the four-parameter logit life table system
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The four-parameter logit life tablesystemBasia Zaba aa Centre for Population Studies , LondonPublished online: 08 Nov 2011.
To cite this article: Basia Zaba (1979) The four-parameter logit life table system, PopulationStudies: A Journal of Demography, 33:1, 79-100
To link to this article: http://dx.doi.org/10.1080/00324728.1979.10412778
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The Four-Parameter Logit Life Table System
BASIA Z A B A * t
INTRODUCTION
It has long been recognized that model life table systems, if they are to provide a reasonable repre- sentation for the range of observable mortality schedules, must incorporate ways of allowing both the level and the pattern of mortality to vary. Lederman and Breas ~ showed by a factor analysis of mortality schedules that at least three factors were needed to explain about 93 per cent of ob- served variation between the 157 mortality schedules which they studied. Bourgeois-Pichat 2 interpreted these as: (i) a factor governing the level of mortality; (ii) a factor governing the re- lationship between mortality in youth and adult fife; and (iii) a factor governing mortality patterns at extreme ages (especially old age, 70+). He postulated the need for two more factors to explain the rest of the observed variation; (iv) a factor governing infant mortality; and (v) a factor govern- ing the differences between male and female mortality schedules.
BRASS'STWO-PARAMETER SYSTEM
The two-parameter logit life table system developed by Brass 3 caters for the first two of these factors. Briefly Brass's system rests on the assumption that different mortality schedules can be related to each other by a linear transformation of the logits 4 of their respective survivorship values. Thus, given two observed series of survivorship values, l(x) and i(x) for ages x = 1 . . . . , ~ ; it is possible to find constants a and/3 such that
logit [i(x)] ~ a +/3 logit [l(x)]
for all x between 1 and ~ .
In fact, by choosing a 'standard' series of l(x) values, and using the above relationship with values of a ranging from -1 .5 to +1.0, and/3 ranging from 0.5 to 1.5 it is possible to generate a wide range of mortality schedules amongst which it is possible to find reasonably accurate repre- sentations of most observed mortality schedules.
The parameter a allows one to vary the 'level' of the standard, ~ allows variation of the 'slope' of the standard-i.e, it controls the relationship between child and adult mortality. Figure 1 illustrates diagramaticaUy the action of the a and 13 parameters on a 'linearized' series of l(x) values. Table 1 shows the standard ls(x) values chosen by Brass.
Brass s has devised a very simple averaging procedure for fitting model life tables to observed
* Basia Zaba is a Research Fellow at the Centre for Population Studies, London. ~" My sincere appreciation is expressed to Professor W. Brass for his advice and encouragement throughout the
preparation of this paper. John Hobcraft, Ken Hill, John Barrett, and John Blacker also made helpful suggestions. i S. Lederman and J. Breas, 'Les dimensions de la mortalit6', Population, No. 4 (1959). 2 j. Bourgeois-Pichat, 'Factor analysis of sex-age specific death rates', Population Bulletin of the United
Nations, No. 6 (1962). s W. Brass, 'On the scale of mortality', in W. Brass (ed.), Biological Aspects of Demography (London: Taylor
& Francis, 1971); N. H. Carrier and J. Hobcraft, 'Appendix 1, "Brass Model Life Table System"', in Demo- graphic Estimation for Developing Societies (London: Population Investigation Committee, 1971).
1 - l ( x ) 1 4 logit l(x) - ~-1 l o g d ~ - y, and the inverse of this function is l(x)
l(x) 1 + e ~y"
s W. Brass, 'Use of the Logit System', in Methods for Estimating FerttTity and Mortality from Limited and Defective Data (University of North Carolina: POPLAB Occasional Publications, 1975).
Population Studies, 33, 1. Printed in Great Britain 79
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80 BASIA ZABA
O
r ~
Is(x) ,
r~x)
age a varies the "level" of the
standard
Figure 1.
1'
0
E r.Cj
/s(x),
/'(x)
age -~ varies the "slope" of the
standard
Table 1. The general standard ls(x) values and their logits*
x ts(x) Ys(x) Age Survivorship Logits [l(x) ]
1 0.8499 -0 .8670 5 0.7691 -0 .6105
10 0.7502 -0 .5498 15 0.7362 -0.5131 20 0.7130 -0.4551 25 0.6826 -0 .3829 30 0.6525 -0 .3150 35 0.6223 -0 .2496 40 0.5898 -0 .1817 45 0.5534 -0.1073 50 0.5106 -0 .0212 55 0.4590 0.0821 60 0.3965 0.2100 65 0.3221 0.3721 70 0.2380 0.5818 75 0.1521 0.8591 80 0.0776 1.2377 85 0.0281 1.7717 90 0.0060 2.5573 95 0.0006 3.7424
* This is the version of the general standard which has been smoothed at ages 45 and over by John Hobcraft- the full table is shown in the Appendix.
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 81
series of survivorship values, which does not require the use of a computer or lengthy sets of tables. The fits obtained in this way compare favourably on the whole with, for example, fits obtained from the Coale-Demeny regional model life tables, but occasionally relatively large discrepancies occur between fitted and observed values at the extreme end of the age distribution (i.e. ages under 5 and over 70). It is possible to get over such difficulties by a judicious choice of standard (e.g. Brass's use of a special 'African' standard for fitting data from countries where the level of early childhood mortality is relatively high), but up till now there has been no systematic way of choosing a 'special' standard.
BASIC FORM OF THE FOUR-PARAMETER SYSTEM
Recently Brass suggested a method for generating a set of standards by adding multiples of two sets of 'age-specific deviations' to his general standard. The two multiplying factors for the sets of deviations needed to specify the 'new' standard thus derived, coupled with the two parameters of a linear transformation of the logits of the 'new' standard effectively turn the logit life table sys- tem into a four-parameter model of mortality.
If ls(x) are the survivorship values for age x of the general standard, and k(x) and t(x) are schedules of deviations from this general standard, a 'new' standard l~x) may be chosen by speci- fying two constants ~ and • such that:
lN(x) = ls(x) + r + •
and further life tables,/(x), can be derived from lN(x) by choosing a and/3 such that
logit [/(x)] = a +/~ logit [lN(x)].
For such a system to perform well as a description of the different mortality patterns ob- served in actual populations, it is necessary that k(x) and t(x) should correspond in some way to those of the factors identified by the analysis of Lederman and Breas and Bourgeois-Pichat which the ordinary two-parameter system could not represent-that is k(x) and t(x) should be capable, in particular, of altering mortality patterns in infancy and old age. Furthermore, if this extended system is to be a useful analytical tool for demographers, then k(x) and t(x) should be such, that for any observed mortality schedule it should be possible (and not too difficult) to compute values (preferably unique values!) of the four parameters: a,/3, $, and X; which would allow the general standard life table to be transformed into a model life table which fitted the observed mortality pattern well.
Theoretical development of the system
The life table function 1 - l(x) can be interpreted as the cumulative distribution function for the probability of dying by age x, and Brass conceived the problem of finding suitable functions to represent k(x) and t(x) as the theoretical equivalent of finding functions which alter the 'tails' of a probability distribution without affecting the 'middle' of the distribution too much.
This type of problem is not uncommon in statistics-for example, an observed probability distribution, F(x), that is 'nearly' normal close to its mean value, but diverges from normality at its tails can be approximated by its Gram--Charlier expansion:
F(x) ~- r + cl r + c2 r (X) + . . .
where r is the normal distribution which has the same mean and variance as F(x); r and r (x) are its third and fourth derivatives, and cl and c2 are constants which can be evaluated from a knowledge of the skewness and kurtosis of F(x).
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82 BASIA ZABA
A similar result holds for the theoretical probability distribution underlying the logit life- table system, the logistic distribution.
The general form of the distribution function linearized by the logit transformation is
e a + bz
X(z) = 1 + e a + b~ (1)
(so that logit [X(z)] = ~;(a + bz)). This function has mean-a /b and variance rr2/3b 2. It can be
shown that the function
~k(Z) - - ~k(Z) + C~k'"(Z) n t- dX iv (z) (2)
(where c and d are (small) constants and X'"(z) and kiV(z) are the third and fourth derivatives with respect to z of X(z)), has the same mean and variance as X(z), but a different skewness and
kurtosis. The variable z in Equations (1) and (2) can be defined as a transformation z (x,a,b) of the
natural age scale, x, which linearizes the logits of the survivorship values of the general standard life table, ls(x), that is:
xo, d z ) = is(X) (3)
The set of survivorship functions represented by X(z)obtained from Equations (1)and (2) by allowing a, b, c and d to vary could then be used to define a four-parameter life-table system.
