Chapter 10 Two Stage Sampling (Subsampling)
In cluster sampling, all the elements in the selected clusters are surveyed. Moreover, the efficiency in cluster sampling depends on the size of the cluster. As the size increases, the efficiency decreases. It suggests that higher precision can be attained by distributing a given number of elements over a large number of clusters and then by taking a small number of clusters and enumerating all elements within them. This is achieved in subsampling.
In subsampling - divide the population into clusters. - Select a sample of clusters [first stage} - From each of the selected cluster, select a sample of the specified number of elements [second stage]
The clusters which form the units of sampling at the first stage are called the first stage units and the units or group of units within clusters which form the unit of clusters are called the second stage units or subunits.
The procedure is generalized to three or more stages and is then termed as multistage sampling.
For example, in a crop survey - villages are the first stage units, - fields within the villages are the second stage units and - plots within the fields are the third stage units.
In another example, to obtain a sample of fishes from a commercial fishery - first, take a sample of boats and - then take a sample of fishes from each selected boat.
Two stage sampling with equal first stage units: Assume that - the population consists of NM elements. - NM elements are grouped into N first stage units of M second stage units each, (i.e., N clusters, each cluster is of size M )
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 1
- Sample of n first stage units is selected (i.e., choose n clusters) - Sample of m second stage units is selected from each selected first stage unit (i.e., choose m units from each cluster). - Units at each stage are selected with SRSWOR.
Cluster sampling is a special case of two stage sampling in the sense that from a population of N clusters of equal size mM , a sample of n clusters are chosen. If further, Mm1, we get SRSWOR. If nN , we have the case of stratified sampling.
th th yij : Value of the characteristic under study for the j second stage units of the i first stage unit; iNjm1,2,..., ; 1,2,.., .
M 1 nd th st Yyiij : mean per 2 stage unit of i 1 stage units in the population. M j1
11NM N YyyYij i MN : mean per second stage unit in the population MNij11 N i 1
m 1 th yiij y : mean per second stage unit in the i first stage unit in the sample. m j1
11nm n y yyyij i mn : mean per second stage in the sample. mnij11 n i 1
Advantages:
The principle advantage of two stage sampling is that it is more flexible than the one-stage sampling. It reduces to one stage sampling when mM but unless this is the best choice of m , we have the opportunity of taking some smaller value that appears more efficient. As usual, this choice reduces to a balance between statistical precision and cost. When units of the first stage agree very closely, then consideration of precision suggests a small value of m . On the other hand, it is sometimes as cheap to measure the whole of a unit as to a sample. For example, when the unit is a household and a single respondent can give as accurate data as all the members of the household.
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 2
A pictorial scheme of two stage sampling scheme is as follows:
Population (MN units)
Cluster Cluster Cluster Population M units M units … … … M units N clusters
(large in number) N clusters
Cluster Cluster First stage Cluster M units M units … … … sample n M units clusters (small
in number)
n clusters
Second stage sample m units n clusters (large Cluster Cluster Cluster number of … … … m units m units m units elements from each cluster) mn units
Note: The expectations under two stage sampling scheme depend on the stages. For example, the expectation at second stage unit will be dependent on first stage unit in the sense that second stage unit will be in the sample provided it was selected in the first stage.
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 3
To calculate the average - First average the estimator over all the second stage selections that can be drawn from a fixed set of n units that the plan selects. - Then average over all the possible selections of n units by the plan.
