Alternative estimates of the volume of emigration from Canada, 1961-71 *

JOHN J . KELLY /Statistics Canadat

L’apport de I’immigration a I’accroissement de la population au Canada a souvent fait I’objet d’une attention toute particuliere. Cependant, on est beaucoup moins bien renseigne sur I’impact qu’a pu avoir I’emigration, puisqu’il n’existe aucune donnee sur le nombre reel de personnes quittant le Canada. Cet expose presente d’abord trois evaluations du nombre d’emigrants du Canada au cours de la periode intercensitaire 1961-71, obtenues a partir de trois methodes differentes, et on tente de determiner quelle evaluation est la plus fiable. On montre que le processus d’elaboration des politiques canadiennes d’immigration se doit de tenir compte de I’emigration.

The contribution of immigration to Canada’s population growth has received considerable analytical attention. However, much less is known about the impact which emigration has had on the nation’s population growth, particularly since actual data are not available on the number of persons emigrating from Canada. This paper presents three different emigration estimates for Canada for the 1961-71 intercensal period derived from different estimation procedures, endeavours to determine which estimate is the most reliable one, and demonstrates the importance of considering emigration in the continuing reformulation of Canada’s immigration policies.

During the past few years, researchers and the general public in Canada have expressed a marked increase in interest in the nation’s past and projected future population growth. This increase in interest is partly due to the United Nations World Population Conference which was held in Bucharest in 1974, and partly due to the publication of the Green Paper on Immigra- tion and Population in February 1975 by Cana- da’s Department of Manpower and Immigra- tion. Historically, much attention has been de- voted to the contributions made by natural in- crease and immigration to population change in Canada, but emigration from Canada has re- ceived comparatively little attention from re- searchers.

How many people emigrate from Canada each year? Although a specific answer to this question is required to permit one to accurately assess the relative importance of each of the four components of Canada’s population change, few researchers have endeavoured to answer this question. Like most other countries in the world, Canada has scanty data on the number of people who emigrate each year to reside in other nations. The major reason for this lack of statistics is the fact that emigrating residents are not required to notify government officials of their departure from the country. Nevertheless, despite the absence of actual data on emigration, it is possible to generate estimates of the volume of emigration, and in

* 1 would like to express my gratitude to my colleagues in the Population Estimates and Projections Division of Statistics Canada for their helpful comments and suggestions on earlier versions of this paper. I am particularly indebted to Alex McMiUan for the assistance he provided me on statistical techniques, and to Anatole Romaniuk for his comments on the conceptual and methodological portions of the paper. Of course, 1 assume full responsibility for the paper’s content. t The views expressed in this paper are personal and are not to be construed as reflecting those of Statistics Canada.

Rev. canad. SOC. &Anth./Canad. Rev. SOC. & Anth. 14(1)1977

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Alternative estimates of the volume of emigration / 59

the remainder of this paper several different emigration estimates will be presented for the 1961-71 intercensal period.

I N T E R C E N S A L E S T I M A T E S O F E M I G R A T I O N

The most common method of generating an es- timate of emigration is by the residual method. Basically, this method involves (i) computing the total population change which occurred be- tween two successive censuses, (ii) determining how much of that total population change can be accounted for by natural increase and immi- gration, and (iii) obtaining through the arith- metical difference of these two items the ‘resid- ual,’ which is often referred to as the volume of emigration for the intercensal years. The esti- mates of emigration obtained in this fashion by the residual method are presented below in Table I for 1961-6, 1966-71, and 1961-71. Con- sequently, researchers who have used this first type of estimation procedure - which in this paper will be referred to as Residual Method ‘A’

-have concluded that an estimated 706, I 12 per- sons emigrated from Canada during the I June 1961 to 31 May 1971 intercensal period.

Although many researchers have utilized Re- sidual Method ‘A’ to generate intercensal esti- mates of emigration (for example, refer to George, 1 9 7 4 : 5 ) , it is regrettable that relatively few have explicitly concerned themselves with the limitations of this estimation device, par- ticularly in the actual calculation of the emigra- tion residual. When one considers the manner in which the emigration residual is calculated through this method, it becomes readily appa- rent that the ‘residual’ is not actually an esti- mate of emigration at all, but rather a mixture of several different things. In fact, the magnitude and accuracy of the residual is affected not only by the reliability of the available statistics on births, deaths, and immigration, but also by the reliability of the population size as determined by the two successive censuses. Consequently, before proceeding to calculate alternative esti- mates of emigration by a variant of Residual Method ‘A, ’ it is advisable to comment on the reliability of data on births, deaths, immigra- tion, and census population counts within the framework of the residual method of emigration estimation.

