THEORETICAL
POPULATION
A General
BIOLOGY
16, 13-24
Method for Constructing Increment-Decrement Life Tables That Agree with the Data* FRANK
Child
(1979)
WM.
OECHSLI
Health and Dmelopent Studies, School of Public University of California, Berkeley, California+ Received
December
Health,
12, 1977
A general method for making increment-decrement life tables is presented. The method involves the finding of probabilities of transition between states, graduated to small intervals of time and age, that are consistent with (i.e., can reproduce) the data, whether the data consist of central age-state specific rates, or some other feature, such as state distributions of a real cohort. The method is then illustrated with a fetal loss life table.
Papers on life tables tend to be either highly abstract, mathematical modeIs which are relatively unconcerned with the practical problems of life table construction, or very much concerned with the concrete problems of methodology of construction with little regard for the mathematicaI niceties that characterize papers of the first sort. The first kind of paper is recognizable by its density of integral/differential equations in contrast to an almost complete absence of numerical illustrations and real data. The second kind, conversely, is characterized by its concentration on details of methods of construction accompanied by one or more examples of the method as carried out with real (or sometimes hypothetical) data, in contrast to a sometimes complete absence of mathematics more complex than simple algebra. An exchange of views by Hoem (1975) and Schoen and Nelson (1975) following the publication of a paper by Schoen (1975) which has many similarities to the present effort, illustrates that the differences between the mathematical and the constructional approaches to the life table can generate misunderstandaings. This paper is of the second kind. In deference to the mathematically motivated, the basic model of an increment-decrement life table is presented as a set of differential equations, but the exposition is primarly directed toward methodo* This research was supported by National Institutes of Health, under NICHD Grant HD-07256 and NICHD Contract HD-S-2816, Center for Population Research. The author thanks anonymous referees for helpful suggestions. + Reprint address: Child Health and DeveIopment Studies, 3867 Howe Street, Oakland, Calif. 94611.
13 0040-5809/?9/040013-12$02.00/O Copyright 0 1979 by AcademicPress, Inc. AU rights of reproduction in any form reserved.
14
FRANK WM. OECHSLI
logical problems of life table construction. Moreover, the life table is treated deterministically in terms of expected values, largely following Key&z (1968a, pp. 22-24) rather than as a stochastic process as in Chiang (1968) or Hoem (1972). In the following, we will (1) outline some basic problems of incrementdecrement life table construction; (2) discussrecent literature on methods for constructing increment-decrement life tablesthat are claimed to be of a general nature, or which can be easily stated asgeneral methods; (3) present a general method for making increment-decrement life tables; and (4) illustrate that method using data on outcome of pregnancy as observedin women who come under medical observation at variable times following conception, and who may be lost to supervision before the outcome is known.
1. BASIC PROBLEMS OF INCREMENT-DECREMENT
LIFE TABLE
CONSTRUCTION
We may conceive of the life table as a mechanism for following a real or hypothetical population (cohort) over time and age in transition between two or more states.In the simplest situation, the ordinary life table, there are two states, living and dead, and we are interested in the transition, which is not reversible, from one to the other. In a somewhat more complex situation, say in a multiple decrement table incorporating causeof death, we must consider transitions to more than one final, absorbing state-the competing risk model. An even more complex situation might need to consider multiple transitions, such as advancesthrough a number of parity classses,or migrations between regionswhere all possibletransitions are possible. To demonstratethe issuesinvolved, it is useful to consider the caseof the single decrement mortality table. The usual data from which one constructs such a table is a set of age-specificmortality rates,
where ,M, is the central age-specific mortality rate for the age interval x to x + % ,a is the number of deaths at ages x to x + R (usually during the calendar year) and &YZ is the population aged x to x + n at risk (usually at midyear). Following Key&z (1968a, pp. 7-9) the life table may be represented by a differential equation,
44 - dx = -44 PM where I(X) is the number of living at exact age x and p(x) is the instantaneous
INCREMENT-DECREMENT
LIFE
15
TABLES
force of mortality at age x. A solution to the differential equation is a set of values of p(x) over the age range 0 to W, but in practice, one deals with finite . . approxrmatrons and the usual method of construction of the table is to derive, by one of several means available, a set of ,,qs, probabilities of dying in the age interval x to x + n, where R is some finite number (usually 5). The usual form of the data for an increment-decrement life table is a tabulation of events (transitions between states) classified by n-period age intervals (usually 5 years) and state (symbolized as ,E, ‘k*i), the transition from state k to state i in the age interval x to x + n), and a table of population exposed to risk of transition classified by n-period age intervals and state (symbolized as nKik’, population in state k at ages x to x + n). (A table of notation is given in Table 1.) From these we can compute age-state-specific transition rates: n
R(k.i) r
=
n
E(k.i),&k). x
TABLE
1
(1-l)
Notation Time
and
a finite
age-parenthesized,
a continuous
A following superscript; the reference state A leading which
both
subscript; a broad data are defined
A leading subscript; age- and state-specific Central ageage interval
as a following
from n (an
Population (an item
at risk of data)
of leaving
table
time
to states, interval,
when usually
both the
interval; the time of state transition
number number
living living
k to state
state item
i) occurring
are time
and
and state-specific rate of transition from from age 3c to x + n (an item of data)
(transitions x to x +
Cohort
refer
a narrow time probabilities
Events group
Life
variable;
subscript,
variable used, or age
k refers interval
age interval are graduated
in one
time
interval
for
to which
k to state
state
to
i in
the
for
age
of data)
k (in state
state
k at exact
in state in state
k) in the age interval
k at exact
s to x +
n
age x to x +
n
age x age .X (an
k in the
item
of data)
Life
table
person-periods
lived
~(~.“‘(s) h*y
The
force
of transition
(instantaneous)
A finite approximation to state i in the such a probability
to the preceding, the probability of transition from state k age interval from age x to x + h (when asterisked, refers to at a pivotal age through which a graduation curve is passed)
pi’ ax ,L,i, Gz
Adjustment
to correct
Life
factor
in state
from
:9&S,
from
age interval
state
k to state
from
i at exact
age JC
iteratively
table central age- and state-specific rate of transition from state k to state i in the age interval x to x + n; omitting suband superscripts, as long as R and r differ, p is adjusted by S, and I is recomputed; eventual agreement of R and r is a sign that a set of graduated p’s has been found that is consistent with (can reproduce) the central rates of the data
16
FRANK
WM.
