Chapter 11 Systematic Sampling, Exams of Survey Sampling Techniques

Download Chapter 11 Systematic Sampling and more Exams Survey Sampling Techniques in PDF only on Docsity! Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 1 Chapter 11 Systematic Sampling The systematic sampling technique is operationally more convenient than simple random sampling. It also ensures, at the same time that each unit has an equal probability of inclusion in the sample. In this method of sampling, the first unit is selected with the help of random numbers, and the remaining units are selected automatically according to a predetermined pattern. This method is known as systematic sampling. Suppose the N units in the population are numbered 1 to N in some order. Suppose further that N is expressible as a product of two integers n and k , so that .N nk To draw a sample of size n , - select a random number between 1 and k . - Suppose it is i . - Select the first unit, whose serial number is i . - Select every thk unit after thi unit. - The sample will contain , ,1 2 ,..., ( 1)i i k k i n k    serial number units. So the first unit is selected at random and other units are selected systematically. This systematic sample is called kth systematic sample and k is termed as a sampling interval. This is also known as linear systematic sampling. The observations in the systematic sampling are arranged as in the following table: Systematic sample number 1 2 3  i  k Sample composition 1 2  n 1y 1ky   ( 1) 1n ky   2y 2ky   ( 1) 2n ky   3y 3ky   ( 1) 3n ky       iy k iy   ( 1)n k iy       ky 2ky  nky Probability 1 k 1 k 1 k  1 k  1 k Sample mean 1y 2y 3y  iy  ky Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 2 Example: Let 50N  and 5.n  So 10.k  Suppose first selected number between 1 and 10 is 3. Then systematic sample consists of units with following serial number 3, 13, 23, 33, 43. Systematic sampling in two dimensions: Assume that the units in a population are arranged in the form of m rows, and each row contains nk units. A sample of size mn is required. Then - select a pair of random numbers ( , )i j such that i   and j k . - Select the ( , )thi j unit, i.e., thj unit in thi row as the first unit. - Then the rows to be selected are , , 2 ,..., ( 1)i i i i m      and columns to be selected are , , 2 ,..., ( 1) .j j k j k j n k    - The points at which the m selected rows and n selected columns intersect determine the position of mn selected units in the sample. Such a sample is called an aligned sample. An alternative approach to select the sample is - independently select n random integers 1 2, ,..., ni i i such that each of them is less than or equal to  . - Independently select m random integers 1 2, ,..., mj j j such that each of them is less than or equal to k . - The units selected in the sample will have the following coordinates: 1 1 2 1 3 1 1( , ), ( , ), ( , 2 ),..., ( , ( 1) )r r r n ri r j i r j k i r j k i r j n k              . Such a sample is called an unaligned sample. Under certain conditions, an unaligned sample is often superior to an aligned sample as well as a stratified random sample. Advantages of systematic sampling: 1. It is easier to draw a sample and often easier to execute it without mistakes. This is more advantageous when the drawing is done in fields and offices as there may be substantial saving in time. 2. The cost is low, and the selection of units is simple. Much less training is needed for surveyors to collect units through systematic sampling. 3. The systematic sample is spread more evenly over the population. So no large part will fail to be represented in the sample. The sample is evenly spread and cross-section is better. Systematic sampling fails in case of too many blanks. Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 5 The intraclass correlation between the pairs of units that are in the same systematic sample is 2 1 ( ) 1 1 2 ( )( ) 1 ; 1 ( ) 1 1 ( )( ) ( 1) . 