National birth and death statistics for Canada can suffer from a number of deficiencies, such as incomplete registrations (especially for In- dians, Eskimos, and some immigrant groups) and late reporting. Although Canada’s national vital statistics contain errors because of such deficiencies, these errors are considered to be comparatively minor, particularly for the pur- poses for which the statistics on births and deaths are being utilized in this paper for the calculation of emigration residuals.

The statistics on immigration to Canada are somewhat more deficient than the statistics on births and deaths, but because the residual method employs census data in the calculation procedure, these deficiencies in the immigra- tion statistics are considered to have little influence on the reliability of the intercensal emigration residuals. Canada’s immigration statistics can be considered deficient for a number of reasons. For example, the Depart- ment of Manpower and Immigration adminis- tratively defines an immigrant as an alien who is granted landed immigrant status in agiven year, and consequently this definition excludes per- sons on student or employment visas temporar- ily admitted to Canada. Fortunately, persons in these categories, as temporary foreign residents of Canada, are also intentionally excluded from Canada’s censuses, and therefore individuals in these excluded categories have no real impact on the reliability of the emigration estimates derived by intercensal residual methods. Sec- ondly, Canadian citizens returning to Canada after having resided abroad are also excluded from the official immigration statistics, but de- pending on their dates of departure and return to Canada, may be included in one or more sets of the census data employed to calculate the emigration residuals. Unfortunately, data on re- turning residents who have resided abroad for one year or more and who returned to Canada to reside during the 1961-6 and 1966-71 intercen- sal periods are not available, and thus no objec- tive methods exist which would enable one to reliably measure either the total number of these return migrants, or, more importantly, that portion of them which effects the inter- censal emigration residual obtained.* Con- sequently, for the purposes of the present paper, it is assumed that this latter subgroup of returning residents is comparatively small in

I Within other frameworks, one might arrive at different conclusions about the reliability of these demographic items. z Of course, even if reliable data on return migration were available. one would not introduce adjustments into

60 / John J. Kelly

volume, and thus that it would have only a very minor influence on the quality of emigration estimates derived by the residual methods. A third factor in the immigration statistics should be noted, and this is the fact that the annual immigration statistics refer to the year in which an immigrant is granted landed status rather than to the actual year of entry to Canada. Al- though these two time reference points do not always coincide for all immigrants, this does not present a problem for the intercensal emigration residual estimation procedure because these persons, when entering Canada, would be clas- sified as temporary foreign residents until such time as they received landed status, and there- fore would not be included in the census counts until they actually had received landed status. Consequently, this measurement problem does not affect the emigration estimates generated herein by the intercensal residual methods. Fi- nally, the statistics on immigration can be deficient for some purposes because of incom- pleteness of coverage resulting from illegal im- migration, but since the vast majority of illegal immigrants would probably intentionally re- frain from completing federal census question- naires, they would be largely excluded from the census data utilized to compute the emigration residual, and therefore it has been concluded that illegal immigration would have little or no effect on the quality of the emigration estimates derived by the residual methods. Consequent- ly, although the immigration statistics can be considered deficient in some ways, these same deficiencies also tend to exist in the census data, and therefore are considered to have little or no impact on the quality of the intercensal emigra- tion residuals contained in this paper.

In addition to being affected by the reliability of the available statistics on births, deaths, and immigration, the magnitude of the residual cal- culated by Residual Method ‘A’ can also be affected by the differential amount of under- enumeration in the two censuses. Even if there are no errors in the published statistics on the

above three components of national population change, the intercensal residual would consti- tute a reasonably accurate approximation of the volume of intercensal emigration only if the two censuses had identical amounts of under-enumeration (us Bureau of the Census, 1973:594). Despite the fact that it is known that Canada’s three most recent censuses had dif- ferent estimated amounts of under-enumera- tion, most researchers have failed to take this crucial factor into account in the actual calcula- tion of estimates of emigration by the residual m e t h ~ d . ~ Accordingly, in this paper alternative estimates of emigration are presented which have been calculated by adjusting the census base data for the estimated amount of under- enumeration in each census (Residual Method ‘BO.

