-------
number of deaths in the i Interval, jr., are recorded; and
2) the yi are preassigned and the boundaries of the Intervals, x,
to x.., are observed.
"hh
The classical estimate of mortality at the midpoint of the i in-
terval is
,co . ifi ._!£..
*4 i I Iff -i. IT
JL o \*»j •* •"
where T. « x, - - x. is the interval width and N. is the number alive at
time x.. Another estimate is
+ T . log
v 94 T J-IT>.
Xi 2 i T±
A numerical study of these estimates showed the bias to be a function of all
the parameters, assuming a Gorapertz mortality function.
For the second type of sampling, the estimate used is
and it is unbiased for M-. if M- + T » M-^ for 0 < i < T, i.e., constant
JL ***4 ~~
mortality in the interval. If |i. increases linearly over the interval,
50
-------
the estimate
where: A. and B. are complicated functions of the statistics is proposed.
The four estimates are applied to statistics on time- to- death of rats
exposed to radiation. Differences of up to 8 percent between the four
estimates are found. Comparison due to different partitions is not dis-
cussed. Plots of p. and M- are given and show several dips.
In his paper, "A New Response Time Distribution," Kbdlin [1967] gives
an increasing mortality rate, M-(t) as
Mt) m c + let
with probability density function, f(t), as
f(t) - (c+ kt)e-c, t,c,k>0 ,
for the time-to-death of an individual under a specific treatment. The mean
and variance are obtained. The maximum likelihood equations are given for
grouped data with censored trails, using: n = number of individuals in a
sample; d. » number dying at t.; s. ° number alive at f .; and, T =
time of loss or end of follow-up.
31
-------
The model is applied to 121 breast cancer patients to compute the
excess risk by vising only those that die from causes other than cancer. The
overall mortality is decreasing and cannot be used with the model.
A second model for mortality, Mt), is proposed
Mt) - - ±- - + c + kt
8 e * +1-8
as are some "crude" estimators.
Interpretation of f(t) for various parameters is given, as applied
to medical follow-up studies.
Mantel [1966] proposes the overall comparison of two survival func-
tions, requiring a value function for rating the duration of survival. A
chi-square procedure is proposed with an implicit value function. The study
begins with a population of N.^ homogeneous patients. In the i inter-
val of the study, r^ . die and t.. are lost track of. The estimated prob-
ability of surviving to the J period, then, is
N - r
li rli
where N - R^ - rn
A chi-square test, based on a contingency table is proposed.
-------
In their paper, "Life Tests Under Competing Causes of Failure and the
Theory of Competing Risks," Moeschberger, et al [1971] use the nonnegative
random variables, y,, to denote the lifetime of a subject due to cause-of-
failure, Ct, I ** 1,...,k. It is assumed that y. » min. y. and C. are
* 1 K * *
observed. Parametric distributions are assumed, in particular, a Weibull
distribution with ungrouped data are desired. Results with this method are
smoother than similar tests made using nonparametrlc or partially nonpara-
metric methods.
Murthy [1965a] in a paper entitled "Estimation of Probability Density,"
defines a function k(x) as a window, if
1) k(x) > 0;
2) k(x) » k(-x);
3) lia x fc(x) - 0; and
k) r^ k(x)dx - 1.
It is shown that the estimate
fn(t) -J Bn k(Bn(x-t)) dFn(x)
where F (x) is the empirical distribution and 1) is a sequence of non-
negative constants with B -» °° and B_/m -» •» as n -» °° is consistent
at a point of continuity of f(x).
In another paper reviewing the "Estimation of Jumps, Reliability and
-------
Hazard Rate," Murthy [I965b] defines the reliability estimator
as having the large sample mean, R(t), and variance R(t)(l - (t)), which
is asymptotically normal. The hazard rate estimator is defined as
fn(t)
Zjt) - n
R (
where R (t) = - times the number of observations greater than t among
x., . ..,XQ, and where fn(t) is as defined by Murthy in [1965a] and is
shown for large samples to be consistent. The estimator
«?<*> ' WTtT
n
is shown to be consistent and asymptotically normal.
Murthy and Swartz [1971] in their report "Contributions to Reliabil-
ity," present various realistic models for life distributions in Chapter IV
that correspond to density functions and that can be expressed in closed
form. Some methods of estimation, using order statistics, are also
suggested.
