5  The NZ Household Economic Survey

This section applies the above approaches to the New Zealand Household Economic Survey 2000/01, which is the latest survey available. The aim is to illustrate the application of the approach in a practical context, and to compare the performance of the alternative distance functions. At this point it may be useful to stress that reweighting may cause non-calibrated variables to change in undesirable ways, so that various other checks need to be made.[15]

The variation in the survey weights provided by Statistics NZ for the period 2000/01 is illustrated in Figure 3, where the weights are arranged in ascending order for a sample of 2808 households.[16] It can be seen that the majority of these weights are within a fairly narrow range, although some are substantially higher, suggesting a considerable degree of under-representation of these household types in the sample.

For present purposes, new weights were obtained using calibration values for 2003/4, therefore allowing for population changes. A total of 36 calibration equations were used, covering the total numbers in the following categories for: 11 family composition types; 16 age/sex types; 2 unemployment benefits; 2 Domestic Purpose Benefits; 2 invalidity benefits; 2 sickness benefits; and 1 widow’s benefits.[17]

Figure 3 – Survey weights

Figure 4 – Calibrated weights: Chi-Square

Figure 5 – Ratio of calibrated to survey Weights: Chi-Square

The calibrated weights obtained using the basic chi-square distance function are shown in Figure 4, where households are arranged in the same order as in Figure 3 (although the vertical axis has been truncated at 2,500). The corresponding ratios of calibrated to survey weights, , are displayed in increasing order in Figure 5. This clearly shows considerable variability in the weights, with some negative weights resulting. Using the modified chi-square distance function, allowing the ratio of weights, , to be restricted with the limits and resulted in values displayed in Figure 6. The effects of the adjustment can clearly be seen in the extent to which the new weights are restricted to a range of variation around the initial profile. The top and bottom of the profile in Figure 6 are substantially ‘smoothed’; clearly a significant number are placed at the limits, particularly at the lower limit.

The calibrated weights obtained using the distance function in (33), allowing for upper and lower limits to of and are shown in Figure 7, and the ratios are shown in ascending order in Figure 8. In both cases where limits were imposed, the range shown is the narrowest for which a solution was obtained (that is, for which the iterative method used to obtain the Lagrange multipliers converged).[18] The main difference between the modified chi-square case and the distance function in (33) is that the former appears to push more values to the lower edge of the profile.

Figure 6 – Calibrated weights: Modified Chi-Squared function

Figure 7 – Calibrated weights: Deville-Särndal function

Figure 8 – Ratio of calibrated to survey weights: Deville-Sarndal function

The use of the other two distance functions failed to produce solutions. Comparing the results for the distance functions producing solutions, it seems that the only serious contenders are the two cases imposing constraints on the proportionate changes in weights. The numerical values of the limits which can be imposed, while still obtaining a solution, appear to be the same for the adjusted chi-squared function and the function in equation (33). Where solutions are available, there seems little to choose between those two cases. However, in further experiments using a larger number of calibration equations, it was found that no solution was available using the distance function in (33), however wide the range of variation allowed. Nevertheless a solution could be obtained using the modified chi-squared distance function. The standard chi-squared function also gave a solution, as expected, but this produced a number of negative weights.