There are several different methods which are available for measuring the completeness of population coverage in national censuses, and the method which has been used by Statistics Canada for the 1961, I+, and 1971 censuses is the Reverse Record Check (RRC). Unfortu- nately, Canada employed the RRC to measure census coverage for the first time only in the 1961 Census, and consequently comparable data on census coverage are not available for previous censuses. Basically, the RRC proce- dure involves tracing a sample of persons from a group of different lists4 which, among them, should include every person who lives in Canada at the time of the census and who should be enumerated in the census. This sam- ple of persons is traced to their current address, and then census documents are examined to ascertain whether these persons were enumer- ated at the same address. If they were not enumerated there, they are contacted or traced to determine whether they were enumerated at a different address, or whether they were not enumerated in the census at all (Winkworth, 1973).

The estimated national population under- coverage rates (excluding Yukon and the

the residual estimation procedure for those who both emigrated and returned within the 1961-6 and 1966-71 intercensal periods. This is because these individuals would have been included in each of the two successive censuses, and therefore would have no effect on the quality of the emigration estimate generated by the residual methods. 3 One notable exception to the general practise of largely ignoring the problems of enumeration errors in migration estimation is a recent article by R. Marvin Mclnnis (1974). Refer also to George (1976:6), who also explicitly discusses the importance of census errors. 4 For example, for the 1971 Census this group of lists was comprised of all persons enumerated in the 1966 Census, persons missed in the 1966 Census, persons born during the 1966-71 intercensalperiod, and persons who immigrated to Canada during the 1966-71 intercensal period. These four groups constituted the four sampling frames in the Reverse Record Check for the 1971 Census.

Alternative estimates of the volume of emigration / 61

TABLE I1

ADJIJSTED POPULATION COUNTS €OR CANADA AT CENSUS DATES BASED ON ESTIMATES OF CENSUS UNDER- ENUMERATION, 1961, 1966, A N D 1971

Mean Estimare Standard estimate Mean

Census of under- error of of persons Estimated range aa’jusfed enumerated enumera- under-enum. missed in of persons missed population population tion* estimate* census in census? counts

1961 18,238,247 3.30% 0.50% 622,905 525,225-720,585 18,861,152 1966 20,014,880 2.62% 0.13% 538,532 5 I1,090-565,974 20,553,412 1971 21,568,310 1.93% 0.09% 424,477 404,292-444,662 21,992,787

* Winkworth, 1973; Muirhead, 1969; and Brackstone and Gosselin, 1974. Note that these estimates of census under-enumeration are for Canada, excluding the Yukon and Northwest Territories. t 68 per cent confidence interval.

Northwest Territories) which were obtained from the Reverse Record Check, with the cor- responding standard errors shown in brackets, were 3.30 per cent (k0.50 per cent) for the 1961 Census, 2.62 per cent (50.13 per cent) for the 1966 Census, and I .93 per cent (20.09 per cent) for the 1971 Census (Winkworth, 1973; Muir- head, 1969; Brackstone and Gosselin, 1974). These estimated population undercoverage rates were generated from Reverse Record Check sample sizes of approximately 6000 per- sons, 26,5oopersons, and 27,popersons for the evaluation of the 1961,1966, and 1971 Censuses respectively .

As can be seen from Table 11, one can obtain alternative national population counts for Canada’s three most recent census dates by taking into account the estimated amounts of population undercoverage in the censuses .5

This is done by equating the estimated rate of census coverage and its standard error to the mean and standard error of a Beta random vari- able and using the properties of a Beta variate to compute the mean and standard deviation of the reciprocal of the coverage rates.6 Since one can define the population corrected for census under-enumeration as the actual census count divided by the rate of census coverage, this procedure yields estimates of the mean and standard deviation of the ‘corrected’ popula- tion. Note that this corrected population is a

random variable (because of the variability in the estimated coverage rate). The estimated mean of this random variable is the census count multiplied by the estimated mean of the reciprocal of the coverage rate. In the interest of brevity, this estimate is referred to as the ad- justed population. These adjusted populations can then be employed to generate alternative sets of emigration residuals. This latter method of computing intercensal emigration residuals has been referred to as Residual Method ‘B,’ and the results which were obtained are pre- sented in Table I I I .