Life tables in this system could be sub-divided into families derived from groups of X(z) which had the same a and b, but varying c and d. These families of life tables would then be characterized by having a particular mean and variance, as measured on the transformed age scale, z. However, this theoretical system would not be a convenient practical representation for life tables. The 'family' property of equal means and variances would not necessarily hold if measured on the natural age scale, x, but working with an empirical transformation of the age scale (as defined by Equation (3)), would be very cumbersome. In any case, means and variances require additional computation if the life table is presented in terms of survivorship values, whereas other measures of location and dispersion, such as the median and other quantiles, can be found directly from the survivorship values themselves. Moreover, quantiles are independent of order-preserving transfor- mations made to the age scale, such as that represented by Equation (3).
A more practical system might be found, therefore, by constructing functions, multiples of which, when added to the survivorship function X(z), would leave unaltered its median, and the range between a pair of quantiles.
Practical adaptation o f the theoretical system
The logistic distribution has some interesting properties which, when used in conjunction with certain symmetries present in the general standard life table, allow the theoretical system outlined above to be adapted in a convenient way for generating a four-parameter life table system.
By writing the logistic distribution function, X(z) in its simplest form (with a = 0, b = 1), some of its properties may be conveniently illustrated.
e z If X(z)-
1 + e z
then
and thus
X'(z) = X(1 - X), , (4)
x"(z) = x'(1 - 2x) (5)
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 83
similarly
and
k'"(z) = ~'(1 - 6X') (6)
x iv (z ) - xt(l -- 2X)(1-- 12X') (7)
-i.e. the higher derivatives of ~,(z) can be expressed as simple power series in ~(z). This means that by using the relationship of Equation (3), with a = 0 and b - 1, it is simple to define numerically
an equivalent series of functions, 'quasi-derivatives', l's(x), l'~(x) . . . . . etc., based on the ls(x) values of the general standard life table, for all the age points x for which the standard survivorship ratio, ls(x), is defined. The five-yearly values of these 'quasi-derivatives' are shown in Table 2.
Table 2. The general standard and its 'quasi-derivatives' . . . . . - - - _ _ - . . . . . . . . . . . . . . . . . . . - -
" l '"(x iv Age Is (x) l'8(x) I s (x) 8 t. ) I s ( x ) x l ( I - / ) l'(1 -2/) l'(1 - 6l') l" (1 - 12/')
1 0.8499 0.1276 -0.0893 0.0299 0.0474 5 0.7691 0.1776 -0.0956 -0.0117 0.1081
10 0.7502 0.1874 -0.0938 -0.0233 0.1171 15 0.7362 0.1942 -0.0917 -0.0321 0.1221 20 0.7130 0.2046 -0.0872 -0.0466 0.1269 25 0.6826 0.2167 -0.0791 -0.0650 0.1266 30 0.6525 0.2267 -0.0692 -0.0817 0.1190 35 0.6223 0.2350 -0.0575 -0.0964 0.1046 40 0.5898 0.2419 -0.0435 -0.1093 0.0827 45 0.5534 0.2471 -0.0264 -0.1193 0.0519 50 0.5106 0.2499 -0.0053 -0.1248 0.0106 55 0.4590 0.2483 0.0203 -0.1217 -0.0403 60 0.3965 0.2393 0.0495 -0.1043 -0.0927 65 0.3221 0.2183 0.0777 -0.0677 -0.1259 70 0.2380 0.1814 0.0950 -0.0160 -0.1118 75 0.1521 0.1289 0.0897 0.0292 -0.0491 80 0.0776 0.0716 0.0605 0.0408 0.0085 85 0.0281 0.0273 -0.0258 0.0228 0.0173 90 0.0060 0.0059 0.0059 0.0057 0.0054 95 0.0006 0.0006 0.0006 0.0006 0.0006
The logits of the ls(x ) values do not increase linearly with x, so a life table survivorship func- tion defined by:
, I , , ~ dliV i(x) = is(x) + cts ix) + ,s (x)
would not necessarily have the same mean and variance as ls(x ) . But it is possible to construct, from a combination of the 'quasi-derivatives' of Is(x), functions defined for the same x values as the standard life table which, when added to it, do not affect measures of location and dispersion based on certain quantiles of the general standard. A simplified description of two functions selected for this system follows. A note on technical considerations influencing the choice of these functions is contained in Appendix I.
Consider the function k(x) , defined by
k(x) = l'"(x) + �89
By expressing ir'"(X) in terms of l'(x), we have
(8)
k ( x ) = l '(x)[ 1.5 - 61'(x)] (9)
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84 B AS IA ZABA
But l'(x)=/(x)[1 - / ( x ) ] (from Equation (4)), so that when l(x)= �89 (i.e., at the median), l'(x) = ~, and thus k(x) = 0. Thus, if the median is used as a measure of the levd of mortality for a life table, the addition of k(x) to l(x) will not alter its position.
Another noteworthy property of the function k(x), is that its value is the same for a pair of points whose l(x) values are equidistant from the median-i.e., if x~ and x2 are a pair of ages such that 1 - l ( x l ) = l(x2), then k(xO = k(x2). Thus the addition of k(x) to l(x) over the whole age range of a life table, will leave the differences in surivivorship values between such pairs of ages unaltered. In other words, if a new life table, l~x), is derived from the general standard life table
Is(x) by:
IN(x) =/s(x) + ~k(x) (10)
where qJ is a (small) constant, the new life table will have the same median age as the general stan- dard and will have the same range of mortality between ages whose l(x) values in the general stan-
dard are equidistant from the median. Values of k(x) for the usual five-year age groups are shown in Table 3. Figure 2(a) illustrates
the effect of adding a multiple of k(x) to the standard survivorship pattern. The function t(x), defined by:
t(x) =/iV(x) + rl"(x) (11)
(where r is a constant, yet to be def'med) also has the property of being zero at the median, as both/iv(x) and l" (x) are exactly zero when l(x) = �89 This function will also be zero when
-/iV(x)
l"(x) - r.
For a pair of points, x l and x2 such that l(xl)= 1 - / (x2) , it is easy to show that t(x 1) = -t(x2).
In the general standard life table,/s(65) -~ 1 - / s (25) ,
iv iv(25) -Is (65) -Is and - ~ ~ ~ 1.61.
1~'(65) " /s(25)
Thus, putting r = 1.61 in Equation (11), will ensure that the function t(x), derived from the general standard will be zero at ages 25 and 65 as well as at the median. Thus, a new life table, IN(x), defined by
IN(x) = Is(x) + Xt(X) (12)
(where X is a (small) constant, and t(x) is defined by Equation (11), with r = 1.61), will also have the same median age as the general standard, and the same range between survivorship values at ages 25 and 65.
Figure 2(b) illustrates the effect of adding a multiple of t(x) to the general standard; and t(x) values for five-year age groups are shown in Table 3.
Figure 3 shows the relative magnitudes of ls(x), k(x) and t(x) on an age scale which linear- ises the Is(x) values.
The quantities r and X in Equations (9) and (11) may be regarded as parameters charac- terizing a system of new standard life tables, l~x) , defined by:
IN(X) = ls(X) + q)k(x) + Xt(X) (13)
All these new standards lN(x) will be related to the general standard life table, ls(x), by having the same median age, and the same range between survivorship values at ages 25 and 65. If qJ and • are small (in most practical applications they would be less than 0.5) these new stan- dards will differ relatively little from the general standard in the central age range, but more
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85
1 0.8499 0.093 7 -0 .0964 5 0.7691 0.0771 -0.0458
10 0.7502 0.0704 -0 .0339 15 0.7362 0.0650 -0 .0256 20 0.7130 0.0557 -0.0135 25 0.6826 0.0433 -0 .0008 30 0.6525 0.0316 0.0077 35 0.6223 0.0211 0.0121 40 0.5898 0.0117 0.0127 45 0.5534 0.0042 0.0094 50 0.5106 0.0002 0.0021 55 0.4590 0.0025 -0.0075 60 0.3965 0.0154 -0 .0130 65 0.3221 0.0415 -0.0008 70 0.2380 0.0747 0.0412 75 0.1521 0.0937 0.0954 80 0.0776 0.0766 0.1059 85 0.0281 0.0365 0.0588 90 0.0060 0.0087 0.0149 95 0.0006 0.0008 0.0015
j , , . . . . . , ,, J , , , , , , , , , , , , , , , , , ,, , , ,
(Single year values of k(x), t(x), Is(x) and Y$(x) are shown in Appendix I.)
t
E
(a) (b)
Is(x)
\ Is(x)
.,,7
age -~
qJ allows the standard to be "curved"-deviations in infancy and old age are in the same direction.
7
t
0 >.
E t,,
R'ir
Is(x)
/s(x)
F O U R - P A R A M E T E R L O G I T L I F E TABLE SYSTEM
Table 3. Values of the general standard survivor- ship ratios (x ) and the two sets of deviations k(x)
and t(x) at five-yearly intervals
x Zs(x) k(x) t(x)
age -* X allows the standard to be "twisted" -
d e v i a t i o n s in infancy and old age
are in opposite directions.
Figure 2. The effects of the third and fourth parameters on the standard.