1                              ij i w w ij k n n ij i i j E y Y y Y E y Y nk y Y y Y nk n nk S nk   So substituting 2 1 ( ) 1 1 ( )( ) ( 1)( 1) k n n ij i w i j y Y y Y n nk S               in ( )iVar y gives     2 2 1 ( ) 1 ( 1) 1 1 ( 1) . sy w w nk S Var y n nk n N S n N n           Comparison with SRSWOR: For a SRSWOR sample of size n , 2 2 2 ( ) 1 . SRS N n Var y S Nn nk n S Nn k S N       Since 2 2 2 2 2 2 1 1 ( ) 1 1 1 ( ) ( ) 1 ( ). sy wsy SRS sy wsy wsy N n Var y S S N n N nk k N n Var y Var y S S N N n n S S n                   Thus syy is - more efficient than SRSy when 2 2 wsyS S . - less efficient than SRSy when 2 2 wsyS S . - equally efficient as 2 2when .SRS wsyy S S Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 6 Also, the relative efficiency of syy relative to SRSy is   2 2 ( ) ( ) 1 1 ( 1) 1 1 1 ( 1) ( 1) 1 1 ; 1. ( 1) 1 ( 1) 1 SRS sy w w w Var y RE Var y N n S Nn N S n Nn N n N n n k nk n nk                              Thus syy is - more efficient than SRSy when 1 1w nk     - less efficient than SRSy when 1 1w nk     - equally efficient as SRSy when 1 1w nk     . Comparison with stratified sampling: The systematic sample can also be viewed as if arising as a stratified sample. If the population of N nk units is divided into n strata and suppose one unit is randomly drawn from each of the strata. Then we get a stratified sample of size n . In doing so, just consider each row of the following arrangement as a stratum. Systematic sample number 1 2 3  i  k Sample composition 1 2  n 1y 1ky   ( 1) 1n ky   2y 2ky   ( 1) 2n ky   3y 3ky   ( 1) 3n ky       iy k iy   ( 1)n k iy       ky 2ky  nky Probability 1 k 1 k 1 k  1 k  1 k Sample mean 1y 2y 3y  iy  ky Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 7 Recall that in the case of stratified sampling with k strata, the stratum mean 1 1 k st j j j y N y N    is an unbiased estimator of the population mean. Considering the set up of stratified sample in the set up of a systematic sample, we have - Number of strata = n - Size of strata = k (row size) - Sample size to be drawn from each stratum = 1 and sty becomes 1 1 1 1 n st j j n j j y ky nk y n       2 1 2 2 2 1 2 2 1 2 2 1 ( ) ( ) 1 1 using ( ) .1 1 1 n st j j n j SRS j n j j wst wst Var y Var y n k N n S Var y S n k Nn k S kn k S nk N n S Nn                     where 2 2 1 1 ( ) 1 k j ij j i S y y k      is the mean sum of squares of units in the thj stratum. 2 2 2 1 1 1 1 1 ( ) ( 1)        n k n wst j ij j j i j S S y y n n k is the mean sum of squares within strata (or rows). Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 10 Under systematic sampling Earlier ijy denoted the value of study variable with the thj unit in the thi systematic sample. Now ijy represents the value of  ( 1) th i j k  unit of the population, so     2 1 1 1 ( 1) , 1, 2,..., ; 1, 2,..., . 1 ( ) ( ) 1 1 ( 1) 1 2                              ij sy i k sy i i n i ij j n j y a b i j k i k j n y y Var y y Y k y y n a b i j k n n a b i k 2 2 1 1 2 2 1 2 2 2 1 1 2 2 2 2 1 1 ( ) 2 2 1 2 1 1 2 2 2 ( 1)(2 1) 1 ( 1) ( 1) 6 2 2 ( 1) 12                                                            k k i i i k i k k i i n nk y Y a b i k a b k b i k k b i k i k k k k k k b k b k k 2 2 2 2 1 ( ) ( 1) 12 ( 1). 12 sy b Var y k k k b k     Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 11 Under stratified sampling   1 ( 1) , 1,2,..., , 1, 2,..., 1 ij k st i i i y a b i j k i k j n y N y N          2 21 ( )st wst wst N n k Var y S S Nn nk       2 2 1 2 1 1 2 1 1 22 1 1 2 2 2 1 where 1 ( ) ( 1) 1 1 ( 1) ( 1) ( 1) 2 1 ( 1) 2 ( 1) ( 1) 12 ( 1) 12 n wst j j k n ij j i j k n i j k n i j S S n y y n k k a b i j k a b j k n k b k i n k b nk k n k k k b                                           2 2 2 1 ( 1) ( ) 12 1 12 st k k k Var y b nk b k n           If k is large, so that 1 k is negligible, then comparing ( ), ( ) and ( ),st sy SRSVar y Var y V y ( )stVar y : ( )syVar y : ( )SRSVar y or 2 1k n  : 2 1k  : ( 1)(1 )k nk  or 1k n  : 1k  : 1nk  or 1 ( 1) k n k   : 1 1 k k   : 1 1 nk k    1 n 1 : n Thus 1 ( ) : ( ) : ( ) :: : 1 : st sy SRSVar y Var y Var y n n So stratified sampling is best for linearly trended population. Next best is systematic sampling. Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 12 Estimation of variance: As such, there is only one cluster, so the variance in principle, cannot be estimated. Some approximations have been suggested. 1. Treat the systematic sample as if it were a random sample. In this case, an estimate of variance is  2 1 2 2 0 1 1 ( ) 1 where ( ) . 