The comparison of the intercensal emigration residuals in Tables I and 111 reveals that the different amounts of under-enumeration in Canada’s censuses have a major effect on the estimates of intercensal emigration obtained by the residual method. For example, the estimate of intercensal emigration for the 1961-71 period obtained from Residual Method ‘A ’ which does not adjust the census population counts for under-enumeration is approximately 706,100; whereas that for the same period obtained from Residual Method ‘B’ in which the census popu- lation counts are adjusted for under-enu- meration is approximately 904,500, 599,744 at the 68 per cent confidence level.’ Thus, it can be concluded that the 1961-71 intercensal emi- gration residual generated from Residual Meth- od ‘B’ ranges from approximately 804,800 to

5 Unfortunately, there are no estimates of national population undercoverage for these censuses for the population of Canada including the Yukon and Northwest Territories, and consequently for purposes of this paper these adjustments were made for the entire population of Canada by using the national undercoverage rates for Canada excluding the Yukon and Northwest Territories. 6 An illustrative example of this procedure is provided in the appendix. 7 I t should be noted that all confidence levels in this paper are approximate (refer to the Appendix). At the 95 per cent confidence level, Residual Method ‘B’ generates an intercensal emigration residual for the 1961-7 1 period of 904.500 2 199.488.

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Alternative estimates of the volume of emigration / 63

I ,004,300 at the 68 per cent confidence level.s When the upper and lower limits of this range are compared with the unadjusted intercensal emigration residual in Table I for the same period, it is apparent that failure to adjust for census under-enumeration can result in an un- derstatement of the volume of emigration dur- ing this decade by as little as 98,700 and as much as 298,200.

The comparison of the data in the preceding tables indicates that researchers who attempt to measure intercensal emigration by the residual method should give more careful consideration to the effect of different amounts of under- enumeration in the two censuses employed in the calculation. Alternatively, if they elect to ignore the amount of under-enumeration in each census, they should explicitly caution readers of this fact to ensure that misleading conclusions are not reached. These cautionary remarks pertain not only to the question of in- tercensal emigration, but to the larger issues of the relative importance of each of the compo- nents of Canada’s population change as well. For example, researchers who follow the cus- tomary procedure of calculating the intercensal population change in the manner shown in Table I would conclude that between the 1961 and 1971 censuses, Canada’s population in- creased by 3,330,063, of which natural increase accounted for 78 per cent, and net international migration for 22 per cent. However, the data in Table 111 in which the census populations were adjusted for under-enumeration provide a diffe- rent impression of Canada’s population change, for they indicate that during the same period Canada’s population increased by 3,13 I ,635 (299,700), with natural increase accounting for 83 per cent (k2.5 per cent) and net migration accounting for only 17 per cent (k2.5 per cent) of this growth. Consequently, the data in Table 1 1 1 suggest that net international migration may actually play a less important role in Canada’s contemporary population growth than is usu- ally assumed. This finding could have definite policy implications for the federal govern- ment’s continuing discussion of the advisability of setting an annual target for the number of landed immigrants to be admitted to Canada in a given year, particularly since the recently ta- bled Special Joint Committee’s third report to Parliament concerning the proposed new im- migration policy stated that ‘a principal objec-

tive of the new policy should be the regulation of immigration flow to achieve desired popula- tion growth’ (Canada, 1975: paragraph 45). Thus, if such a target were quantified without taking into account the fact that emigration from Canada has been considerably higher than had previously been thought, the target might fail to have its intended relationship to Canada’s actual or desired population growth.

A N N U A L P O S T C E N S A L E S T I M A T E S O F

E M I G R A T I O N

In addition to the two different emigration esti- mates discussed above which were obtained by residual methods, the Population Estimates and Projections Division of Statistics Canada has produced annual emigration estimates for utili- zation in its production of postcensal national population estimates. Its annual emigration es- timates have been generated by adding to the actual annual data on the number of persons previously residing in Canada who migrated to the USA and the U K , an assumed volume of 20,000 persons per year who migrate from Canada to all other countries. The annual post- censal emigration estimates which were gener- ated from this procedure for the 1961-2 through 1970-1 period are shown in Table I V below where it can be seen that for the 1961-71 inter- censal period this procedure yielded an emigra- tion estimate of 694,700 persons.. Like the pre- vious two emigration estimates obtained by the residual methods, the postcensal emigration es- timate is subject to error. The most important source of error in this third intercensal emigra- tion estimate is that component of the estimate represented by the assumed volume of 20,000 persons per year who are believed to have emi- grated from Canada to countries other than the USA and the U K . Unfortunately, the absence of actual immigration data for these other coun- tries of persons previously residing in Canada makes it impossible to measure the reliability of this assumption.

