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UW-Madison STAT 411 - rand_modS15

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Nordheim Statistics 411 Spring 2015 Model vs Randomization Approach for Ratio Estimation Suppose we have a population with N=100. We take an SRS with n=10 and are interested in estimating the population total. Suppose we have an auxiliary variable, X, with € tx= 2600. Consider the following data: > x [1] 21 22 23 24 25 26 27 28 29 30 > y [1] 11.22 11.63 12.58 12.79 14.15 13.58 14.87 14.59 16.23 15.58 ˆty=ˆR * tx=yx* tx=13.72225.5* 2600 = .538118 * 2600 = 1399.11 Using the randomization approach, € Vˆ a r(ˆ t y) =N2n(x U)2x (1−nN) *sr2x =10021026225.5(1−nN) *sr2x = 26509.802 * (1 −nN) *sr2x . where sr2=1n −1(yii=1n∑−ˆRxi)2= 0.201489 . Thus, € Vˆ a r(ˆ t y) = 26509.802 * (1−nN) *sr2x = 26509.802 * (.9) *.20148925.5= (13.730)2 . Using the model-based approach, € Vˆ a r(ˆ t y) =N2n(x U)2x (1−xii=1n∑tx) *ˆ σ 2= 26509.802 * (1−xii=1n∑tx) *ˆ σ 2 . Now € ˆ σ 2 must be obtained by using the appropriate weighted regression (through the origin). With “x” and “y” as above, let > wt=1/x and then fit the regression: > out=lm(y~x -1,weights=wt) > summary(out) Coefficients: Estimate Std. Error t value Pr(>|t|) x 0.538118 0.005409 99.49 5.31e-15 *** --- Residual standard error: 0.08637 on 9 degrees of freedom Multiple R-squared: 0.9991, Adjusted R-squared: 0.999 F-statistic: 9898 on 1 and 9 DF, p-value: 5.313e-15 Note that the slope estimate is identical to ˆR. > anova(out) Response: y Df Sum Sq Mean Sq F value Pr(>F) x 1 73.841 73.841 9897.8 5.313e-15 *** Residuals 9 0.067 0.007The mean square error is the sum of the “weighted” residuals. wri= (yi−ˆRxi) / xi; € ˆ σ 2=1n −1(wrii=1n∑)2 . Now € ˆ σ 2= .00746 . [[R rounds the Mean Sq much too much!!]] Thus, € Vˆ a r(ˆ t y) =N2n(x U)2x (1−xii=1n∑tx) *ˆ σ 2= 26509.802 * (1−xii=1n∑tx) *ˆ σ 2= 26509.801* (1−2552600) * .00746 = (13.355)2 . The variance for the model-based approach is slightly lower in this case. However, by altering the values of the x’s and y’s, one can get quite different results. The model-based approach sometimes results in variance estimates that are substantially smaller than the randomization estimates but can also lead to larger variance


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UW-Madison STAT 411 - rand_modS15

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