ESTIMATION


OVERVIEW

Weighting is the process of adjusting results from a sample survey to infer results for the total in-scope population. The weight can be considered an indication of how many population units are represented by a sample unit. Essentially, the PES weighting methodology attaches weights to each responding person in the PES, to enable estimates of undercount to be produced.

The representation of PES weighting and estimation has been simplified in this section, to provide a broad overview. It should be considered illustrative, with the methods applied discussed in more detail in Research Paper: An Estimating Equation Approach to Census Coverage Adjustment, May 2007 (cat. no. 1351.0.55.019).

PES weighting began with a 'selection weight' for each dwelling, which was essentially equal to the inverse of the probability that a dwelling was selected in the PES sample. Dwelling weighting adjustments were then made to compensate for the different rates of PES non-response in different dwelling types and areas of Australia. Next, person weighting adjustments were applied, allowing for undercoverage and non-response within dwellings. Larger adjustments were made for categories of people (such as young adult males) who were harder to contact and interview successfully in the PES, as is generally the case with most surveys.


DWELLING WEIGHTING

Each dwelling in the PES sample was given a 'dwelling weight' so that the PES sample represented all private dwellings in Australia. The PES sample was designed to ensure each private dwelling in a state or territory had a known non-zero probability of selection. The inverse of this probability is the dwelling's selection weight. In practice, certain types of dwellings are more likely to be missed in the PES. Adjustment for this was made according to a dwelling's post-stratum, where dwellings were assigned to a particular post-stratum (i.e. one of many different groups) based on the following variables:

• Census response category at the start of the PES enumeration period (responding, non-responding or unoccupied on Census night), plus a category "missed" for PES dwellings that were not successfully matched to a Census dwelling (with a small degree of category collapsing required, based on the matching outcome outlined in the People and dwellings missed Technical Note [in Explanatory Notes]);
• Dwelling structure; and
• Region (six states and NT divided into capital city and rest of state, plus ACT, that is, 15 regions).
• Separate post-strata were formed for dwellings sampled in discrete Indigenous communities (ICF).

Selection-weighted estimates of dwelling numbers were obtained for each post-stratum as the total of the selection weights for PES dwellings in that post-stratum. These will typically understate the actual Census counts. Initial dwelling weights were obtained by multiplying the selection weights by a factor to adjust for this, For dwellings matched to a Census- responding dwelling, the factor used was the Census count divided by the selection-weighted estimate for the dwelling's post-stratum. For other dwellings, a factor from similar Census-responding dwellings (those with the same region and dwelling structure) was applied. These initial weights are applied to all dwellings in the PES sample, even those that were not fully responding in the PES.

Initial dwelling weight (IDW) =

Selection weight (S) x Factor for similar Census-responding dwellings (F)
F = Census count for post-stratum (C) / Sum of selection weights for post-stratum (E)
IDW = S x F = S x C / E

A final dwelling weight was derived by weighting up responding dwellings in the PES in each post-stratum to represent non-responding dwellings deemed to be occupied:

Final dwelling weight (FDW) =

Initial dwelling weight (IDW) x Sum of initial dwelling weights for occupied dwellings (T) / Sum of initial dwelling weights for responding PES dwellings (R)
FDW = IDW x T / R

For example, consider a Census-responding post-stratum in which a group of 27 dwellings were selected by PES, each with a selection probability of 0.25. The selection weight (S) for each dwelling is 4 (i.e. 1 / 0.25). The total selection weight of the dwellings counted in the Census is E = P * S (i.e. 27 * 4) which equals 108. If 135 dwellings were counted in the Census (C), the factor F to be applied is 135/108. The initial dwelling weight (IDW) for all these dwellings is S * F (i.e. 4 * 135/108), which is equal to 5.

IDW = (1/0.25) x 135/(27 x 4)

=4 x 135/108
=5

Furthermore, suppose that of the 27 dwellings selected by the PES in this post-stratum (total weight T=27 x 5=135), only 25 dwellings responded in the PES (total weight R=25 x 5=125). The cumulative weight of the 2 non-responding dwellings will be redistributed to the other dwellings in the sample in order for the sample to reflect the independently estimated distribution of the population. The adjustment factor to account for non-response is T/R (i.e. 135/125), which is 1.08. This factor is applied to the initial weight (IDW) to derive a final weight (FDW) for this dwelling.

