# Sampling and Weighting Technical Report, Census of Population, 2016 4. Estimation from the census long form sample

Any sampling process requires an associated estimation procedure for scaling sample data up to the population level and for ensuring that survey estimates are representative of the population. The choice of an estimation procedure is generally governed by both operational and theoretical constraints. From the operational viewpoint, the procedure must be feasible within the processing system of which it is a part, and from the theoretical viewpoint, the procedure should minimize the statistical error of the estimates it produces.

The estimation procedure produces a set of weights, and the weight for each sample unit corresponds to the number of units in the population that the sample unit represents. These weights are applied to the sample data to produce millions of estimates from the census long-form sample. Estimates are summary measures such as totals, averages, proportions and medians calculated from the sample for various characteristics of interest.

## 4.1 Considerations in the choice of an estimation procedure

### 4.1.1 Operational considerations

Mathematically, an estimation procedure can be described by an algebraic formula, or estimator, that shows how the estimate for the population is calculated as a function of the observed sample values and other information from the sample design or external data sources. Most of the time, this estimator is a simple function of weights and of the variable of interest for the responding units. Using a unique set of weights to produce all estimates guarantees a certain level of consistency among the different estimates of the survey.

Therefore, the approach taken for the census long-form sample (and in most sample surveys) was to split the estimation procedure into two steps: (a) the calculation of weights (known as the weighting procedure) and (b) the use of weights to produce estimates, such as the estimation of a particular population count by summing the weights of those persons or households with the characteristic of interest. Most of the mathematical complexity is contained in step (a), which is performed just once. Meanwhile, step (b) is reduced to a simple process, such as summing weights whenever tabulation is required. Since the weight attached to each sample unit is the same for any tabulation involving that unit, consistency between different estimates based on sample data is assured.

### 4.1.2 Theoretical considerations

For a given sample design and a given estimation procedure, one can, from sampling theory, make a statement about the chances that a certain interval will contain the unknown population value being estimated. A primary criterion in the choice of an estimation procedure is the minimization of the width of such intervals for a given level of confidence so that these statements about the unknown population values are as precise as possible. A common measure of precision for comparing estimation procedures is known as the standard error. Provided that certain conditions are met, intervals of plus or minus two standard errors from the estimate will contain the true population value for approximately 95% of all possible samples.

As well as minimizing standard error, a second objective in the choice of an estimation procedure for the long-form sample is to ensure, as far as possible, that sample estimates for census characteristics are consistent with the corresponding known census values. Fortunately, these two objectives are usually complementary in the sense that sampling error tends to be reduced by ensuring that sample estimates for certain basic characteristics are consistent with the corresponding population figures. However, while this is true in general, forcing long-form sample estimates for census characteristics to be consistent with corresponding census figures for very small subgroups can have a detrimental effect on the standard error of estimates for the sample characteristics themselves. For example, if in several dissemination areas only a few subjects have a given characteristic, such as birth in a certain country, ensuring consistency between the sample estimates and the census counts for that place of birth would unduly increase the standard error for the rest of the characteristics.

In cases where no information about the population being sampled is available other than that collected for sample units and unit non-response has not occurred, the estimation procedure would be restricted to weighting the sample units inversely to their probability of selection. For example, if a unit had a one-in-four chance of selection, then that selected unit would receive a weight of 4. When unit non-response is observed, the weight must be further adjusted according to the estimated probability of response of the unit, for example. In practice, some supplementary knowledge about the population (e.g., its total size and possibly its breakdown by a certain variable—perhaps by province and territory) is often available. Such information can be used to improve the estimation formula so as to produce estimates with a greater chance of being close to the unknown population value. In the case of the census long-form sample, a large amount of very detailed information about the population being sampled is available from the census short-form data at every geographic level. This wealth of population information is used in the coverage, non-response and calibration adjustments to improve the estimates made from the long-form sample.

Nevertheless, the long-form sample estimates for census characteristics cannot be made consistent with all the census counts at every geographic level. Differences between sample estimates and census counts become visible when a cross-tabulation of a sample variable and the corresponding census variable is produced. The tabulation of sample-based estimates of totals for particular characteristics will not necessarily agree with the equivalent census count tabulations for those characteristics.

Adjusting the weights, by the minimal amounts possible, to achieve perfect agreement between long-form estimates and census counts for certain characteristics and subgroups is known as "calibration."