Two factors are considered in Sampford's [1952] paper on the estima-
tion of response-time distributions. They are:
-------
I. fundamental concepts and general methods, and
II. multi-stimulus distributions.
A normal distribution of the response time is assumed and the experi-
ment is carried through to completion. Maximum likelihood methods are de-
valorped for grouped data. A second part, considers estimation of parameters
in the distribution of response time when two causes of death are operating
on the subject. The main model assumes that the joint action of the two
causes of death is independent and the density of the primary cause is normal•
Various densities for the secondary cause of death are considered, including
those of the exponential and normal types.
Swartz, in his paper "The Mean Residual Lifetime Function," [19731
establishes conditions and properties on a function which characterize that
function as a mean residual lifetime function.
The elementary properties of hazard functions are discussed by Watson
and Leadbetter in "Hazard Analysis, I" [1964]. Estimation of hazard rates
using windows is presented with the estimator being
n
Z
n-r+1
and
p 00 p 00
E[h*(x)l - / 5(x-y)h(y)dy - / 8 (x - y) F°(y) h(y)dy
n J0 n JQ n
35
-------
with an expression for Var (h*) given also. 8 is a window function.
Some less sophisticated estimators are given: a graphical derivative method,
a histogram estimator, and the classical actuarial estimator. Death dates
on mice, exposed to gamma radiation, are used to obtain hazard estimates by
various methods.
Zelen [1966] explores the application of statistics and the role it
can play by the increased use of probability models to guide the researcher
in the types of data to collect. Suggestions for further experiments and as
a frame of reference are discussed. The time-to-failure is assumed to have
a survival function
r e-Mt-G), t > G
S(t) - \ ,
L 1 , t < G
where G is the guarantee time. The mean is G + (l/A). This model is
applied to animal tumor systems and acute leukemia, destruction of tumor
cells with laser energy and the analysis of survival data, with concomi-
tant information. In the last case, it is assumed that time-to-failure has
density
-At
» A. e for all t > 0
for the i patient, and that l/^ » a + bx. where x. is the
36
-------
concomitant variable, such as the logarithm of white "blood count, as in the
sample subjects with acute myelogenous leukemia.
37
-------
CHAPTER III
Realistic Models for Mortality Rates and
Their Estimation
Introduction
In this chapter mathematical models are advanced which, as special
cases, represent constant, increasing and decreasing mortality rates, along
with combinations of these properties. Bath-tub shaped mortality rate curves
are the general shape of these models. The first part of the tub corresponds
to infantile mortality, the second part (more or less, constant mortality)
corresponds to useful life, and finally, the last part, which is increasing
corresponds to decay, aging, etc., culminating in death. Their corresponding
probability distributions and survivorship functions are obtained in closed
form. To the best of our knowledge, the closed form representations of sur-
vivorship functions corresponding to the wide class of forces of mortality,
considered herein are new to the literature relevant to mortality rates and
their estimation. The variety of mortality shapes derived are illustrated
and applied. Estimation is performed by several methods: empirical, various
window-smoothing, and maximum likelihood techniques.
Estimation of the Hazard Rate Function
Observed times-to-failure can be used to estimate the hazard rate
function (mortality rate or risk will also be used, interchangeably). Non-
parametric methods for estimating hazard rates were considered by Murthy [196
and Grenahder [1956]. However, the most they can do is to throw light on the
shape of the underlying hazard rate function and cannot be used for decision-
making and prediction. Thus, parametric procedures based on realistic models
-------
for the hazard rate function are the only suitable) approaches for estimation
and prediction. It is generally accepted that the most general hazard rate
curve consists of an early decreasing phase, followed by a section of constant
failure (corresponding to the useful life period), followed by a rapidly in-
creasing failure rate as death approaches.
