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2. The Sen-Yates-Grundy variance estimators are generally very close to the actual variation in the estimates over repeated sampling. 3. The jackknife variance estimators are inappropriate in business surveys with high sampling fractions, and do not seem to be corrected by application of the finite population correction. 4. The use of software packages for estimators for which they are not designed, or the use of “naïve variance estimators” through using the right weights in the wrong formula both produce variance estimates which are very biassed. These approaches are not recommended. Hence an appropriate package must be used when ratio, regression or more complex estimators are in use. 32
4 Variances in STATA/SUDAAN compared with analytical variances 4.1 Expansion estimator The usual estimator of a total in stratified sampling is David Holmes, University of Southampton H ∑ tˆ = N y (4.1) y h= 1 h sh where ys h = n − 1 h ∑ s h y k , and the variance is given by V ( tˆ ) y ( − f ) 2 N 2 h 1 h = ∑ S h yU n h (4.2) h where where 2 S yU h is the stratum variance. This variance is estimated by ( tˆ ) ( − f ) N h h Vˆ y = ∑ 2 1 2 S ys (4.3) h n 2 2 −1 S ys h is the sample variance in stratum h , ys = ( n −1) ∑ ( − ) h h y s k ys h h h S . 2 Note: (4.1) can be written as N tˆ = h y = ∑∑ whk yk , where whk . n h s h h 4.2 Ratio estimator The separate ratio estimator of a total in stratified sampling (used by ABI) is where x h given by where ∑ −1 = nh x s k h ∑ y sh t ˆ y rat = t xh (4.4) , h xsh and t xh is the stratum total of the x k . The variance of the estimator is ( 1− f ) 2 N h h 2 V ( tˆ y, rat ) = ∑ S yUh , rat (4.5) n h h S 2 yU , rat h = 1 ∑ N −1 = S h 2 yU h − 2R U h h ( y − R x ) S k xyU h h + R 2 h k S 2 2 xU h (4.6) is the stratum variance of the variable section 6-10. This variance can be estimated by y − R x and hk h hk R h t yh = . See Cochran (1977), t xh 33
Page 1 and 2: Model Quality Report in Business St
Page 3 and 4: Outline of Model Quality Report Vol
Page 5 and 6: Contents 1 Introduction ...........
Page 7 and 8: The requirements for a variance est
Page 9 and 10: where y k is the value of the study
Page 11 and 12: where n hq is the size of the part
Page 13 and 14: The parameters (2.1)-(2.4) are tota
Page 15 and 16: previous sections. However, in orde
Page 17 and 18: The finite population correction (f
Page 19 and 20: esponse homogeneity groups in such
Page 21 and 22: The four main packages investigated
Page 23 and 24: There is a two-volume manual which
Page 25 and 26: 2.2.6.1 Initial reactions of new us
Page 27 and 28: knowledge from using the software i
Page 29 and 30: log ( y ) = β + β log( x + 1) +
Page 31 and 32: The main conclusion seems to be tha
Page 33 and 34: The jackknife estimators are extrem
Page 35: produce identical solutions for the
Page 39 and 40: where δ hi ⎧1 if sample unit k i
Page 41: SÄRNDAL, C.-E. & SWENSSON, B. (198

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