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
\(\text {Margin of Error} (y)=13.8 \times 1.96\)
\(\text {Margin of Error} (y)=27.05\)