OECHSLI
An increment-decrement life table allowing for complex transitions between m states may be represented as a set of m differential equations, one for each possible state K, k ranging from 1 to m:
i # k,
(1.2)
where (x) dx represents time and age, Fk)(x) represents the number living in state k at exact age X, and po.i)( x ) re p resents the force impelling transition from state i to state j at age x. The left-hand term on the right side of (1.2) represents entries to state k, and the right-hand term represents exits from state k. The mathematical task in constructing the table is to derive, somehow, a set of forces of transition, ~(“3~) (x), from the data provided, the set of n,Lk,i’. In practice, mathematical derivation is complicated, even for simple, single decrement mortality tables, and one deals with finite approximations, such as the probabilities of transition in the interval from age x to x + n, from state i to state j, ,p, oP’) . In the single decrement table, this is precisely the problem of deriving nq2 from the set of age-specific mortality rates, ,M, . Several ways of doing this are given in the literature, but note that it is not an acceptable solution to substitute ,M, directly for ,lqz, nor is an analogous procedure acceptable for an increment-decrement table, though this seems to have been done (Das Gupta, 1976). The three methods of increment-decrement life table construction to be described in the next sections take three different routes to the solution of the basic problem, but give similar results.
2. RECENT
WORK
ON
GENERAL LIFE
TABLE
METHODS
OF INCREMENT-DECREMENT
CONSTRUCTION
The first increment-decrement life table considered here is Andrei Rogers’ Multiregional Life Table (1973; 1975, Chap. 3). Rogers makes no explicit claim for generality of his method, the presentation being directed entirely to the ad hoc problems of devising a model which allows migration between m regions with different mortality conditions. However, the generalizability of the model is apparent if one merely substitutes the more general term “state” for “region”; the migration need not be strictly geographical. The kinship of Rogers’ model to the other two discussed here, which are presented as general, is borne out by the fact that he has compared his model with Schoen’s (Rogers and Ledent, 1976) and with Oechsli’s (Ledent and Rogers, 1972), using migration data. Rogers’ (1975, Chap. 3) methods are largely multi-region analogues of those of the standard actuarial literature with a strong influence from Chiang’s (1968)
INCREMENT-DECREMENT
LIFE
17
TABLES
life table methods. Chiang’s influence is apparent by Rogers’ use of ,agzi’, &), the average number of years spent in state i by those (a) remaining in i, ln3odthose (b) going to state j, in deriving transition probabilities. The use of the average number of years lived in the interval poses a serious limitation on the underlying assumptions required. Since the values of the a’s were not known, Rogers was required to assume that multiple transitions in the n-year interval were not permitted. This, indeed, is the chief drawback to his method, as he points out. Fortunately, at least as applied to migration data, it seems to make little difference, since comparison of his method with Schoen’s (Rogers and Ledent, 1976) and with Oechsli’s (Ledent and Rogers, 1972) showed minor differences. Schoen and Nelson (1974) presented a life table which differentiates a population by marital status, allowing for divorce and widowhood (as well as remarriage). Schoen (1975) generalized the method to apply to any increment-decrement table. The essence of the method is rather simply stated, though the solution of even fairly simple (three and four state) models involves some rather intricate, though straightforward, algebra. Schoen starts with the actuarial identity (notation as in Table 1): 8tk’zi-12= 1;’ + f &A)
- il ,e$‘)
for
i # k,
(2.1)
i=l
which may be expressed in words as: The number of living in state k at exact age x + ft is the number living in state k at exact age x plus the number of transitions from states i (fk), to state k in the age interval x to x + n minus the number of transitions from state k to states i (#k) in the same age interval (mortality is ignored, as it may be expressed as one of the states of interest). He then requires that the age-state transition rates (that are the data shown above in (1.1)) equal the age-state transition rates computed from the life table: ,$k,i) ,R(k,i) 2
=nz. n
L’k’ z
(2.2)
This requirement is Key&z (1968b) specification for a life table that agrees with the data. When one adds the definition of nLf$:
&’=s x+t * I(‘) dt
0
(2.3)
one gets a set of equations with a like number of unknowns which can be solved (somewhat laboriously) for the unknown Zkk’ once one has specified some analytic form for the curve in IL”. If the curve for Zkk’ is chosen as a sample trapeziodal interpolation, the number of equations and unknowns for each age interval is m3 + 2m, where m is the number of states permitted.
18
FRANK
WM.
OECHSLI
Two features of the logically elegant (though perhaps algebraically complicated) solution call for comment. The solution is entirely for a set 2;‘) without ever requiring a “force of transition,” such as the p(i*j)(~) given in (1.2) above, or of a finite approximation of it. It is as if, in construction of a simple singledecrement mortality table we solved directly for 1, without ever computing nqx Y the probability of dying in the age interval from x to x + n. The other feature of note is that the model does not, and apparently need not, explicitly take into account the possibility of multiple transitions in an age interval. Thus, if in real life a person may change states from single, to married, to divorced, and finally, to married, all within the age range x to x + n, that fact will not be (generally) observable in the data. In general, if the data do not permit identifying groups of individuals who make multiple transitions within a limited age interval, a model which attempts to make this distinction is perhaps overelaborate (and may well be misleading). The two features just referred to are probably related. It seems unlikely that one could work with n-year transition probabilities without being required to make restrictive assumptions on the number of transitions allowed. We have seen that Rogers’ use of the average number of years spent in the state by those ending in the state and those leaving the state required just such assumptions because the data were not available for estimating the a’s reasonably. Oechsli (1972, 1975) h as p resented a life table which follows a hypothetical cohort of women through marriage and, once married, through parity progression, allowing throughout for the decrements from mortality. As with Schoen, Keyfitz’ (1968b) principle of the life table that agrees with the data is applied. Except for that similarity, the methods differ. The following section gives the details of a general method for construction along the lines of Oechsli (1975) so those details need not be elaborated here. We can merely say that where Schoen works with algebraic relationships over broad age ranges, we work with detailed numeric operations on a finite approximation to the force of transition between states, p(“*Q(x), using interpolation to permit the use of time intervals small enough so that one can ignore the possibility of the occurrences of more than one event in the time interval. It was stated in Oechsli (1975) that the method is easily generalizable for the construction of any life table for which requisite data are available. This paper is that generalization.
3. A
GENERAL
METHOD
FOR
MAKING
INCREMENT-DECREMENT
LIFE
TABLES
Recall that the data available for the construction of the life table are a set of age-state-specific rates as in (1.1). The table itself is represented as a set of differential equations as in (1.2). A solution to the set of equations (1.2) is a set of curves of po*J)(x) for 1 < i, j < m given a distribution of P(x) at x = 0. It is usually difficult to derive,
INCREMENT-DECREMENT
LIFE
19
TABLES
by mathematical means, the po*j)(~) from the nEk’Vi) and the ,JCF’ (or nR~‘~l’), as in (l.l), but the data given impose a set of conditions on the solution to the differential equation. We may estimate, if we have some set of z&j)(x), the life table analogues of nEf95’, and nKz’, and ,,Rz*j’; that is: (3.1) Thus, we have a solution to the set (1.2) if we can find a set of @j)(x) that meet the condition that the expression in (3.1) equals (1.1). To simplify things, we now deal with finite approximations. A finite approximation to (1.2) is a difference equation for the finite short age interval x to x + h linking lLk’ and hphk>i’ where hpLkli’ is the probability of going from state K to state i in the fine increment of age and time, x to x + h: lzh
=
12)
+
1
l~)kp~.k)
_
C
lf)hpz,i),
i # k,
To convert (3.1) to a finite approximation, hefC’i) = l~)hPy). (k.i) ??a!