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86 BASIA ZABA
Is(X)
.1 __L~ (x)
0.0
20 30 40 50 60 70 80
X ~
-.1
Figure 3. The general standard survivorship values, ls(x), and the two sets of age-specific deviations, k(x) and t(x), on an age scale which linearizes ls(x).
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 87
noticeably at very young and very old ages. The effects of the k(x) deviations, the magnitude of
which is determined by the parameter if, will be to 'curve' the mortality pattern of the general
standard, in the same direction in old age as in infancy (i.e. either decrease mortality at both ends
of the age scale, or increase it at both ends). The effects of the t(x) deviations, on the other hand,
as controlled by the parameter X, will be to 'twist' the pattern of mortality in opposite directions at extreme ends of the age scale.
Relationship between this system and the factors identified by Bourgeois-lh'chat
The magnitudes of the k(x) and t(x) deviations in infancy and old age are broadly similar, but
they are opposite in direction in infancy and have the same direction in old age. Thus, by making
and X equal, it is possible to make the deviations at younger ages cancel each other and to con-
centrate the change from the general standard life table at the older ages. By making ff and • equal
in magnitude, but opposite in sign, the reverse can be achieved, i.e. the deviations due to these two
parameters will cancel each other out at the older ages, but will act in the same direction in in-
fancy, concentrating the changes from the general standard at the younger ages.
Thus, it is possible to reproduce to some extent, using combinations of k(x) and t(x), the
effects of the third and fourth parameters described by Bourgeois-Pichat.
EXAMPLES OF THE 'NEW STANDARD' LIFE TABLES
This small selection of 'new standard' life tables has been chosen to illustrate some of the possible
tables resulting from various combinations of ff and X.
Table 4. Examples of some 'new standards'
-0.4 0.4 0.0 0.0 -0.2 0.2 -0.2 0.2 0.0 • 0.0 0.0 -0.4 0.4 -0.2 -0.2 0.2 0.2 0.0
age, x Survivorship values, lN(X) ls(x)
1 0.8124 0.8874 0.8885 0.8114 0.8505 0.8879 0.8119 0.8494 0.8499 5 0.7392 0.7999 0.7874 0.7508 0.7628 0.7936 0.7445 0.7753 0.7691
10 0.7220 0.7783 0.7637 0.7366 0.7429 0.7710 0.7293 0.7575 0.7502 15 0.7102 0.7622 0.7464 0.7259 0.7283 0.7543 0.7181 0.7441 0.7362 20 0.6908 0.7353 0.7184 0.7076 0.7046 0.7269 0.6992 0.7215 0.7130 25 0.6653 0.7000 0.6829 0.6823 0.6741 0.6914 0.6738 0.6911 0.6826 30 0.6398 0 . 6 6 5 1 0.6494 0.6556 0.6446 0.6573 0.6477 0.6604 0.6525 35 0.6138 0.6307 0.6174 0.6271 0.6156 0.6241 0.6205 0.6289 0.6223 40 0.5851 0.5945 0.5847 0.5949 0.5849 0.5896 0.5900 0.5947 0.5898 45 0.5518 0.5551 0.5497 0.5572 0.5507 0.5524 0.5545 0.5562 0.5534 50 0.5105 0.5107 0.5098 0.5114 0.5102 0.5102 0.5110 0.5110 0.5106 55 0.4580 0.4600 0 . 4 6 2 1 0.4560 0.4600 0.4610 0.4570 0.4580 0.4590 60 0.3904 0.4027 0.4017 0.3913 0.3960 0.4022 0.3909 0.3970 0.3965 65 0.3055 0.3387 0.3224 0.3218 0.3140 0.3305 0.3136 0.3302 0.3221 70 0.2081 0.2679 0.2215 0.2545 0.2148 0.2447 0.2313 0.2612 0.2380 75 0.1146 0 . 1 8 9 5 0.1139 0.1902 0.1142 0.1517 0.1524 0.1899 0.1521 80 0.0470 0.1083 0.0353 0.1200 0.0411 0.0718 0.0835 0.1141 0.0776 85 0.0135 0.0427 0.0046 0.0516 0.0090 0.0236 0.0325 0 . 0 4 7 1 0.0281 90 0.0025 0.0095 0.0004 0.0119 0.0013 0.0047 0.0072 0.0107 0.0060 95 0.0002 0.0009 0.0 0.0011 0.0001 0.0004 0.0007 0.0010 0.0006
Expectation of life at selected ages ~x
0 41.89 44.98 43.07 43.80 42.48 44.02 42.84 44.39 43.44 10 47.55 47.29 45.83 49.06 46.67 46.57 48.31 48.15 47.43 50 17.54 20.28 17.78 20.04 17.66 19.03 18.79 20.16 19.43 70 6.58 8.94 5.77 9.76 6.16 7.50 8.33 9.34 7.95
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88 BASIA ZABA
FITTING TABLES FROM THE FOUR-PARAMETER SYSTEM TO OBSERVED DATA
The relationship between the 'new' standards, IN(x)(of having a fixed median age and survivorship range between a pair of ages) leads naturally to a method of fitting model life tables from this system to observed data. A linear transformation of logits of an observed life table allows it to be transformed into the same general form as a model table from the set of new standards-i.e, having its median age at 51 and a difference in survivorship ratios at ages 25 and 65 of 0.3616. The easiest way to visualize this transformation is in two steps: the translation of logits of the observed life table to bring its median into line with that of the general standard; followed by a rotation about the median to equalize the range.
Thus, if the logits of the observed life table are denoted by Y(x), and those of the general standard by Ys(x), then the logits of the transformed life table are given by ~'(x);
Y(x) = p[r + Y(x)] (14)
where
~ = Ys(51) - Y(51)=-Y(51) (15)
(because Ys(51) = logit (�89 = 0) and
Ys(65)- Ys(25) 0.7550 p = = (16)
Y(6S)- r(2s) Y(65)- r (2s)
In practice, l(51) and, therefore, Y(51) are often not available directly-a good estimate of is then given by:
Ys(50) r(55) - rs(55) r(50) = (17)
rs(55)- rs(50)
0.0212Y(55) + 0.0821Y(50)
0.1033
If the transformed life table is denoted by i(x) (where logit [i(x)] = I?(x)), then [(x)is approxi- mately of the same general form as tables from the set of 'new standards' IN(x). An appropriate 'new' standard l~v(x), can then be chosen by selecting ff and X so as to minimize the differences between lN(x) and l(x).
This could be done in a variety of ways (e.g. by least squares)-but the use of a complex or lengthy technique to obtain a close fit at this stage is not justifiable for a number of reasons" on practical grounds, if the four-parameter system is to be as easy to handle as the two-parameter system, one does not want to have to resort to fitting procedures which cannot be done efficiently with just a desk calculator and/or a set of tables; besides, choosing ~b and X on the basis of mini- mizing a 'goodness of fit' measure such as the sum of squares of differences between IN(x) and i(x) will not necessarily yield that particular standard which, when transformed linearly on the logit scale, provides a model fit to the observed life table for which the same criterion of 'good- ness of fit' is also a minimum.
Many different methods of estimating ~ and X from the set of transformed survivorship
values i(x)were tried, and the resulting new standards, lN(x), were fitted to the observed l(x) values by linear transformations of their logits, the two parameters a and/3 being estimated by the usual 'averaging' process as recommended by Brass. 6 The resulting model fits were compared with two-parameter fits, and with the 'best possible' fits obtainable using this four-parameter system. The latter were found by using a computer program for general function minimization, which searched the four-parameter space for a combination of a,/5, r and X corresponding to the mini-
op. cit. in footnote 5.
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 89
mum of a chosen measure of 'goodness of fit'. Several measures of fit were used: weighted and unweighted sums of squares of differences; sums of absolute differences; and the maximum absolute difference between corresponding model and observed survivorship values.
In this way, it was discovered that estimating ~ and X from just two values of i(x), at ages x - 1 and x - 75, consistently gave good results-that is, it produced fits which were a consider- able improvement on those obtained by using the two-parameter system, and which were pretty close to the 'best possible' fits identified by the computerized searching procedure both in terms of parameter values and fitting criteria.
Estimating ~k and X from just the two i(x) values at ages 1 and 75 is an attractively simple procedure-and it can be justified on theoretical grounds, for, as noted in the previous section, it is precisely at those ages that the functions k(x) and t(x) have their maximum absolute values, and that the fits obtainable from the two-parameter system perform poorly.
