hkt h h h h hkt h h h h 

2 −1 2 −1 Var (u) = ∑ N m (1− m / N )[S + 2S ]+ ∑ N n (1− n / N )S (4.5) p h h h h h hu u hu u h h h h h hu u 1 1 1 2 2 2 
 

 


 
 
 
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 
 


 

 

 

 

 

 

 

 

where S = ∑ J (x − X )( y −Y ) /(N −1) , X = ∑ J x / N and J is the stratum membership indicator hxy hk k h k h h h hk k h hk 

variable for element k . 

4.2 Optimal Sample Si es: STSRS

Consider the complete response case with p characteristics of interest y1,..., y p, and assume that the estimator Uˆis 

unbiased under the sampling design. Then, under STSRS, it follows from (4.5) that we can express Varp

(Uˆ 

j

) , for 

the j

th variable y j

, as 

Var (Uˆ ) = v + ∑ (v / m + v / n ) , j =1,..., p (4.6) p j j 0 h jh1 h jh2 h 

where v = − ∑ N S , v = N 2(S + 2S ) , v = N

2

S , u = u + u . We determine the optimal m j 0 h h hjuu hj1 h hju u hju u hj 2 h hju u jk j1k j 2k h 1 1 1 2 2 2 

and nh by mathematical programming methods such that the cost 

C = c + ∑ (c m + c n ) (4.7) 0 h h1 h h2 h 

is minimized subject to constraint on the variances 

Var (Uˆ ) ≤ V , j =1,..., p , (4.8) p j j 

and constraint on samples sizes and population sizes 

2 ≤ n ≤ m ≤ N , h = 1,..., H , (4.9) h h h 

where c is the overhead cost, c and c are the cost per element in stratum h for the first-phase and second-phase 0 h1 h2 

sample respectively, and Vj is the specified tolerance. For example, one could specify upper limit, φ j, on the 

2 coefficient of variation of θ

ˆ so that V = (φ E θˆ ) . We assume that c << c . j j j p j h1 h2 

We consider the bias-adjusted estimator (1) * ˆ Y

ˆ 

j k jk k k jk j = ∑d y + ∑(d − d ) yˆ of the total Y , given by (2.6) with β

obtained as solution to the estimating equations l

ˆ (β) = ∑d I Φ (β)( y −  (β)) = 0 . In this case jβ k jk jk jk jk 



y u * 

jk  j k 1 

u =  ≡  (4.10) jk y − y

* u  jk jk  j 2k 

with y

* 

= (1− I )t + I  (β ) , and β is the solution to l (β) = ∑ I Φ (β)( y −  (β)) = 0 . jk jk jk jk jk jN jN jβ jk jk jk jk 

4.3 Empirical Study

We considered the case of a single characteristic, y , and generated a finite population {y1,..., yN} of size N = 393 

from the model 

y =  + x

1/ 2σε , k k k k 

with independent errors ε generated from N(0,1) ,  = E ( y ) =α + β x and specified constants (α,β,σ ) , k k m k k 

where the fixed xk are the “number of beds” in hospital k for the hospitals population. We stratified the population 

into two strata with 272 units k having xk ≤ 350 in stratum 1 and 121 units k with xk > 350 in stratum 2. For the 

cost, we set c = 0 , c = c = 1 , and two different costs for c = c : 0.1 and 0.5. We set φ = 0.05 for the 0 h2 2 h1 1 

ˆ ˆ (1) (1) ˆ tolerances. Two estimators are considered Y = ∑d y + ∑(d − d )x and Y = ∑d y + ∑(d − d )x β with S k k k k k R k k k k k 

ˆ Y

ˆ Yˆ β = ∑d y / ∑d x . We have u = (x , y − x )

T for and u = (x β , y − x β )T for , where k k k k k k k k S k k N k k N R 

β = ∑ y / ∑ x . N k k 
 

 

 

 

 

 

 

 

 


 
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We repeated the optimization process for different value of σ,(0,..,15) keeping (α,β) = (0,1) and (0,1.2). Figures 1 

and 2 provide the minimum cost for β = 1.0 under the two-phase design for c1 = 0.5 and c1 = 0.1 respectively. The 

results under substitution and ratio imputation are similar. 

For comparison, the minimum cost under stratified single phase sampling, using the unbiased Horvitz-Thompson 

(HT) estimator are also presented. It is seen from Figure 1 that when the cost ratio c / c is small ( c / c = 2 ), 2 1 2 1 

minimum cost under two-phase is smaller than under one-phase sampling only for small values of σ ( ≤ 2 ). On the 

other hand, Figure 2 shows that when the cost ratio is large ( c2 / c1 =10 ), minimum cost is smaller under two-phase 

sampling for a much wider ranger of σ ( ≤10 ). Figures 3 and 4 report the minimum cost for the case of β = 1.2 

under substitution and ratio imputation for c1 = 0.5 and c1 = 0.1 respectively. 