,
(3.2)
x = 0, h, 2h ,..., i2.
we note that we can get:
hLp = O.Sh[Z~’+ Z$]; x =
O,n, 2n,...,
(3.3) il.
The denominator of (3.3) is a simple trapezoid integration of ZAk’summedover the n-period interval to give ,LLk’, person-periods exposed to risk of leaving state K. Our task now is to find a set of hpik*i’ such that the central age-state-specific transition rates on an n-period interval as computed in (3.3) ,Y:~**) equal the data nRLkti) as computed in (1.1). For a simple table, this may often be done directly, given somesimplifying assumptions,but, generally, in an incrementdecrement table one cannot find easy meansto derive the probabilities from the rates. However, note that if one assumesa set of rates, one can tell from a comparisonof (3.3) and (1.1) whether the probabilities assumedagreewith the data (i.e., whether (1.1) and (3.3) agree), and by successiveapproximation, finalli arrive at a set of probabilities, hpLkfi), consistent with the n-period age-event-specific rates of (1.1). Following is a method for doing that. First, we select a pivotal age, x*, for each age interval x to x + n such that x < x* < x + n, and at each pivotal age choose an initial arbitrary pivotal probability of transition from state K to state i in the short interval x to x + h, X W,i) Next, we choose an interpolating function connecting up the pivotal hpm * probabilities. Such an interpolation function can be assimple asa step function, with constantprobabilities over the interval x to x + n, or ascomplex asingenuity (or theory) will allow.
20
FRANK
WM.
OEGHSLI
Given the interpolated transition probabilities, we may build up a whole table of I:’ using (3.2) and in the process accumulate the n-period central rates rCkli). These may be compared with the data central rates, ,R~kli’, and a correcyitn to the pivotal $pikPi’, may be found and applied: (3.4) where the second equation signifies replacement of $pkk,i’ by an improved value, and the function which defines the correction term is left unspecified at this general level of exposition. Given an improved set of transition probabilities, the process may be repeated as many times as is necessary to achieve a set of correction terms, 6kkYi’, that are negligible. Thus, we have derived a set of probabilities that are consistent with (can reproduce) the age-specific transition rates that are the data. In that sense, we have solved the difference equation (3.2) that is the finite approximation to the differential equation (1.2). In the foregoing, we have not discussed the issue of multiple transitions in an n-year interval; it was unnecessary since the transitions occur over h-year intervals in the model. If h is chosen small enough, the probability of the joint occurrence of two or more transitions may be neglected. However, we may easily derive an expression for the n-year interval probability of transitions between states that takes into account in an aggregate way, the possibilities of multiple transition. Consider a matrix, *Pz, elements of which are hp~V”; this is the h-year transition matrix. The n-year transition matrix is, simply, x+n-h 2,
=
n t=x
hpt.
This matrix, multiplied onto the vector of number living at age x by state lk”, will give the number living state vector at age x + 12. There are several increment-decrement table analogues of simple life table functions that may be computed in addition to l!$, ,L!$, hpLkli’, and ,P, , though these will not be discussed here. Moreover, if the life table is supplemented by a birth function, the stationary population of the life table is readily extended to a stable population allowing for births. The interested reader may find details for specific applications in Oechsli (1972, 1975) and Rogers (1975). ’ One more thing should be noted. The discussion so far assumes that the data that form the basis for the increment-decrement life table are in the form of age-state-specific central rates, as in Eq. (1.1). This presentation is in accord with the usual period data available, from which one usually makes a life table. However, data may come in other forms. One such form is to be given the state distribution of a real cohort at exact ages x, say at ages 0, n, 272,3n,..., w. In that case, one could define 8kkyi’ as a function of [lz’ - ~$1, where cc) is the observed cohort number in state i at exact age X, for use in (3.4) with no need at all to
INCREMENT-DECREMENT
LIFE
TABLES
compute (3.3). There are other possibilities, which will not be explored For a concrete example, the data for the table we will now use to illustrate principles, are cohort state distributions at exact ages X.
4. A
FETAL
Loss LIFE
21 here. these
TABLE
The life table chosen to illustrate the principles just discussed has been selected because a solution by more conventional means has been found, and it is possible to show explicitly how the two compare. The application is a life table analysis of fetal loss by Taylor (1970) with illustrative data on 16.170 pregnancies from the Child Health and Development Studies (CHDS). Women entred supervision when interviewed by a study interviewer at the time of the first visit during the pregnancy to the Ob-Gyn clinic of the Kaiser-Permanente Health Plan in Oakland, California, and were followed until they either suffered fetal loss, delivered a live-born child or children, or left supervision. A detailed discussion of the population can be found in Taylor (1970). Formally, a woman in the analysis may be in one of five states: (1) pregnant, not under supervision; (2) pregnant, under supervision; (3) not pregnant, having had a fetal loss; (4) not pregnant, having had a live birth; or (5) lost to supervision. We note that some women potentially eligible are never ascertained; a woman must be in state 2 at some time to enter the table. The data available in CHDS records allow complete daily reconstruction of all events to women who entered supervision and Taylor used probabilities of loss and survival on a daily basis. For our purposes, we will consider two data sets: state distributions at 7-day and also at 2%day intervals, dating from the first day of the last menstrual period (LMP) before pregnancy. Data for the 28-day interval distributions are shown in Table 2 in the C!$ columns. The only transitions allowed are from state 1 to state 2, and from state 2 to one of the states 3, 4, or 5. We will graduate the probabilities of transition to l-day intervals using a step function. That is, we will assume constant daily probabilities over the n-day interval, where n is 7 or 28 days. Consider first the transition from state 1 to state 2. In any interval from day x to day x + n, we know the number of women in state 1 on day x and the number on day x + n; the difference is the number of transitions in the interval. The problem of finding a daily probability, &*‘, to apply to daily I:” to give the correct Z,$ at the end of the interval, is directly soluble without any iteration; a complete set of correct ip,$“’ can be immediately applied to get the set of lf2’ z on a daily basis. From state 2, a woman can progress to any one of three states, but to simplify the problem, we will deal with the probability that a woman remains in state 2, IPZ(2y2) where 1Pz
(2,2)
=
1 -
1p$‘3) _ 1p$4) _ 1p;‘5).
(4.1)
22
FRANK
WM.