If ff and X are estimated at ages 1 and 75, then they must satisfy:
i(1) = I s ( l ) + @k(1) + Xt(1)
and (18)
/(75) = 1s(75) + ~k(75) + Xt(75)
The survivorship values at ages 1 and 75 in the general standard life table happen to be related by the approximation: ls(1) -~ 1 - l s (75) . This allows us to use the approximations: k(1) ' " k(75); t (1) -~- t (75) to simplify Equation (18) when solving for ff and X. Substituting the numerical values for the 'known' terms in the solution of this pair of equation yields:
i(1) + 1(75) - 1.002 4,=
0.1874 (19)
1(75) - i(1) + 0.6978 X =
0.1918
(where the i(x) terms can be evaluated from the observed data using Equations (14)--(17)). A further simplification can be introduced into the procedure for estimating the coefficients
and • of k(x) and t(x) which eliminates the need for calculating transformed survivorship values, i(x), provided that the transformed logits,j~(x) do not differ from those of the general standard at the fitting points (ages 1 and 75) by a very large amount.
If these differences are small, a Taylor series expansion of the logit function leads to the relationship:
l(x) - is(x) -~ 2ts(x)[ 1 - ts(x)] I t s ( x ) - Y(x)] (20)
This allows us to solve Equation (18) in terms of the transformed logits, Y(x), at ages 1 and 75. As these can be expressed in terms of the original logits, Y(x), and the transformation coeffi- cients ~ and p, (given by Equations (16) and (17)) there is no need to evaluate I7(1) and Y(75). In numerical terms, we have, simply:
= -1.3693{ 0.0077 + p[2r + Y(1) + Y(75)] }
X = 1.3379 { 1.7263 + p [Y(1) - Y(75)]}
(21)
(The full derivation of the estimating Equations (18) to (21) contained in Appendix 1.) In practice, there appears to be little to choose between the two methods of estimating
and • in terms of goodness of fit of the resulting models. If l(x) values with four figures are used, consideration of the magnitude of k(x) and t(x) shows that ~k and X need only be specified correct to two places of decimals. In fact, the values of ~ and X found by minimizing different indices of
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90 B AS IA ZAB A
goodness of fit, often differ from each other by up to -+ 0.05 ;and it has been found that rounding ~b and • to the nearest 0.05 does not have a large effect on the aggregate measures of goodness of fit. If the estimates of ff and X obtained by the two methods differ by more than -+ 0.05 each, it is probably worth calculating model fits from both the implied 'new standards'. An example of the fitting procedure is shown in Appendix III, applied to data from the Seychelles.
Evaluation o f the fitting procedure
To evaluate the performance of the model system and the fitting procedure, a series of com- parisons were made between fits obtained using Brass's two-parameter system and those obtained using the four-parameter system, employing both the estimation procedures outlined above, and a function minimization computer program. 7 The program was used to locate the four-parameter models for which the sum of squares of differences from the observed data was a minimum, and also those with the smallest sum of absolute differences. (The computerized minimization pro- cedure was also used to minimize the sum of absolute differences, and various weighted sums of squares, 8 but the results were so similar (in terms of parameter values) to those found by minimiz- ing the sum of squares of differences that they are not presented here.) Thus, for each observed life table, four model fits were compared: (i) a life table obtained from the general standard by a linear transformation of its logits only, the parameters a and/3 being found by the averaging method suggested by Brass; (ii) a life table obtained from a linear transformation of the logits of a 'new' standard, ff and • being calculated as described above, a and/3 by the same averaging tech- nique as in method (i); (iii) a four-parameter fit obtained using a general function minimization routine to minimize the sum of squares of differences between observed and fitted survivorship values; (iv) as in (iii), but minimizing the maximum absolute difference between corresponding pairs of survivorship values. 'Best fits' for each observed life table from the Coale-Demeny system were also found, but as these were, on the whole, worse than even the two-parameter fits, the results are not shown here.
The data to which these tests were applied were selected mainly on the grounds that they had hitherto been difficult to fit. Some had been rejected by Coale and Demeny 9 in the construc- tion of their regional life tables on the grounds of having atypical mortality patterns due to the prevalence of tuberculosis (Finland, Hungary, Switzerland). Some had been singled out by Brass 1~ as having an extreme relationship between child and adult mortality which the Coale/Demeny tables could not reflect accurately (Mauritius, Guyana, Philippines, U.S.S.R.). Others were chosen because they were better represented by tables from the Coale/Demeny regional system than by fits from the two-parameter logit life-table system (Ceylon, Sweden, Italy, Japan). Also included were tables which various members of the Centre for Overseas Population Studies came across in the course of their work, and noted for the oddities in their patterns (Turkey, Seychelles, Egypt). A few more were included to make the data set more representative historically and geographically (England and Wales, United States, South Africa (coloured)). The sources are listed with the other references, the life tables and their model fits are shown in full in the Appendix. (The results shown here are for ages 1-75 inclusive-very similar results obtain for an extended age range, but data were not available for the extended range for some of the countries, so for comparability only ages 1-75 are shown.)
7 James and Roos, 'Minuits'-Program D506 from the CERN Program Library (run through the CDC 200 user terminal at London School of Hygiene and Tropical Medicine, connected to CDC 6000 series computers at the University of London Computing Centre).
8 N. H. Carrier and T. J. Goh, 'The Validation of Brass's Model Life Table System', Population Studies, 26, 1 (March 1972).
9 A. J. Coale and P. Demeny, Regional Model Life Tables and Stable Populations (Princeton University Press, 1966).
~o loc. cit. in footnote 3.
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 91
Table 5 shows two indexes describing 'goodness of fit' (sum of squares of differences and maximum absolute difference) for each of the four model fits for all the life tables. Broadly speaking, the fits obtained by the estimating procedure outlined earlier give quite good improve- ments on the two-parameter fits, in terms of both indices, the only exception being Switzerland, which is already so well fitted by the two-parameter method that it hardly needs the extra sophisti- cation of a four-parameter fit. When compared with the fits obtained by mimimization, the esti- mation procedure still performs quite well, for the sums of squares of differences are of a com- parable order (and often smaller than) those of the fit obtained by minimizing the maximum absolute difference; and the maximum absolute differences in the fits obtained by the estimation procedure are frequently smaller than those in the fits obtained by minimising the sum of squares.
USES OF THE FOUR-PARAMETER SYSTEM
The main uses for the four-parameter logit life table system will probably be in theoretical work, where a flexible method of varying mortality patterns is required-for example, to assess the robustness of indirect estimation techniques to non-standard mortality patterns. Model life tables from this system fitted to observed data could also be used for smoothing and interpolation, and for projection purposes. The nature of the system and the suggested fitting procedure, however, presuppose fairly detailed knowledge of the mortality pattern, particularly at extreme ages, so where estimates of infant mortality and mortality at ages over 70 are unknown or thought to be very unreliable, the use of this system is not justified.
Another possible use might be in the development of single-parameter sets of life tables, for studying data sets from particular geographical regions, or from historical time series. The variation between related life tables from such sets might be adequately described by a single parameter- that is, it may be possible to fix the values of three of the parameters and allow just one to vary. Alternatively, it may be possible to specify functions describing the interdependence of all four parameters, so that given the value of one of them, the other three can be determined. However, preliminary work in this direction has not revealed any useful groupings of life tables or inter- relationships between the parameters.
SOURCES FOR LIFE TABLE DATA
Ceylon, males 1952 Guyana, males 1945-47 Philippines, males 1946-49 Sweden, females 1956-60 Mauritius, males 1942-46 Japan, females 1909-13 Finland, males 1941-45 Hungary, males 1920-21 Italy, females 1901-11 Switzerland, females 1920-21 U.S.S.R., females 1926-27 England and Wales, females 1973 U.S., males 1940
South African (coloured), males 1960
U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. U.N. Demographic Yearbook. Report of the Registrar General, HMSO, 1973. S. M. Preston, Causes of Death-Life Tables for National
Populations. S. M. Preston, Causes of Death-Life Tables for National
Populations.
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92 BASIA ZABA
r
k .
r~
~D
6 ~
t~
6~
~L
o. o. o. o. o. o. o. o. o. o. o. o. o. o. o. o. o.
�9 0 , . 0 0 ~ 0 0 . . . . . . . r
. . . . . . 0 0 0 . . . . . . . - -
.q- .q- tr~ ~0 ,-~ r r r .q- r r ~.O 0 0 O~ r O
r t '~ r .q- .q- ,.-~ r r r r tr~ t'xl r r r O r o. o. o. o. o. o. o. o o. o. o. o. o. o. o. o. o.
~D r r... r r OO OO r'.. O ~,~ r ~.O O .-4 t,,3 tr~ ,..~ r tr ~ r ~ r ,q- r ~ O r OO r ,--4 r r OO
O. o o. o. o. o. o. o. o. o. o. o. o. o. o. o o.
r r r ~..~ ~O r... ,q. 0 0 r .q- r ~ ,q- tr 0 0 t ~ r 0 0
o. o o. o. o. o. o. o. o. o. o. o. o. o. o. o. o.
. . . . . . . . . . . . . o .o . o o o o o o o o o o o o o , o.