4.4 Missing Responses

In the case of missing responses, y , in the second-phase sample, the variance of the estimator of general form 

Uˆ = ∑d (1)u + ∑d o u is given by k 1k k k 2k 

Var(Uˆ

) =V (u) +V (u ) , 

s r 2 

with 

(1) (1) V (u) = ∑ ∑[(1−ω ) /ω ]u u s kt kt 1k 1t 

+ ∑ ∑[(1−ω )/ω ]ξ u u (4.11) kt kt kt 2k 2t 

(1) (1) + 2∑ ∑[(1−ω ) /ω ]ξ u u , kt kt t 1k 2t 

and 

V (u ) = ∑ ∑(ξ −ξ ξ )u u . (4.12) 

r 2 kt k t 2k 2t 

We assume independent response mechanism. Under stratified simple random sampling at both stages, we have 

V (u) = ∑ N 2/ m (1− m / N )[S + 2S ] s h h h h h hu u hu u&& 1 1 1 2 (4.13) 

+ ∑ N 2/ n (1− n / N )[S + ∑ J ξ (1− ξ )u u ] h h h h h hu&& u&& hk k k h1k h2k 2 2 

r

where &x& = x ξ , u = u + u , and u = u + u&& . It follows from (4.11), (4.12) and (4.13), that we can express k k k k 1k 2k k 1k 2k 

Var(Uˆ

) as (4.6) with, 

v = N 2(S + 2S ) , hj1 h hu u hu u&& 1 1 1 2 

v = N v

2

[S v + ∑ J u u ξ (1−ξ )] , hj 2 h hu u hk h1k h2k k k 2 2 

and v = −∑ N [S r + ∑ J u u ξ (1−ξ )]+ ∑ξ (1−ξ )u

2 

. j 0 h h huu hk h1k h2k k k k k 2k 

References

Demnati, A. and Rao, J.N.K. (2004), “Linearization Variance Estimators for Survey Data (with discussion)”. Survey 

Methodology, 30, 17-34. 

Demnati, A. and Rao, J.N.K. (2007), “Linearization Variance Estimators for Survey Data: Some Recent Work (with 

comments)”. Third International Conference on Establishment Surveys, Montréal, Canada, 916-925. 

Demnati, A. and Rao, J.N.K. (2008), “Linearized Variance Estimation from Simulated Census Data”. Proceedings of 

the Survey Methods Section, Statistical Society of Canada. 

Valliant R., Dorfman, A.H. and Royall, R.M. (2000), “Finite Population Sampling and Inference: A Prediction 

Approach”, Wiley. 
I I I I I __________________ L _________________ J __________________ L _______________________ L _________________ _ 

I I I I 

' ' 

' ' ' ' 

I0

12345 789

101112131415

012345 789

101112131415

Table 1: Simulation Results 

Imputation Method Estimator Variable 

Substitution Naïve 1 

2 

3 

Design-based 1 

2 

3 

Ratio Naïve 1 

2 

3 

Design-based 1 

2 

3 


 



 


250

200

(N ) RB(θ

ˆ

) MSE(Y

ˆ )/ MSE(Yˆ

) RB{θ(θ

ˆ

)}

.0001 .94 -.01 

-.1181 9.85 -.93 

-.1228 10.66 -.93 

.00004 1 -.01 

 .0003 1 -.001 

 .0005 1 -.001 

-.0004 .99 -.010 

.0004 .99 -.011 

-.0023 .99 -.006 

.00004 1 -.011 

.0003 1 -.010 

.0006 1 -.002 

minimum
cost

150

100

50

0

Sigma

TwophaseSubstitution Onephase TwophaseRatio

Figure
1:

minimum
cost
for (β =1.0, c1 = 0.5)

250

200

minimum
cost


150

100

50

0

Sigma

TwophaseSubstitution Onephase TwophaseRatio

Figure
2:
minimum
cost
for (β =1.0, c1 = 0.1)

 
 
 
 
 



 
 
 
 
 


 

012345 789

101112131415

012345 789

101112131415

0

20

40

60

80

100

120

140

160

180

minimum cost

Sigma

TwophaseSubstitution Onephase TwophaseRatio

Figure 3: minimum cost for (β =1.2, c1 = 0.5) 

0

20

40

60

80

100

120

140

160

180

minimum cost

Sigma

TwophaseSubstitution Onephase TwophaseRatio

Figure 4: minimum cost for (β =1.2, c1 = 0.1)