Ratio estimation using stratified ranked set sample

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1 METRON - International Journal of Statistics 003, vol. LXI, n. 1, pp HANI M. SAMAWI MAHMOUD I. SIAM Ratio estimation using stratified ranked set sample Summary - Ratio estimation metod is used to obtain a new estimator wit iger precision for estimating te population mean or total. Tere are two metods for estimating ratios wen te sampling design is simple stratified random sampling (SSRS), namely te combined and te separate ratio estimates. We study te performance of te combined and te separate ratio estimates using stratified ranked set sample (SRSS) introduced by Samawi (1996). Teoretical and simulation study as well as real data example are presented. It appears tat using SRSS is more efficient tan using SSRS for ratio estimation bot in te case of combined and separate metods. Key Words - Concomitant variables; Order statistics; Ranked set sample; Ratio estimators, Stratified sample. 1. Introduction In many applications te quantity tat is to be estimated from a random sample is te ratio of two variables bot of wic vary from unit to unit. Te ratio of bilirubin level in jaundiced babies wo stay in neonatal intensive care; to teir weigt at birt or te ratio of acres of weat to total acres on a farm are two examples. Also, tis metod is used to obtain increased precision for estimating te population mean or total by taking advantage of te correlation between an auxiliary variable X and te variable of interest Y. In te literature, ratio estimators are used in case of simple random sampling (SRS) as well as in case of SSRS. (See for example Cocran, 1977). Tere are two metods for estimating ratios tat are generally used wen te sampling design is SSRS, namely te combined ratio estimate and te separate ratio estimate. McIntyre (195) used te mean of n units based on a ranked set sample (RSS) to estimate te population mean. Samawi and Muttlak (1996) used RSS Received September 001 and revised November 00.

2 76 H. M. SAMAWI M. I. SIAM to estimate te population ratio and sowed tat it provided a more efficient estimator compared to te one obtained by using SRS. Also, Samawi (1996) introduced te concept of stratified ranked set sampling (SRSS), to improve te precision of estimating te population mean. In tis paper we use te idea of SRSS instead of SSRS to improve te precision of te two metods for estimating ratios. Also, we study te properties of tese estimators and compare tem wit oter estimators. In Section and 3, we obtain te separate and combined ratio estimators using SRSS sample respectively. We also derive te asymptotic mean and variance of tese estimators. Te comparison between te two estimators (separate and combined) is discussed in terms of bias and efficiency in Section 4. Also, te results of our simulation study and te use of te two metods using real data about bilirubin level of baby s, wo stay in neonatal intensive care, is discussed in Section 5 and 6 respectively Stratified ranked set sample For te -t stratum of te population, first coose r independent samples eac of size r = 1,,...,L. Rank eac sample, and use RSS sceme to obtain L independent RSS samples of size r, one from eac stratum. Let r 1 + r r L = r. Tis complete one cycle of stratified ranked set sample. Te cycle may be repeated m times until n = mr elements ave been obtained. A modification of te above procedure is suggested ere to be used for te estimation of te ratio using stratified ranked set sample. For te -t stratum, first coose r independent samples eac of size r of independent bivariate elements from te -t subpopulation, = 1,,...,L. Rank eac sample wit respect to one of te variables say Y or X. Ten use te RSS sampling sceme to obtain L independent RSS samples of size r one from eac stratum. Tis complete one cycle of stratified ranked set sample. Te cycle may be repeated m times until n = mr bivariate elements ave been obtained. We will use te following notation for te stratified ranked set sample wen te ranking is on te variable Y.Forte k-t cycle and te -t stratum, te SRSS is denoted by {(Y (1)k,X [1]k )(Y ()k,x []k ),...,(Y (r )k,x [r ]k) : k =1,,...,m; =1,,...,L}, were Y (i)k is te i-t order statistic from te i-t set in te -t stratum and X [i]k is te corresponding concomitant variable (see Stokes, 1977). 1.. Ratio estimate using SSRS Te parameter of interest to be estimated in tis paper is R = µ Y µ X. Using SSRS, we ave two types of ratio estimators:

3 Ratio estimation using stratified ranked set sample 77 In separate case. Following Levy and Lemesow (1991), we ave te following definition of separate ratio estimator. For te -t stratum ( = 1,,...,L), te ratio is defined by ˆR srs = Y. Terefore, assuming tat te X subpopulations totals of te variable X are known, te separate ratio estimator using SSRS is given by ˆR SSRS(s) = T T ˆR srs = Y X T T = W µ X µ X Y X (1.1) were W = N N, N is te stratum size, N is te total population size, r r T = N µ X, T = Nµ X, Y = Y i, X r = X i,µ r X, = 1,,...,L are te known subpopulation specific means for te random variable X and µ X = L W µ X. Note tat te subpopulations totals of te variable X need to be known in order to compute ˆR SSRS(s) for estimating te population mean or total of te variable Y. It can be sown (see Hansen, 1953) tat 1 E( ˆR SSRS (s)) = R + O Min (mr ) and te variance of ratio estimator can be approximated by Var( ˆR SSRS(s) ) W µ X R } µ {C X Xn + C Y ρ XYC X C Y (1.) were n = mr, C X = σ X, C µ Y = σ Y,ρ X µ XY = Y µ Y N (X i µ X )(Y i µ Y ) Nσ X σ Y, R =, and σ µ X, and σ Y are te standard deviations of te variable X and X Y, respectively in te -t stratum. Note tat, equation (1.) also can be derived easily from equation (6.45) in Cocran (1977) by dividing (6.45) by te population total of te variable X and simple algebra. In combined case. Combined ratio estimator using SSRS is defined by ˆR SSRS(c) = Y L SSRS = W Y X L SSRS W. (1.3) X It can be sown (see Hansen (1953) tat E( ˆR SSRS(c) ) = R + O(Max(n 1 )) and tat te variance of te estimator is given by Var ( ˆR SSRS(c) ) R W n { σ Y µ Y + σ X µ X σ Y µ Y σ X µ X ρ XY }. (1.4)

4 78 H. M. SAMAWI M. I. SIAM. Separate ratio estimation using SRSS.1. Ratio estimation wen ranking is on variable Y Te separate ratio estimate, requires knowledge of te stratum totals T (see Levy and Lemesow, 1991) to estimate te population mean or te total. Using te notation introduced in Section (1.1) ratio is estimated as follows. Let Y (r ) = 1 n X [r ] = 1 n m k=1 m r k=1 r Y (i)k, X [i]k, and n = mr. Ten te separate ratio estimator using SRSS wen te ranking on variable Y,isgivenby ˆR SSRS1(s) = It can be sown using Taylor expansion tat 1 E( ˆR SSRS1 (s)) = R + O Min (mr. ) µ X Y (r ) W. (.1) µ X X [r ] Also, te approximate variance of separate ratio is given by Var ( ˆR SSRS1(s) ) W µ X µ X R n { C X + C Y ρ XYC X C Y m r } (.) (M X[i] M Y(i) ) n were, M X[i] = µ X[i] µ X µ X, M Y(i) = µ Y(i) µ Y µ Y, E(Y (i) )=µ Y(i) and E(X [i] ) = µ X[i]. For evaluating te terms in te variances based on RSS see Stokes (1980). For evaluating te expectation and te variance of te concomitant variable of te order statistics see Stokes (1977).

5 Ratio estimation using stratified ranked set sample 79.. Ratio estimation wen ranking is on variable X By canging te notation ( ) of perfect ranking by [ ] of imperfect ranking for X and Y (i.e. canging te role of te order statistics wit te role of te concomitant variable of te order statistics) ratio is given by µ X Y [r ] ˆR SSRS(s) = W (.3) µ X X (r ) were Y [r ] = 1 m r Y [i]k and X (r ) = 1 m r X (i)k. n n k=1 k=1 Again we get te following results: E( ˆR SSRS(s) ) = µ Y 1 + O µ X Min (mr ) and Var ( ˆR SSRS(s) ) W µ X µ X R n { C X + C Y ρ XYC X C Y m r } (.4) (M X(i) M Y[i] ) n were, M X(i) = µ X(i) µ X, M µ Y[i] = µ Y[i] µ Y, E(Y X µ [i] )=µ Y[i] and E(X (i) ) = Y µ X(i). Teorem.1. Assume tat te approximation to te variance of te ratio estimators in (1.), (.) and (.4) are valid and te bias of te estimators can be ignored for large m. Ten Var ( ˆR SRSS1(s) ) Var ( ˆR SSRS(s) ) and Var ( ˆR SRSS(s) ) Var ( ˆR SSRS(s) ). Proof. Take { Var ( ˆR SSRS(s) ) Var ( ˆR SRSS1(s) )=m W µ X µ X R n [ r ]} ( ) MX[i] M Y(i) >0. Terefore, Var( ˆR SRSS1(s) ) Var( ˆR SSRS(s) ). Similarly, Var( ˆR SRSS(s) ) Var( ˆR SSRS(s) ).

