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A Note on the Measurement of Digital Preference in Age Recordings

Published online by Cambridge University Press:  18 August 2016

N. H. Carrier
N. H. Carrier
Affiliation:
Reader in Demography in the University of London
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Extract

1. The field of biased and unbiased errors in recorded ages is vast, and there are many opportunities for research in it. This note is restricted to a consideration of a small part of the field. First, only errors from digital preference are considered, i.e. errors to move the age to one with a preferred end digit such as 0 or 5 or in general any even number. Secondly, concentrations at isolated ages which are a genuine feature of the population are not considered. Thirdly, the use of supplementary data (such as birth registrations) or supplementary calculations (such as graduation) is ignored.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 85 , Issue 1 , June 1959 , pp. 71 - 85
Copyright
Copyright © Institute and Faculty of Actuaries 1959

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References

page 71 note * I am advised by Dr Benjamin that recent (as yet unpublished) researches of his suggest that, in some circumstances, the traditional concept of digital preference may be wide of the mark, and that the conventional techniques may often measure no more than the cyclical element in the totality of error. I have retained the usual terminology in this paper, but without prejudice to reviewing the situation when Dr Benjamin has completed his researches.

page 72 note * R. J. Myers, Errors and Bias in the Reporting of Ages in Census Data. T.A.S.A. 41, 411-15.

page 72 note † In the text of his paper, Myers recognizes that his theory may be applied to any age range, but in his worked examples he carries on to the highest ages where the population virtually vanishes.

page 73 note * Substituting the coefficients of Myers's Test, viz. A(r) = r/10 = 1 - C(r), B(r) = 1, it will be found that only the first two equations are satisfied, i.e. an error enters the formula if the population second-order differences are other than zero. Just as the T(x)'s of Myers's Test were obtained by averaging ten S(x)'s starting from progressive ages, a set of aggregates, say W(x)'s, may be obtained by averaging ten T(x)'s starting from progressive ages. These will have the property of satisfying all three of the equations (2), (3) and (4). Mathematically their form is amusing, but they are more complex and probably of less practical value than functions that will be developed later.

page 76 note * From one solution, say A(r), B(r), C(r) a whole range of other solutions may be derived. For instance, A'(r) = K'A(r), B'(r) = K'B(r), C'(r) = K'C(r) is also a solution, and so is A”(r) = A(r)-K”, B”(r) = B(r) + 2K”/(n-1), C”(r) = C(r)-K”, where K' and K” are any constants.

page 76 note † It may be noted that these coefficients tend to those of the Myers Test when n tends to infinity.

page 77 note * A similar adjustment to that described in paragraph 6.3 is therefore required.

page 78 note * 2 is the least value of n for which equations (2), (3) and (4) can be satisfied when B(r) ≠ 1. Using the technique of paired comparisons, however, equations (2), (3) and (4) can be satisfied with n = 1.

page 83 note * Strictly speaking the techniques under discussion may not properly be applied to this population because of its derivation from an uneven birth incidence. Apparent irregularities, in fact true features of the age structure, will be shown incorrectly by the indices as due to digital preference.

page 83 note † In his paper, Myers gives me the impression that he objects to the examination of the aggregates S(x) because in general they are not constant, even in the absence of digital preference. Prima faciet would be thought that the adjustment by R(x) takes care of this, and that the ability readily to take note of second-order differences (ignored by Myers) and to deal with a pure unweighted mean of digital preference over the age range covered (unlike Myers's and all the other tests considered in this paper) together made this test as good as or better than any others considered. In the event, however, it is seen that Myers's fundamental idea of the weighted aggregate opens the door to a whole field of superior tests.