The postcensal method of emigration estima- tion, then, provides one with a third estimate of emigration for the 1961-71 intercensal period. Although it is only approximately I I ,400 less than the emigration estimate generated by Re- sidual Method ‘A,’ it is approximately 209,800 (k99,800 at the 68 per cent confidence level) less than the emigration estimate generated by

8 At the 95 per cent confidence level, the 1961-71 emigration residual ranges from approximately 705,100 to 1,104,000.

64 / John J. Kelly

TABLE IV

ANNUAL POSTCENSAL ESTIMATES OF EMIGRATION FROM CANADA, 1961-2 TO 1970-1

Emigration from Canada to Estimated

June 1 - May 31 United States* United Kingdom? Other countriest emigration total

1961-2 44,400 9,500 20,000 73,900 1962-3 50,000 6,800 20,000 76,800 1963-4 5 1,200 5,900 20,000 77,100 1964-5 50,000 1 1 , 1 0 0 20,000 81,100 1965-6 39,300 9,400 20,000 68,700

19667 33,800 9,000 20,000 62,800 1967-8 41,200 1 1,700 20,000 72,900 1968-9 30,200 12,400 20,000 62,600

1970-1 22,900 15,700 20,000 58,600 1 966-7 1 155,300 61,800 100,000 317,100 1961-71 390,200 104,500 200,000 694,700

* Data obtained from the us Department of Justice, Immigration and Naturalization Service. t Data obtained from the UK Office of Population Censuses and Surveys, based on its International Passenger Survey. 3 An assumed volume of emigration to all other countries.

1961-6 234,900 42,700 1 00,000 377,600

1969-70 27,200 13,000 20,000 60,200

Residual Method ‘B,’ in which an endeavour had been made to adjust the census data for the extent of census under-enumeration. It is in- teresting to note that the 1961-71 intercensal emigration estimate of 694,700 persons which was generated by the postcensal method of emigration estimation fails to fall within the range of 804,800 to 1,004,300 emigrants (68 per cent confidence limits) obtained from Residual Method ‘ B . ’ ~

C O M P A R I S O N O F T H E T H R E E D I F F E R E N T

E S T I M A T E S O F I N T E R C E N S A L

E M I G R A T I O N

It must be recognized, of course, that each of these three different sets of intercensal emigra- tion estimates is subject to error. It would be preferable, of course, to have reliable data on the actual number of emigrants leaving Canada each year instead of a number of different esti- mates of emigration. However, since such ac- tual data are not available, recourse must be made to generating estimates of emigration, even if such estimates are not completely satis- factory. Of the three different emigration esti- mates presented in this paper, which one is the most reliable? Despite the fact that the absence of actual data on emigration from Canada pre-

vents one from ‘answering this question defini- tively, the intercensal emigration estimate of 9 4 , 5 4 0 persons generated by Residual Method ‘B’ is considered by the present author to be the most accurate measure of emigration for the 1961-71 period. The reasons for this are sum- marized below.

First, the intercensal emigration estimate of 706,112 for the 1961-71 period derived by Re- sidual Method ‘A’ is considered too low because this method fails to make the logically neces- sary adjustments for the estimated number of persons who were not enumerated in the cen- suses. As has been shown, when these adjust- ments are made in Residual Method ‘B,’ one obtains emigration estimates which are consid- erably higher than the estimate of 706,112 gen- erated from Residual Method ‘A.’

Second, if the emigration estimate of 706, I 12 persons for the 1961-71 period is considered too low, it follows that the estimate of 694,700 for the same period which was obtained by Statis- tics Canada’s postcensal method of emigration estimation is also too low. Moreover, the con- clusion that this postcensal method of emigra- tion estimation underestimates the volume of emigration from Canada receives additional support from unpublished data, most of which are from an ongoing pilot project on interna-

9 Indeed, this estimate of 694,700 also fails to fall within the range of 705.100 to I , 104,000 emigrants obtained from Residual Method ‘B’ when a 95 per cent confidence level is utilized.