## 4.2 Weighting areas

The various adjustments to design weights were made independently by weighting area. The geographic areas used for this purpose were aggregate dissemination areas (ADAs) and super aggregate dissemination areas (SADAs). ADAs are a new dissemination geography created for the 2016 Census. SADAs were created specifically for the weighting procedures by ADA aggregation. The geographic subdivisions used in 2011 and earlier were constructed differently.

### 4.2.1 Aggregate dissemination areas

ADAs are a new dissemination geography created for the 2016 Census. Their purpose is to enable the dissemination of a greater quantity of data at a detailed geographic level across the country. In total, Canada was divided into 5,386 ADAs, and households were selected for the long-form sample in 5,143 ADAs. Of the 243 ADAs without sampled households, 235 consisted solely of out-of-scope households. The other eight ADAs had only a handful of in-scope households, and none of them were selected.

ADAs satisfy the following delineation criteria:

1. ADAs cover the entire country and, where possible, have a population count of 5,000 to 15,000 (based on the population counts from the previous census).
2. ADAs respect provincial and territorial borders, as well as the boundaries of census divisions (CDs), census metropolitan areas (CMAs) and census agglomerations (CAs) subdivided into census tracts (CTs) in effect for the 2016 Census.
3. ADAs are based on one of three 2016 Census dissemination geographic areas: dissemination areas (DAs), census subdivisions (CSDs) or census tracts (CTs):
• Within CMAs and CAs with CTs, adjacent CTs are combined to meet the ADA population criterion.
• In areas without CTs (areas outside CMAs and the largest CAs) where CSDs have a population of fewer than 15,000, adjacent CSDs are combined to meet the ADA population criterion.
• In areas without CTs where CSDs have a population of over 15,000, adjacent DAs are combined within these CSDs to meet the ADA population criterion.
4. Every CSD that consists of an Indian reserve and a small number of other areas where the canvasser method is required constitute distinct ADAs.

"For more information about Aggregate Dissemination Areas, refer to the Dictionary, Census of Population, 2016, Catalogue no. 98-301-X."

Table 4.2.1.1 shows the degree to which ADAs with households in the long-form sample were properly adjusted to CSDs. The first scenario occurred in most cases, since ADAs were designed above all to respect the boundaries of CTs and CSDs. Scenario 4 is the only one where CSD boundaries were not respected. CTs were not included in the table because they were all in the first scenario except one, which was in scenario 3.

Table 4.2.1.1
Number of census subdivisions within the boundaries of ADAs with households in the long-form sample, 2016 Census
Table summary
This table displays the results of Number of census subdivisions within the boundaries of aggregate dissemination areas with households in the long-form sample. The information is grouped by Scenario (appearing as row headers), Description and Census subdivision, calculated using number and percent units of measure (appearing as column headers).
Scenario Description Census subdivision
number percent
1 The CSD was small enough to be fully contained in an ADA, and this ADA only had complete CSDs. No CSDs in the ADA were part of another ADA. 4,512 92.40
2 The CSD was small enough to be fully contained in an ADA, but another CSD in the same ADA was part of a different ADA. 81 1.66
3 The CSD was large enough to contain full ADAs. No ADAs were also part of another CSD. 261 5.35
4 The CSD was part of two or more ADAs. 29 0.59
Total 4,883 100.00

Table 4.2.1.2 shows the distribution of ADAs with households in the long-form sample by province or territory.

Table 4.2.1.2
Number of ADAs with households in the long-form sample, by province or territory
Table summary
This table displays the results of Number of aggregate dissemination areas with households in the long-form sample. The information is grouped by Region (appearing as row headers), Number of aggregate dissemination areas (appearing as column headers).
Prince Edward Island 21
Nova Scotia 146
New Brunswick 124
Quebec 1,118
Ontario 1,655
Manitoba 216
Alberta 516
British Columbia 916
Yukon 28
Northwest Territories 40
Nunavut 26

Table 4.2.1.3 shows the number of ADAs by the number of in-scope households in the census. The majority of ADAs with households in the long-form sample had from 2,000 to 4,999 households. A considerable number of ADAs had small populations.

