Chapter 8 Double Sampling (Two Phase Sampling)

1 Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 1 Chapter 8 Double Sampling (Two Phase Sampling ) The ratio and regression methods of estimation require the knowledge of population mean of an auxiliary variable ()X to estimate the population mean of study variable ( ).Y If the information on the auxiliary variable is not available, then there are two options one option is to collect a sample only on study variable, and use sample mean as an estimator of the population mean. An alternative solution is to use a part of the budget for collecting information on an auxiliary variable to collect a large preliminary sample in which ix alone is measured. The purpose of this Sampling is to furnish a good estimate of X. This method is appropriate when the information about ix is on file cards that have not been tabulated. After collecting a large preliminary sample of size 'n units from the population, select a smaller sample of size n from it and collect the information on y.

2 These two estimates are then used to obtain an estimator of the population mean .Y This procedure of selecting a large sample for collecting information on auxiliary variable x and then selecting a sub-sample from it for collecting the information on the study variable y is called Double Sampling or two- Phase Sampling . It is useful when it is considerably cheaper and quicker to collect data on x than y and there is a high correlation between x and .y In this Sampling , the randomization is done twice. First, a random sample of size 'n is drawn from a population of size N and then again a random sample of size n is drawn from the first sample of size 'n. So the sample mean in this Sampling is a function of the two phases of Sampling . If SRSWOR is utilized to draw the samples at both the phases, then the - number of possible samples at the first Phase when a sample of size n is drawn from a population of size N is 0'NMn , say.

3 - number of possible samples at the second Phase where a sample of size n is drawn from the first Phase sample of size 'n is 1'nMn , say. Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 2 Then the sample mean is a function of two variables. If is the statistic calculated at the second Phase such that 01,1, 2,..,,1, 2,..,withijijiMjMP being the probability that ith sample is chosen at the first Phase and jth sample is chosen at the second Phase , then 12( )( )EE E where 2()E denotes the expectation over the second Phase and 1E denotes the expectation over the first Phase . Thus 01010111/11/111 stage2 stage()(using ()( ) ( / )) stndMMij ijijMMij iijijMMij iijijEPPPP A BP A P B APP Population of X (N units) Sample (Large) 'nunits Subsample(small) n units 0samples M 0samples M 1samples M Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 3 Variance of 2222222222122222122122221212121212( )( )(( )) (( )( ))( )( )( )0( )][( )( )( )( )( ) constant for( )( )(( ))( )( )VarEEEEEEEEEEE EEEEE EEE E EEEE VE EE EE VV E Note: The two- Phase Sampling can be extended to more than two phases depending upon the need and objective of the experiment.

4 Various expectations can also be extended on similar lines. Double Sampling in ratio method of estimation If the population mean X is not known, then the Double Sampling technique is applied. Take a large initial sample of size 'n by SRSWOR to estimate the population mean X as '11 ''niiXxxn . Then a second sample is a subsample of size n selected from the initial sample by SRSWOR. Let and yx be the means of y and x based on the subsample. Then ( '), ( ), ( ).E xX E xX E yY The ratio estimator under Double Sampling now becomes 'RdyYxx . The exact expressions for the bias and mean squared error of RdY are difficult to derive. So we find their approximate expressions using the same approach mentioned while describing the ratio method of estimation. Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 4 Let 012', , y Yx XxXYXX 01222112212221222222( )( )( ) 011()1()()( ')1()( ') | '1( ')11'11'( ).

5 XxxEEEECnNEE x X xXXE E x X xX nXExXXSnN XCnNE 021()( , ')11( | '), ( ' | ')( , ') | '11,( ', ')1( ', '11'11'11'xyyxxyECov y xXYCov E y n E x nE Cov y xnXYXYCov Y XE Cov y xXYXYCov y xXYSnN XYSSnNX YCCnN where 'y is the sample mean of 'ys based on the sample size '.n Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 5 011()( , )1111 11 xyyxxyECov y xXYSn N X YSSn NX YCCnN 20212122'' 21122222221( )( )1( | ')(| ')111()'11111''1111nnyyyyyEVar yYV E y nE V y nYV yEsYn nSSYnNn nSn N YCnN where '2ys is the mean sum of squares of y based on an initial sample of size '.n 122221()( , ')1( | '), ( ' | ')01( ')ECov x xXCov E x n E x nXVar XX where (')Var X is the variance of mean of x based on an initial sample of size 'n. Estimation error of RdY Write RdY as 0211021202112020 2111 21(1) (1)(1)(1)(1)(1)(1)(1)( )(1)RdoYYXXYYY up to the terms of order two.)