6 80 H. M. SAMAWI M. I. SIAM.3. Wic variable to rank? Since we can not rank on bot variables at te same time and some time it is easier to rank on one variable tan te oter, we need to decide to rank on variable X or Y. Teorem.. Let us assume tat tere are L linear relationsips between Y and X, i.e., ρ > 0 and it is easy to rank on te variable X. Also, assume tat te approximation to te variance of te ratio estimators ˆR SRSS1(s) and ˆR SRSS(s) as given in equations (.) and (.4) respectively are valid and te bias of te estimators can be ignored. Ten Var ( ˆR SRSS(s) ) Var ( ˆR SSRS1(s) ). Proof. By looking at te two variances in equations (.) and (.4) we need oly to compare r (M X[i] M Y(i) ) wit r (M X(i) M Y[i] ). At tis end, we note firstly tat µ X[i] = { µx(i) for perfect ranking of X in te i-t set µ X for imperfect ranking of X in te i-t set. (.5) Consider te simple linear regression model of Y on X Y i = α + β X i + ε i, (.6) µ Y = α + β µ X (.7) were α and β are parameters and ε i is a random error wit E(ε i ) = 0, Var (ε i ) = σ, and Cov(ε i,ε j ) = 0 for i j, i = 1,,...,r. Also, ε i and X i are independent. Case 1. If we are ranking on te Y variable we get te following model from equation (.6) Y (i) = α + β X [i] + ε [i], (.8) were ε [i] is a random error wit, Var (ε [i] ) = σ [r ]i and Cov(ε [i],ε [ j] ) = 0 for i j, i = 1,,...,r also ε [i] and X [i] are independent. Ten, From (.7) and (.9) we get µ Y(i) = α + β µ X[i]. (.9) M Y(i) = β M X[i] R (.10)

7 and now (.) can be written as Var ( ˆR SRSS1(s) ) Ratio estimation using stratified ranked set sample 81 W µ X µ X R n { C X + C Y ρ XYC X C Y m r ( 1 µ X n M X[i] β ) }. µ X µ Y Case. If we are ranking on te variable X we get te following model: Te expected value of Y [i] is Similarly, we can sow tat and now (.4) can be written as Var ( ˆR SRSS(s) ) Y [i] = α + β X (i) = ε [i]. (.11) µ Y[i] = α + β µ X(i) + E(ε [i] ). (.1) W µ X µ X M Y[i] = β M X(i) R (.13) R n { C X + C Y ρ XYC X C Y m r ( 1 µ X n M X(i) β ) }. µ X µ Y Terefore, from (.5) it is clear tat Var ( ˆR SRSS(s) ) Var ( ˆR SRSS1(s) ). 3. Combined ratio estimation using SRSS 3.1. Ratio estimation wen ranking on variable Y Te combined ratio estimate using SRSS is defined by ˆR SRSS1(c) = Y (SRSS) X [SRSS] (3.1)

8 8 H. M. SAMAWI M. I. SIAM were and Y (SRSS) = X [SRSS] = W Y (r ) W X [r ] Terefore, ˆR SRSS1(c) = Y L (SRSS) = W Y (r ) X L [SRSS] W. (3.) X [r ] For fixed r, assume tat we ave finite second moments for X and Y. Since te ratio is a function of te means of X and Y, i.e., R = µ Y, and µ X ence R as at least bounded second order derivatives of all types in some neigborood of (µ Y,µ X ) provided tat µ X 0. Ten, assuming large m, and by using te multivariate Taylor series expansion, we can approximate te variance and get te order of te bias of te ratio estimator as follows: and E( ˆR SRSS(c) ) = R + O(Max(n 1 )) Var ( ˆR SRSS1(c) ) R W n { σ Y µ Y + σ X µ σ Xσ Y ρ XY X µ X µ Y ) } (3.3) (D Y(i) D X[i] m n r were, D X[i] = µ X[i] µ X µ X and D Y(i) = µ Y(i) µ Y µ Y. 3.. Ratio estimation wen ranking on variable X In tis case te estimate is given by: ˆR SRSS(c) = Y [SRSS] X (SRSS) were Y [SRSS] = W Y (r )

