Assignment 7: Variance Estimation for Complex Surveys
due: April 30, 1996
 
     	Most surveys taken by the Census Bureau or large survey 
organizations such as Gallup or Research Triangle Institute are complex, 
with several stages of clustering as well as stratification.  In 
addition, weights are often used make the demographic information of the 
sample agree with information for the population from the U.S. Decennial 
Census, and to adjust for nonresponse.  In such a situation, calculating 
variances for each response would be extremely tedious, requiring a new 
computer program to calculate the variance for each response and survey.  
Since many of the quantities of interest are nonlinear functions of 
population means (e.g. mu_y/mu_x), we need a flexible method for 
calculating variances.
	Several general methods for the estimation of variances of complex 
functions of the population means have been derived, methods that can be 
applied to almost any theory.  These are described in Wolter (1985).  One 
of the simplest is the random groups method, also known as 
interpenetrating subsampling.  In this method, the basic survey design is 
replicated k times.  This may be done by drawing k different samples, or 
by drawing one sample, and later splitting it into k parts, each part 
being a miniature version of the basic sampling design.
	The jackknife is also useful for estimating the variance of 
nonlinear quantities.  If we are interested in the parameter theta, then 
we can also estimate theta by dividing the sample into k subgroups, as in 
the random group method, but instead of estimating theta in each subgroup 
separately, we estimate theta using all of the data except  that in the 
jth subgroup, for j = 1,...,k.  The estimate obtained by omitting the jth 
subgroup is theta-hat(j).  Let  theta-hat(.) be the average of all of the 
theta-hat(j).  Then theta-hat(.) also estimates theta, and the variance 
of theta-hat, the estimate of theta using all of the data, can be 
estimated by (k-1) \sum (theta-hat(j) - theta-hat(.))2/k.

For these exercises, draw a simple random sample of size 200 from 
Lockhart City.  We want to estimate R = mu_y/mu_x, the ratio of the price
a household is willing to pay for cable TV (Y) to the assessed value of
the house (X).  We will apply them to a  simple random sample and an
estimated ratio; however the techniques are usually applied to more
complicated designs.
 

1.     Randomly divide your sample into 10 different subsamples, each of 
size 20.  Be sure to explain how you randomly divided your sample.  Find 
the ratio r_i = ybar_i/xbar_i for each group.  The r_i are 10 (almost)
independent observations; if we use rbar to estimate R, then the estimated
variance of rbar is sum (r_i - rbar)^2/90.  For the purpose of grading,
rearrange your output file from SURVEY so that the first group is the first
20 observations, the second group is the next 20 observations, etc.

2.     Calculate 200 different estimates of R, each using all but one of 
the 200 data points.  One way of doing the calculations is to define two 
new variables, SUMX and SUMY, to be the sum of all 200 observations for X 
and Y, respectively.  Then the variable YJACK = (SUMY - Y)/199 contains 
the 200 values of ybar(j).  Use the variables YJACK and XJACK to 
calculate the jackknife estimate of the variance of r = ybar/xbar.

3.     How do your variance estimates from 1 and 2 compare with the usual 
Taylor series-based estimate of the variance of this ratio?