OECHSLI
TABLE Cohort
State Distribution Fetal Loss on Day
2
at 28-Day Intervals, and Probability x or Later, given Pregnant on Day x
of
pcs,s, s
Oechsli Day x
6
II,
(8, C,
(3) cc
(4, CL?
(61 c,
Taylor
0
0
0
0
-
16,159
11
0
0
0
15,553
606
9
0
2
84
10,783
5,214
154
0
112
6,835
8,970
324
140
4,408
11,252
168
2,842
196 224
No. 1 -____
No. 2
-
-
160.1
156.6
128.6
105.3
105.4
102.1
19
59.3
59.3
57.5
0
41
35.1
35.1
34.9
436
3
71
24.4
24.4
24.2
12,670
528
18
112
16.8
16.8
16.7
1,810
13,559
571
78
152
13.6
13.6
13.6
1,010
14,157
621
193
189
10.1
10.1
10.1 7.1
0
16,170
28 56
252
375
14,142
669
764
220
7.1
7.1
280
40
8,350
729
6,809
242
5.1
5.1
5.1
308
3
602
765
14,554
246
11.6
11.6
11.7
336
I
52
771
15,096
250
w
0
0
772
15,147
251
19.2 -
19.2 -
19.2 -
We need to find a set of daily probabilities of remaining in state 2 (constant over the n-day interval) from day x to x + n that will leave the correct number of women in state 2 on day x + n; that is, such that l& = C& . This could probably be solved directly without an inordinate amount of trouble, but it is just as convenient to find the answer by successive approximation. For successive n-day intervals, we assume arbitrary initial Tp$a’ and applying these day by day to the daily $a) estimate the terminal lz+a (‘) for each interval. We then compute a correction, 8~~” for opt” and repeat until 12:’ - et’ 1 < 0.1. In the present instance, we arbitrarily set all initial I*pc*‘) to 0.5 and set the initial correction factor for each n-day interval to 0.25, and after each iteration, we set ,y
= (0.5) t-p’;
Sgn = sign of [I,$’ - C?‘];
* (2.2) _ * (2.2) - Sgn[8S(2*2)] IA - IPZ 5 ?
(4.2)
INCREMENT-DECREMENT
LIFE
TABLES
23
where Sgn is -1 if 1:’ is greater than Cf) and +I if ZL2’is less than CL’). In other words, within each x to x + tt interval, we perform a binary search for a daily probability that gives the proper result. Once the values of *pc2*‘) are determined, the values of the transition probabilities to states 3,4, id25 from state 2 are straightforwardly obtained using the fact that they sum to 1 - TP’,~~’ and must be, respectively, proportional to the number of event transitions involved, nEF9s’, ,EF?*) and ,,EFp5’. Once the whole set of probabilities is defined, one would want, following Taylor, probabilities of fetal loss over the interval from day x to x + n, ,JJ~~~‘, given pregnant and supervised at day x, with the probability of being lost to supervision removed. We redefine the probabilities of remaining or leaving state 2 on the assumption that 1pF15’ is 0 for the states 2, 3, and 4 as ;+2A z
_
1Ps Cl
(2.i)
,
for
j = 2, 3,4.
Pk2*j'
That is, we remove the competing risk of loss to supervision, and using redefined probabilities of survival and fetal loss, compute: (2.3) A P2
Note that where A is large enough to span the whole pregnancy the probability given is the total probability of fetal loss during the pregnancy given that a woman is pregnant and supervised at day x. For purposes of comparison, we show in Table 2 the probability of fetal loss on day x or later, given pregnant and supervised on day x as computed by Taylor, and as computed by the present method, with constant probabilities over 7and 28-day periods (labeled Oechsli, No. 1, and Oechsli, No.2, respectively). There are only minor differences between Taylor’s results and the present method using constant probabilities over 7-day intervals; the differences are negligible after the second lunar month, and are zero after the third lunar month. Even when we assume constant probabilities over a lunar month, the comparison with Taylor’s results shows small differences after the second lunar month. It is clear from the numbers at risk in the second lunar month that the probabilities of fetal loss will be highly sensitive to the exact timing of observed fetal loss. The values of weekly probabilities of fetal loss in the second month as given by Taylor, for example, are 0, 40.4, 4.9, and 16.8 as against the present method (using weekly constant probabilities), which gives 0.0, 37.0, 6.8, and 14.3. The method here reported clearly has the effect of smoothing out daily and weekly fluctuations, when the numbers at risk are low. Hence it is conservative.
24
FRANK
WM.
OECHSLI
REFERENCES CHIANG, C. L. 1968. “Introduction to Stochastic Processes in Biostatistics,” Wiley, New York. DAS GUPTA, P. 1976. Age-parity-nuptiality-specific stable population that recognes births to single women, /. Amer. Statist. Assoc. 71, No. 354, 308-314. HOEM, J. M. 1972. Inhomogeneous semi-Markov processes, select actuarial tables, and duration-dependence in demography, in “Population Dynamics” (T. N. E. Greville, Ed.), Academic Press, New York. HOEM, J. M. 1975. The construction of increment-decement life tables: A comment on articles by R. Schoen and V. Nelson, Demography 12, No. 4, 661-662. KEYFITZ, N. 1968a. “Introduction to the Mathematics of Populations,” Addison-Wesley, Reading, Mass. KEYFITZ, N. 196813. A life table that agrees with the data, J. Amer. Statist. Assoc. 63, No. 324. LEDENT, J. AND ROGERS, A. 1972. “An Interpolative Iterative Procedure for Constructing a Multiregional Life Table,” MFMP Working Paper No. 9, Department of Civil Engineering, Northwestern University, Evanston, Ill. OECHSLI, F. W. 1972. “The Parity and Nuptiality Problem in Demography,” Ph. D. Thesis, University of California, Berkeley, Calif. OECHSLI, F. W. 1975. A population model based on a life table that includes marriage and parity, Theor. Pop. Biol. 7, No. 2, 229-245. ROGERS, A. 1973. The multiregional life table, J. Math. Sot. Japan 3, 127-137. ROGERS, A. 1975. “Introduction to Multiregilnal Mathematical Demography,” Wiley, New York. ROGERS, A. AND LEDENT, J. 1976. Increment-decrement life tables: A comment, Demography 13, No. 2, 287-290. SCHOEN, R. 1975. Constructing increment-decrement life tables, Demography 12, No. 2, 303-312. SCHOEN, R. AND NELSON, V. E. 1974. Marriage, divorce, and mortality: A life table analysis, Demography 11, No. 2, 267-290. SCHOEN, R. AND NELSON, V. E. 1975. Reply to Hoem, Demography 12, No. 4, 663-664. TAYLOR, W. F. 1970. The probability of fetal death, in “Congenital Malformations,” Proceedings of the Third International Conference. Excerpta Medica, Internation Congress Series No. 204.
