With the exception of analysis of Table 5.1, all the standard errors in this paper are based on the ‘jackknife method’, which calculates the effect of each unit on the estimate. If there are n units in the sample, then n estimates are calculated from the sample where a single different unit is removed each time from the total sample. The variance estimate is then based on the difference between these estimates and the estimate obtained from the total sample (Wonnacott & Wonnacott 1984:250–1). The advantage of this method is that, even if the original estimate of variance is slightly biased (but asymptotically unbiased), the ‘jackknife method’ will often eliminate the bias and produce consistent estimates of standard errors.

Given that jackknife errors were not obtained for the estimates of the proportion of the sample without health expenditure (Table 5.1), it is possible to derive some using the conventional binomial formula and adjusting for a design effect. That is, in order to account for the fact that the NHS is a complex sample rather than a simple random sample, the standard errors from the binomial formula need to be adjusted by a design effect.

The design effect was estimated as the factor by which the binomial standard errors must be scaled up to equate them with the (consistent) jackknife estimates of standard errors. This was done for both Indigenous and non-Indigenous population using the Table 6.2 jackknife estimates as the benchmark. For example, the jackknife estimates of the standard errors in that table were divided by the binomial estimates to derive a design effect specifically for all the estimates in Table 5.1.

Approximation of standard errors using the binomial formula and a design effect

Under simple random sampling (SRS), the estimate of variance of a proportion is given by

B1:

p = esitmate of proportion

n = sample size

Vâr_SRS = estimate of the variance of the estimate of proportion, assuming SRS

N = sample size

For large N, as in our case, we can simplify this to:

B2:

Under a complex desigh the variance of the estimate can be estimated by:

(B3)

where deff=design effect

As indicated above, the design effect was calculated using the jackknife estimates for Table 6.2 and then used to scale up the estimated variances for a simple random sample using equation B3.