Theory
Let the nonnegative random variable, T, 'be the time to death of the
subject. Let T have a cumulative distribution function (cdf)
F(t) - P(T < t) (l)
and assume that T has a mean, JA, variance, * „ and a probability density
function
f(t)dt - dF(t) . (2)
The mortality rate, Mt), is the instantaneous chance of dying at time t,
given that the subject has survived to time t and: is defined as
where
H(t) - 1 - F(t) 00
is/the survivorship function. That Mt) is a mortality rate can be seen by
finding the probability of death in the time interval (t, t + At), where
-------
At is a small positive time increment, given survival to time t. We have
from the definition of conditions! probability that
P(tt) F(t+At) F(t)
P(T > t) " ~~Rft}*
By the mean value theorem of calculus, F(t + At) - F(t) « f U)At , for some
6 between t and t + At, hence
P(t < T < t + Y > *) - fjlj A*> t < 5 < t + At . (6)
In the limit, as At —> 0, the conditional probability is A(t)dt.
The mortality rate also uniquely defines the survivorship function,
as can be seen by writing the difference equation for the probability of
death by time t + At:
F(t-t- At) - F(t) + [l-F(t)] Mt)dt . (7)
The right-hand side is the sum of the probabilities of two mutually exclusive
events,
l) death by time t with probability F(t); and
2) no death until time t and death in the interval (t, t + At),
with probability [l-F(t)]X(t)At.
Dividing both sides by At and taking the limit as At —> 0 gives
-------
- [1 - F(t)] Mt) (8)
uv
which can be written as
Mt) - - £ In R(t) (9)
Solving this differential equation for R(t) and using the initial condition
R(o) =1 and R(«) - 0, we obtain
-/* Mx)dx
R(t) - e ° (10)
The integral A(t) « /Q A(x)dx is also called the accumulated, or cumulative
hazard by time t; its derivative Mt) is the hauard rate.
Another interpretation of the mortality rates is to compute the ratio
of expected deaths in the interval (t, t + At) to the expected number of
subjects alive at time t. Let n(t) be the expected number alive at time
t from an original population of N « n(0), then
n(t + At) - N[R(t + At) - R(t)] ; (ll)
dividing both sides by At, and taking the limit as At —> 0, we have
» - -Nf(t) » -N 1 [1 - F(t)] - -n(t) Mt) (12)
or
-------
This differential equation may be solved for n(t), producing
-/* Mx)dx
n(t) » N e .
The mortality rate is sometimes referred to as a conditional density;
00
however, X(t) is not a density since /_ Mx)dx * *• ^(*) is> in fact,
f(t/(T > t)) which is a conditional density only when x » t, but is not a
conditional density as t varies.
We are concerned in this chapter with selecting a mortality function
that models a medical follow-up study and that also can be integrated to give
a survivorship function in closed form. Murthy and Swartz [1971] show that
any function, h(x), which is positive and for which
rt
lim / h(x)dx (15)
t -»* 0
is unbounded, can be used as a mortality rate which will give a unique surviv-
orship function
-/* h(x)dx
R(t) » e u . (16)
Three models are discussed in the following section.
Realistic Models of the Mortality Rate
MODEL I; This class of distributions is characterized by the hazard
1*2
-------
rate
Mt) - - + c dt*"1, t > 0 (17)
where the parameters, a,b,c > 0 and d > 1. Some special cases of this
model are: when c - 0,
a , a decreasing function of time . (18)
1 + b
The mortality rate reduces to a constant when a » 0, d • 1, or c « 0,
b » 0 and we have an exponential distribution. When b - 0, d = 2, we have
the linear rate used by Kodlin [1967] to model death from breat cancer.
Finally,
Mt) c c dt1"1 when a - 0 , (19)
corresponds to a two-parameter Weibull distribution.
The cumulative distribution function for Model I is
(1 + bx)f
and the density function is
f (t) - ( r^ + c dt*-1 ) ( 3^-75 ectd ) . (21)
\1+bt /\ (H- bt)a/b /
-------
MODEL II; This class of distributions has the hacard rate
Mt) - ab e"bt -i- c dt*"1; d > 1, a,b,c > 0 . (22)
The corresponding cumulative distribution function is
F(t) = 1 - e'a(l'e )"ct (23)
and the density is
f(t) » (ab e" + c dt " ) e ~e
n
The special case of (24), where a « 1/b , may be of interest since it is a
three parameter family of distributions with the bath-tub property. This
class of distribution is given by
---
F(t) » 1 - e b (25)
with density function
f(t) - V-+ cdt*'1 ( eb . (26)
Note that the two-parameter Weibull and Kbdlin's models are also special cases
of this general model.
