SydU STAT3014 (2015) Second semester Dr. J. Chan 18

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1 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Stratified Random Samping.1 Introduction Description The popuation of size N is divided into mutuay excusive and exhaustive subpopuations caed strata of N 1, N,, N L units where N s are known. A simpe random sampe of size n is drawn from the -th stratum, = 1,, L. Tota sampe size : n = n. Reasons for Stratification: N 1 N N 3 n 1 n n 3 SRS 1 SRS SRS 3 ȳ 1 ȳ ȳ 3 1. Aow sub-estimates which can then be combined to give an overa estimate, e.g. district and state income eves.. Provide administrative convenience as strata may exist in natura, e.g. counties and branches. 3. Aow different samping fractions and methods in different stratum, e.g. sma/arge business, private/government housing, urban/rura househods. 4. Enabe efficient estimates if a heterogeneous popuation is divided into strata that are internay homogeneous. ȳ ȳ 1 ȳ ȳ 3 Gp 1 Gp Gp 3 s 1 s s 3 s i < s SydU STAT ) Second semester Dr. J. Chan 18

2 Notation Popuation N STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Sampe n Y i i = 1,, N y i, i = 1,, n i-th unit from stratum h Ȳ = 1 N Y i ȳ = 1 n y i N n i=1 S s i=1 Aso W = N N is the stratum weight and f = n is the samping fraction N for stratum.. Estimators of mean and variance For the popuation mean Ȳ = 1 N N Ȳ : Ȳ = ȳ st = 1 N N ȳ = W ȳ. For the popuation tota Y = NȲ : Ŷ = N ȳ st = N ȳ = N W ȳ. Ceary Eȳ st ) = Ȳ since Eȳ ) = Ȳ and the ȳ s are independent. Hence var Ȳ ) = W 1 n ) s = N n W s n 1 N W s SydU STAT ) Second semester Dr. J. Chan 19

3 varŷ ) = N STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe W 1 n ) s = N N n W s n N W s Estimators and variance estimates for a stratified SRS Parameter Estimator Variance Ordinary estimator s y = 1 n n 1 yi n ȳ ), W = N N i=1 Ratio R Rst = 1 X W ȳ var R st ) = 1 X W 1 n ) s y N n Mean Ȳ Ȳ st = L W ȳ var Ȳ st ) = 1 n ) s y Tota Y W Ŷ st = N L W ȳ varŷst) = N W N n 1 n ) s y N n SydU STAT ) Second semester Dr. J. Chan 0

4 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Exampe Cochran, 1963) A sampe of U.S. coeges was drawn to estimate the enroments in 1947 from 196 Teachers Coeges. These coeges were separated into 6 strata. Suppose the foowing sampe information is obtained. Compete the tabe and estimate tota enroment, its s.e. and coefficient of variation. N n ȳ N ȳ s N 1 n N ) s n = )31 9 = = )31 7 = = )15 11 = = )105 7 = = )9 14 = = ) = Tota Soution: We have N = 196 and Ŷ st = N ȳ = = varŷst) = i=1 i=1 N 1 n N ) s = ) n sdŷst) = = cvŷst) = sd Ȳ st ) Ȳ st = ) = = Read Tutoria 10 Q4c and Tutoria 11 Q1a,b. SydU STAT ) Second semester Dr. J. Chan 1

5 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe.3 Optima Aocation We consider 1. Choice of n = nw s to obtain maximum precision, for a given tota sampe size n or, more generay, a given tota cost. Here w is the desire aocation.. For a specified precision choice of the minimum n and corresponding n s. 1. Optimization Let K denote a cost function where C = C 0 + C n, C 0 0, C > 0, = 1,, L. Let V = Varȳ st ) = W S n 1 N W S. Note that if we take C 0 = 0, C = 1, 1 then C reduces to the tota sampe size n.. Theorem For a fixed vaue of C, V is minimized by the choices of n where n N S C, = 1,, L. The proportionaity constant is chosen in accordance with the fixed vaue of C. We shoud take a arger sampe from a statrum if SydU STAT ) Second semester Dr. J. Chan