Table 4.2.1.3
Distribution of ADAs with households in the long-form sample, by number of in-scope households
Table summary
This table displays the results of Distribution of aggregate dissemination areas with households in the long-form sample. The information is grouped by In-scope households (appearing as row headers), Number of aggregate dissemination areas and Percent (appearing as column headers).
In-scope households Number of ADAs Percent
0 to 499 976 18.98
500 to 999 117 2.27
1,000 to 1,999 366 7.12
2,000 to 2,999 1,339 26.04
3,000 to 3,999 1,229 23.90
4,000 to 4,999 664 12.91
5,000 to 5,999 300 5.83
6,000 to 6,999 98 1.91
7,000 to 7,999 32 0.62
8,000 to 8,999 14 0.27
9,000 to 9,999 3 0.06
10,000+ 5 0.10
Total 5,143 100.00

Table 4.2.1.4 presents the number of ADAs by range of numbers of households that responded to the 2016 Census long-form questionnaire. For the ADAs with the fewest respondents, a specific type of processing was applied to have enough households for weighting purposes (see section 4.5). Overall, ADAs had more respondents than the weighting areas used in 2011.

Table 4.2.1.4
Distribution of ADAs with households in the long-form sample, by number of respondent households for the long-form questionnaire
Table summary
This table displays the results of Distribution of aggregate dissemination areas with households in the long-form sample, by number of respondent households for the long-form questionnaire. The information is grouped by Respondent households (appearing as row headers), Number of aggregate dissemination areas and Percent (appearing as column headers).
Respondent households Number of ADAs Percent
0 to 99 605 11.76
100 to 199 265 5.15
200 to 299 158 3.07
300 to 399 142 2.76
400 to 499 311 6.05
500 to 599 537 10.44
600 to 699 628 12.21
700 to 799 604 11.74
800 to 899 518 10.07
900 to 999 400 7.78
1,000 to 1,099 304 5.91
1,100 to 1,199 229 4.45
1,200 to 1,299 162 3.15
1,300 to 1,399 101 1.96
1,400 to 1,499 73 1.42
1,500+ 106 2.06
Total 5,143 100.00

### 4.2.2 Super aggregate dissemination areas

SADAs were created specifically for weighting 2016 Census data, so that certain weighting procedures for which a large number of observations is desirable could be conducted.

SADAs were created according to the following rules (in order of priority):

2. SADAs respect provincial and territorial borders (mandatory).
3. SADAs have a population of 50,000 to 150,000 persons (except for CDs with a population of 40,000 to 50,000 persons that constitute their own SADA.
4. SADA population excluding persons living in canvasser CUs).
5. SADAs respect the boundaries of CDs.
6. SADAs respect the boundaries of CMAs and CAs.
7. SADAs respect the boundaries of CSDs.
8. SADAs are single contiguous entities.
9. SADA are as compact as possible.

The first two rules were mandatory, and rules 3 to 9 were followed where possible. A total of 409 SADAs were created.

Table 4.2.2.1 shows the distribution of SADAs by province or territory.

Table 4.2.2.1
Number of SADAs, by province or territory
Table summary
This table displays the results of Number of super aggregate dissemination areas, by province or territory. The information is grouped by Region (appearing as row headers), Number of super aggregate dissemination areas (appearing as column headers).
Prince Edward Island 2
Nova Scotia 13
New Brunswick 8
Quebec 97
Ontario 150
Manitoba 15
Alberta 44
British Columbia 55
Yukon 1
Northwest Territories 1
Nunavut 1

Table 4.2.2.2 shows the degree to which SADAs were properly adjusted to CDs and CMAs. SADAs respected the boundaries of the majority of CDs (scenarios 1 and 3) and the boundaries of three-quarters of CMAs. The other CMAs were part of at least two SADAs (scenario 4).

Table 4.2.2.2
Number of census divisions and census metropolitan areas within SADA boundaries, 2016 Census
Table summary
This table displays the results of Number of census divisions and census metropolitan areas within super aggregate dissemination area boundaries, 2016 Census. The information is grouped by Scenario (appearing as row headers), Description, Census divisions and Census metropolitan areas, calculated using number and percent units of measure (appearing as column headers).
Scenario Description Census divisions Census metropolitan areas
number percent number percent
1 The CD or CMA was small enough to be fully contained within a SADA, and the SADA included only complete CDs or CMAs. No CDs or CMAs in the SADA were part of another SADA. 249 84.98 2 5.71
2 The CD or CMA was small enough to be fully contained within a SADA, but another CD or CMA in the same SADA was also part of another SADA. 2 0.68 0 0.00
3 The CD or CMA was large enough to contain complete SADAs. No SADAs were also part of another CD or CMA. 40 13.65 25 71.43
4 The CD or CMA was part of two or more SADAs. 2 0.68 8 22.86
Total 293 100.00 35 100.00

Table 4.2.2.3 shows the number of SADAs by the number of in-scope persons.

