Consultant
Normal user
28 Mar 2014, 00:40
I would like to request assistance in calculating variance for stratified PPS sampling.
Let's say we do a survey with 3 strata with the following populations and prevalence of malnutrition in a 30 X 7 survey:
Strata Population Sample Prevalence Weight
1 9,870 225 81.3% 0.4997
2 33,599 219 54.8% 1.7475
3 14,130 212 31.1% 0.7590
I calculated the weights from the population and sample for each of the strata, and also calculated the weighted prevalence which is 53.22%.
I would like assistance in calculation of the variance and design effect in this case.
Thanks in advance!
Consultant Epidemiologist
Frequent user
28 Mar 2014, 12:08
I think you want a confidence interval around a point estimate.
Here is a simple approach ...
You data:
Strata N n P
------ ----- --- -----
1 9870 225 0.813
2 33599 219 0.548
------ ----- --- -----
3 14130 212 0.311
Add weights as:
w = N / sum(N)
giving:
Strata N n P w
------ ----- --- ----- -----
1 9870 225 0.813 0.171
2 33599 219 0.548 0.583
3 14130 212 0.311 0.245
------ ----- --- ----- -----
The point estimate is:
sum(P * w) = 0.813 * 0.171 + 0.548 * 0.583 + 0.311 * 0.245 = 0.535
The variance is:
sum((w^2 * p * (1 - p)) / n)
From your data:
Strata N n p w x$w^2*x$p*(1-x$p))/x$n
------ ----- --- ----- ----- -----------------------
1 9870 225 0.813 0.171 1.975795e-05
2 33599 219 0.548 0.583 3.844253e-04
3 14130 212 0.311 0.245 6.067027e-05
------ ----- --- ----- ----- -----------------------
SUM = 0.00046485
-----------------------
SQRT(SUM) = 0.02156046
-----------------------
The 95% CI is then:
Lower 95% CL = 0.535 - 1.96 * 0.02156046 = 0.4927 (49.27%)
Upper 95% CL = 0.535 + 1.96 * 0.02156046 = 0.5773 (57.73%)
Is this what your need?
BTW : The design effect is the ratio of the variance (calculated above) and the variance calculated from a simple random sample. With your data:
c = round(225 * 0.813 + 219 * 0.548 + 212 * 0.245)
= 355
n = 225 + 219 + 212
= 656
p = 355 / 656
= 0.541
var = (p * (1 - p)) / (n - 1)
= (0.541 * (1 - 0.541)) / (656 - 1)
= 0.000379113
DEFF = 0.00046485 / 0.000379113
= 1.23
Avoid rounding early (as I have done above).
I hope this helps.
You should check my arithmetic.
Consultant
Normal user
28 Mar 2014, 12:44
Many thanks Mark, very very helpful!
Consultant Epidemiologist
Frequent user
28 Mar 2014, 14:31
Just to be clear ... rounding errors can accumulate and end up becoming quite large over a series of calculations ... I have been lazy above ... keep all numbers to full precision throughout and round only the final results. If you do calculations in a spreadsheet with raw numbers then you should be OK as the full precision is often retained "behind the scenes".