To calculate the standard error of any statistic derived from the survey data, the method is as follows:
- Calculate the estimate of the statistic of interest using the main weight.
- Repeat the calculation above for each replicate weight, substituting the replicate weight for the main weight and creating G replicate estimates. In the example where there are 60 replicate weights, you will have 60 replicate estimates.
- Use the outputs from step 1 and 2 as inputs to the formula below to calculate the estimate of the Standard Error (SE) for the statistic of interest.
\normalsize SE (y)=\sqrt{\frac{G-1}{G} \sum_{g=1}^{G}(y_{(g)}-y)^{2}}
[Equation 1]
- G = Number of replicate groups
- g = the replicate group number
- y_{(g)} = Replicate estimate for group g, i.e. the estimate of y calculated using the replicate weight for g
- y = the weighted estimate of y from the sample
From the replicate variance you can then derive the following measures of sampling error: relative standard error (RSE), or margin of error (MoE) of the estimate.
\text{Relative Standard Error (RSE)} = \frac{\text{SE}}{\text{Estimate}}
[Equation 2]
\text{Margin of Error (MoE)} = 1.96 \times \text{SE}
[Equation 3]
An example in calculating the SE for an estimate of the mean
Suppose you are calculating the mean value of earnings, y, in a sample. Using the main weight produces an estimate of $500.
You have 5 sets of Group Jackknife replicate weights and using these weights (instead of the main weight) you calculate 5 replicate estimates of $510, $490, $505, $503, $498 respectively.
To calculate the standard error of the estimate you will substitute the following inputs to equation [1]
- G = 5
- y = 500
- g = 1, y_{(g)} = 510
- g = 2, y_{(g)} = 490
- …
\normalsize SE (y)=\sqrt{\frac{5-1}{5} \sum_{g=1}^{5}(y_{(g)}-500)^{2}}
\normalsize SE (y)=\sqrt{\frac{4}{5} ((510-500)^{2} + (490-500)^{2} + (505-500)^{2} + (503-500)^{2} + (498-500)^{2}})
\normalsize SE (y)=\sqrt{\frac{4}{5} \times 238}
\normalsize SE (y)=13.8
To calculate the RSE you divide the SE by the estimate of y ($500) and multiply by 100 to get a %
\normalsize RSE (y)=\frac{13.8}{500} \times 100
\normalsize RSE (y)=2.8\%~
To calculate the margin of error you multiply the SE by 1.96
\text {Margin of Error} (y)=13.8 \times 1.96
\text {Margin of Error} (y)=27.05