O O O ~ O O O O O O O O O
O O O O O O O O O O O O ~ ~ O O
~ . , I , ~ w . , 4
�9 ' tq r./~
Tab
le 5
. Ind
ices
of
'goo
dnes
s o
f fit
' fo
r m
od
el li
fe ta
bles
obt
aine
d b
y di
ffer
ent f
itti
ng p
roce
dure
s
Inde
x S
ums
of s
quar
es o
f dif
fere
nces
M
axim
um a
bsol
ute
diff
eren
ces
(i)
(ii)
(iii
) (i
v)
(i)
(ii)
(i
ii)
Min
imiz
ing
Fou
r-M
inim
izin
g m
axim
um
Fou
r-M
inim
izin
g T
wo-
para
met
er
sum
s o
f ab
solu
te
Tw
o-pa
ram
eter
su
ms
of
Fit
tin
g m
etho
d pa
ram
eter
(e
stim
ates
) sq
uare
s di
ffer
ence
pa
ram
eter
(e
stim
a tes
) sq
uare
s
Eng
land
and
Wal
es, f
emal
es 1
973
0.00
167
0.00
018
0.0
00
12
0.
0001
6 0.
0244
0.
0087
0.
0051
U
.S.,
mal
es 1
940
0.00
247
0.00
034
0.00
023
0.00
033
0.02
34
0.01
24
0.01
11
So
uth
Afr
ican
(co
lour
ed),
mal
es 1
960
0.0
06
40
0.
0005
1 0.
0003
3 0.
0005
7 0.
0305
0.
0107
0.
0112
E
gypt
, mal
es 1
962
0.00
843
0.00
336
0.00
271
0.00
363
0.04
98
0.02
45
0.03
38
Tu
rkey
, m
ales
196
5-66
0.
0054
2 0.
0002
4 0.
0001
8 0.
0002
3 0.
0491
0.
0070
0.
0076
S
eych
elle
s, m
ales
197
1-75
0
.00
08
4
0.00
009
0.00
007
0.00
008
0.01
43
0.00
72
0.00
44
Cey
lon,
mal
es 1
952
0.00
239
0.00
135
0.0
00
74
0.
0012
8 0.
0302
0.
0161
0.
0165
G
uyan
a, m
ales
194
5-47
0.
0051
3 0.
0002
2 0
.00
01
8
0.00
047
0.03
02
0.00
81
0.00
59
Phi
lipp
ines
, mal
es 1
946-
49
0.00
359
0.00
173
0.00
133
0.00
170
0.03
34
0.01
66
0.02
15
Sw
eden
, fe
mal
es 1
956-
60
0.00
217
0.00
007
0.0
00
04
0.
0000
6 0.
0223
0.
0051
0.
0038
M
auri
tius
, mal
es 1
942-
46
0.00
767
0.00
055
0.00
027
0.00
032
0.05
82
0.01
68
0.00
79
Jap
an, f
emal
es 1
909-
13
0.00
511
0.00
069
0.0
00
64
0.
0008
9 0.
0280
0.
0129
0.
0133
F
inla
nd,
mal
es 1
941-
45
0.0
01
17
0.
0005
7 0.
0002
5 0.
0003
6 0.
0236
0.
0100
0.
0101
H
unga
ry,
mal
es 1
920-
21
0.00
096
0.00
009
0.00
008
0.00
010
0.02
08
0.00
49
0.00
46
Ital
y, f
emal
es 1
901-
11
0.00
262
0.00
023
0.00
017
0.00
021
0.03
39
0.0
07
0
0.00
61
Sw
itze
rlan
d, f
emal
es 1
920-
21
0.00
013
0.00
034
0.00
012
0.00
025
0.00
72
0.00
78
0.00
82
U.S
.S.R
., f
emal
es 1
926-
27
0.00
179
0.00
060
0.00
048
0.00
085
0.0
34
0
0.01
17
0.01
12
(iv)
M
inim
izin
g m
axim
um
abso
lute
di
ffer
ence
0.00
53
0.00
59
0.0
08
4
0.02
14
0.00
52
0.00
35
0.01
23
0.00
77
0.01
37
0.0
03
0
0.00
59
0.0
10
8
0.0
08
3
0.00
36
0.0
05
4
0.00
58
0.00
96
\0
N
t:I:'
;I>
U>
- ;I> N
;I>
t:I:'
;I>
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Egypt, males 1962
FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM
Turkey, males 1965-66
Seychelles, males 1971-75
93
New data on deaths from World Health Statistics, 1962, Vol. 1, WHO Life table calculated by M. Husein, un- published Ph.D. thesis, London School of Hygiene and Tropical Medicine, 1977.
Turkish Demography: Proceedings of a Conference. Hacettepe University Publications, No. 7 (1969).
J. G. C. Blacker and J. N. Hobcraft, Fertility, Mortality and Population Growth in the Seychelles, unpublished report submitted to Ministry of Overseas Development, 1977.
APPENDIX I
TECHNICAL NOTES
CONSIDERATIONS GOVERNING THE CHOICE OF THE k(x) AND t(x) FUNCTIONS
(This note is included by way of a sketchy explanation as to how the functions used in this system were chosen. If it were desired to construct this type of four-parameter system based on a dif- ferent standard, similar considerations would apply.)
An examination of the derivatives, d n X/dz n, of X(z) = eZ/1 + e z, shows that the odd order derivatives are symmetric about the median, whilst the even order derivatives are skew-symmetric -that is, if z l and z2 are two points such that:
then
X ( z , ) - �89 = �89 - X(z
X tn) (zl) = X tn) (z2) for n odd
- X tn) (z2) for n even.
Thus, by making k(x) a linear combination of the odd order quasi-derivatives, and t(x) a combina- tion of those of even order, we can obtain sets of deviations with certain useful symmetries.
Confining our attention to the first four quasi-derivatives, we have the following general forms for the two functions:
k(x) = l"'(x) + pl'(x) + q
t(x) =/iV(x) + rl"(x)
where p, q and r are constants. The practical problem is then to choose the constants p, q and r in such a way as to make
k(x) and t(x) yield 'useful' sets of deviations from the standard l(x).
In this case, it was desired that:
(1) The functions be zero at the median. (2) They leave undisturbed the range between a convenient pair of quantiles. (3) They should be relatively small near the median and relatively large for l(x) values correspond-
ing to old age and infancy.
Some of these requirements are satisfied automatically, the others can be written down as
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94 BASIA ZABA
conditions for the roots and stationary values of k(x) and t(x). Using the general standard as the base, these three requirements can be expressed thus:
(1) k(x) should have a root at Is(x) = 0.5. (2) t(x) should have a pair of roots in the ranges 0.75 > ls(x) > 0.65 and 0.35 > ls(x) > 0.25. (3) Both functions should attain non-zero maximum or minimum values in the ranges0.9 > ls(x) >
0.8 and 0.2 > I s ( x ) > 0 . 1 ; and the absolute values of the functions outside these ranges should be relatively small.
By writing k(x) and t(x) in terms of powers of Is(x) it should be possible to find an optimal solution (in terms of p, q and r) to the set of equations which express the above conditions. Further constraints may be introduced by requiring that the roots and/or stationary values be located at (or very near to) x values for which ls(x) is tabulated.
SOLUTION OF THE ESTIMATING EQUATIONS FOR ~ AND x
If, in the simultaneous equations (18), we made the substitutions ls(75) = 1 - ls(1); k(75) = k(1 ); t(75) - t( 1 ) we get:
/ (1)- Is(l)= ffk(1) + Xt(1)
i(75) + I s ( l ) - 1 = ffk(1)= Xt(1)
from which
i(1) + i(75) - 1
2k(1)
,r - i(75) + 1 - 21s(1)
2t(1)
However, the relationships between ls(x), k(x) and t(x) at ages 1 and 75 are only approxi- mate. Solving for qJ and X in terms of the exact values of these functions at age 1 thus yields model life tables which fit better at age 1 than at age 75. To eliminate this bias, average values of k(1) and k(75); t(1) and - t (75) , etc. can be used instead. Thus, the numerical solution quoted in the text (Equation (19)) is in fact:
2(1) + [(75) - Is(l) - 1s(75)
k(1) + k(75)
i(75)- i(1) + 1s(75)- ls(1) X -
t(75) - t(1)
The second method of estimation is based on the fact that for two proportions p and p + 6p which differ by the small amount 6p, we can write:
d logit (p + 8p) = logit (p) + 6p "7-__ logit (p) + . . . (terms in higher powers of 6p) ap
and
d logit p = �89 = dp dp p 2(1 - p )p
thus
6p -~ -2(1 - p)p [logit (p + 6p) - logit p]
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FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 95
so, putting
we have
ls(x) = p, i(x) = p + 6p,
i(x) - ls(x) : 2ls(x) [1 - ls(x)] [Ys(x) - Y(x)]
That is, we can express the difference between the general standard ls(x) values and those of the transformed life table, i(x), in terms of their respective logits.