In the case of two stage sampling, ˆˆ EEE ( ) 12 [ ( )] average average average over over over all all possible 2nd stage
all 1st stage selections from samples samples a fixed set of units
In case of three stage sampling,
EEEE()ˆˆ () . 123
To calculate the variance, we proceed as follows: In case of two stage sampling, Var()ˆˆ E ( )2
ˆ 2 EE12() Consider
ˆˆˆ22 2 EEE222() ()2() 2 . EV(()2()ˆˆ E ˆ2 22 2
Now average over first stage selection as
2 EE()ˆˆˆˆ2 E E () E V ()2()() EE E 2 12 1 2 1 2 12 1 2 EE()ˆˆ2 EV () 12 12 Var (ˆˆ ) V E ( ) E V ( ˆ ) . 12 12
In case of three stage sampling, ˆˆ ˆ ˆ Var() V13 E E () E 123123 V E () E E V () . 2
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 4
Estimation of population mean:
Consider y ymn as an estimator of the population mean Y . Bias: Consider
Ey() E12 E ( ymn ) nd st EEyi12(im ) (as 2 stage is dependent on 1 stage)
EEyi12(im ) (as y i is unbiased for Y i due to SRSWOR)
1 n = EY1 i n i1 1 N Yi N i1 Y .
Thus ymn is an unbiased estimator of the population mean.
Variance
Var() y E12 V ( y i ) V 12 E ( y /) i 11nn EV12 yiii VE 1 2 y/ i nnii11 11nn EVyiVEyi() (/) 1122 ii nnii11 111nn 1 ESVY2 112 ii nmMii11 n 111n ES()2 Vy () 2 11ic nmMi1
(where yc is based on cluster means as in cluster sampling)
11122Nn 2 nSSwb nmM Nn
11 122 1 1 SSwb nm M n N
NNM2 2211 where SSwi YY iji NNM(1) iij111 N 221 SYYbi() N 1 i1
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 5
Estimate of variance
22 An unbiased estimator of the variance of y can be obtained by replacing SSbwand by their unbiased estimators in the expression of the variance of y.
Consider an estimator of
N 221 SSwi N i1 M 2 2 1 where SyYiiji M 1 j1 n 221 as sswi n i1 m 221 wheresyyiiji ( ) . m 1 j1 So
22 Es()ww EE12 s i
n 1 2 EE12 si i n i1 1 n 2 EEsi12 ()i n i1 n 1 2 ES1 i (as SRSWOR is used) n i1 n 1 2 ES1()i n i1 NN 112 Si NNii11 N 1 2 Si N i1 2 Sw
2 2 so sw is an unbiased estimator of Sw .
Consider
n 221 syybi() n 1 i1 as an estimator of
N 221 SYYbi(). N 1 i1
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 6
So n 221 Es()bi E ( y y ) n 1 i1 n 222 (1)()nEsEbi yny i1 n 22 E yi nE() y i1 n 2 EE y2 nVary() Ey () 12 i i1 n 222211 1 11 EEyin12 ())ibw S SY i1 nN mMn n 2 11222 1 11 E1 Var( yii)(Ey n S b S w Y i1 nN mMn n 1122 11 2 111 22 ESYnSSY1 ii b w i1 mM nN mMn n 11122 111 21122 nE1 Sii Y n S bSYw nmMi1 nN mMn NN 11122 1 11 2 111 22 nSYnSSYii b w mMNii11 N nN mMn N 1122 1 11 2 111 22 nSYnSwi b SY w mM Ni1 nN mMn N 11222n 11 2 (1)n Swi YnYn S b mM Ni1 nN N 112222n 11 (1)nSYNYnS wi b mM Ni1 nN
1122n 11 2 (1)nSNSnS wb ( 1) b mM N nN
11 22 (1)nSnS wb (1). mM
22211 Es()bwb S S mM
22211 or Esbwb s S. mM Thus
11 1ˆ 22 1 1 ˆ Var() y S Sb nm M n N
11 1222 1 1 1 1 ssswbw nm M n N m M
11 122 1 1 sswb . Nm M n N
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 7
Allocation of sample to the two stages: Equal first stage units: The variance of the sample mean in the case of two stage sampling is
11 122 1 1 Var() y Swb S . nm M n N
22 It depends on SSbw,,and. n m So the cost of survey of units in the two stage sample depends on nmand .
Case 1. When cost is fixed We find the values of n and m so that the variance is minimum for given cost.