Alternative estimates of the volume of emigration / 65

tional migration which is being conducted by member countries of the United Nations Economic Commission for Europe (ECE). In this pilot project, the ECE member countries exchanged data for 1972 on immigration tabu- lated by previous ECE country of residence, and these data, in combination with published data for selected other countries, suggest that Statis- tics Canada’s assumption of 20,000 which was discussed in the preceding section may be too low. For example, this combination of pub- lished and unpublished immigration data indi- cates that approximately 24,600 persons emi- grated from Canada to ZI selected ‘other’ coun- tries in 1972, and this total for these few coun- tries is considerably more than the assumed volume of ~ 0 , 0 0 0 persons who migrate from Canada each year to all ‘other’ countries in the world. lo

Finally, officials at Statistics Canada have utilized the Reverse Record Check (RRC) to generate an independent estimate of the volume of emigration from Canada during the 1966-71 intercensal period (Gosselin, 1975). and this procedure yielded a five-year intercensal emi- gration estimate of 530,300 (226,000) for this period.” As can be seen, this independent emigration estimate compares very favourably with the emigration estimate of 540,352 (534,066) for the same five-year period shown in Table 111 which had been calculated by Re- sidual Method ‘B. ’

It should also be noted that Gosselin’s RRC estimate of 530,300 emigrants for the 1966-71 intercensal period is particularly interesting when examined by RRC sampling frames, for this indicates that 167,900 (26,500) of the 530,300 emigrants were persons who had im- migrated to Canada during the 1966-71 inter- censal period (Gosselin, 1975). Consequently, of the 890,340 immigrants admitted to Canada during this five-year period, an estimated 167,900 persons (18.9 per cent) had either re- turned to their country of origin or had immi- grated to another country from Canada before I June 1971. This strongly suggests that Canada’s ability to retain its immigrants is not as strong as has frequently been assumed. This in turn adds

insight to an earlier finding of this paper, name- ly, that net international migration has played a less important role in Canada’s recent popula- tion growth than has previously been thought. Consequently, unless factors such as these are taken into account by policy-makers in their formulation of annual immigration targets, they may not succeed in their intended desire of re- lating future immigration levels to desired levels ofpopulation growth for Canada.

C O N C L U S I O N

Despite the fact that accurate information is not available on the actual number of Canadian res- idents who emigrate from Canada each year to establish residence in other countries, it is pos- sible to generate estimates of emigration in a number of different ways. Three different esti- mates of emigration for the 1961-71 intercensal period have been discussed in this paper, and reasons have been given why the estimate of 904,540 emigrants generated by Residual Method ‘B’ is considered to most closely ap- proximate the actual volume of emigration from Canada during the 1961-71 intercensal period. However, all estimates are by definition subject to error, and therefore must be viewed with caution. In view of the public discussion and the continuing governmental review of the impor- tance of international migration to Canada’s present and future population. growth, addi- tional research should be conducted in this area to ensure that a more definitive response may be obtained to the fundamental question ‘How im- portant are immigration and emigration to Canada’s population growth?’ If this is done, policy-makers will be in an improved position for making objective decisions regarding plan- ned targets for not only the future volume of immigration to Canada, but for the future popu- lation size of Canada as well.

A P P E N D I X : M E T H O D F O R A D J U S T I N G

T H E C E N S U S P O P U L A T I O N S I Z E F O R U N D E R - E N U M E R A T I O N

The method employed to adjust the census

10 The ECE has not released the data from this project yet because at this stage it remains a pilot project. Nevertheless, advances are being made, and it is the present author’s hope that the ECE will make sufficient progress in the future to enable it to commence publishing internationally comparable data on the exchange of migrants among ECE countries in order that researchers may use these data for further analysis. This project, if successful, should be able to provide researchers with separate data on migration entries and exits of nationals and of persons on employment or student visas. II This estimate is exclusive of emigration from the Yukon and Northwest Territories.

66 / John J. Kelly

enumerated population for the extent of under- enumeration in a given census is outlined be- low, using data from the 1971 Census as an illustration.