Substituting for/s(1) - / ( 1 ) and/s(75) - / ( 7 5 ) in Equation (18) we have:
ffk(1) + Xt(1) = 21s(1)[1 - / s (1 ) ] [Ys(1) - ]"(1)]
ffk(75) + Xt(75) = 2ls(75)[ 1 - / s (75) ] [Ys(75) - Y(75)]
Once again, the simultaneous equations can be simplified by putting: /s(75)= 1 - / s ( 1 ) ; Ys(75) = - Y s ( 1 ) ; k ( 7 5 ) = k(1); t(75) = - t (1); and solved to give:
ls(1)[1 - l s (1) ] I - Y ( 1 ) - I7"(75)]
= k(1)
ls(1)[(1 - Is(i)] [2Ys(1)- ]"(1) + ~'(75)] x - t(1)
and as Y ( x ) = p I dp + Y(x) I, we have:
-ls(1)[1 - ls(1)] [Y(1) + Y(75) + 2@]p
= t(1)
l s (1 ) [1 - I s ( i ) ] {2Ys(1)+ plY(75) - Y(1)] / X = k(1)
In order to avoid the bias resulting from the approximation ls(75) = 1 - ls(1); we can once again use average values of the appropriate functions at ages 1 and 75, giving:
[l'(75) + l'(1)] Ys(1)+{Ys(75)- p[2r + Y(1) + Y(75)]}
= k(1) + k(75)
[l'(75) + l'(1)1 Ys(75) -{Ys(1) + p [Y(7S) - Y(1)]} x = t(75) - t(1)
which is the basis for the numerical estimates given in the text (Equations (21)).
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Oct
ober
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4
Z ~
,,..,
.A.
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 I I I I 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0
0 0 0 0 0 0 0 ~ 0 0 0 0 0 ~ 0 0 ~ 0 0 0 0 ~ 0 0
I I I I I I 1 1 I I I I I I I I I
0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0
I I I I I I I I I I I I I I I I I I I I I
e~O0~ '~[" . , t ~- I~r),~.t 'Q,,~ O 0'~00~'-%1~ Lr) e ~ l ~ O 0 ~ c ' ~ O 0 0 L r ) e,'~ 0 '~0000[ ' . . - ~ L'~[".-["--I "-,, ~ ~ ' ~ ) ~ w ~ (:~ ~)~, ,O~LI '~ U'~ lr) Lr) , ,~-~-~l .
0 0 0 0 0 C ~ O 0 ~ 0 0 0 0 0 0 0 0 0 0 C~ 0 0 0 0 0
I I I I l l l l l l l l l l l l l l l l l l l l l
AP
PE
ND
IX I
I
The
gen
eral
sta
ndar
d li
fe ta
ble,
its
log
its,
and
th
e se
ts o
f dev
iati
ons,
by
sing
le y
ears
of a
ge
x Is
Vc)
.
Ys(x
) k(x
) t(
x)
x ls
(x)
Ys(
x)
k(x
) tV
c)
x ls
(x)
Ys(x
) k(x
) t(
x)
x ls
(x)
Y s(
x)
k(x
) tV
c)
1 0.
8499
-0
.86
70
0.
0937
-0
.09
64
26
0.
6764
-0
.36
86
0.
0409
0.
0013
51
0.
5010
-0
.00
21
0.
0000
0.
0002
76
0.
1358
0.
9253
0.
0934
0.
1027
2
0.80
70
-0.7
15
2
0.08
81
-0.0
70
8
27
0.67
04
-0.3
54
9
0.03
85
0.00
31
52
0.49
12
0.01
77
0.00
01
-0.0
01
7
77
0.12
00
0.99
62
0.09
15
0.10
78
3 0.
7876
-0
.65
52
0.
0830
-0
.05
80
28
0.
6643
-0
.34
13
0.
0361
0.
0048
53
0.
4809
0.
0383
0.
0005
-0
.00
37
78
0.
1050
1.
0714
0.
0880
0.
1100
4
0.:]
692
-0.6
01
9
0.07
72
-0.0
45
8
29
0.65
84
-0.3
28
0
0.03
38
0.00
63
54
0.47
01
0.05
98
0.00
13
-0.0
05
6
79
0.09
09
1.15
13
0.08
30
0.1
09
4
0.76
91
-0.6
01
5
0.07
71
-0.0
45
8
30
0.65
25
-0.3
15
0
0.03
16
0.00
77
55
0.4
59
0
0.08
21
0.00
25
-0.0
07
5
80
0.07
76
1.23
77
0.07
66
0.10
59
6 0.
7642
-0
.58
79
0.
0755
-0
.04
26
31
0.
6466
-0
.30
20
0.
0295
0.
0089
56
0.
4474
0.
1055
0.
0041
-0
.00
93
81
0.
0654
1.
3298
0.
0693
0.
0997
7
0.76
01
-0.5
76
6
0.07
40
-0.0
40
0
32
0.64
06
-0.2
88
9
0.02
73
0.00
99
57
0.4
35
4
0.12
99
0.00
62
-0.0
10
8
82
0.05
43
1.42
87
0.06
12
0.09
13
8 0.
7564
-0
.56
66
0.
0727
-0
.03
77
33
0.
6346
-0
.27
59
0.
0252
0.
0108
58
0.
4229
0.
1554
0.
0087
-0
.01
20
83
0.
0444
1.
5346
0.
0528
0.
0812
9
0.75
32
-0.5
57
8
0.07
15
-0.0
35
7
34
0.62
84
-0.2
62
7
0.02
31
0.01
15
59
0.40
99
0.18
21
0.0
1l8
-0
.01
28
84
0.
0356
1.
6496
0.
0445
0.
0702
10
0.
7502
-0
.54
98
0.
0704
-0
.03
39
35
0.
6223
-0
.24
96
0.
0211
0.
0121
6
0
0.39
65
0.21
00
0.01
54
-0.0
13
0
85
0.02
81
1.77
17
0.03
65
0.05
88
11
0.74
77
-0.5
43
1
0.06
94
-0.0
32
3
36
0.61
60
-0.2
36
4
0.01
91
0.01
25
61
0.38
25
0.23
94
0.01
96
-0.0
12
5
86
0.02
17
1.90
43
0.02
91
0.04
77
12
0.74
52
-0.5
60
6
0.06
85
-0.0
30
8
37
0.60
97
-0.2
23
0
O.o
t72
0.01
28
62
0.36
81
0.27
01
0.02
43
-0.0
11
1
87
0.01
63
2.05
01
0.02
25
0.03
76
13
0.74
25
-0.5
29
6
0.06
75
-0.0
29
3
38
0.60
32
-0.2
09
4
0.01
53
0.0
13
0
63
0.3
53
2
0.30
24
0.02
95
-0.0
08
8
88
0.01
20
2.20
54
0.01
69
0.02
86
14
0.73
96
-0.5
22
0
0.06
63
-0.0
27
6
39
0.59
66
-0.1
95
6
0.01
35
0.01
29
64
0.33
79
0.33
64
0.03
53
-0.0
05
4
89
0.00
86
2.37
37
0.01
23
0.02
10
15
0.73
62
-0.5
13
1
0.06
50
-0.0
25
6
40
0.
5898
-0
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16
0.
0117
0.
0127
65
0.
3221
0.
3721
0.
0415
-0
.00
08
9
0
0.00
60
2.55
50
0.00
87
0.01
49
16
0.73
27
-0.5
04
3
0.06
36
-0.0
23
7
41
0.58
29
-0.1
67
4
0.01
00
0.0
12
4
66
0.30
59
0.40
97
0.04
80
0.00
51
91
0.00
40
2.75
87
0.00
59
0.0
10
2
17
0.72
87
-0.4
94
1
0.06
21
-0.0
21
5
42
0.
5759
-0
.15
30
0.
0084
0
.01
19
6
7
0.28
93
0.44
94
0.05
48
0.01
24
92
0.00
26
2.97
48
0.00
39
0.00
67
18
0.72
41
-0.4
82
4
0.06
02
-0.0
19
0
43
0.
5686
-0
.13
81
0.
0069
0
.01
l2
68
0.27
24
0.49
12
0.06
16
0.02
09
93
0.00
16
3.21
81
0.00
24
0.00
42
19
0.71
89
-0.4
69
4
0.05
81
-0.0
16
3
44
0.56
11
-0.1
22
9
0.00
55
0.01
04
69
0.25
53
0.53
53
0.06
83
0.03
06
94
0.
0010
3.
4534
0.
0014
0.
0025
20
0.
7130
-0
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51
0.
0557
-0
.01
35
45
0.
5534
-0
.10
73
0.
0042
0.
0094
70
0
.23
80
0.
5818
0.
0747
0.
0412
95
0.
0006
3.
7090
0.
0008
0.
0015
21
0.
7069
-0
.44
01
0.
0532
-0
.01
06
46
0.
5454
-0
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11
0.
0031
0.
0082
71
0.
2206
0.