9 and Ratio estimation using stratified ranked set sample 83 X (SRSS) = Terefore, in combined case, we get W X [r ]. ˆR SRSS(c) = Y [SRSS] X (SRSS) = L W Y [r ] L W. (3.4) X (r ) Using te same argument as in Section (3.1), E( ˆR SRSS(c) ) = R + O(Max(n 1 )) and Var ( ˆR SRSS(c) ) R W n { σ Y µ Y + σ X µ X m n r σ Xσ Y µ X µ Y ρ XY ( D Y[i] D X(i) ) }, (3.5) were, D X(i) = µ X(i) µ X and D µ Y[i] = µ Y[i] µ Y. X µ Y Teorem 3.1. Assume tat te approximations to te variance of te ratio estimators in (1.4), (3.3) and (3.5) are valid and te bias of te estimators can be ignored for large m. Ten Var ( ˆR SRSS1(c) ) Var ( ˆR SSRS(c) ) and Var ( ˆR SRSS(c) ) Var ( ˆR SSRS(c) ). Proof. Similar to tat for Teorem Ranking on wic variable? Again, since we cannot rank bot variables at te same time we need to decide wic variable we sould rank. Terefore, we need to compare te variance of ˆR SRSS1(c) in equation (3.3) and variance ˆR SRSS(c) in equation (3.5). Teorem 3.. Assume tat tere are L linear relationsips between Y and X, i.e., ρ > 0 and it is easy to rank variable X. Also assume tat te approximation to te variance of te ratio estimators ˆR SRSS1(c) and ˆR SRSS(c) given in equations (3.3) and (3.5) respectively are valid and te bias of te estimators can be ignored for large m. Ten Var ( ˆR SRSS(c) ) Var ( ˆR SRSS1(c) ). Proof. Is similar to tat of Teorem (.).

10 84 H. M. SAMAWI M. I. SIAM 4. Comparison of te combined and separate estimates Consider te case wen ranking is on variable X. Equations (.4) and (3.5) can be written respectively as W Var ( ˆR SRSS(s) ) = m [ n µ X R r r r ] σ X(i) + σ Y[i] R σ X(i)Y[i] and W Var ( ˆR SRSS(c) ) = m n µ X [ r r r ] R σ X(i) + σ Y[i] R σ X(i)Y[i] were σ X(i) =Var ( X (i) ),σ Y[i] =Var ( Y [i] ) and ) σ X(i)Y[i] =Cov (X X(i), Y Y[i]. Tus, Var( ˆR SRSS(c) ) Var ( ˆR SRSS(s) ) = m µ X W n [ (R R ) = m W µ X n ( r [ (R R ) r σ X(i) r ] σ X(i)Y[i] (R R ) r σ X(i) +(R R) σ X(i)Y[i] R r σ X(i) As in te case of SSRS (see Cocran, 1977), if te ratio estimate is valid, te last term on te rigt is usually small. (It vanises if witin eac stratum te relationsip between Y [i] and X (i) is a straigt line troug te origin). Also, as in Cocran (1977), unless R is constant from stratum to stratum, te use of a separate ratio estimate in eac stratum is likely to be more precise if te sample in eac stratum is large enoug so tat te approximate formula for Var ( ˆR SRSS(s) ) is valid and te cumulative bias tat can effect te ratio estimate is negligible. Wit only a small sample in eac stratum, te combined estimate is to be recommended. Similarly, we can sow tat similar conclusions old in te case wen ranking on Y is perfect and ranking of X is not perfect. )].