POPULATION
A General
BIOLOGY
16, 13-24
Method for Constructing Increment-Decrement Life Tables That Agree with the Data* FRANK
Child
(1979)
WM.
OECHSLI
Health and Dmelopent Studies, School of Public University of California, Berkeley, California+ Received
December
Health,
12, 1977
A general method for making increment-decrement life tables is presented. The method involves the finding of probabilities of transition between states, graduated to small intervals of time and age, that are consistent with (i.e., can reproduce) the data, whether the data consist of central age-state specific rates, or some other feature, such as state distributions of a real cohort. The method is then illustrated with a fetal loss life table.
Papers on life tables tend to be either highly abstract, mathematical modeIs which are relatively unconcerned with the practical problems of life table construction, or very much concerned with the concrete problems of methodology of construction with little regard for the mathematicaI niceties that characterize papers of the first sort. The first kind of paper is recognizable by its density of integral/differential equations in contrast to an almost complete absence of numerical illustrations and real data. The second kind, conversely, is characterized by its concentration on details of methods of construction accompanied by one or more examples of the method as carried out with real (or sometimes hypothetical) data, in contrast to a sometimes complete absence of mathematics more complex than simple algebra. An exchange of views by Hoem (1975) and Schoen and Nelson (1975) following the publication of a paper by Schoen (1975) which has many similarities to the present effort, illustrates that the differences between the mathematical and the constructional approaches to the life table can generate misunderstandaings. This paper is of the second kind. In deference to the mathematically motivated, the basic model of an increment-decrement life table is presented as a set of differential equations, but the exposition is primarly directed toward methodo* This research was supported by National Institutes of Health, under NICHD Grant HD-07256 and NICHD Contract HD-S-2816, Center for Population Research. The author thanks anonymous referees for helpful suggestions. + Reprint address: Child Health and DeveIopment Studies, 3867 Howe Street, Oakland, Calif. 94611.
13 0040-5809/?9/040013-12$02.00/O Copyright 0 1979 by AcademicPress, Inc. AU rights of reproduction in any form reserved.
14
FRANK WM. OECHSLI
logical problems of life table construction. Moreover, the life table is treated deterministically in terms of expected values, largely following Key&z (1968a, pp. 22-24) rather than as a stochastic process as in Chiang (1968) or Hoem (1972). In the following, we will (1) outline some basic problems of incrementdecrement life table construction; (2) discussrecent literature on methods for constructing increment-decrement life tablesthat are claimed to be of a general nature, or which can be easily stated asgeneral methods; (3) present a general method for making increment-decrement life tables; and (4) illustrate that method using data on outcome of pregnancy as observedin women who come under medical observation at variable times following conception, and who may be lost to supervision before the outcome is known.
1. BASIC PROBLEMS OF INCREMENT-DECREMENT
LIFE TABLE
CONSTRUCTION
We may conceive of the life table as a mechanism for following a real or hypothetical population (cohort) over time and age in transition between two or more states.In the simplest situation, the ordinary life table, there are two states, living and dead, and we are interested in the transition, which is not reversible, from one to the other. In a somewhat more complex situation, say in a multiple decrement table incorporating causeof death, we must consider transitions to more than one final, absorbing state-the competing risk model. An even more complex situation might need to consider multiple transitions, such as advancesthrough a number of parity classses,or migrations between regionswhere all possibletransitions are possible. To demonstratethe issuesinvolved, it is useful to consider the caseof the single decrement mortality table. The usual data from which one constructs such a table is a set of age-specificmortality rates,
where ,M, is the central age-specific mortality rate for the age interval x to x + % ,a is the number of deaths at ages x to x + R (usually during the calendar year) and &YZ is the population aged x to x + n at risk (usually at midyear). Following Key&z (1968a, pp. 7-9) the life table may be represented by a differential equation,
44 - dx = -44 PM where I(X) is the number of living at exact age x and p(x) is the instantaneous
INCREMENT-DECREMENT
LIFE
15
TABLES
force of mortality at age x. A solution to the differential equation is a set of values of p(x) over the age range 0 to W, but in practice, one deals with finite . . approxrmatrons and the usual method of construction of the table is to derive, by one of several means available, a set of ,,qs, probabilities of dying in the age interval x to x + n, where R is some finite number (usually 5). The usual form of the data for an increment-decrement life table is a tabulation of events (transitions between states) classified by n-period age intervals (usually 5 years) and state (symbolized as ,E, ‘k*i), the transition from state k to state i in the age interval x to x + n), and a table of population exposed to risk of transition classified by n-period age intervals and state (symbolized as nKik’, population in state k at ages x to x + n). (A table of notation is given in Table 1.) From these we can compute age-state-specific transition rates: n
R(k.i) r
=
n
E(k.i),&k). x
TABLE
1
(1-l)
Notation Time
and
a finite
age-parenthesized,
a continuous
A following superscript; the reference state A leading which
both
subscript; a broad data are defined
A leading subscript; age- and state-specific Central ageage interval
as a following
from n (an
Population (an item
at risk of data)
of leaving
table
time
to states, interval,
when usually
both the
interval; the time of state transition
number number
living living
k to state
state item
i) occurring
are time
and
and state-specific rate of transition from from age 3c to x + n (an item of data)
(transitions x to x +
Cohort
refer
a narrow time probabilities
Events group
Life
variable;
subscript,
variable used, or age
k refers interval
age interval are graduated
in one
time
interval
for
to which
k to state
state
to
i in
the
for
age
of data)
k (in state
state
k at exact
in state in state
k) in the age interval
k at exact
s to x +
n
age x to x +
n
age x age .X (an
k in the
item
of data)
Life
table
person-periods
lived
~(~.“‘(s) h*y
The
force
of transition
(instantaneous)
A finite approximation to state i in the such a probability
to the preceding, the probability of transition from state k age interval from age x to x + h (when asterisked, refers to at a pivotal age through which a graduation curve is passed)
pi’ ax ,L,i, Gz
Adjustment
to correct
Life
factor
in state
from
:9&S,
from
age interval
state
k to state
from
i at exact
age JC
iteratively
table central age- and state-specific rate of transition from state k to state i in the age interval x to x + n; omitting suband superscripts, as long as R and r differ, p is adjusted by S, and I is recomputed; eventual agreement of R and r is a sign that a set of graduated p’s has been found that is consistent with (can reproduce) the central rates of the data
16
FRANK
WM.