-------
MODEL III: Ibis class has the hazard rate!
Mt) - j~+bt* cde J a,b,c,d>0 . (27)
The corresponding cdf and density are, respectively, given by
F(t) -x' ^rt^' (28)
and
f(t) - -£_+,*. e-'e- . (29)
V1+bt A(i+bt)a/tl /
When a » 0 and d > 0, Mt) » cde , which is a monotonlcally increasing
hazard rate called the Gompertz rate by Broadbent [1958] and applied to the
lifetimes of milk bottles. This rate is studied by Garg, et al [1970] and is
used to model mortality of mice in oral contraceptive studies.
Application to Medical Follov-Up Studies
The mortality rates corresponding to many real life situations fall
within the domain of our models. From the practical, point of view, we dis-
tinguish among three categories of mortality rates: BTR, DHR and ME,
depending on their respective shape. Category BTR is marked by a decreasing
risk of mortality function, followed by an increasing one. Category DHR is
purely decreasing, while that of IHR is increasing. The classical example of
-------
a bath-tub hazard curve is the survival data of humans (as well as other
animals). It Is characterised by a high Infant mortality during the first
year of life, slowly decreasing risk from ages 1 through k, a low mortal-
ity rate from ages 5 to around 35, and increasing risk for older ages, as
shown In Figure 1.
Usually in statistical analysis of data a cumulative distribution
function is assumed. The random variable under study, in this case, the
time-to-death, can in many situations be described by a mathematical model
which also determines the functional form of the distribution function.
Zelen [1966] has given several such models for death due to specific types of
cancer. These models lead him to consider the exponential distribution.
Davis [1952] showed that the exponential model gave a good fit In the case of
thirty nonhuman systems. Unfortunately, the exponential distribution has a
constant mortality rate which may be a reasonable assumption in the case of
certain short periods of time. That it is not the case for time periods over
five years, in medical follow-up studies on humans, was shown by Berg and
Bobbins [1967]* In the absence of a mathematical model to guide us in the
selection of a distribution, the concept of the mortality rate has been
adopted. Buckland [1964] discusses many examples of hazard rates that cor-
respond to the Weibull, gamma, normal, log normal, and extreme value distri-
butions. Boag [19^9] considered the Weibull, gamma, log normal and exponen-
tial distributions in a study on deaths from cancer after treatment. He
found that the best fit to be given by the log normal distribution; and, for
1*6
-------
a large group, suggested that the survivorship be modeled by a mixture of c
percent completely cured and (l-c) percent whose survival times follow a
log normal distribution. Gehan [1969] showed that the mortality rate of the
log normal distribution was inappropriate for medical follow-up studies and
that a bath-tub rate was necessary.
The DHR class of distributions characterize situations where there is
an Initial high risk which decreases with time. Over a limited time period,
this might describe survival data of patients after transplants or implanta-
tions of mechanical devices, besides the usual infantile mortality in a
"normal" situation.
The IHR class of distributions fit situations where a very high mor-
tality rate is reduced and the patient is given a longer time to live, but is
not "cured." This class may be a useful approximation to the mortality rate
over the 100 odd years of human life, because the high mortality rate of the
final five years is several orders of magnitude greater than the initially
high infant death rate. Broadbent [1958], Garg, et al [1970], Kbdlin [1967]
and Grenander [1956] consider this type of hazard.
To demonstrate the versatility of the three models proposed in this
chapter, with respect to their ability to describe the various mortality
rates that one encounters in real life, an interactive graphics program was
written so that a person at the console can select one of the models and the
parameter values. The corresponding hazard function is displayed on the
screen. If desired, increments of the parameters can be typed in, and plots
-------
«J
>s
o
s
1
Infantile
mortality
Useful life period
(Polsson failures)
Chance and
fatigue
mortality
B
-»• Length of life T (age)
Typical mortality rate as a function of age
"Figure 1.
-------
of these curves displayed, one after the other, as the parameters Increase or
decrease. Thus, the characteristics of each model can be fully illustrated.
Pictures of some of these plots are shown in Chapter k of this report.
