4.4 Further Numerical Examples
The application of the distance functions presented in the previous subsection to the hypothetical sample used earlier gives the results shown in Table 4, where the simple (unrestricted) chi-squared results are added for comparison. In cases where limits are imposed on the degree of adjustment of the weights, it cannot be expected that a solution will always be available. For this reason, care is needed in the choice of 
and 
, as discussed below.
The values of 
and 
were initially selected as being well outside the range of ratios obtained using the other distance functions. When the range was reduced to the potentially restrictive values of 
and 
, none of the ratios obtained was actually at the limits specified. Nevertheless, the change to the weighting function produces a different set of weights, as shown by comparisons in Table 4: some actually move further away from their initial, or survey, weights.
|
|
A | B |
|
|
|
Chi-squared |
|---|---|---|---|---|---|---|---|
| 1 | 3.000 | 2.674 | 2.654 | 2.706 | 2.513 | 2.483 | 2.753 |
| 2 | 3.000 | 2.228 | 2.260 | 2.178 | 2.408 | 2.400 | 2.109 |
| 3 | 5.000 | 5.998 | 6.012 | 5.976 | 6.162 | 6.187 | 5.945 |
| 4 | 4.000 | 3.944 | 3.926 | 3.974 | 3.951 | 4.019 | 4.005 |
| 5 | 2.000 | 2.514 | 2.521 | 2.501 | 2.534 | 2.493 | 2.484 |
| 6 | 5.000 | 4.456 | 4.423 | 4.510 | 4.189 | 4.138 | 4.589 |
| 7 | 5.000 | 5.729 | 5.717 | 5.747 | 5.911 | 6.094 | 5.752 |
| 8 | 4.000 | 3.944 | 3.926 | 3.974 | 3.951 | 4.019 | 4.005 |
| 9 | 3.000 | 2.228 | 2.260 | 2.178 | 2.408 | 2.400 | 2.109 |
| 10 | 3.000 | 3.086 | 3.074 | 3.106 | 3.213 | 3.325 | 3.120 |
| 11 | 5.000 | 5.998 | 6.012 | 5.976 | 6.162 | 6.187 | 5.945 |
| 12 | 4.000 | 3.814 | 3.762 | 3.897 | 3.645 | 3.769 | 3.985 |
| 13 | 4.000 | 5.108 | 5.136 | 5.065 | 5.094 | 4.990 | 5.019 |
| 14 | 3.000 | 3.490 | 3.487 | 3.494 | 3.604 | 3.680 | 3.490 |
| 15 | 5.000 | 4.665 | 4.666 | 4.665 | 4.442 | 4.314 | 4.678 |
| 16 | 3.000 | 2.370 | 2.380 | 2.355 | 2.428 | 2.408 | 2.345 |
| 17 | 4.000 | 5.191 | 5.232 | 5.128 | 5.115 | 4.993 | 5.070 |
| 18 | 5.000 | 4.603 | 4.604 | 4.600 | 4.366 | 4.237 | 4.614 |
| 19 | 4.000 | 5.028 | 5.043 | 5.001 | 5.069 | 4.986 | 4.967 |
| 20 | 3.000 | 2.228 | 2.260 | 2.178 | 2.408 | 2.400 | 2.109 |
The choice of and 
shown in the penultimate column of the table, actually places some adjustments to the weights at the lower limit of the range: for individuals 2, 9 and 20, the value of 
is equal to 
. However, no adjustments are at the upper range specified. If is raised to 
(with unchanged), unreported results show that individual 16 is placed at the lower limit, along with 2, 9 and 20 as before; in addition individuals 5, 13, 17, 19 are pushed to the upper limit of 
. The attempt to raise to 
means that no solution is possible. However, if is set to the higher value of 
then 
is found to be the highest value (where the range of variation is limited to the second decimal point) of for which a solution is possible. The two highest ratios needed in this case are for persons 13 and 17, who have values of 
and 
respectively. If is kept at this value of 
the lowest value of for which a solution exists is 
. In this case individuals 1, 2, 6, 9, 16 and 20 are placed at the lower limit and individuals 5, 13, 17 and 19 are placed at the upper limit. Clearly, some care needs to be exercised in the choice of upper and lower limits.
While these examples help to explore the characteristics of the different approaches, it is necessary to examine the practical implementation of the method. This is carried out in the following section.