Table 4.2.2.3
Distribution of SADAs with households in the long-form sample, by number of in-scope individuals
Table summary
This table displays the results of Distribution of super aggregate dissemination areas with households in the long-form sample, by number of in-scope individuals. The information is grouped by In-scope individuals (appearing as row headers), Number of super aggregate dissemination areas and Percent (appearing as column headers).
In-scope individuals Number of SADAs Percent
30,000 to 39,999 2 0.49
40,000 to 49,999 26 6.36
50,000 to 59,999 23 5.62
60,000 to 69,999 45 11.00
70,000 to 79,999 106 25.92
80,000 to 89,999 67 16.38
90,000 to 99,999 46 11.25
100,000 to 149,999 94 22.98
Total 409 100.00

## 4.3 Design weights

The design weight for each household in the long-form sample was calculated differently, depending on the collection method of the CU where the corresponding dwelling was located.

If the method of collection was:

• mail-out, the design weight was equal to the inverse of the survey fraction, giving a weight of 4
• list/leave, the design weight was equal to the ratio of the number of private dwellings enumerated to the number of private dwellings sampled in the CU, and this gave a weight of approximately 4 for 98% of the selected households, although the weights varied from 1 to 7
• canvasser, the design weight was 1.

Households living in private dwellings attached to collective dwellings were an exception to the rule. As mentioned in section 2.2, all of these households were included in the sample. They were considered take-all, so their design weight was 1.

### 4.3.1 Weights for households counted in the sample

Sampled households with a design weight of 1 did not have their weight adjusted. These households kept their weight of 1 after the weighting procedures were completed (coverage and non-response, as well as calibration to census counts). They either were located in canvasser CUs or were private households that were attached to a collective dwelling.

Total non-response and partial non-response for these households were addressed by imputation. Once the missing data were imputed, these households were considered to be respondents for estimation purposes (although they were considered to be non-respondents for the calculation of response rates in section 3.11).

## 4.4 Coverage and total non-response adjustment

The several ways of treating non-response in surveys can be divided into two main categories: imputation and reweighting. The former is usually applied for the treatment of item missing values and the latter for the treatment of total non-response. A household was considered to be a respondent to the long form when it answered at least one of the long-form questions. With the high response rate to the long form, any non-response adjustment method would have had, for the most part, only a modest impact on the final survey weights and estimates. Coverage and total non-response for households in Indian reserve and canvasser enumeration CUs were compensated for with imputation procedures and, for the most part, with whole household imputation (WHI) as described in section 3.6. In the rest of the country, reweighting procedures were used. The rest of this chapter describes those weighting procedures.

The main purpose of coverage and non-response adjustments is to minimize the impact of any potential biases from lack of complete coverage (or from duplicates) and from unit non-response. For the adjustment to actually reduce the potential bias, a rich set of information about the non-respondents is very useful. Otherwise, the non-response adjustment that can be applied is limited, and the potential bias will not be greatly lessened. Geographical information was known for every non-responding household and long-form sample non-respondent (i.e., respondents in the long-form sample who answered the short-form questions but not the long-form questions). The information on non-respondents was thus somewhat limited. Fortunately, before the coverage and non-response adjustments, the process of WHI occurred. An important part of WHI is to impute the short-form characteristics for all non-respondents to the short form. This included long-form sample non-respondents. This additional information served as the basis for the long-form sample non-response adjustment.

The method used to adjust for coverage and total non-response in the long-form sample was a reweighting calibration-based procedure applied to the design weights. The procedure can be divided into the following main steps:

1. selection of calibration constraints for steps 2 and 3
3. estimation of a non-response propensity based on non-linear calibration for non-response
4. application of a score method based on the propensity of step 3.

The first step consisted of a forward selection of calibration constraints in the SADA. It was performed as follows:

• The set of potential constraints was derived from the variables common to both the short form and the long form, as well as from some administrative data obtained with record linkage strategies (where all units of the long-form population undergo the linkage procedures). The requirements of the non-linear calibration method used in the second and third steps meant that only constraints at the SADA level, and the number of households and persons in each ADA of the SADA, were considered.
• In each SADA, two mandatory constraints were selected first: the number of households in the SADA (TOTHHLD) and the number of persons in the SADA (TOTPERS).
• All other potential SADA constraints were evaluated; priority was given to the ones that split the SADA population as closely as possible into halves.