6 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe 1) the stratum size N is arger; and/or ) the variabiity in terms of s.d. S is greater; and/or 3) the square root of the cost of samping a unit C is ower. 3. Best precision for specified tota sampe size Optima aocation: For fixed tota sampe size n, precision is greatest if N S / C n = N S / W S / ) C n = C W S / n, = 1,, L. C Neyman aocation: C 1 = C = = C L First determined by Aexander Chuprov 193), rediscovered by Neyman 1934). n = N ) S W S n = n, = 1,, L. N S W S Proportiona aocation: If S 1 = S = = S L, then n = N N n = W n, = 1,, L. SydU STAT ) Second semester Dr. J. Chan 3

7 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe 4. Minima Tota Sampe Size for Specified Precision If we require that is Pr{ ȳ st Ȳ δ µ} 1 α, { } ȳ st Pr Ȳ V arȳst ) δ µ 1 α V arȳst ) or δ µ V arȳst ) z α/ so Varȳ st ) δ µ zα/. For optima aocation, n = n L Varȳ st ) = 1 n n/ ) L W C S V + 1 N W S n 1 N W S / C W S /. C =1 W S ast eqt in P.19) W S = ) W S / C ) nw S / 1 W S C N ) = 1 W S W S C) 1 W S δ µ n C N z α/ W S C) W S C) W S C ) W S or n = n ) W S C ) W S C =1 W S C W S C 1 N 1 N W S C W S + δ µ z α/ 1 W S + δ µ z α/ W S C =1 V + 1 N. W S SydU STAT ) Second semester Dr. J. Chan 4 =1

8 where V = δ µ L STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe z α/ ) L W S C V is the bound for variance. Ignoring fpc, ) W C S or n W S C For Neyman aocation, n = n W S / L 1 N ) W S W S + V or n = n Ignoring fpc, the formua is L ) W S V or n W S W S =1 =1 W S C =1 1 N L ) W S W S =1 V V W S. W S W S =1, = 1,, L., = 1,, L. W S + V =1, = 1,, L. For proportiona aocation, n = n W. Equating S fais because S in V arȳ st ) are not a equa. Instead we shoud consider Varȳ st ) = W S n 1 N W S = W S nw 1 N W S V W S W W S V + 1 N W S or n = n W V + 1 N =1, = 1,.., L. W S =1 SydU STAT ) Second semester Dr. J. Chan 5

9 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Sampe size determination for a SRS and stratified SRS SRS for mean Ŷ s.t. Pr y Y δ µ) = 0.95 and S = p1 p) for prop. P Sam. n NS Nδ µ/z α/ + S f=0 z α/ S δ µ = S V Stratified SRS for mean Ŷ s.t. Pr y st Y δ µ ) = 0.95 and V = δ µ zα/ Sizes Optimum Neyman Proportiona Strat. n Sam. n n = n L W S C i=1 W i S i Ci ) L W S C i=1 V + 1 N W S ) W Ci i S i n = n W S W i S i i=1 L ) W S V + 1 N W S n = nw W S V + 1 N W S 5. Confidence Intervas for specified tota sampe size Provided each n is moderatey arge ȳ st µ Varȳst ) N 0, 1), at east approximatey. Optima aocation for a given tota sampe size wi give the shortest 1 α) 100% CI with ength: z 1 α ) Var ȳst = z 1 α ) Varopt ȳ st SydU STAT ) Second semester Dr. J. Chan 6

10 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Exampe Cochran, 1963) The standard deviation for coege enroments, s, in a given stratum is estimated by the 1943 vaue. Stratum N S The tota enroment in 1943 was 56,47. For the 1947 study the aim is to obtain a coefficient of variation CV) which is a ratio of standard error to estimate to be approximatey 5%. Determine the sampe size n required and the Neyman aocation n of these vaues to strata as given previousy. Soution: We have N = 196, Ŷ = 56, 47 and seŷ ) = cv Ŷ = ) = 84 varŷ ) = 84 = 7, 974, 976 = N V We assume that C are a equa. Then Strat. N S N S N S ω = N S N S n = ω n roundn ) Tota SydU STAT ) Second semester Dr. J. Chan 7

11 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Take n = 58. = ) W S V + 1 N W S = ) N S V N + N S 6841) = Read Tutoria 10 Q4a,b and Tutoria 11 Q1e. SydU STAT ) Second semester Dr. J. Chan 8

