Appendix A: List of regions and their census divisions
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New Brunswick
Quebec
Ontario
Methodological note 1
The information presented in this report relies on the data provided by the 2B Census questionnaire (long form) administered to one out of five Canadian households. It is a large sample, of about 20% of the Canadian population (6.2 million people).
Sampling is an integral part of census-taking. Its use can lead to substantial reductions in costs and respondent burden associated with a census, or alternatively, can allow the scope of a census to be broadened at the same cost. The price paid for these advantages is the introduction of sampling error to census figures that are based on the sample. The effect of sampling is most important for small census figures, whether they are counts for rare categories at the national or provincial level or counts for categories in small geographic areas. Within the current study, specific provinces, territories and sub-provincial areas with low numbers of health care professionals whose first official language spoken is a minority language or who use the minority language at work do not generally provide statistically reliable estimates of these professionals. Therefore, statistical tests have not been applied in the provinces, territories or sub-provincial areas with less than 400 health care professionals, where the total number of official-language minority health care professionals or those who know or use the official minority language at work is less than 20.
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. The primary criterion in the choice of an estimation procedure is minimization of the width of such intervals so that these statements about the unknown population values are as precise as possible. The usual measure of precision for comparing estimation procedures is known as the standard error. Provided that certain relatively soft conditions are met, intervals of plus or minus two standard errors from the estimate will contain the population value for approximately 95% of all possible samples.
Sampling Variance
Sampling error can be divided into two components: variance and bias. The variance measures the variability of the estimate about its average value in hypothetical repetitions of the survey process, while the bias is defined as the difference between the average value of the estimate in hypothetical repetitions and the true value being estimated. Even with a perfectly unbiased sampling method, the results would still be subject to variance, simply because the estimates are based only on a sample. The variance may be estimated using the data collected by the sample survey. The Sampling Variance Study was carried out to estimate the effect of the sampling and estimation procedures on those census figures that are based on sample data.
On the basis of the 2B sample data, thousands of tables are produced by Statistics Canada. Conceptually, a measurement of precision, the estimated sampling variance, can be associated with every estimate calculated in these tables. This measurement takes into account both the sample design and the estimation method. In practice, however, it cannot be calculated for every census estimate because of high data processing costs. Sampling variance is thus estimated for only a subset of census estimates. From this, the combined effect of the sample design and the estimation method on the sampling variance can be estimated. Simple estimates of sampling variance, which are inexpensive to calculate, can then be adjusted for this impact to produce estimates of sampling variance for any census estimates.
The square root of the sampling variance is known as the standard error. The following formula may be used to calculate the non-adjusted standard errors (NASE) for any estimated total for an area of any size:
where NASE is the non-adjusted standard error, E is the estimated total and N is the total number of persons, households, dwellings or families in the area. For example, for an estimated total of 750 persons in an area with a total of 9,000 persons, the non-adjusted standard error would be:
The 2001 Census Technical Report provides adjustment factors by which the non adjusted standard errors should be multiplied to adjust for the combined effect of the sample design and the estimation procedure. To calculate these adjustment factors, sampling variance estimates were calculated for regression estimates for different categories of all of the characteristics given in Table 9.2 of the 2001 Census Technical Report. This was done for each sampled Weighting Area (WA). The provincial- and national-level sampling variance estimates were obtained by summing up the WA-level estimates. The adjustment factors were calculated for each characteristic in each category by dividing the square roots of these estimates by the non-adjusted standard errors. Adjustment factors were calculated at the provincial and national levels for each characteristic by averaging the adjustment factors for all of its categories. For further information, see the 2001 Census Technical Report available on the Statistics Canada web site.
To estimate the standard error for a given census sample estimate, the user should determine the adjustment factor, found in the 2001 Census Technical Report, that applies to the chosen characteristic and multiply this factor by the non adjusted standard error obtained. The Technical Report provides adjustment factors for a wide range of characteristics, both at the national and provincial levels, as well as for Weighting Areas.
