178 SAMPLING TECHNIQUES TABLE 6.7 RESULTS FOR DIFFERENT ESTIMATES OF Y Method Variance (Bias)2 MSE Simple expansion 820.3 0.0 820.3 Combined ratio 262.8 6.5 269.3 Separate ratio 35.9 24.1 60.0 Separate Hartley-Ross 153.6 0.0 153.6 Separate Lahiri 19.6 0.0 19.6 Separate Quenouille 42.9 1.1 44.0 Separate Beale 28.9 8.0 36.9 Separate Tin 28.6 5.7 34.3 with the consequences that this unit has a high probability of being drawn and that samples containing this unit give good estimates of Rn. The Quenouille, Beale, and Tin methods all produced substantial decreases in bias as compared with the separate ratio estimate, and all had smaller MSE's, so that in this example they achieved their principal purposes. The study by J. N. K. Rao (1969) of natural populations cited in section 6.9 compared the Quenouille, Beale, and Tin methods for n = 2, 4, 6, 8, on 15 such populations. For n = 2, the most severe test, the medians and the upper quartiles of the quantities | bias|/MSE were as follows: Ro, 3%, 7%; B, 8%, 12%; T, 8%, 19%; as against 15%, 20% for . The more complex methods appear to help materially as regards bias in these tiny samples. The same study compared the MSE's of five of the estimates in this section with that of R. (Lahiri's method was omitted, since the study was confined to simple random samples.) For each method the ratio 100 MSE (RO)/MSE(R), and so forth, was calculated for each population. For n=4, Quenouille's and Mickey's estimates were slightly inferior to in these populations but, for n 6, all methods had average MSE's very close to those of R. For a very small sample from a single population, this study suggests that these more complex methods have no material advantage in accuracy over . But the fact that they reduce bias with little or no increase in MSE in a single stratum should give them an advantage in a separate ratio estimate with numerous strata having small samples. Under a linear regression model, comparisons of the MSE's of these methods for small n by P.S.R.S. Rao and J. N. K. Rao (1971), Hutchinson (1971), and J. N. K. Rao and Kuzik (1974) gave results in general agreeing with those from the natural populations. 6.16 COMPARISON OF THE METHODS Example. The artificial population in Table 6.6 contains three strata with N = 4, nn = 2 in each stratum. The population was deliberately constructed so that (a) R, varies markedly from stratum to stratum, thus favoring a separate ratio estimate Rs, and (b) n overestimates in each stratum, with the threat of a serious cumulative bias in Rs. The TABLE 6.6 A SMALL ARTIFICIAL POPULATION I Stratum II III 4 6 11 = 20 22592 824 223 y x 1 3 4 7 8 9 24 23 25 81342 Totals 20 32 40 36 44 20 Rn 0.625 1.111 2.200 following methods of estimating the population total Y were compared. Simple expansion: Nyn Combined ratio: (st/xst)X Separate ratio: (n/xx)X = nXn The remaining estimates, the separate Hartley-Ross, Lahiri, and Quenouille, Beale, and Tin methods, have the same form as the separate ratio estimate, except that (HR)h, R (L)h, and so on, replace . (For n = 2, the Hartley-Ross and Mickey methods are identical.) There are 63=216 possible samples. Results are exact apart from rounding errors. For help in some computations I am indebted to Dr. Joseph Sedransk. The results in Table 6.7 show some interesting features. For the combined ratio estimate, the contribution of the (bias) to the mean square error is trivial, despite the extreme conditions, but this estimate does poorly as regards variance because of the wide variation in the R. As judged by the MSE, the separate ratio estimate is much more accurate than the combined estimate, but it is badly biased. Of the unbiased methods, Hartley-Ross shows relatively high variance, as it has been found to do for n = 2 in some studies on natural populations. The Lahiri method does particularly well. This population suits the Lahiri method because one unit in each stratum has unusually high values of both y, and xi, kalai clear all; format short; N=2 2 2 1 3 1 3 4 5 4 7 3 4 6 9 8 9 4 11 20 24 23 25 12 ]; Nh=4; nh=2; fh=nh/Nh; n=6; totN=4*3; f=n/totN; Y=[N(1) N(:,3); N(:,5);]; X=[N(2) N(:,4); N(:,6);]; T1 [N(1) N(:,3) N(:,5)]; T2 [N(2) N(:,4) N(:,6)]; meansstrat=mean (N); totals=sum(N); Rh=[totals (1)/totals (2) totals (3)/totals (4) totals (5)/totals (6)]; R=sum(Y)/sum(X); Sarvar (Y); Ssgrx=var (X); varxh-var (T1); Varxh var (T2); Actualy=sum(Y) %%%%%%standard deviation Sch Syh=sqrt(varxh) Sxh=sqrt(yarth) %%%%%Rho rhol cork (N (:1), N(:,2)); rho2 corr(N (:,3), N(:,4)); rho3 cork (N (4), N(:,6)); rhorhol rho2 rho3] %%%% simple expansion variance Varyhat-(((totN^2) * (1-f))/n) * Ssxxx %%%%% simple expansion Bias That Nh*meansstrat (1) +Nh*meansstrat (3) +Nh*meansstrat(5); Bias That Actualy %%%%% simple expansion MSE SEMSE-Varyhat+Bias Combined Ratio % Xhatst Nh*meansstrat (1)Nh* meansstrat (3) +Nh*meansstrat (5); % Xhatst Nh*meansstrat (2)+Nh*meansstrat (4) +Nh*means strat (6); % xbarst Xhatst/totN xbarst Yhatst/totN % WhatRc=(Thatst/Xhatst) * sum (X) %%YhatRc1= (xkarst/xkarst)*sum(x) %%%%%%%%Variance for Combined Ratio % CF=(((Nh^2)*(1-fh))/nh); % VaryhatRc=CF* ((varyh (1) + (R^2) *varxh (1) - 2*R*rhol*Syh (1) *Sxh (1)) + (varyh (2)+(R^2) *varxh (2)- 2*R*rho2*Syh(2)*Sxh(2))+(varYh(3)+(R^2)*varXh(3)-2*R*rho3*Syh(3)*Sxh(3))) Combined Ratio Bias Biascomb-XhatRc-Actual