12 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe.4 Doube samping for stratification In stratified SRS, the auxiiary variabe X for stratification can be known ony after samping, e.g. income eve or education eve. In other words, W = N N, = I,..., L are unknown before samping. Doube samping is a two-phase samping. For exampe, we may ca many voters in an opinion po to identify income eve phase 1 sampe), when ony a few coud be interviewed phase sampe) for purposes of competing a detaied questionnaire. Phase 1: Take a arge sampe of size n for obtaining preiminary information such as income eve of a voter) to identify strata with Ŵ = n n, = 1,..., L denote the proportion of the first sampe faing into stratum. Then Ŵ is an unbiased estimator of W = N N. Phase : A much smaer sampe of n eements are randomy samped from the n eements identified in stratum for coecting information on Y. Then the subsampe mean and variance, ȳ and s, are cacuated for each stratum. N X, Y, Sy SRS s s n n n 1 n L x, Ŵ Post-stratification n SRS n 1 n L x, ȳ, s SydU STAT ) Second semester Dr. J. Chan 9

13 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe An estimator of the popuation mean if the phase samping fractions n N are a sma and N is arge is Y st,ds = varŷ st,ds) = Ŵ ȳ n n 1 [ Ŵ Ŵ n ) ] s + Ŵȳ Ŷ st,ds) n n If n is so arge that Ŵ n varŷ st,ds) = is negigibe, this variance estimate reduces to ] [Ŵ s + Ŵȳ Ŷ st,ds) n n Ignoring fpc terms, the first part takes the usua form with w repacing N N. The second part is the additiona variance since we estimate the weights W from a sampe. It is not necessariy to make the second term sma because it means making the ȳ about equa which is not desirabe because stratification is more efficient than SRS when the ȳ are quite different. Thus, choosing strata that produce different ȳ sti may be better than a SRS, even though doube samping might have to be empoyed to estimate the W. SydU STAT ) Second semester Dr. J. Chan 30

14 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Exampe: Coege enroment) From a ist of enroments and facuty sizes for American 4-year coeges and universities, it is desired to estimate the average enroment for the academic year). Private institutions tend to be smaer than pubic ones, so stratification is in order. Whie the data indicate the type of coege or university, the ist is not broken up according to type. A 1-in-10 systematic sampe was done to obtain information on type of coege and the resuts are: Private Pubic Tota n 1 = 84 n = 57 n = 141 Subsampes of 11 private and 1 pubic coeges gave the foowing data on enroments and facuty size. Estimate the average enroment for American coeges and universities in Private Pubic n 1 = 11 n = 1 Enroment Facuty Enroment Facuty ȳ 1 = 1, s 1 = 1, 77.7 ȳ = 5, 85.7 s 1 = 5, SydU STAT ) Second semester Dr. J. Chan 31

15 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Soution: From the data given above, Y [ 84 st,ds = ȳ st,ds = Ŵ ȳ = 141 = 3, varŷ st,ds) = 1 n 1 Ŵ1s 1 ) + 1 n Ŵs ) + 1, ] 5, n [Ŵ1ȳ 1 ȳ st,ds ) + Ŵȳ ȳ st,ds ) ] = 1 ) ) [ , , 367.4) + 57 ] 5, , 367.4) 141 = 553, , = 583, seŷ st,ds) = 583, = The second part of the variance is due to estimating the true stratum weights. The standard deviation of Ŷ st,ds is sti quite arge due to the sma sampe sizes and arge variation among coege enroments, but it is much smaer than the error associated with a singe SRS of 3 coeges from the ist. Y srs = varŷ srs) = n n ȳ = 1 n N [ 11 3 seŷ srs) = 965, = ] 5, , = 3, ) s n = 1 3 ), 576, 503 = 965, where s is cacuated using the combined 1 SRS) sampe and N = SydU STAT ) Second semester Dr. J. Chan 3

16 STAT3014/3914 Appied Stat.-Samping C-Stratified rand. sampe Doube samping estimator based on -stage samping Parameter Estimator Variance [ ] Mean Y Y w st,ds = w ȳ varŷ st,ds) = s + w ȳ Ŷ st,ds) n n w = n n If n & N are arge and n N are a sma SydU STAT ) Second semester Dr. J. Chan 33