The selection process excluded constraints that occurred in fewer than 250 households in the SADA and constraints that were redundant or almost redundant in terms of collinearity with those constraints or with constraints already selected. Constraints that were redundant with constraints already selected were excluded since they did not add any new information. Given those filters, the order of priority used in the evaluation of constraints ensured that the constraints selected complemented each other and corrected for any potential coverage differential between the long form and the short form, as well as for census total non-response.

The second step applied a coverage non-linear calibration adjustment to the whole sample in the SADA (i.e., respondents and non-respondents). The long-form sample weighted counts, for the constraints selected in the first step, were made to coincide with the corresponding population counts. The purpose of this step was to correct for any potential coverage differential between the long-form sample and its complement (i.e., the set of households receiving only the short form). One way in which overcoverage can occur is if some individuals are counted in two different households. The coverage for the two populations could also be different if, for example, occupied dwellings were more likely to be incorrectly treated as unoccupied dwellings for the long form than for the short form. Another objective of this step was to isolate as much as possible the sampling error. Without this step, the non-response calibration carried out in the next step would confound the non-response error with the sampling error. This step makes the sample estimates coincide with the population estimates. In addition, the same control totals are used in both calibration procedures. As a result, the non-response propensity estimation done next does not have to correct (directly or indirectly) for the sampling error. Combining a correction for the sampling error and for the non-response error in the next step would have been inappropriate. The calibration procedure would have failed if the weight of any respondent was required to decrease to match the census counts, because the estimated propensity would have been greater than 1. Moreover, the score method applied in the last step required an estimate of the response propensity alone. To the extent that the variable of interest was related to the selected constraints, the sampling variance was also reduced by this step.

After these two steps, the main non-response adjustment took place. The weights, adjusted in the previous step, of non-respondents were set to 0 and the weights of respondents were increased so that the weighted sums in the SADA coincided with the corresponding population counts for the selected constraints. A logistic link function between the response propensity and the characteristics used in calibration enabled the implicit estimation of the response propensity. Folsom and Singh (2000) proposed this non-linear calibration method as a way of adjusting for non-response while ensuring that both estimates coincided with selected population counts and that the estimated response probabilities were between 0 and 1. This last condition does not necessarily hold when linear calibration is used for non-response adjustment. To the extent that the response propensity was related to the selected constraints, this step reduced the potential non-response bias without increasing the variance.

The inverse of the estimated response probabilities obtained in the previous step could be directly used to adjust the weights for non-response. However, the score method was used for the last step of the non-response adjustment to smooth the estimated probabilities from the previous step. This further ensured the quality of the non-response adjustment and avoided too extreme adjustments. For each ADA, homogeneous weighting classes were formed according to the estimated response probabilities. In each class, the weighted harmonic mean of the response probabilities was calculated. The harmonic mean was used because it is less affected by outliers in the estimated response probabilities. The inverse of this mean was applied to the weights of respondents in the class as the non-response adjustment.

In summary, the coverage and total non-response adjustment was a product of two quantities: the coverage adjustment and the inverse of the score-method harmonic mean.

## 4.5 Final calibration

Final calibration is a linear calibration that was done to minimize the sampling variability of estimates derived from long-form questionnaire responses, while ensuring consistency between estimated totals and Census of Population totals. This weighting step was necessary, since ensuring consistency between estimated totals and Census of Population totals was important for a large number of variables and geographic areas, i.e., satisfying calibration constraints.

Only the weights for households in mail-out or list/leave CUs were calibrated, since these households were sampled. Exceptions to this rule were households in these CUs that lived in a private dwelling attached to a collective dwelling. Since all these households were included in the long-form sample and all the long-form questionnaire responses for these households were imputed, no calibration was done. The final weights for these households were therefore equal to 1. The weights produced by the calibration process were the final weights used to calculate the long-form estimates, and these weights applied to households as well as families and persons. In other words, all families and persons from the same household received the household weight. For this final adjustment, the variability of the calibrated weights needed to be limited to avoid having an excessive portion of the weight applied to a single household or person. Therefore, weights were constrained to range from 1 to 20.