(I) When cost function is C = kmn Let the cost of the survey be proportional to sample size as Cknm where C is the total cost and k is constant. C When cost is fixed as CC . Substituting mVary 0 in ( ), we get 0 kn
22 1122SSwbkn Var() y Sbw S nMNnC 0
222 1 2 SSkSwbw Sb . nMNC0
2 2 Sw This variance is a monotonic decreasing function of n if Sb 0. The variance is minimum when M n assumes maximum value, i.e., C nmˆ 0 corresponding to 1. k
2 2 Sw If Sb 0 (i.e., intraclass correlation is negative for large N) , then the variance is a monotonic M C increasing function of n , It reaches minimum when n assumes the minimum value, i.e., nˆ 0 (i.e., no kM subsampling).
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 8
(II) When cost function is Cknkmn12
Let cost CCknkmnbe fixed as 01 2 where kk12and are positive constants. The terms kk12and denote the costs of per unit observations in first and second stages respectively. Minimize the variance of sample mean under the two stage with respect to m subject to the restriction Cknkmn01 2.
We have
22 22 SSbw222 SkS ww1 CVary0122 ( ) kSbwb kS mkS . NM Mm 2 2 Sw WhenSb 0, then M 2 2 22 22 SSbw222 SkS ww1 CVary0122 ( ) kSbwb kS mkS NM Mm which is minimum when the second term of right-hand side is zero. So we obtain
k S 2 mˆ 1 w . 2 k2 2 Sw Sb M
The optimum n follows from Cknkmn01 2 as C nˆ 0 . kkm12 ˆ
2 2 Sw When Sb 0then M
22 22 SSbw 222 SkS ww1 CVary0122() kSbwb kS mkS NM Mm is minimum if m is the greatest attainable integer. Hence in this case, when
C0 CkkMmM012;and.ˆˆ n kkM12
Ck01 If CkkM012 ; then mnˆˆand 1. k2
22 If N is large, then SSw (1 )
S 2 SS22w w M
k1 1 mˆ 1 . k2
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 9
Case 2: When variance is fixed
Now we find the sample sizes when variance is fixed, say as V0.
11 122 1 1 VSS0 wb nm M n N 2211 SSbw mM n S 2 V b 0 N So
2 2 Sw Sb 2 M kSw Ckmnkm22 . SSbb VV 00NN
2 2 Sw If SCb 0, attains minimum when m assumes the smallest integral value, i.e., 1. M
2 2 Sw If Sb 0 , C attains minimum when mMˆ . M
Comparison of two stage sampling with one stage sampling One stage sampling procedures are comparable with two stage sampling procedures when either (i) sampling mn elements in one single stage or mn (ii) sampling first stage units as cluster without sub-sampling. M We consider both the cases.