11 5. Simulation study 5.1. Design of te simulation study Ratio estimation using stratified ranked set sample 85 We did computer simulation to gain insigt in te properties of te ratio estimator. Bivariate random observations were generated from a bivariate normal distribution wit parameters µ X,µ Y,σ X,σ Y, = 1,,...,L and correlation coefficient ρ. Also we divide te data in te sample into tree strata and in some cases into four strata. Te simulation was performed wit r = 10, 0, 30 and wit m = 1 for te SRSS, SSRS, RSS and SRS data sets. Te ratio of te population means were estimated for tese sampling metods. Using 000 replications, estimates of te means and mean square errors were computed. We considered ranking on eiter variable Y or X. Results of tese simulations are summarized by te relative efficiencies of te estimators of te population ratio and by te bias of estimation for different values of te correlation coefficient ρ. In order to reduce te size of tis paper we present two tables only. Tables 1 gives efficiency wen ranking is perfect on variable X and Y respectively. Tables gives te bias in estimation wen ranking is perfect on variable X and Y respectively. 5.. Results of te simulation study We conclude tat te largest gain in efficiency is obtained by ranking te variable X and wit large values of negative ρ. (For example in Table 1, te relative efficiency wen ρ = 0.90 and r = 30 is 1.97 wile it is 4.16 wen ρ =.90 and r = 30. Te results of simulation indicates tat, wen ranking is on te variable X or Y, te efficiency will decrease wit decreasing values of ρ from 0.99 to 0, and start to increase as ρ decreases from 0 to However, in te separate case tis conclusion may be canged wen r = 10 as we indicated in Section 4, wen te sample size is small we cannot use te separate ratio estimate. Moreover, te efficiency will increase wen te set size (r) is increased. Also, tere will be no cange in te efficiency if te sample size is increased by increasing te number of cycles. Also, we note tat in combined case for any values of r or ρ MSE( ˆR SRSS ) MSE( ˆR RSS ) MSE( ˆR SSRS ) MSE( ˆR SRS ) wen R = 1.45, W 1 = 0.3, W = 0.3 and W 3 = 0.4 and ave equals variances witin strata. Tis is not completely true for different cases, e.g., wen R = 1.17 or wen variances witin strata are not equal.

12 86 H. M. SAMAWI M. I. SIAM Table 1. Relative effeciency of ratio estimators using SRSS relative to SSRS. W :.3/.3/.4 µ X : /3/4 σ X : 1/1/1 R = 1.45 µ X : 3/4/6 σ Y : 1/1/1 ρ r Ranking on Variable X Ranking on Variable Y Combined Separate Combined Separate From Tables it appears tat te bias of ˆR SRSS is iger wen ρ is negative tan wen it is positive. For example, te bias wen ρ = 0.99 and r = 30 is wile te bias wen ρ = 0.99 and r = 30 is However, in most cases te bias is less tan 0.01 but for small r te bias in separate case exceeds 0.01.

13 Ratio estimation using stratified ranked set sample 87 Table. Bias of ratio estimators using SRSS and SSRS. W :.3/.3/.4 µ X : /3/4 σ X : 1/1/1 R = 1.45 µ X : 3/4/6 σ Y : 1/1/1 ρ r Ranking on Variable X Ranking on Variable Y Combined Separate Combined Separate SRSS SRSS SRSS SRSS SRSS SRSS SRSS SRSS Also, te bias will decrease wen te sample size is increased by increasing r. Te bias in te combined case is always less tan te corresponding bias in te separate case. Similar conclusions can be drawn wen ranking on te variable Y. However, te bias wen ranking is on Y is sligtly lower tan