Calibration constraints were defined at the person, household and census family levels. In 2016, the notion of constraints was also expanded by the addition of two levels to the hierarchy of geographic units, i.e., ADAs and SADAs. These two levels were added to maximize the overall consistency between estimated totals and Census of Population totals, while minimizing the number of calibration constraints. This should help to reduce the variability of estimates. Appendix C lists all of the ADA and SADA constraints that were taken into consideration during the calibration process. Characteristics available from the census, administrative sources and the long-form questionnaire and for which consistency was attempted included age, sex, marital status, common-law status, household size, dwelling type and official language spoken.

The first step in the process to select calibration constraints was to categorize each of the constraints into one of three groups:

Mandatory constraints: These constraints had to be used in the calibration because the census counts had to agree with the long-form estimates at the geographic levels that are usual aggregates of ADAs and SADAs (e.g., Canada, provinces and territories). The number of persons and the number of households in the ADAs and SADAs were the two mandatory constraints.

Low-response constraints: Constraints evaluated for a population of 200 or fewer households were not used in the calibration because they can make survey estimates unstable.

All other constraints: These constraints were examined further to see whether they should be used in the calibration.

The second step was to determine which constraints from the third group should be used in the calibration process, in addition to the mandatory constraints. The constraints from the third group were added one by one, by repeatedly choosing the constraint that divided the population of the SADA or ADA in two as evenly as possible. Constraints that were too linearly dependent were excluded. To avoid introducing a bias in the point estimates and to avoid increasing their variance, the number of selected constraints was limited. Evaluations determined that this number had to be smaller than the square root of the number of respondent households involved in the constraint.

After the calibration constraints to be used were selected, a final edit was done to check whether the set of constraints chosen at the ADA and SADA levels was free of collinearity.

The calibration itself was then carried out for the final set of constraints from the second step. The weights adjusted for coverage and non-response were modified as little as possible, so that the weighted estimates would be equal to census totals for these constraints. Statistics Canada's Generalized Estimation System (GES) was used to carry out the calibration.

Sample estimates can differ from census counts for a few reasons, particularly for small areas, even after the calibration step. A few of these reasons are given below.

• Constraints excluded during the constraint selection process: As described above, possible constraints could be excluded for having low counts, for being linearly dependent (or overly dependent) on other chosen constraints or for being linearly dependent (or overly dependent) on low response constraints. This led to some differences between census counts and long-form estimates for these variables when a perfect linear dependency with the chosen constraints was not present.
• Sub-weighting area: In 2016, the ADA was the smallest weighting area for which agreement was attempted between the census counts and the long-form estimates. Any entity smaller than an ADA, such as the majority of DAs, is referred to as a sub-weighting area. These sub-weighting areas could have discrepancies between the census counts and the long-form estimates.

## 4.6 Details on the selection of constraintsNote 1

Constraints were selected twice during the weighting process: during the coverage and non-response adjustment, which requires the use of non-linear calibration techniques, and during the final calibration. The variables making up the constraints were essentially the same, but the inclusion or exclusion of constraints varied between the two weighting steps, since their respective objectives were different. Basically, constraints were not selected using the exact same criteria, and the weighting areas varied depending on the weighting step.

### 4.6.1 Process for the coverage and non-response adjustment

The coverage and non-response adjustment procedure uses calibration to adjust the survey weights. The rationale is that, if estimates based on respondents match as much as possible the census counts for auxiliary variables, then the non-response bias of estimates associated to those variables will be reduced. See section 4.4 for more information on the coverage and non-response adjustment.

A constraint that is excluded frequently usually has a larger difference between the census count and the non-response adjusted sample estimate than a constraint that is used more often. This can be seen by comparing Appendix C with Table 5.2.1. Appendix C lists all the potential variables or constraints, the number of times each constraint was used for calibration and the number of times that constraints were excluded for one of the reasons enumerated in Table 4.6.3.1. A constraint was excluded from the coverage and non-response adjustment under five criteria: "No population," "Small population," "Linearly dependent," "High collinearity" and "Explanatory redundancy." A constraint may have been excluded from calibration for one of the reasons above and yet been calibrated at the end of the process. This occurs for example when the constraint is collinear with the selected constraints. In this case, Appendix C shows that constraint as "Calibrated."

### 4.6.2 Procedures for the final calibration adjustment

The purpose of the final calibration is to adjust the household weights so that the long-form estimates are as close as possible to the census counts for many common characteristics. In addition to producing agreement between the estimates, an appropriate choice of constraints reduces variance. Appendix C provides the complete list of possible constraints, and section 4.5 describes the calibration performed on the long-form estimates. The criteria applied to the selection of constraints are similar to those applied to the selection of constraints for the coverage and non-response adjustment, with a few differences as presented in Table 4.6.3.1.