Case 1: Sampling mn elements in one single stage The variance of sample mean based on - mn elements selected by SRSWOR (one stage) is given by
112 Vy()SRS S mn MN - two stage sampling is given by
11 122 1 1 Vy()TS S w S b . nm M n N
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The intraclass correlation coefficient is
2 N 1 2 Sw Sb 22 NMMN(1) Sbw NS 1 2 ; 1 (1) NM 1 2 (1)MN S M 1 S NM and using the identity
NM NM NM 222 ()yYij () yY ij i () YY i ij11 ij 11 ij 11 22 2 (NM 1) S ( N 1) MSbw N ( M 1) S (2) 11NM M whereYyYyij , i ij . MNij11 M j 1
2 2 2 Now we need to find Sb and Sw from (1) and (2) in terms of S . From (1), we have
222MN11 N SMSMSwb . (3) NN Substituting it in (2) gives
22NMN11 2 2 (1)(1)(1)NM S N MSbb N M MS MS NN 22 2 (NMSM 1)bb ( 1)( NS 1) MMMNS ( 1)( 1) 22 (NMSM 1)b [1 ( 1)] MMMNS ( 1)( 1) 2 2 (NMS 1) b MM(1)(1) MN S (1)MN S 2 SM2 1( 1) b MN2 (1) Substituting it in (3) gives
222 NM(1)( Swb NM 1)(1) S N MS 2 2 (1)MN S (NM 1) S ( N 1) M 2 1 ( M 1) MN(1) MM1( 1) (NM 1) S 2 M (NM 1) S2 ( M 1)(1 )
22MN 1 SSw (1 ). MN
22 Substituting SSbwand in Var() yTS
MNS1(1)2 mn Nnm Mm Vy()TS 1 ( M 1) . MN mn M(1) N N 1 M M
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 11
m When subsampling rate is small, MNMNMM 1 and 1 , then M S 2 Vy() SRS mn
SNn2 Vy()TS 1 m 1. mn N 1 The relative efficiency of the two stage in relation to one stage sampling of SRSWOR is
Var() yTS Nn RE11 m . Var() ySRS N 1 Nn Nn If NN1 and finite population correction is ignorable, then 1, then NN1 RE1(1). m
Case 2: Comparison with cluster sampling mn Suppose a random sample of clusters, without further subsampling is selected. M The variance of the sample mean of equivalent mn/ M clusters is
M 1 2 Var() ycl S b . mn N The variance of the sample mean under the two stage sampling is
11 122 1 1 Var() yTS S w S b . nm M n N
So Var( ycl ) exceedes Var ( y TS ) by
11M 22 1 SSbw nm M which is approximately
2 1 M 2 2 Sw 1 S for large N and Sb 0. nm M MN1 S 2 where SM2 1 ( 1) b MN(1) M
MN 1 SS22 (1 ) w MN So smaller the mM/ , larger the reduction in the variance of two stage sample over a cluster sample.
2 2 Sw When Sb 0 then the subsampling will lead to loss in precision. M
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Two stage sampling with unequal first stage units: Consider two stage sampling when the first stage units are of unequal size and SRSWOR is employed at each stage. Let
th th yij : value of j second stage unit of the i first stage unit.
th M i :number of second stage units in i first stage units (iN 1,2,..., ) .
N M 0 M i : total number of second stage units in the population. i1
th mi :number of second stage units to be selected from i first stage unit, if it is in the sample.
n mm0 i : total number of second stage units in the sample. i1
1 mi yyimi() ij mi j1 1 Mi Yyiij M i j1 1 N YyY i N N i1
N Mi N y ij MYii N ij11 i1 1 YuYN ii MN N i1 M i i1 M u i i M 1 N MM i N i1
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The pictorial scheme of two stage sampling with unequal first stage units case is as follows:
Population (MN units)
Cluster Cluster Cluster Population M M … … … 1 2 MN N clusters units units units
N clusters
Cluster Cluster Cluster First stage M1 M2 … … … Mn sample n n clusters units units units clusters (small)
Second stage Cluster Cluster Cluster sample n m units m units … … … 1 2 mn units clusters (small)
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 14
Now we consider different estimators for the estimation of the population mean. 1. Estimator based on the first stage unit means in the sample:
ˆ 1 n YySimi2() y n i1 Bias:
1 n Ey()Simi2() E y n i1 1 n EEy12() ()imi n i1 1 n EY1 iii[Since a sample of size m is selected out of M units by SRSWOR] n i1 1 N Yi N i1
Y N Y .