FDW = IDW x (27 x 5)/(25 x 5)

= 5 x 1.08
=5.4

PERSON WEIGHTING

In estimation, person weights of those responding in the PES were adjusted so that when summed across those persons counted in the Census, the totals correspond to the actual Census counts within a number of benchmark categories. The benchmark categories were based on personal characteristics including age, sex, state/territory, country of birth and Indigenous status.

Estimates of the number of people who should have been counted in the Census based on dwelling weights would only represent the population of people who were in private dwellings at the time of the PES, given the scope of the PES was limited to private dwellings only. That is, they would underestimate the private dwelling population at the time of the Census because some people in private dwellings on Census night would have been in non-private dwellings, overseas, or may have even been deceased at PES time. Such estimates would also not represent people living in non-private dwellings.

To represent all in-scope people on Census night, adjustments were made to the dwelling weights in order to give a person weight. The initial person weight adjustments were chosen to ensure that the PES estimates of people counted in private dwellings (other than late return or imputed dwellings - see Components of net undercount) in a set of benchmark categories matched the actual Census counts for these categories. The weight adjustment applied was such that all persons with identical PES category values received the same weight adjustment, whether or not they were counted in the Census.

The variables used to form these benchmark categories were:

• Region;
• Sex;
• Age (by 16 age groups);
• Country of birth;
• Marital status;
• Indigenous status;
• Whether sampled in an discrete Indigenous community (ICF) dwelling;
• Whether sampled in the Northern Australia region (see Glossary for definition of Northern Australia region); and
• Whether sampled in a 'hard to enumerate' area (see Glossary for definition of 'hard to enumerate').

For information on the resolution of Census not-stated values for Indigenous status and Country of birth for use in benchmarking, see the Explanatory Notes.

As a final step in weight adjustment, the initial person weights were adjusted so that the PES estimates represented not only people in private dwellings but also people in non-private dwellings, to match the scope of the Census. This final step used only region, age and sex as information on other items was not reliable for non-private dwellings.

Intuitively, a 'good' set of weights for the PES should ensure that if the PES were used to estimate the actual Census count, the PES would get the 'right' answer. The above step ensured this was the case. Technically, this is a desirable property for a set of PES weights to have since there is a very strong relationship between the actual Census count and the count that the Census should have made.

For example, consider the benchmark category consisting of Queensland females aged 35 to 39 years. Suppose that the dwelling-weighted estimate of persons counted in the Census in this category (in dwellings that were not imputed or late returns) was 10,000, but the actual Census count of such persons was 10,500. This would lead to adjusting the person weights of all Queensland females aged 35 to 39 in the PES sample to exceed their dwelling weights by about 5 percent (=10,500/10,000). This adjustment is applied regardless of whether they were counted in the Census. In practice, persons contribute to a number of benchmark categories, so the actual weight change for an individual person in this benchmark category could be lower or greater than 5 percent.

In 2011, the ABS again used the Prediction Regression (PREG) estimator, which was developed and used as an estimator in the 2006 PES, and which extended the Dual System Estimator approach to account for overlapping benchmark categories and the situation where people gave different responses between PES and Census. A detailed description of the PREG estimator can be found in Research Paper: An Estimating Equation Approach to Census Coverage Adjustment, May 2007 (cat. no. 1351.0.55.019).


ESTIMATES OF NET UNDERCOUNT

The weighted estimate of population for a category of persons is obtained as the sum of the person-weights of persons who should have been counted in that category in the Census. The final PES estimate adjusts this figure for consistency with Census counts, by adding on the Census count for the category from the responding Census dwellings and subtracting the weighted estimate of this (i.e. the sum of the person-weights of persons who were actually counted in the category in responding Census dwellings).

Net undercount for any category of person is the difference between the final PES estimate of population (i.e. the number of people who should have been counted in the Census) and the actual Census count (including imputed persons in non-responding Census dwellings). This calculation takes into account the components that are described in Components of net undercount.