# Calculating variance and DEFF for stratified sampling

This question was posted the Assessment and Surveillance forum area and has 4 replies.

### BLESSING T MUREVERWI

Consultant

Normal user

28 Mar 2014, 00:40

### Mark Myatt

Consultant Epidemiologist

Frequent user

28 Mar 2014, 12:08

```
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.
### Mark Myatt

Consultant Epidemiologist

Frequent user

28 Mar 2014, 14:31

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