OECHSLI
An increment-decrement life table allowing for complex transitions between m states may be represented as a set of m differential equations, one for each possible state K, k ranging from 1 to m:
i # k,
(1.2)
where (x) dx represents time and age, Fk)(x) represents the number living in state k at exact age X, and po.i)( x ) re p resents the force impelling transition from state i to state j at age x. The left-hand term on the right side of (1.2) represents entries to state k, and the right-hand term represents exits from state k. The mathematical task in constructing the table is to derive, somehow, a set of forces of transition, ~(“3~) (x), from the data provided, the set of n,Lk,i’. In practice, mathematical derivation is complicated, even for simple, single decrement mortality tables, and one deals with finite approximations, such as the probabilities of transition in the interval from age x to x + n, from state i to state j, ,p, oP’) . In the single decrement table, this is precisely the problem of deriving nq2 from the set of age-specific mortality rates, ,M, . Several ways of doing this are given in the literature, but note that it is not an acceptable solution to substitute ,M, directly for ,lqz, nor is an analogous procedure acceptable for an increment-decrement table, though this seems to have been done (Das Gupta, 1976). The three methods of increment-decrement life table construction to be described in the next sections take three different routes to the solution of the basic problem, but give similar results.
2. RECENT
WORK
ON
GENERAL LIFE
TABLE
METHODS
OF INCREMENT-DECREMENT
CONSTRUCTION
The first increment-decrement life table considered here is Andrei Rogers’ Multiregional Life Table (1973; 1975, Chap. 3). Rogers makes no explicit claim for generality of his method, the presentation being directed entirely to the ad hoc problems of devising a model which allows migration between m regions with different mortality conditions. However, the generalizability of the model is apparent if one merely substitutes the more general term “state” for “region”; the migration need not be strictly geographical. The kinship of Rogers’ model to the other two discussed here, which are presented as general, is borne out by the fact that he has compared his model with Schoen’s (Rogers and Ledent, 1976) and with Oechsli’s (Ledent and Rogers, 1972), using migration data. Rogers’ (1975, Chap. 3) methods are largely multi-region analogues of those of the standard actuarial literature with a strong influence from Chiang’s (1968)
INCREMENT-DECREMENT
LIFE
17
TABLES
life table methods. Chiang’s influence is apparent by Rogers’ use of ,agzi’, &), the average number of years spent in state i by those (a) remaining in i, ln3odthose (b) going to state j, in deriving transition probabilities. The use of the average number of years lived in the interval poses a serious limitation on the underlying assumptions required. Since the values of the a’s were not known, Rogers was required to assume that multiple transitions in the n-year interval were not permitted. This, indeed, is the chief drawback to his method, as he points out. Fortunately, at least as applied to migration data, it seems to make little difference, since comparison of his method with Schoen’s (Rogers and Ledent, 1976) and with Oechsli’s (Ledent and Rogers, 1972) showed minor differences. Schoen and Nelson (1974) presented a life table which differentiates a population by marital status, allowing for divorce and widowhood (as well as remarriage). Schoen (1975) generalized the method to apply to any increment-decrement table. The essence of the method is rather simply stated, though the solution of even fairly simple (three and four state) models involves some rather intricate, though straightforward, algebra. Schoen starts with the actuarial identity (notation as in Table 1): 8tk’zi-12= 1;’ + f &A)
- il ,e$‘)
for
i # k,
(2.1)
i=l
which may be expressed in words as: The number of living in state k at exact age x + ft is the number living in state k at exact age x plus the number of transitions from states i (fk), to state k in the age interval x to x + n minus the number of transitions from state k to states i (#k) in the same age interval (mortality is ignored, as it may be expressed as one of the states of interest). He then requires that the age-state transition rates (that are the data shown above in (1.1)) equal the age-state transition rates computed from the life table: ,$k,i) ,R(k,i) 2
=nz. n
L’k’ z
(2.2)
This requirement is Key&z (1968b) specification for a life table that agrees with the data. When one adds the definition of nLf$:
&’=s x+t * I(‘) dt
0
(2.3)
one gets a set of equations with a like number of unknowns which can be solved (somewhat laboriously) for the unknown Zkk’ once one has specified some analytic form for the curve in IL”. If the curve for Zkk’ is chosen as a sample trapeziodal interpolation, the number of equations and unknowns for each age interval is m3 + 2m, where m is the number of states permitted.
18
FRANK
WM.
OECHSLI
Two features of the logically elegant (though perhaps algebraically complicated) solution call for comment. The solution is entirely for a set 2;‘) without ever requiring a “force of transition,” such as the p(i*j)(~) given in (1.2) above, or of a finite approximation of it. It is as if, in construction of a simple singledecrement mortality table we solved directly for 1, without ever computing nqx Y the probability of dying in the age interval from x to x + n. The other feature of note is that the model does not, and apparently need not, explicitly take into account the possibility of multiple transitions in an age interval. Thus, if in real life a person may change states from single, to married, to divorced, and finally, to married, all within the age range x to x + n, that fact will not be (generally) observable in the data. In general, if the data do not permit identifying groups of individuals who make multiple transitions within a limited age interval, a model which attempts to make this distinction is perhaps overelaborate (and may well be misleading). The two features just referred to are probably related. It seems unlikely that one could work with n-year transition probabilities without being required to make restrictive assumptions on the number of transitions allowed. We have seen that Rogers’ use of the average number of years spent in the state by those ending in the state and those leaving the state required just such assumptions because the data were not available for estimating the a’s reasonably. Oechsli (1972, 1975) h as p resented a life table which follows a hypothetical cohort of women through marriage and, once married, through parity progression, allowing throughout for the decrements from mortality. As with Schoen, Keyfitz’ (1968b) principle of the life table that agrees with the data is applied. Except for that similarity, the methods differ. The following section gives the details of a general method for construction along the lines of Oechsli (1975) so those details need not be elaborated here. We can merely say that where Schoen works with algebraic relationships over broad age ranges, we work with detailed numeric operations on a finite approximation to the force of transition between states, p(“*Q(x), using interpolation to permit the use of time intervals small enough so that one can ignore the possibility of the occurrences of more than one event in the time interval. It was stated in Oechsli (1975) that the method is easily generalizable for the construction of any life table for which requisite data are available. This paper is that generalization.
3. A
GENERAL
METHOD
FOR
MAKING
INCREMENT-DECREMENT
LIFE
TABLES
Recall that the data available for the construction of the life table are a set of age-state-specific rates as in (1.1). The table itself is represented as a set of differential equations as in (1.2). A solution to the set of equations (1.2) is a set of curves of po*J)(x) for 1 < i, j < m given a distribution of P(x) at x = 0. It is usually difficult to derive,
INCREMENT-DECREMENT
LIFE
19
TABLES
by mathematical means, the po*j)(~) from the nEk’Vi) and the ,JCF’ (or nR~‘~l’), as in (l.l), but the data given impose a set of conditions on the solution to the differential equation. We may estimate, if we have some set of z&j)(x), the life table analogues of nEf95’, and nKz’, and ,,Rz*j’; that is: (3.1) Thus, we have a solution to the set (1.2) if we can find a set of @j)(x) that meet the condition that the expression in (3.1) equals (1.1). To simplify things, we now deal with finite approximations. A finite approximation to (1.2) is a difference equation for the finite short age interval x to x + h linking lLk’ and hphk>i’ where hpLkli’ is the probability of going from state K to state i in the fine increment of age and time, x to x + h: lzh
=
12)
+
1
l~)kp~.k)
_
C
lf)hpz,i),
i # k,
To convert (3.1) to a finite approximation, hefC’i) = l~)hPy). (k.i) ??a!