In 1971, the overall rate of under-enu- meration obtained from the Reverse Record Check was 1.93 per cent with a standard error (s) of .09 per cent. Therefore, the rate of enu- meration ( r ) of the 1971 Census was

I’ = (1 - .0193) = .9807 and

s = .0009. Let R be the realized rate of coverage in the Census, and C be the population count obtained in the census. Thus, theactual population (T)is, by definition, equal to C / R . In this procedure, since R is constrained to lie in the interval o to I , a Beta distribution is fitted with two unknown parameters, a and b. A Beta (a , b) variate R on the interval (0 , I ) has mean:

U E R = - a + b

and variance: ab

(a + b)’(u + b + 1) VR =

(See, for example, Wilks, 1962; Section 7.6.) Setting r = ER and s 2 = VR, we can solve the system of equations ( I ) and (2) to find a and b in terms of r and s. Thus

a = (1 - - I’

= (.0193) .9807 - .9807 (.0009)

= 22915.3282, and

= (.9807) (.O009) -

= 450.9677. Now we can exploit the relation between the Beta distribution and the Snedecor F-distri- bution (also known as the Variance-ratio dis- tribution). I t can be shown that if F has a Sned- ecor F distribution F (m, n), then

H = 1 1 + mF/n

has a Beta (:,:) distribution. (See Wilks,

1962: Section 7.8). It follows that

m 1 R n - 1 +-F _ - (3)

Thus, ifwe set n/z = a , and m/z = b, we see that I

has a distribution of

b 1 + - F U

where F has the Snedecor F distribution F(zb, 24. The mean of a Snedecor F(m, n ) variate is

n n - 2 E F = -

and its variance is

2n2(n2 + ii - 2) m(n - 2)*(n - 4) VF =

(4)

Using equations (3), (4). and (5) and replacingm and n by za and zb respectively, we obtain

b E(1IR) = 1 + (-) a - 1

= 1.0196806, and

b(a + b - 1) (a - I)’(a - 2) V(l/R) =

= .87581908 x

Therefore, the standard deviation of r/R =

d.87581908 x 1 0 - 3 = .0009359. Thus, a one standard deviation interval for IIR is 1.0196806 -t .0009359. For large values of both a and b , F(a , b) is approximately normal so that an approximate 68 per cent confidence in- terval is 1.0187447 s I/R s 1.0206165, and an approximate 95 per cent confidence interval is

The appropriate values may now be utilized in the formula

T = CIR to calculate the adjusted actual population of Canadaat the 1971 Census. Thus, Tcan be said to fall within the range 21,972,602 to 22,012,972 at the 68 per cent confidence level, with the best

1.0178088 C I/R Q 1.0215524.*

* I t should be noted that in most statistical applications, the parameters of the Beta distribution are integers or half integers ( i . e . , of the form K / z where K is an integer) and those of the Fare integer valued. However, the properties li\ted here are valid for non-integer parameters as well.

Alternative estimates of the volume of emigration / 67

estimate of T being set at 21,992,787 (see Table 1 1 ) . At the 95 per cent confidence level, Tcan be said to fall within the range of 21,952,417 to 22,033,157, with the best estimate of T also being set at 21,992,787.

R E F E R E N C E S

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verse Record Check.’Ottawa: Statistics Canada

Canada 1975 Minutes of Proceedings and Evidence of

the Special Joint Committee of the Senate and of the House of Commons on Immi- gration Policy, Third Report to Parlia- ment. Ottawa: Information Canada

George, M.V. 1974 ‘Population Growth.’ I n The Population

ofCanada. UnitedNations CICRED Series 1976 Population Growth in Canada. 1971 Cen-

sus Profile Study. Ottawa: Statistics Canada(Bul1etin 5.1-1, Catalogue99-701)

Gosselin, J.F. 1975 ‘Estimation of intercensal emigration de-

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Mclnnis, R.M. 1974 ‘Census survival ratio estimates of net

migration for Canadian regions.’ Cana- dian Studies in Population 1:93-116

Muirhead, R.C. 1969 ‘Census Evaluation Program, 1966: Re-

verse Record Check.’ Ottawa: Statistics Canada

u s . Bureau ofthe Census 1973 The Methods and Materials of Demog-

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Wilks, S.S. 1962 Mathematical Statistics. New York: John

Wiley and Sons Inc. (Sections 7.6and 7.8) Winkworth, A.V. 1973 ‘Evaluation of Canadian Censuses.’ Ot-

tawa: Statistics Canada