Calibration was performed simultaneously for SADAs and calibration ADAs. In each SADA, calibration constraints were defined at the SADA and ADA levels. All constraints were evaluated in each SADA and excluded only if necessary. The total number of persons (TOTPERS) and the total number of households (TOTHHLD) were the only two mandatory constraints. This meant that they could not be excluded for any of the ADAs.

Statistics Canada's GES was sent a total of 132,777 preselected constraints at the national level, so that it could perform the final linear calibration. This represents an average of 27 constraints per ADA and an average of 31 constraints per SADA. The mandatory constraints were selected in all the weighting areas. The process selected income constraints the most often, particularly the household income constraint and the low-income household constraint. The constraints selected the least often were primarily the year of immigration and the country of origin.

### 4.6.3 Comparison of the procedures for the two adjustment and selection steps

Criteria were applied in the selection of constraints at each weighting step. These criteria are indicated in Table 4.6.3.1 by weighting step.

Table 4.6.3.1
Criteria applied in selecting coverage, non-response and final calibration adjustment constraints
Table summary
This table displays the results of Criteria applied in selecting coverage. The information is grouped by Criteria (appearing as row headers), Adjustment for coverage and non-response and Final calibration (appearing as column headers).
Criteria Adjustment for coverage and non-response Final calibration
No population according to the census counts: If the constraint had no population in the weighting area, then the estimate after adjustment must also be 0 for that constraint. These constraints are not classified as excluded but rather as ineligible to the adjustment process. Applied at the SADA/ADA level. Applied at the SADA/ADA level.
Small population according to the census counts: If a constraint involves less than a certain number of households in the population of the weighting area, then it is considered small and the constraint is excluded. Including such a constraint would unduly increase the variance. However, constraints with small population can be implicitly calibrated and in this case are included in the total number of calibrated constraints. Applied at the SADA/ADA level. The number of households in the population is larger than 0 but less than 250 in the weighted area. Applied at the SADA/ADA level. The number of households in the population is more than 0 but less than 200 in the weighted area.
Linearly dependent: If the value of a constraint can be calculated by combining the values of other constraints, one of these constraints is not necessary and must be deleted during the adjustment process because of its linear dependency. However, constraints that are excluded because of their linear dependency are implicitly calibrated. They are therefore included in the total number of calibrated constraints.  Applied at the SADA level. The selection of constraints can be compared to the selection of explanatory variables in a linear regression. The variance inflation factorTable 4-6-3-1 Note 1 (VIF) and the condition numberTable 4-6-3-1 Note 2 are thus used to detect high collinearity. Applied at the SADA/ADA level. Two dependency checks are conducted to identify linearly dependent constraints. The first check is done when the constraints at the SADA/ADA level are selected, and the second check includes all the constraints chosen at both levels of the geographic  hierarchy (SADAs and ADAs).
High collinearity: If a constraint value can be almost calculated by the combination of other constraint values, then at least one of those constraints must be avoided in the adjustment process. Such a constraint is not perfectly calibrated. Applied at the SADA level. The selection of constraints can be compared to the selection of explanatory variables in a linear regression. The variance inflation factorTable 4-6-3-1 Note 1 (VIF) and the condition numberTable 4-6-3-1 Note 2 are thus used to detect high collinearity. Applied at the SADA/ADA level. Two linear dependency checks are conducted to identify constraints that are close to being linearly dependent. The first check is done when the constraints at the SADA level and the ADA level are selected, and the second check includes all the constraints chosen at both levels of the hierarchy simultaneously (SADAs and ADAs).
Explanatory redundancy: If a constraint explains the non-response (almost) as well as other constraint(s) already selected, then the non-response calibration procedure would fail. This is equivalent to saying that if a constraint does not add any information about the non-response mechanism, beyond what is explained by the already-selected constraints, then it should not be included. Applied at the SADA. A sequential procedure is applied (a form of logistic regression) to test the convergence of the logistic regression. N/A

Appendix C indicates the status of each constraint selected in at least one of the geographic areas once the selection of constraints was carried out at the weighting step. The geography column indicates the geographic level to which the constraint was applied. When a constraint was applied to both geographic levels, the totals reflect both levels with no distinction. A constraint could have been excluded from the calibration process and still have been calibrated. In that case, the constraint was considered to be calibrated.