So yS 2 is a biased estimator of Y and its bias is given by
Bias() ySS22 E () y Y 11NN YMYiii NNMii11 11NNN MYii Y i M i NMiii111 N 1 N ()().MMYYiiN NM i1 This bias can be estimated by
n N 1 Bias() ySiimiS2()2 ( M m )( y y ) NM(1) n i1 which can be seen as follows: N 11n EBiasy() E E ( M my )( y )/ n SiimiS212()2 NM n 1 i1 N 11n EMmYy (iin )( ) NM n 1 i1 1 N (MMYYiiN )( ) NM i1
YYN 1 n where yni Y . n i1
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 15
An unbiased estimator of the population mean Y is thus obtained as N 11 n ySiimiS2()2()()Mmy y. NM n 1 i1 Note that the bias arises due to the inequality of sizes of the first stage units and probability of selection of second stage units varies from one first stage to another.
Variance:
Var() ySS22 Var E () y n E Var () y S 2 n 11nn Var y E Var() y i iimi2 () nnii11 11 1n 1 1 SE22 S bi2 nN ni1 mii M N 1122 1 1 1 SSbi nN Nnmi1 ii M
1 N 2 where SYY2 biN N 1 i1 2 Mi 2 1 SyYiiji. M i 1 j1 The MSE can be obtained as
2 MSE() ySS22 Var () y Bias (). y S 2
Estimation of variance: Consider mean square between cluster means in the sample
1 n 2 syy2 . bimS()i 2 n 1 i1 It can be shown that
N 22111 2 Es()bb S S i. NmMi1 ii
mi 221 Also syyiijimi (() ) mi 1 j1
Mi 221 2 Es()ii S ( y iji Y ) M i 1 j1
nN 11122 1 11 SoEsii S . nmMii11ii NmM ii
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Thus
n 22111 2 Es()bb S E s i nmMi1 ii 2 and an unbiased estimator of Sb is
n ˆ 22111 2 Ssbb s i. nmMi1 ii 22 So an estimator of the variance can be obtained by replacing SSbiand by their unbiased estimators as
N 11ˆˆ22 1 1 1 Var() ySb2 S S i. nN Nnmi1 ii M
2. Estimation based on first stage unit totals: 1 n My Yyˆ * iimi() S 2 nM i1 1 n uyiimi() n i1 M where u i . i M Bias n * 1 Ey()Siimi2() E uy n i1 1 n EuEyi iimi2()() n i1 1 n EuY ii n i1 1 N uYii N i1 Y . * Thus yS 2 is an unbiased estimator of Y . Variance:
** * Var() ySS22 Var E () y n E Var () y S 2 n 11nn Var u Y E u2 Var() y i ii2 i imi() nnii11 N 11*2 1 2 1 1 2 Subi S i nN nNi1 mii M
1 Mi whereSyY22 ( ) iijiM 1 i j1 N *21 2 SuYYbii(). N 1 j1
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3. Estimator based on ratio estimator:
n Myiimi() ˆ ** i1 YyS 2 n M i i1 n uyiimi() i1 n ui i1 y * S 2 un
n M i 1 where uuuini,. Mni1 This estimator can be seen as if arising by the ratio method of estimation as follows:
* Let yuyiiimi () M x* i ,iN 1,2,..., i M be the values of study variable and auxiliary variable in reference to the ratio method of estimation. Then
n ***1 y yyiS2 n i1 n **1 x xuin n i1 N 1 * XX*1. i N i1 The corresponding ratio estimator of Y is
* ˆ y * yS 2 ** YXRS*1 y2 . xu* n
** So the bias and mean squared error of yS 2 can be obtained directly from the results of ratio estimator. Recall that in ratio method of estimation, the bias and MSE of the ratio estimator upto second order of approximation is Nn Bias() yˆ Y ( C2 2 C C ) RxxyNn Var() x Cov (,) x y Y XXY2 ˆ 2 MSE() YR Var () y R Var ()2 x RCov (,) x y Y where R . X Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 18
Bias:
** The bias of yS 2 up to second order of approximation is
*** ** Var() xSSS222 Cov (, x y ) Bias() yS 2 Y 2 XXY
n * * * 1 where xS 2 is the mean of auxiliary variable similar to yS 2 as xxSimi2() . n i1