,
(3.2)
x = 0, h, 2h ,..., i2.
we note that we can get:
hLp = O.Sh[Z~’+ Z$]; x =
O,n, 2n,...,
(3.3) il.
The denominator of (3.3) is a simple trapezoid integration of ZAk’summedover the n-period interval to give ,LLk’, person-periods exposed to risk of leaving state K. Our task now is to find a set of hpik*i’ such that the central age-state-specific transition rates on an n-period interval as computed in (3.3) ,Y:~**) equal the data nRLkti) as computed in (1.1). For a simple table, this may often be done directly, given somesimplifying assumptions,but, generally, in an incrementdecrement table one cannot find easy meansto derive the probabilities from the rates. However, note that if one assumesa set of rates, one can tell from a comparisonof (3.3) and (1.1) whether the probabilities assumedagreewith the data (i.e., whether (1.1) and (3.3) agree), and by successiveapproximation, finalli arrive at a set of probabilities, hpLkfi), consistent with the n-period age-event-specific rates of (1.1). Following is a method for doing that. First, we select a pivotal age, x*, for each age interval x to x + n such that x < x* < x + n, and at each pivotal age choose an initial arbitrary pivotal probability of transition from state K to state i in the short interval x to x + h, X W,i) Next, we choose an interpolating function connecting up the pivotal hpm * probabilities. Such an interpolation function can be assimple asa step function, with constantprobabilities over the interval x to x + n, or ascomplex asingenuity (or theory) will allow.
20
FRANK
WM.
OEGHSLI
Given the interpolated transition probabilities, we may build up a whole table of I:’ using (3.2) and in the process accumulate the n-period central rates rCkli). These may be compared with the data central rates, ,R~kli’, and a correcyitn to the pivotal $pikPi’, may be found and applied: (3.4) where the second equation signifies replacement of $pkk,i’ by an improved value, and the function which defines the correction term is left unspecified at this general level of exposition. Given an improved set of transition probabilities, the process may be repeated as many times as is necessary to achieve a set of correction terms, 6kkYi’, that are negligible. Thus, we have derived a set of probabilities that are consistent with (can reproduce) the age-specific transition rates that are the data. In that sense, we have solved the difference equation (3.2) that is the finite approximation to the differential equation (1.2). In the foregoing, we have not discussed the issue of multiple transitions in an n-year interval; it was unnecessary since the transitions occur over h-year intervals in the model. If h is chosen small enough, the probability of the joint occurrence of two or more transitions may be neglected. However, we may easily derive an expression for the n-year interval probability of transitions between states that takes into account in an aggregate way, the possibilities of multiple transition. Consider a matrix, *Pz, elements of which are hp~V”; this is the h-year transition matrix. The n-year transition matrix is, simply, x+n-h 2,
=
n t=x
hpt.
This matrix, multiplied onto the vector of number living at age x by state lk”, will give the number living state vector at age x + 12. There are several increment-decrement table analogues of simple life table functions that may be computed in addition to l!$, ,L!$, hpLkli’, and ,P, , though these will not be discussed here. Moreover, if the life table is supplemented by a birth function, the stationary population of the life table is readily extended to a stable population allowing for births. The interested reader may find details for specific applications in Oechsli (1972, 1975) and Rogers (1975). ’ One more thing should be noted. The discussion so far assumes that the data that form the basis for the increment-decrement life table are in the form of age-state-specific central rates, as in Eq. (1.1). This presentation is in accord with the usual period data available, from which one usually makes a life table. However, data may come in other forms. One such form is to be given the state distribution of a real cohort at exact ages x, say at ages 0, n, 272,3n,..., w. In that case, one could define 8kkyi’ as a function of [lz’ - ~$1, where cc) is the observed cohort number in state i at exact age X, for use in (3.4) with no need at all to
INCREMENT-DECREMENT
LIFE
TABLES
compute (3.3). There are other possibilities, which will not be explored For a concrete example, the data for the table we will now use to illustrate principles, are cohort state distributions at exact ages X.
4. A
FETAL
Loss LIFE
21 here. these
TABLE
The life table chosen to illustrate the principles just discussed has been selected because a solution by more conventional means has been found, and it is possible to show explicitly how the two compare. The application is a life table analysis of fetal loss by Taylor (1970) with illustrative data on 16.170 pregnancies from the Child Health and Development Studies (CHDS). Women entred supervision when interviewed by a study interviewer at the time of the first visit during the pregnancy to the Ob-Gyn clinic of the Kaiser-Permanente Health Plan in Oakland, California, and were followed until they either suffered fetal loss, delivered a live-born child or children, or left supervision. A detailed discussion of the population can be found in Taylor (1970). Formally, a woman in the analysis may be in one of five states: (1) pregnant, not under supervision; (2) pregnant, under supervision; (3) not pregnant, having had a fetal loss; (4) not pregnant, having had a live birth; or (5) lost to supervision. We note that some women potentially eligible are never ascertained; a woman must be in state 2 at some time to enter the table. The data available in CHDS records allow complete daily reconstruction of all events to women who entered supervision and Taylor used probabilities of loss and survival on a daily basis. For our purposes, we will consider two data sets: state distributions at 7-day and also at 2%day intervals, dating from the first day of the last menstrual period (LMP) before pregnancy. Data for the 28-day interval distributions are shown in Table 2 in the C!$ columns. The only transitions allowed are from state 1 to state 2, and from state 2 to one of the states 3, 4, or 5. We will graduate the probabilities of transition to l-day intervals using a step function. That is, we will assume constant daily probabilities over the n-day interval, where n is 7 or 28 days. Consider first the transition from state 1 to state 2. In any interval from day x to day x + n, we know the number of women in state 1 on day x and the number on day x + n; the difference is the number of transitions in the interval. The problem of finding a daily probability, &*‘, to apply to daily I:” to give the correct Z,$ at the end of the interval, is directly soluble without any iteration; a complete set of correct ip,$“’ can be immediately applied to get the set of lf2’ z on a daily basis. From state 2, a woman can progress to any one of three states, but to simplify the problem, we will deal with the probability that a woman remains in state 2, IPZ(2y2) where 1Pz
(2,2)
=
1 -
1p$‘3) _ 1p$4) _ 1p;‘5).
(4.1)
22
FRANK
WM.