1 sy wc n wc i jk i j Var y s n nk s y y n              This estimator under-estimates the true variance. 2. Use of successive differences of the values gives the estimate of variance as    21 ( 1) 0 1 1 1 ( ) . 2( 1) n sy i jk i j k j Var y y y n nk n              This estimator is a biased estimator of true variance. 3. Use the balanced difference of 1 2, ,..., ny y y to get the estimate of variance as   22 2 1 24 4 1 2 3 1 1 1 ( ) 5( 2) 2 2 or 1 1 1 ( ) . 15( 4) 2 2 n i i sy i i n i i sy i i i i y y Var y y n nk n y y Var y y y y n nk n                                     4. The interpenetrating subsamples can be utilized by dividing the sample into C groups each of size . n c Then the group means are 1 2, ,..., .cy y y Now find  1 2 1 1 1 ( ) ( ) . ( 1) c t t c sy t t y y c Var y y y c c         Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 15 Example: Let 17N  and 5.n  Then 3q  and 2r  . Since , 3. 2 n r k q   Then sample sizes would be * 1 5 with probability 1 3 2 1 6 with probability . 3 r r r n q q q n r r r n q q q                                               This can be verified from the following example: Systematic sample number Systematic sample Probability 1 2 3 1 4 7 10 13 16, , , , ,Y Y Y Y Y Y 4 5 8 11 14 17, , , , ,Y Y Y Y Y Y 3 6 9 12 15, , , ,Y Y Y Y Y 1/3 1/3 1/3 We now prove the following theorem which shows how to obtain an unbiased estimator of the population mean when .N nk Theorem: In systematic sampling with sampling interval k from a population with size ,N nk an unbiased estimator of the population mean Y is given by ' ˆ n i k Y y N        where i stands for the thi systematic sample, 1, 2,..., and 'i k n denotes the size of thi systematic sample. Proof. Each systematic sample has a probability 1 k . Hence ' 1 ' 1 1ˆ( ) . 1 . k n i i k n i i k E Y y k N y N                   Now, each unit occurs in only one of the k possible systematic samples. Hence ' 1 1 , k n N i i ii y Y            which on substitution in ˆ( )E Y proves the theorem. Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 16 When ,N nk the systematic samples are not of the same size and the sample mean is not an unbiased estimator of the population mean. To overcome these disadvantages of systematic sampling when N nk circular systematic sampling is proposed. Circular systematic sampling consists of selecting a random number from 1 to N and then selecting the unit corresponding to this random number. After that, every thk unit in a cyclical manner is selected until a sample of n units is obtained, k being the nearest integer to . N n In other words, if i is a number selected at random from 1 to N , then the circular systematic sample consists of units with serial numbers , if 0,1,2,..., ( 1). , if i jk i jk N j n i jk N i jk N           This sampling scheme ensures an equal probability of inclusion in the sample for every unit. Example: Let 14N  and 5.n  Then, k  nearest integer to 14 3. 5  Let the first number selected at random from 1 to 14 be 7. Then, the circular systematic sample consists of units with serial numbers 7,10,13, 16-14=2, 19-14=5. This procedure is illustrated diagrammatically in the following figure. 13 12 1 2 3 4 5 6 7 8 9 10 11 12 Sampling Theory| Chapter 11 | Systematic Sampling | Shalabh, IIT Kanpur Page 17 Theorem: In circular systematic sampling, the sample mean is an unbiased estimator of the population mean. Proof: If i is the number selected at random, then the circular systematic sample mean is 1 , n i y y n         where n i y        denotes the total of y values in the thi circular systematic sample, 1, 2,..., .i N We note here that in circular systematic sampling, there are N circular systematic samples, each having probability 1 N of its selection. Hence, 1 1 1 1 1 ( ) N n N n i ii i E y y y n N Nn                   Clearly, each unit of the population occurs in n of the N possible circular systematic sample means. Hence, 1 1 , N n N i i ii y n Y            which on substitution in ( )E y proves the theorem. What to do when N nk One of the following possible procedures may be adopted when .N nk (i) Drop one unit at random if the sample has ( 1)n  units. (ii) Eliminate some units so that .N nk (iii) Adopt circular systematic sampling scheme. (iv) Round off the fractional interval k .