6311
0.
0805
0.
0525
96
0.
0003
4.
0557
0.
0005
0.
0008
22
0.
7005
-0
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48
0.
0506
-0
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78
4
7
0.53
72
-0.0
65
5
0.00
21
0.00
69
72
0.20
32
0.68
32
0.08
56
0.06
41
97
0.00
02
4.25
85
0.00
02
0.00
04
23
0.69
44
-0.4
10
3
0.04
81
-0.0
05
2
48
0.52
87
-0.0
57
4
0.00
12
0.0
05
4
73
0.18
59
0.73
85
0.08
96
0.07
55
98
0.00
01
4.60
51
0.00
01
0.00
02
24
0.68
84
-0.3
96
3
0.04
57
-0.0
02
9
49
0.51
98
-0.0
39
6
0.00
06
0.0
03
8
74
0.16
88
0.79
71
0.09
23
0.08
61
99
0.00
00
5.12
70
0.00
01
0.00
01
25
0.68
26
-0.3
82
9
0.04
33
-0.0
00
8
50
0.51
06
-0.0
21
2
0.00
02
0.00
21
75
0.15
21
0.85
91
0.09
37
0.09
54
100
0.00
00
5.55
55
0.00
00
0.00
00
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07
Oct
ober
201
4
FOUR-PARAMETER LOGIT LIFE TABLE SYSTEM 97
A P P E N D I X III
FITTING A FOUR-PARAMETER MODEL LIFE TABLE TO OBSERVED DATA: ILLUSTRATED WITH DATA FROM THE SEYCHELLES, FOR MALES, 1971-75
The Seychelles survivorship data shown here are based on average annual death rates for 1971-75.
The data are of high enough quality to warrant a four-parameter fitting but based on insufficient numbers for single year estimates to be made directly. In this case, the four-parameter system pro-
vided a useful tool for graduation and interpolation enabling single-year estimates of life table functions to be made.
The observed survivorship ratios and their logits �9 , , , . . . . .
x l(x) Y(x) x l(x) Y(x)
1 0.9641 - 1.6452 40 0.8655 -0.9309 5 0.9409 - 1.3838 45 0.8424 -0.8381
10 0.9366 - 1.3464 50 0.8045 -0.7073 15 0.9343 -1.3273 55 0.7584 -0.5720 20 0.9227 -1.2398 60 0.6967 -0.4158 25 0.9108 -1.1617 65 0.6028 -0.2086 30 0.8981 - 1.0881 70 0.4770 0.0460 35 0.8843 - 1.0169 75 0.3451 0.3203
from Equation (16):
from Equation (17):
from Equation (14)"
and
Y s ( 6 5 ) - Ys(25) P = = 0.7922
Y(65) - Y(25)
rs(50) , r(55) - rs(55) , r(50) = = 0.6795
Y s ( 5 5 ) - Ys(50)
I7(1)= 0[r + Y(1)] = -0 .7650
Ir(75) = p[q~ + Y(75)] = 0.7920
Using the inverse of the logit transformation:
[(1) = 0.8220; i(75) = 0.1702
from Equation (19)"
[(1) + i(75) - 1.002 - -~ -0 .05
0.1874
i(75) - i(1) + 0.6978 X = -~ 0.24
0.1918
(Alternatively, we could have found ~O and X directly from the logits of the observed survivorship ratios at ages 1 and 75, using Equation (21):
= - 1 . 3 6 9 3 I0.0077 + 012~ + Y(1) + Y(75)1} - ~ - 0 . 0 4
• = 1.3379 {1.7263 + p[Y(1) + Y(75)] } "" 0.23
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Oct
ober
201
4
98 BASIA ZABA
- the estimates are very dose, so the resulting 'new standard' life tables would be pretty similar,
and only one of them is shown below.)
The 'new standard'life table
lN(X ) = I s (x ) - 0.05k(x) + 0.24t(x); and its logits.
x lN(X) YN(X) x lN(X) YN(X)
1 0.8221 -0.7653 40 0.5923 -0.1867 5 0.7543 -0.5608 45 0.5554 -0.1113
10 0.7385 -0.5191 50 0.5111 -0.0222 15 0.7268 -0.4892 55 0.4571 0.0860 20 0.7070 -0.4404 60 0.3926 0.2182 25 0.6802 -0.3 773 65 0.3198 0.3 773 30 0.6528 -0.3157 70 0.2442 0.5649 35 0.6241 -0.2535 75 0.1703 0.7918
Brass's averaging procedure yields the following estimates of a and/3.
75 35
Y ( x ) - ~Y(x) - 3 . 3 0 6 4 + 10.2092 40 1
= 78 35 1.7234 + 3.7213 - 1.2678 ~' YN(X ) -- ~, YN(X) 40 1
35 35
a = i8[~ Y(x ) - ~ ~Yu(x)] = -~(--10.2092 +/3x3.7213) = --0.6864 1 1
The logits of the fitted model life table, YF(X), are then found using YF(X)= a + ~YN(X)= --0.6864 + 1.2678 YN(X).
These logits are shown below, together with the fitted model life table lF(X), derived from
the YF(X) by the reverse logit transformation.
x IF(X) YF(X) x lF(X) YF(X)
1 0.9649 - 1.6570 40 0.8636 -0.9227 5 0.94 24 - 1.3975 45 0.8395 -0.8271
10 0.9364 -1.3446 50 0.8066 -0.7140 15 0.9317 -1.3066 55 0.7601 -0.5767 20 0.9234 - 1.2447 60 0.6938 -0.4 089 25 0.9113 -1.1646 65 0.6020 -0.2070 30 0.8978 -1.0865 70 0.4845 0.0311 35 0.8824 -1.0075 75 0.3457 0.3191
Dow
nloa
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by [
Uni
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8:29
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Oct
ober
201
4
F O U R - P A R A M E T E R L O G I T L I F E T A B L E S Y S T E M 9 9
•
I I I
I I I I I I I I I I
, - . ; , - -- , ,-.., o ,.--, o o o o , - ,--,,:5 c5,..., ,..., ~ r..- t'-,I t-~ ~ r....
I I I I I I I I I I I I I I I 1 I 1 I I
r.... r...- r.... '~:1- ',~1- ',~1- t ~ t-,I t',,I t .~ t ~ t ~ , ~ t .~ t ~ . ~ '~:1" ' ~ t r ~ , ~ - tr~ t',,I t',,I t",l t ~ t ' ~ t ' ~ r'.-- r"-- r--. ~ ~
O ~ D ~ ' ~ ~ ' ~ : 1 " t ' ~ O'~ ~,O r t ~ t ~ ~ O ~ ' ~ ~- ~ t,e~ t',q t ~ ~.O o o r..- tr~ t ' q t"~ t ~ t"q t '~ t ' q ~ ,,~:I. t ~ t ~
t ' q ,-.-~ t",l t ' q t ' q t",l t ' o ' ~ l " '~" ,...~ t ~ t ~ ~ o o o o ~,.O tr~ ~ o ~ O ~ o o t ' q ,.-~ t"q '~- t"q t",i -..~ ,--~ t",l t"q t ' q t ~
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O ~ O ~ t ~ o O O 0 c ~ ~ ~ tr~ ~,D ~ ~.O ~ ) oO o~ o o t ~ r ~ r .... ~,.o t ~ t ".~ ~ D ~ tT~ t ~ t ~ '~1" '~1" ',~1-
o ' ~ o ~ o 0 o o o o t ~ ~ . - t "..... ~ 4 ~ , D I '~ r"... t "~ o ~ o ~ o o ~ t'-.- o~ ~ t '~ i ~ I "~ t ~ t ~ t ~ tr~ t r ) tr~
e'..- e".. t '~ ~ , . . . ~ ~ tr~ ~ , ~ t ' ~ , O t ~ t ' q t ' q t ' q t ' q t ' q r r t ' q t ' q t ' q t ~ t"q t ~ t",l t ~ r--- t"-- r...- t " q , ~ - t ~
o o o o o 0 ~....~ ~--~ -..~ t.~ t.~ t ..... t ~ t..~ o o t .~ r r t.~ r162 r162 r r r162 tr~ tr~ t r ) tr~ ,~- tr~ oO o o o o ~ g"-- t"..-
O ~ O ~ O ~ O ~ t ~ t ~ c ~ r.-.. t'..., t ' . ~ r -.- t-~ t ~ r ..-. ~ t ~ O ~ o o o 0 ~ o o ~ e.-.. e...- t'-- o 'x t ~ t ~ r ~ t'.~ r..-
4 ~ ~ ~ ~ ~ ~, ~ ~
AP
PE
ND
IX I
V
A c
ompa
riso
n o
f the
fits
obt
aine
d to
rea
l su
rviv
orsh
ip d
ata
usin
g th
e tw
o-an
d fo
ur-p
aram
eter
log
it li
fe t
able
sys
tem
s
Lif
e ta
ble
Fit
Eng
land
and
Wal
es,
12
fem
ales
197
3 0 14
U
nite
d S
tate
s,
12
mal
es 1
940
0 14
Sou
th A
fric
a 12
(c
olou
red)
