STAT 344 Sample Surveys LAB 6 Chao Xiong February 15, 2011 Chao Xiong STAT 344 Sample Surveys LAB 6 Outline 1 Stratification 2 Questions? Slides will be available on the website: http://www.stat.ubc.ca/~alex.xio/ My email address: [email protected] 0 Chao Xiong STAT 344 Sample Surveys LAB 6 Information about strata and samples 1 2 Strata Sizes N1 N 2 Means Y1 Y2 Variances S12 S22 Samples Sizes n1 n2 Means y1 y2 Variances s12 s22 Chao Xiong 3 4 Ng Yg Sg2 Ng YG SG2 ng yg sg2 ng yG sG2 STAT 344 Sample Surveys LAB 6 a) Compute the stratified mean estimator ys t and its estimated variance. G X ¯st = Yˆ Wg y¯g 1 where Wg = Ng /N and v (Yˆ¯st ) = G X Wg2 (1 − fg )/ng · sg2 1 Chao Xiong STAT 344 Sample Surveys LAB 6 b) Construct 95% confidence limits for population total ¯st = Yˆ = N Yˆ G X Ng y¯g i=1 Its estimated variance ¯st ) v (Yˆ ) = N 2 v (Yˆ 95% C.I (Yˆ − 1.96 ∗ s.e(Yˆ ), Yˆ + 1.96 ∗ s.e(Yˆ ) Chao Xiong STAT 344 Sample Surveys LAB 6 c) Test the hypothesis that the average response value of Stratum II is twice of that of Stratum I. H0 : Y¯2 = 2Y¯1 v .s. H1 : Y¯2 6= 2Y¯1 For each stratum, v (¯ yg ) = (1 − fg )/ng · sg2 v (¯ y2 − 2¯ y1 ) = (1 − f2 )/n2 · s22 + 4 · (1 − f2 )/n2 · s22 Under the null hypothesis, y¯2 − 2¯ y1 ∼ N(0, 1) S.E .(¯ y2 − 2¯ y1 ) Reject the null if y¯2 −2¯ y1 S.E .(¯ y2 −2¯ y1 ) > Z1.96 or Chao Xiong y¯2 −2¯ y1 S.E .(¯ y2 −2¯ y1 ) < −Z1.96 STAT 344 Sample Surveys LAB 6 Alternatively, you can use two sample t-test to get more accurate approximation for small samples t= y¯2 − 2¯ y q 1 s12 · n11 + where s12 = 1 n2 ∼ t(n1 + n2 − 2) 4(n1 − 1)s12 + (n2 − 1)s22 n1 + n2 − 2 Chao Xiong STAT 344 Sample Surveys LAB 6 d) Proportional sample size allocation ng = Wg n In practise, you will round these numbers to integers. Under proportional allocation, X X XX Yˆ¯st = Wg y¯g = ng y¯g /n = ygi /n G and i X ¯st ) = 1 − f Wg Sg2 Vp (Yˆ n G the corresponding estimate X ¯st ) = 1 − f Vp (Yˆ Wg sg2 n G Compared to SRSOR, Vs S2 =P Vp Wg Sg2 Chao Xiong STAT 344 Sample Surveys LAB 6 the conclusion is unless there is not meaningful difference between strata (which make the 2nd term in the approximation nearly 0), the stratified srsor with proportional sample size allocation results in more accurate estimate of the population mean through the use of stratified mean. Chao Xiong STAT 344 Sample Surveys LAB 6 Neyman allocation ¯st ) is minimized if If we already know Sg , then the variance of V( Yˆ we set ng Wg Sg =P Wg Sg n the minimized variance is 1 X 1 X VN = ( Wg Sg )2 − Wg Sg2 n N Note: the above are all theoretic results. In practice, you should first compute ng under different allocation methods, and then ¯st and Vp (Yˆ¯st ) using around them into integers. Next, calculate Yˆ v (Yˆ¯st ) = G X Wg2 (1 − fg )/ng · sg2 1 See the example in the books. Chao Xiong STAT 344 Sample Surveys LAB 6 Conclusion In terms of variance of the stratified mean under stratified srsor, proportional allocation is less efficient than Neyman allocation. The advantage increases when the differences between Sg increases. The stratified mean with proportional sample size allocation has lower variance than the sample mean under straight srsor when V (Yg ) are different. In some applications, there are a few units with huge sizes that they form a stratum with huge stratum variance. The optimal allocation may suggest to have more units sampled than the stratum size. We should watch out for such situations and adjust appropriately. Chao Xiong STAT 344 Sample Surveys LAB 6 e) Sample size calculation the variance should not exceed a given value V |Yˆ¯st − Y¯ | should not exceed a given value e except for a probability of α P( ¯st − Y¯ | |Yˆ e ≤ )=1−α ˆ ¯st ) S.E .(Y¯st ) S.E .(Yˆ we have ¯st ) ≤ v (Yˆ e2 1.962 ¯st − Y¯ |/Y¯ should not exceed a given value the relative error |Yˆ a except for a probability of α Similarly we get ¯ ˆ ) ≤ ( aY )2 v (Y¯ st 1.96 Chao Xiong STAT 344 Sample Surveys LAB 6 Questions? Chao Xiong STAT 344 Sample Surveys LAB 6
© Copyright 2024