OECHSLI
TABLE Cohort
State Distribution Fetal Loss on Day
2
at 28-Day Intervals, and Probability x or Later, given Pregnant on Day x
of
pcs,s, s
Oechsli Day x
6
II,
(8, C,
(3) cc
(4, CL?
(61 c,
Taylor
0
0
0
0
-
16,159
11
0
0
0
15,553
606
9
0
2
84
10,783
5,214
154
0
112
6,835
8,970
324
140
4,408
11,252
168
2,842
196 224
No. 1 -____
No. 2
-
-
160.1
156.6
128.6
105.3
105.4
102.1
19
59.3
59.3
57.5
0
41
35.1
35.1
34.9
436
3
71
24.4
24.4
24.2
12,670
528
18
112
16.8
16.8
16.7
1,810
13,559
571
78
152
13.6
13.6
13.6
1,010
14,157
621
193
189
10.1
10.1
10.1 7.1
0
16,170
28 56
252
375
14,142
669
764
220
7.1
7.1
280
40
8,350
729
6,809
242
5.1
5.1
5.1
308
3
602
765
14,554
246
11.6
11.6
11.7
336
I
52
771
15,096
250
w
0
0
772
15,147
251
19.2 -
19.2 -
19.2 -
We need to find a set of daily probabilities of remaining in state 2 (constant over the n-day interval) from day x to x + n that will leave the correct number of women in state 2 on day x + n; that is, such that l& = C& . This could probably be solved directly without an inordinate amount of trouble, but it is just as convenient to find the answer by successive approximation. For successive n-day intervals, we assume arbitrary initial Tp$a’ and applying these day by day to the daily $a) estimate the terminal lz+a (‘) for each interval. We then compute a correction, 8~~” for opt” and repeat until 12:’ - et’ 1 < 0.1. In the present instance, we arbitrarily set all initial I*pc*‘) to 0.5 and set the initial correction factor for each n-day interval to 0.25, and after each iteration, we set ,y
= (0.5) t-p’;
Sgn = sign of [I,$’ - C?‘];
* (2.2) _ * (2.2) - Sgn[8S(2*2)] IA - IPZ 5 ?
(4.2)
INCREMENT-DECREMENT
LIFE
TABLES
23
where Sgn is -1 if 1:’ is greater than Cf) and +I if ZL2’is less than CL’). In other words, within each x to x + tt interval, we perform a binary search for a daily probability that gives the proper result. Once the values of *pc2*‘) are determined, the values of the transition probabilities to states 3,4, id25 from state 2 are straightforwardly obtained using the fact that they sum to 1 - TP’,~~’ and must be, respectively, proportional to the number of event transitions involved, nEF9s’, ,EF?*) and ,,EFp5’. Once the whole set of probabilities is defined, one would want, following Taylor, probabilities of fetal loss over the interval from day x to x + n, ,JJ~~~‘, given pregnant and supervised at day x, with the probability of being lost to supervision removed. We redefine the probabilities of remaining or leaving state 2 on the assumption that 1pF15’ is 0 for the states 2, 3, and 4 as ;+2A z
_
1Ps Cl
(2.i)
,
for
j = 2, 3,4.
Pk2*j'
That is, we remove the competing risk of loss to supervision, and using redefined probabilities of survival and fetal loss, compute: (2.3) A P2
Note that where A is large enough to span the whole pregnancy the probability given is the total probability of fetal loss during the pregnancy given that a woman is pregnant and supervised at day x. For purposes of comparison, we show in Table 2 the probability of fetal loss on day x or later, given pregnant and supervised on day x as computed by Taylor, and as computed by the present method, with constant probabilities over 7and 28-day periods (labeled Oechsli, No. 1, and Oechsli, No.2, respectively). There are only minor differences between Taylor’s results and the present method using constant probabilities over 7-day intervals; the differences are negligible after the second lunar month, and are zero after the third lunar month. Even when we assume constant probabilities over a lunar month, the comparison with Taylor’s results shows small differences after the second lunar month. It is clear from the numbers at risk in the second lunar month that the probabilities of fetal loss will be highly sensitive to the exact timing of observed fetal loss. The values of weekly probabilities of fetal loss in the second month as given by Taylor, for example, are 0, 40.4, 4.9, and 16.8 as against the present method (using weekly constant probabilities), which gives 0.0, 37.0, 6.8, and 14.3. The method here reported clearly has the effect of smoothing out daily and weekly fluctuations, when the numbers at risk are low. Hence it is conservative.
24
FRANK
WM.
OECHSLI
REFERENCES CHIANG, C. L. 1968. “Introduction to Stochastic Processes in Biostatistics,” Wiley, New York. DAS GUPTA, P. 1976. Age-parity-nuptiality-specific stable population that recognes births to single women, /. Amer. Statist. Assoc. 71, No. 354, 308-314. HOEM, J. M. 1972. Inhomogeneous semi-Markov processes, select actuarial tables, and duration-dependence in demography, in “Population Dynamics” (T. N. E. Greville, Ed.), Academic Press, New York. HOEM, J. M. 1975. The construction of increment-decement life tables: A comment on articles by R. Schoen and V. Nelson, Demography 12, No. 4, 661-662. KEYFITZ, N. 1968a. “Introduction to the Mathematics of Populations,” Addison-Wesley, Reading, Mass. KEYFITZ, N. 196813. A life table that agrees with the data, J. Amer. Statist. Assoc. 63, No. 324. LEDENT, J. AND ROGERS, A. 1972. “An Interpolative Iterative Procedure for Constructing a Multiregional Life Table,” MFMP Working Paper No. 9, Department of Civil Engineering, Northwestern University, Evanston, Ill. OECHSLI, F. W. 1972. “The Parity and Nuptiality Problem in Demography,” Ph. D. Thesis, University of California, Berkeley, Calif. OECHSLI, F. W. 1975. A population model based on a life table that includes marriage and parity, Theor. Pop. Biol. 7, No. 2, 229-245. ROGERS, A. 1973. The multiregional life table, J. Math. Sot. Japan 3, 127-137. ROGERS, A. 1975. “Introduction to Multiregilnal Mathematical Demography,” Wiley, New York. ROGERS, A. AND LEDENT, J. 1976. Increment-decrement life tables: A comment, Demography 13, No. 2, 287-290. SCHOEN, R. 1975. Constructing increment-decrement life tables, Demography 12, No. 2, 303-312. SCHOEN, R. AND NELSON, V. E. 1974. Marriage, divorce, and mortality: A life table analysis, Demography 11, No. 2, 267-290. SCHOEN, R. AND NELSON, V. E. 1975. Reply to Hoem, Demography 12, No. 4, 663-664. TAYLOR, W. F. 1970. The probability of fetal death, in “Congenital Malformations,” Proceedings of the Third International Conference. Excerpta Medica, Internation Congress Series No. 204.