, 0
mal
es 1
960
14
Egy
pt,
12
mal
es 1
962
0 14
Thr
key,
12
m
ales
19
65
-66
0 14
S
eych
elle
s,
12
mal
es 1
97
1-7
5
0 14
Ce0
on
, ~
mal
es 1
952
0 14
Guy
ana,
12
m
ales
19
45
-47
0 14
P
hili
ppin
es,
12
mal
es 1
94
6-4
9
0 14
Sw
eden
, 12
fe
mal
es 1
96
5-6
0
0 14
Mau
riti
us,
12
mal
es 1
94
2-4
6
0 14
992
985
988
962
941
949
869
849
841
777
803
781
812
801
794
969
964
965
898
911
902
93
2
902
910
841
874
860
991
986
988
862
804
815
5
984
983
983
930
930
926
802
783
794
714
691
712
756
748
750
944
941
942
856
853
856
871
864
864
777
765
782
983
983
982
727
716
713
10
982
982
981
922
924
921
787
777
784
700
683
698
744
736
741
937
937
936
846
833
846
842
845
844
762
748
764
981
981
981
693
693
688
15
980
980
980
916
919
916
775
772
776
691
676
688
735
730
735
931
934
932
839
828
838
855
852
853
751
740
751
980
980
980
667
676
670
20
977
978
979
904
910
909
756
763
762
675
668
673
721
723
724
922
923
923
827
822
826
821
831
828
734
728
730
976
976
978
625
644
638
Age
25
30
35
40
45
50
55
60
65
70
75
973
976
976
889
898
897
731
747
744
654
658
654
702
712
710
908
911
911
811
812
810
791
809
805
711
707
702
972
975
975
569
597
594
968
963
974
971
972
968
872
853
884
868
883
867
956
948
967
956
962
954
832
806
847
818
847
820
706
680
652
620
729
706
678
646
723
701
675
642
635
616
595
572
646
633
618
599
636
619
601
582
684
666
698
682
695
680
893
877
898
884
898
882
795
779
801
788
794
778
759
726
783
754
779
749
688
666
689
662
676
650
967
962
972
968
971
967
647
626
666
648
663
643
859
836
866
842
864
840
761
740
772
752
761
742
689
645
718
667
712
666
641
614
635
605
623
595
955
947
962
953
961
953
515
462
408
352
545
491
434
369
547
497
44
2
379
937
920
893
849
771
939
915
879
826
746
941
920
885
828
746
773
727
663
574
451
777
720
645
550
43
9
781
726
647
544
427
583
536
477
403
315
606
547
479
379
284
600
545
473
386
291
546
513
473
422
360
573
534
481
417
325
559
530
490
433
350
600
569
529
478
41
4
621
586
536
469
391
617
583
537
474
391
805
763
703
616
491
805
758
697
603
47
7
807
760
694
602
48
4
715
726
718
683
640
691
643
687
646
583
506
576
48
7
588
503
591
523
439
337
225
601
519
424
315
210
605
525
42
4
312
208
581
541
490
42
7
348
570
528
478
423
362
562
525
482
432
372
935
918
939
918
94
0
920
292
228
299
228
308
231
892
847
885
832
886
830
769
747
743
164
104
056
159
097
05
2
156
093
049
63
0
627
637
299
310
308
216
195
199
284
23
4
234
333
284
290
331
345
345
403
373
384
122
121
123
256
290
298
630
613
618
02
4
02
3
023
a
-1.3
20
-1.3
50
-0
.58
7
-0.6
07
-0
.14
7
-0.1
80
-0
.07
8
-0.1
07
-0
.19
0
-0.2
25
-0
.68
4
-0.6
86
-0
.44
4
-0.4
52
-0
.15
6
-0.1
80
-0
.14
7
-0.1
09
-1
.31
0
-1.3
47
0.
478
0.44
4
Par
amet
er v
alue
s
{3
1.22
7
1.40
9 1.
179
1.31
8 0.
920
1.01
5 0.
630
0.60
0 0.
626
0.66
1 1.
205
1.26
9 0.
744
0.72
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329
1.46
4 0.
791
0.74
6 1.
216
1.37
0 1.
609
1.76
4
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9
0.58
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1
0.47
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5
0.34
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2
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5
-0.4
8
0.08
-0.0
5
0.24
-0.0
7
-0.1
6
-0.2
1
0.39
0.45
-0
.03
-0.3
2
0.46
-0.2
3
0.36
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Oct
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AP
PE
ND
IX I
V (
cont
inue
d)
-0 0
Age
P
aram
eter
val
ues
Ufe
tabl
e F
it
1 5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
a {J
'" X
Japa
n,
12
838
763
746
733
712
685
658
631
602
569
531
485
427
358
277
188
-0.0
43
0.
900
fem
ales
190
9-13
0
855
778
758
739
705
668
636
605
574
544
514
477
431
372
298
210
14
869
769
746
730
704
671
640
610
581
549
514
475
428
370
297
208
-0.0
12
0.
820
0.34
-0
.26
F
inla
nd,
12
941
894
881
872
856
833
809
784
755
721
678
622
548
452
333
205
-0.3
47
1.
193
mal
es 1
941-
45
0 93
1 90
3 88
8 87
8 85
7 83
0 80
5 78
0 75
2 71
6 67
1 61
4 53
9 44
6 34
0 22
8 14
93
6 89
3 88
2 87
3 85
7 83
6 81
2 78
7 75
7 72
1 67
5 61
6 53
9 44
3 33
4 22
0 -0
.34
0
1.22
9 0.
03
0.16
H
unga
ry,
12
775
694
676
663
643
616
591
566
539
510
476
436
387
329
261
186
0.06
4 0.
786
mal
es 1
920-
21
0 78
5 69
9 67
3 65
9 63
8 61
1 58
9 56
7 54
3 51
7 48
5 44
5 39
6 33
3 25
6 16
5 14
78
3 69
4 67
5 66
1 64
0 61
4 58
9 56
5 54
0 51
4 48
3 44
5 39
8 33
6 25
7 16
5 0.
050
0.75
7 -0
.12
-0
.24
It
aly,
12
81
4 74
3 72
8 71
6 69
7 67
3 65
0 62
6 60
1 57
3 54
0 50
0 45
0 38
9 31
5 23
0 -0
.06
4
0.77
8 fe
mal
es 1
901-
11
0 84
8 73
9 72
1 70
8 68
9 66
6 64
1 61
7 59
2 56
7 54
1 50
9 46
6 40
3 31
7 21
0 tl
:j
14
842
746
726
712
690
662
637
614
590
566
538
505
463
405
322
213
-0.0
63
0.
701
0.07
-0
.46
>
S
wit
zerl
and,
12
93
7 89
6 88
5 87
7 86
4 84
5 82
6 80
5 78
2 75
5 72
1 67
6 61
5 53
3 42
5 29
4 -0
.45
2
1.03
6 v.
> -fe
mal
es 1
920-
21
0 93
0 90
2 88
9 87
9 86
4 84
6 82
5 80
4 78
0 75
3 72
0 67
6 61
7 53
6 42
6 29
3 >
14
93
4 89
5 88
5 87
7 86
4 84
7 82
8 80
8 78
6 75
8 72
4 67
7 61
5 53
0 42
0 28
8 -0
.45
9
1.05
3 -0
.07
0.
05
N >
U.S
.S.R
., 12
79
4 73
1 71
7 70
7 69
1 67
1 65
1 63
1 61
0 58
6 55
8 52
4 48
2 42
9 36
3 28
4 -0
.10
3
0.65
9 tl
:j
fem
ales
192
6-27
0
828
729
705
696
683
665
646
626
605
582
558
527
489
436
366
275
>
14
818
734
717
705
685
662
640
620
599
578
553
525
489
440
373
281
-0.0
95
0.60
2 0.
16
-0.3
6
Not
es:
0=
Obs
erve
d li
fe t
able
; 12
=
two-
para
met
er f
it; 1
4 =
fo
ur-p
aram
eter
fit
. T
he a
bove
fit
s w
ere
obta
ined
usi
ng t
he
esti
mat
ion
proc
edur
es d
escr
ibed
in
the
text
. V
alue
s sh
own
here
hav
e be
en r
ound
ed d
own
to t
hree
fig
ures
to
fac
ilit
ate
visu
al c
om-
pari
sons
. T
he in
dice
s o
f go
odne
ss o
f fi
t sh
own
in T
able
5 w
ere
deri
ved
from
th
e ab
ove
resu
lts
befo
re r
ound
ing.
Dow
nloa
ded
by [
Uni
vers
ity o
f B
irm
ingh
am]
at 0
8:29
07
Oct
ober
201
4