14 88 H. M. SAMAWI M. I. SIAM wen ranking is on X. Moreover, it is clearly from equation (1.) and (1.4) tat for negativete ρ te variance of te ratio estimators, in case of separate and combied metods, are larger tan for positive ρ. However, our simulation indicated tat te efficiency of using SRSS, for ratio estimation, is iger wen ρ is negative. Tis may be due tat te correlation between te order statistics is always positive. 6. Ratio of bilirubin level to weigt at birt We give a real life example about Bilirubin level in jaundiced babies wo stay in neonatal intensive care. Most birt surveys on live newborns sowed tat jaundice is common. Jaundice in new born can be patological and pysiological. It start on second day of life and it as relationsip wit race, metod of feeding and gestational age. On te oter and if te total serum bilirubin in blood is above 1.5 mg/dl ten we classify it as yper-bilirubin. Neonatal jaundice is define as yellowis discoloration of skin and sclera and it occurs if bilirubin level is more tan 5mg/dl. (see Nelson et al., 1994). Neonatal jaundice usually appears on te second day of life. Most of normal newborn babies leave te ospital after 4 ours of life. Terefore, our primary concern will be on babyies staying in neonatal intensive care. Pysicians are interested in te jaundice, because of te risk on te earing, brain and deat. We will focus on te ratio of te level of bilirubin to te weigt at birt for te newborn babies. Te data was collected from five ospitals in Jordan. Te jaundice is measured by te level of bilirubin in te blood. Tis level is determined according to a blood test (TSB), wic takes nearly 30 minutes. Moreover, ranking on te level of bilirubin in te blood can be done visually by observing te following: (i) Color of te face. (ii) Color of te cest. (iii) Color of te lower parts of te body and (iv) te color of te terminal parts of te wole body. Ten as te yellowis goes from (i) to (iv) te bilirubin level in te blood goes iger. We present below te analysis of te collected data for 10 babies according to teir weigt at birt, sex and bilirubin level. For illustration, assume tat te collected data of 10 babies from te five ospitals is te study population. Denote te bilirubin level by Y and te weigt at birt by X. Since tere are two strata, L =, m = and r = 10, ten n = r.m = 0, W 1 = 7 10 = 0.6 and W = 48 = 0.4. Terefore, for male babies 10

15 Ratio estimation using stratified ranked set sample 89 n 1 = mr 1 = = 1 and or for female babies n 1 = mr = = 8. Also, te parameter of interest to be estimated is R = Using te sampling scemes of SRSS and SSRS, Table 3 contains te two selected samples. Note tat te ranking was on variable X (weigt). Table 3. Te selected samples. SRSS sample SSRS sample Female Male Female Male X Y X Y X Y X Y kg mg/dl kg Mg/dl kg mg/dl kg mg/dl cycle cycle Based on te SRSS and SSRS, Table 4 contains te results of te illustration. Table 4. Summary of te results of te illustration using Bilirubin data. ˆR SSRS(s) ˆR SSRS(C) ˆR SRSS(s) ˆR SRSS(C) Estimate Estimated Variance Finally, we get eff( ˆR SRSS(s), ˆR SRSS(c) ) = 1.08, eff( ˆR SSRS(s), ˆR SSRS(c) ) = 1.07, eff( ˆR SSRS(c), ˆR SRSS(c) ) = 1.1, eff( ˆR SSRS(s), ˆR SRSS(s) ) = 1.1. Acknowledgments Te autors would like to tank te referees for teir comments wic were elpful in improving te paper.

16 90 H. M. SAMAWI M. I. SIAM REFERENCES Cocran, W. G. (1977) Sampling Tecniques, Tird edition, Jon Wiley & Sons. Dell, T. R. and Clutter, J. L. (197) Ranked set sample teory wit order statistics background, Biometrics, 8, Hansen, M. H., Hurwitz., W. N., and Madow, W. G. (1953) Sampling survey metods and teory, Vol.. Jon Wiley & Sons, New York. Levy P. S. and Lemesow S. (1991) Sampling of populations metods and applications, Jon Wiley & Sons, New York. McIntyre, G. A. (195) A metod of unbiased selective sampling using ranked sets. Australian, J. Agricultural Researc, 3, Nelson, W. E., Berman, R. E., Kliegman, R. M., and Vaugan, V. C. (1994) Textbook of Pediatrics, 4-t edn, W.B. Saunders Company Harcourt Barace Jovanovic, Inc. Samawi, H. M. (1996) Stratified ranked set sample, Pakistan J. of Stat., Vol. 1 (1), Samawi, H. M. and Muttlak, H. A. (1996) Estimation of ratio using rank set sampling, Biom. Journal, 38, Stokes S. L. (1977) Ranked set sampling wit concomitant variables, Comm. Statist. -Teor. Met., 1 (6), Stokes S. L. (1980) Estimation of te variance using judgment order ranked set samples, Biometrics, 36, HANI M. SAMAWI Department of Matematics & Statistics Sultan Qaboos University P.O.Box 36 Al-kod 13, Sultanate of Oman samawi@squ.edu.om MAHMOUD I. SIAM Department of Matematics & Statistics Sultan Qaboos University P.O.Box 36 Al-kod 13, Sultanate of Oman