** Now we find Cov(, xSS22 y ).
nn nn ** 11 11 Cov(, xS2 y S 2 ) Cov E u i x i () mi , u i y i () mi E Cov u i x i () mi , u i y i () mi nnii11 nn ii 11 11nn 1 n Cov u E(),() x u E y E u2 Cov (,) x y i i i() mi i i () mi2 i i () mi i () mi nnii11 n i 1
1 nn1111 n Cov uX, uY E u2 S ii ii2 i ixy nnii11 nmM i 1ii N 11*2 1 1 1 Subxy i S ixy nN nNi1 mii M where 1 N SuXXuYY* ()() bxyN 1 i i i i i1 1 Mi SxXyYixy()(). ij i ij i M i 1 j1
* ** Similarly, Var() xS 2 can be obtained by replacing x in place of y in Cov(, xSS22 y ) as
N **22211 1 1 1 Var() xSbxiix2 S u S nN nNi1 mii M 1 N whereSuXX*2 ( ) 2 bxN 1 i i i1 Mi *21 2 SxXix(). ij i M i 1 i1
** * ** Substituting Cov(, xSS22 y ) and Var() xS 2 in Bias(), yS 2 we obtain the approximate bias as
*2* N 2 **11SSbxSSbxy 1 2 1 1 ix ixy Bias() ySi2 Y22 u . nN X XYnN m M X XY i1 ii
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Mean squared error
** * * * * *2 * MSE() ySS22 Var ()2 y R Cov (,) x SSS 222 y R Var () x
N **11 *2 1 2 1 1 2 Var() ySbyiiy2 S u S nN nNi1 mii M N **11 *2 1 2 1 1 2 Var() xSbxiix2 S u S nN nNi1 mii M N ***11 * 1 2 11 Cov(, xSS22 y ) S bxy uiixy S nN nNi1 mMii where N *21 2 SuYYby() i i N 1 i1
Mi *21 2 SyYiy() ij i M i 1 j1 Y RY* . X Thus
N **11 *2 * * *2 *2 1 2 1 1 2 * *2 2 MSE() yS2 S by 2 R S bxy R S bx u i S iy 2 R S ixy R S ix . nN nNi1 mii M Also
NN **11 1 2 *2 1 2 1 1 2 * *2 2 MSE() ySiiiiiyixyix2 u Y R X u S 2 R S R S . nNN1 ii11 nN mii M
Estimate of variance Consider 1 n *** suyyuxxbxy i i() mi S 2 i i () mi S 2 n 1 i1 1 n sxxyy. ixy ij i() mi ij i () mi mi 1 j1
It can be shown that
N **111 2 Esbxy S bxy u i S ixy NmMi1 ii
Es()ixy S ixy .
So
nN 111122 11 Euiixyiixy s u S. nmMNii11ii mM ii
Sampling Theory| Chapter 10 | Two Stage Sampling (Subsampling) | Shalabh, IIT Kanpur Page 20
Thus
n ˆ**111 2 Ssbxy bxy u i s ixy nmMi1 ii n ˆ*2 *2111 2 2 Ssbx bx u i s ix nmMi1 ii n ˆ*2 *2111 2 2 Ssby by u i s iy . nmMi1 ii Also
nN 11111122 2 2 Euiixiix s u S nmMNmM ii11ii ii
nN 11111122 2 2 Euiiyi s u S iy. nmMNmM ii11ii ii
** A consistent estimator of MSE of yS 2 can be obtained by substituting the unbiased estimators of
** respective statistics in MSE() yS 2 as
**11 *2 * * *2 *2 MSE() ySbybxybx2 s 2 r s r s nN n 11122**22 usrsrsiiyixyix2 nNi1 mii M n 11 1 2 yrximi() * imi () nNn1 i1 n * 11122**22* yS 2 usrsrsriiyixyix2 where * . nNi1 mii M x S2
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