6 Other terms of degree higher than two are assumed to be negligible. Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 6 Bias of RdY 20 20 11 2120 20 11 21222 ()1 0 0() 0()()( ) ()()()()()( )11111111''11'RdRdRdxyxyxxxE YYEEEEBias YE YYY EEEEYC CC CCCnNn NnNn NYCCnn 11 ().'xyxxyCYC CCnn The bias is negligible if n is large and relative bias vanishes if 2,xxyCC , the regression line passes through the origin. MSE of :RdY 2`2202122220120 20 11 2222220120 20 1222 ()()()(2222221111 RdRdyMSE YE YYYEYEYEYCn Nn N retaining the terms upto order two) 2222222211111122''11112(2)'112.'xxxyxyxy xyxyxxyxCCC CC CnNnNn NYCCC CYCCCn NnNMSEYC CCnN (ratio estimator) The second term is the contribution of the second Phase of Sampling .

7 This method is preferred over the ratio method if 2 201or2xyxxyC CCCC Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 7 Choice of n and 'n Write ' ()'RdVVMSE Ynn where and'VV contain all the terms containing n and 'n respectively. The cost function is 0''CnC n C where C and 'C are the costs per unit for selecting the samples n and 'n respectively. Now we find the optimum sample sizes n and 'n for fixed cost The Lagrangian function is 022' (' ')' 0' 0'.''VVnC n C CnnVCnnVCnn 2 Thus ' '' '.CnVVnCnCVCn CV C Thus 0''VCV CC and so 0'0'20 Optimum, say'''Optimum ', say'''' ()(' ') optoptoptRdoptoptCVnnCVCV CCVnnCVCV CVVVar YnnVCV CC Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 8 Comparison with SRS If X is ignored and all resources are used to estimate byYy, then required sample size = 220022()/()Relative effiiency = (' ')()yyyoptRdSCSVar yC CCCSVar yVCV CVar Y Double Sampling in regression method of estimation When the population mean of the auxiliary variable X is not known, then Double Sampling is used as follows: - A large sample of size 'n is taken from of the population by SRSWOR from which the population mean X is estimated as 'x, '.

8 Xx - Then a subsample of size n is chosen from the larger sample and both the variables xand y are measured from it by taking 'x in place of X and treat it as if it is known. Then ( '), ( ), ( )E xX E xX E yY . The regression estimate of Y in this case is given by ( ')regdYyxx where 12221()() is an estimator of()niixyxyinxxiixx yysSsSxx based on the sample of size n. It is difficult to find the exact properties like bias and mean squared error of regdY, so we derive the approximate expressions. Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 9 Let 0011223322224421234(1)(1)'' (1)(1)(1)( ) 0, ( ) 0, ( ) 0, ( ) 0 xyxyxyxyxyxxxxxyYyYYxXxXXxXxXXsSsSSsSsSS EEEE Define 221330() ()E x Xy YE x X Estimation error: Then 32124132142232144 ( ')(1)()(1)(1)()(1)(1)()( )regdxyxxyxYyxxSyXSSyXSyX Retaining the powers of 's up to order two assuming 31, (using the same concept as detailed in the case of ratio method of estimation) 22 32 411 31 4 ().

9 RegdYy X Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 10 Bias: 2 32 41 31 4 The bias of upto the second order of approximation is ()()()()() ()()( ')()111 'regdregdregdxyxyYregdE YYXEEEEBias YE YYxX sSXnN NX 222222( ')()111 '()()111 ()()111 xyxXXxyxyxyxxxSxX sSnN NXSx X sSn N NXSx X sSn N NXS 30302121223021211111111 ''11 .'xyxxyxxyxXnN XSnN XSn N XSn N XSn nSS Mean squared error: 222232144 ()() ( ') ()(1)()( )regdregdMSE YE YYyxxYE y YX Retaining the powers of 's up to order two, the mean squared error up to the second order of approximation is Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 11 222 32 411 31 422222121 21222222121 2012 ()()() ()(2) 2[()()] ()(2) 2[ ()] regdMSE YE y YXE y YXEX E y YE y YXEXY E 2222222222111111 ( )2'1111 2'1111 ( )2'' xxxxyxyxxySSSVar yXn N XnN Xn N XSSXYnN XYn N XYVar ySSn nn n 2222422211 ( )2'11 ( )

10 2'1111 '11 xxyxyxyxxyxxxyyxVar ySSnnSSVar ySSn nSSSSn Nn nSnN 222222211(using)'(1) . (Ignoring the finite population correction)' yyxyxyyySSSS SnnSSnn Clearly, regdY is more efficient than the sample mean SRS, when no auxiliary variable is used. Now we address the issue of whether the reduction in variability is worth the extra expenditure required to observe the auxiliary variable. Let the total cost of the survey is 012'CC n C n where 12andCC are the costs per unit observing the study variable y and auxiliary variable x , respectively. Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling ) | Shalabh, IIT Kanpur Page 12 Now minimize the MSE ()regdY for fixed cost 0C using the Lagrangian function with Lagrangian multiplier as 222212022122222(1)(')'10(1)0100''yyyySSC n C n CnnSCnnSCnn Thus 221(1)ySnC and 2'ySnC.