Unless the area is smaller than a province, the "National or Provincial Factor" column should be selected. Adjustment factors are given for different provinces only where they differ significantly from those at the national level; this only occurred for some of the language characteristics. It should be noted that since no sampling occurred in the Northwest Territories or Nunavut, the adjustment factors for all characteristics in these territories should be zero. Since sampling was done in the Yukon, the "Other provinces" adjustment factor should be used, when available. If an adjustment factor is needed for a census estimate associated with an area smaller than a province, then the percentiles of WA-level factors will provide a more accurate value. The percentiles give the spread of all the adjustment factors calculated in the study at the WA level for the different category characteristics. N% of the adjustment factors at the WA level are below the Nth percentile and 100 - N% are above the Nth percentile. For example, 90% of the WA-level adjustment factors are below the 90th percentile and 10% are above it. The choice of which percentile to use will depend on how conservative a standard error estimate is being sought. For example, the 99th percentile would provide a very conservative estimate, while the 75th percentile would provide a somewhat less conservative one. 2
The following example illustrates how to calculate the adjusted standard errors. While referring to the numbers included in Table 2.1, suppose the estimate of interest is the number of persons speaking French as their first official language in Ontario. The 2006 estimate for this characteristic was 537,595. The 2006 Census count for the population of Ontario was 12,028,900. Since neither number is very close to any of the values given in Table 9.1 of the 2001 Census Technical Report, the formula given to calculate the non-adjusted standard error should be used. In this case the result would be 1,433.3. From Table 9.2, the Ontario (Other provinces) adjustment factor for the characteristic "first official language spoken = French" is 1.27. Consequently, the adjusted standard error for this estimate is 1,433.3 x 1.27 = 1,820.3. The variance of this estimate is calculated by squaring the value of the standard error, and is equal to 3,313,321.
The sample estimate and its standard error may be used to construct an interval within which the unknown population value is expected to be contained with a prescribed confidence. The particular sample selected in this survey is one of a large number of possible samples of the same size that could have been selected using the same sample design. Estimates derived from the different samples would differ from each other. If intervals from two standard errors below the estimate to two standard errors above the estimate were constructed using each of the possible estimates, then approximately 19 out of 20 such intervals would include the value normally obtained in a complete census. Such an interval is called a 95% (19 ÷ 20 = 95%) confidence interval. In order to guarantee 95% confidence however, these intervals must be calculated using the true standard errors of the sample estimates.
The adjusted standard errors calculated from tables 9.1 and 9.2 from the 2001 Census Technical Report are only estimates of the true standard errors. For provincial- and national-level sample estimates however, the adjusted standard errors should be close enough to the true standard errors so as to produce approximate 95% confidence intervals of reasonable precision. Below the provincial level, the adjusted standard errors may not be accurate enough for this purpose.
The following example sheds light on how it was determined, from Table 2.1 of the current document, that the difference in proportions between the total population of Ontario whose first official language spoken (4.5%) is French and the number of doctors having French as their first official language and practicing in Ontario (3.5%) is statistically significant.
Using the NASE formula, multiplied by the adjustment factor as described above and then squared (as to obtain the variance of the estimate), one can calculate the variance of the total population having French as their first official language spoken (FOLS) in Ontario (3,313,321). Likewise, the variance of the proportion represented by this population (Pa) results from dividing the variance of the total French-speaking population (3,313,321) by the square of the total population in Ontario (12,028,900). Therefore, the variance of the proportion represented by the French-speaking population is equal to 2.3 x 10-8.
Upon examining the proportion represented by doctors who live in Ontario and whose first official language is French (3.5%), the same process can be applied. Hence, using the same NASE formula multiplied by the adjustment factor of 1.27, after squaring it (to obtain the variance of the estimate), the variance of the number of doctors whose first official spoken language is French comes to 3,451. Likewise, the variance of the proportion represented by this professional group (Pb) results from the operation dividing the variance (3,451) by the squared total number of doctors in the province (15,225 doctors), that is 1.0 x 10-5.
The standard error is then calculated, to determine the interval between the proportion of the French-speaking population (Pa) and that of the French-speaking doctors (Pb). This standard error corresponds to the square root of both variances (Pa and Pb) once added together. Finally, a t-Student test is applied, that yields the following results: (Pa-Pb)/Standard Error (Pa-Pb) = 2.474.
Since the absolute value of 2.474 is superior to two standard errors (1.96), it can be ascertained that the interval between the two proportions is significant at the alpha level of 0.05.
It should be noted that when two characteristics are compared (for instance, the proportion of French-speaking doctors versus the proportion of doctors who use French at work), the highest factor of adjustment for these characteristics must be used.
The tables published in this report do not contain the confidence intervals associated with the total counts. It is therefore possible, for instance, that an equal interval between the proportion or the total count of French-speaking individuals (Pa) and the proportion or the total count of the French-speaking doctors (Pb) in two different geographic areas, would lead in one case to a statistically significant interval, whereas in the other case, the interval would not be statistically significant.
Glossary
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