Hi All,
My question concerns a Hypothesis test to compare proportions between 2
survey samples when they are computed using the ratio estimate. What is the
test statistic?
Since the ratio estimate is biased can this also be factored in?
In the case of a simple random sample I can just apply the standard Z test
to compare 2 proportions with the following formula:
Sample 1: proportion = p1 = r1/n1, sample size = n1, sample fraction = f1
Sample 2: proportion = p2 = r2/n2, sample size = n2, sample fraction = f2
My test statistic Z = (p1 - p2) / sqrt{ p(1-p)*[ (1-f1)/(n1-1) +
(1-f2)/(n2-1) ] }
where p = (n1*p1 + n2*p2)/(n1+n2) is the pooled proportion estimate.
However if I am using the ratio method to estimate p1 and p2 it is not clear
to me how to construct the test statistic and whether it has a t-dist. or
Z-dist. So for example
Sample 1:
- n1 sample units, xi1 experiment units of which yi1 have my attribute of
interest in each sample unit (i=1,...n1)
- sample fraction = f1
- p1 = sum(yi1)/sum(xi1) (i= 1,.....n1). This is a biased estimate.
- var(p1) = (1-f1)/[n1*mean(x1)^2] * {p1^2*var(x1) + var(y1) -
2*p1*cov(x1,y1) }
where var(.) is the variance and cov(.,.) the covariance. Sample 2 would be
defined in a similar manner.
So in this case does Z = p1-p2 / sqrt( var(p1) + var(p2) )??
Or do I need to compute a pooled estimate of var(p1-p2)??
Also since p1 is a biased estimate of P1 the true proportion i.e. E(p1) = P1
+ bias; can I just subtract the bias from my estimate p1 so that is
unbiased?
Many thanks in advance for your help,
Richard
|