### 4.6.4 Analysis of calibration during the coverage and non-response adjustment

This section summarizes the number of constraints selected and excluded. The reasons for not selecting constraints are also summarized. Additionally, the section sheds some light on the reasons why some constraints are frequently excluded.

Persons born in certain places tend to be more concentrated in certain parts of the country, to the point that many SADAs have little or no population with a given place of birth. As a result, the constraints for place of birth were often not selected because they had small census counts. Similarly, the constraint involving the French official language (OLN_FR) has little or no population in certain regions of the country. Consequently, it was often excluded.

The constraints most often excluded because of high collinearity were:

• "Children in a census family" (CHILD)
• "Census families with children" (CHILDFAM)
• "One-person households" (HHSIZE1)
• "Female" (FEMALE)
• "Male" (MALE)
• "Census families without children" (NOCLDFAM)
• "Females aged 14 years and younger" (FEMALELT15)
• "Persons in an economic family" (INEFAM)
• "Persons in a household that are not part of an economic family" (NOINEFAM)
• "Males aged 14 years and younger" (MALELT15)
• "Persons aged 10 to 14 years" (AGE14)
• "Five-or-more-person households" (HHSIZEGE5)
• "Persons not in a census family" (NOTINFAM)
• "Persons aged 5 to 9 years" (AGE9)
• "Persons aged 0 to 4 years" (AGE4)
• "Six-or-more-person households" (HHSIZEGE6)
• "Persons in a couple (spouse, partner)" (COUPLE)
• "Two-person households" (HHSIZE2).

The procedure excluded these constraints automatically, since they could be determined very accurately from a combination of the mandatory constraints, "Households" (TOTHHLD) and "Persons" (TOTPERS), together with other constraints that were selected often, such as some constraints for age, marital status, household size, sex by age and persons in census family. These constraints might also have been explained too well by a combination of the variables selected and of the small constraints.

The actual differences between the census counts and the non-response adjusted estimates are examined in section 5.2.

Table 4.6.4.1 shows the number of times that each reason for dropping or removing a SADA-level constraint occurred. The total number of constraints excluded is the sum of the "Small population," "High collinearity" and "Explanatory redundancy" categories. The "No population" category is not included in the total because it does not actually represent excluded constraints. The average number of constraints excluded per SADA is the total for each category divided by 408, the number of SADAs where coverage and non-response adjustments were done.

Table 4.6.4.1
Table summary
This table displays the results of Summary statistics on super aggregate dissemination area-level constraints in 2016 coverage and non-response adjustment . The information is grouped by Constraint (appearing as row headers), Calibrated, No population, Excluded and Total excluded (appearing as column headers).
Constraint Calibrated No population Excluded Total excluded
Small population High collinearity Explanatory redundancy
Total constraints 30,328 2,865 29,742 11,550 49 41,341
Average number of constraints per SADA 74.3 7.0 72.9 28.3 0.1 101.3

On average, 74.3 SADA-level constraints were calibrated per SADA. An average of 72.9 constraints were discarded per SADA because of small population and 28.3 because of high collinearity.

### 4.6.5 Analysis of the final calibration

Other than cases where the population was nil, each time that a constraint was excluded, the calibration process did not attempt to make census counts and long-form estimates agree for that constraint in that weighting area. The gap between the census count and the long-form estimate was usually larger for a constraint that was excluded frequently than for a constraint that was excluded less often.

Table 4.6.5.1 shows how often a constraint was removed or excluded at the weighting area level according to each criterion. The total number of constraints excluded is equal to the sum of the values for the various removal criteria. The average number of constraints excluded per weighting area is simply equal to the total for the category divided by the number of weighting areas.

Table 4.6.5.1
Summary statistics on constraint selection status at the weighted area level in the final weight adjustment, 2016
Table summary
This table displays the results of Summary statistics on constraint selection status at the weighted area level in the final weight adjustment. The information is grouped by Constraints (appearing as row headers), Weighted area, Calibrated, No population, Excluded, Total excluded and Number of weighted areas (appearing as column headers).
Constraints Weighted area Calibrated No population Excluded Total excluded Number of weighted areas
Small
population
High collinearity
Number of constraints ADA 210,307 112,442 283,947 99,319 383,266 4,180
SADA 39,262 2,698 27,292 10,588 37,880 408