3.2 A small example
The above procedure may be illustrated using a simple example. Suppose there are four variables, 
(for 
), of concern, for which population values 
are available. The hypothetical data, for a sample of 20 individuals, are shown in Table 1. Suppose variable 
refers to age, so that 
for those who are ‘young’ and is zero otherwise, while 
for those who are unemployed, and zero otherwise. Variable 
measures earnings from employment, while variable 
is another categorical variable referring to location (
if the individual lives in a city, and is zero otherwise).[6] Given the sample design weights shown in the penultimate column of Table 1, the estimated population totals are equal to 
.
The symmetric matrix 
and its inverse are given in Table 2. The zero elements reflect the property of the basic data, that only individuals who work (for whom 
) are assumed to receive positive earnings, 
. Suppose that the known population totals are 
reflecting a younger population than in the sample weights and a lower unemployment rate. The resulting calibrated weights are shown in the final column of Table 1.
|
|
|
|
|
|
|
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 0 | 3 | 2.753 |
| 2 | 0 | 1 | 0 | 0 | 3 | 2.109 |
| 3 | 1 | 0 | 2 | 0 | 5 | 5.945 |
| 4 | 0 | 0 | 6 | 1 | 4 | 4.005 |
| 5 | 1 | 0 | 4 | 1 | 2 | 2.484 |
| 6 | 1 | 1 | 0 | 0 | 5 | 4.589 |
| 7 | 1 | 0 | 5 | 0 | 5 | 5.752 |
| 8 | 0 | 0 | 6 | 1 | 4 | 4.005 |
| 9 | 0 | 1 | 0 | 0 | 3 | 2.109 |
| 10 | 0 | 0 | 3 | 1 | 3 | 3.120 |
| 11 | 1 | 0 | 2 | 0 | 5 | 5.945 |
| 12 | 1 | 1 | 0 | 1 | 4 | 3.985 |
| 13 | 1 | 0 | 3 | 1 | 4 | 5.019 |
| 14 | 1 | 0 | 4 | 0 | 3 | 3.490 |
| 15 | 0 | 0 | 5 | 0 | 5 | 4.678 |
| 16 | 0 | 1 | 0 | 1 | 3 | 2.345 |
| 17 | 1 | 0 | 2 | 1 | 4 | 5.070 |
| 18 | 0 | 0 | 6 | 0 | 5 | 4.614 |
| 19 | 1 | 0 | 4 | 1 | 4 | 4.967 |
| 20 | 0 | 1 | 0 | 0 | 3 | 2.109 |
|
|
|||
|---|---|---|---|
| 44.000 | 12.000 | 101.000 | 18.000 |
| 12.000 | 24.000 | 0.000 | 7.000 |
| 101.000 | 0.000 | 981.000 | 101.000 |
| 18.000 | 7.000 | 101.000 | 32.000 |
|
|||
| 0.037 | -0.016 | -0.003 | -0.008 |
| -0.016 | 0.053 | 0.003 | -0.011 |
| -0.003 | 0.003 | 0.002 | -0.005 |
| -0.008 | -0.011 | -0.005 | 0.053 |
The required adjustments to the weights can clearly be seen to be consistent with expectations, given the calibration requirements and the characteristics of the individuals. For example, the weights for individuals 2, 9 and 20 fall by a relatively large amount (from 3 to 2.109), since these individuals are all unemployed, old and living in rural locations, for all of which the aggregates are required to fall. The weights for individuals 1 and 6 do not drop so far because, although these are unemployed and in a rural location, they are young. The weight for person 12 falls by a small amount because, although unemployed, this person is young and in a city. The weights for individuals 13, 17 and 19 increase by relatively large amounts as they are young, employed and living in a city.
Notes
- [6]The number of variables needed is of course one less than the number of categories of each type, otherwise singularity problems arise.