Life Table Model
Assume a homogeneous collection of people. Let the time-to-death of
a person in the group be a positive random variable, T, with hazard z(t)
of either our model I, II or III. At a particular time, we have recorded n
time-to-deaths, t, < ••• < t , of the first n people to die. One problem
which arises is to estimate the parameters in the model. Another is to com-
pare the hazard of one such group with another.
Follow-Up Model
In this model, a fixed number of people, n, at the same age, all
are assumed to have similar mortality rates when observed for a fixed length
of time, L, and their times-to-death are recorded as, t, < t, < ••• < t ,
j. — & — — r
where r is an integer-valued random variable r < n. (if r < n, we refer
to the sample as being truncated.) The problem is to estimate the model para-
meters and to compare the hazard with other groups.
A probabilistic model of the preceding models is obtained by letting a
positive random variable, T, be the time-to-death of an Individual and as-
suming that the individual times-to-death are independent and identically dis-
tributed random variables with common survivorship;, R(t). A random sample
-------
of size n of the random variable T will be denoted by t., . ..,t . In the
first model, the times are observed in order, and hence, correspond to the
order statistics, t/.,\ where t/.\ is the i order statistic. The three
models will be fitted, by nonlinear least-square technique, to the estimated
hazard.
Hazard rate, Mt), defined as
R(t * R(t)
50
-------
suggests the estimator, ^ (t):
z (t ) L— . ±-i (33)
n i n. n n.
where n. « No.Cj : t. > t.) is the number of persons alive at time, t.,
n. . « Wo.(j : t. > t. + h) is the number alive at time, t. + h, and m. =
i+n j i 11
n. - n. . is the number who die in the time interval (t. , t. , ). Because
i n-n i i+h
the ratio of random variables is involved, it has not been possible to evalu-
ate the expected value of this estimator.
Another estimator is provided by the definition,
Mt) - - £ In R(t) , (3^)
X(t) - - lim * - n . (35)
h-*0 h
This suggests the estimator, which we call the empirical estimator:
In n. , - In n. ..
For small m., This estimate approaches Z _(t.) of formula (33). The fol
lowing program segment will compute zn(*.i ) for time-to- death dates, in
days, stored in order in array, T(l), with TOT = n'^ and SUM = n. + h.
51
-------
DO 100 I-1,H
TOT = 0
SUM-0
DO 200 J«I,N
IF(T(J).GT.T(I)) TOT»TOT+I
200 ZF(T(J).GT.(T(I>H)) suM
IP(SUM.LE.O.) SUM«1
100 ZN(I)»ALOG(TOT/SUM)/H.
We compute the expected value of the empirical estimator as
EZ (t.) - - £
n 1 n
(37)
Since n. has a binomial distribution
(n - k)
J ) Rk(ti)
(58)
we have
EZ (t. ,h) » - J
n l n
In k( J )(G(ti
(39)
where
G(t) - Rk(t)
The expected value, as h is made arbitrarily small, is
EZ(t.) = lim EZ(t.,h)
n i
Onkf^G'd^) . (1H)
Evaluating G'(t), we obtain
-------
f (t.)
EZn(ti) - £7—^ b(n, t±t 9^ , ... , 0) (1*2)
1
where b(«) is a bias factor that depends on n, t. and the parameters
G ,...,0 of the survival and is expressed as:
b(n,t1,e1,...,op)
n (U5)
- £ in k ( J ) Kk(t1)(l-H(t1))B"k.(k-iiR(t1)) .
If the time-to-death has an exponential survival, R(t) = e~ ' , t > 0,
it was shown by Epstein and Sobel [1953] that a minimum variance unbiased
estimator of 6 is 0, where
T/.x +•••••• T/.x + (n-i)T/. x
5\ -L / \ 1 / \ l /
B 1 n. i T > ^
Since the hazard A(t) = 1/6, we propose the estimator Z (t.),
Zn(t.) - i . (45
n 1 g
Preliminary investigation of this estimate, under different survivorships, is
not encouraging as the estimator exhibits severe bias.
Estimation of densities by using "windows" has been applied to hazard
rate estimation by Murthy [1965]. A window K(w) is a function satisfying
the conditions
55
-------
1. K(w) » K(-w);
2. K(w) > 0;
3. /"„ K(w) - 1; and
4. lim wK(w) « 0 .
|w| -»«°
Murthy [1965] shows that the estimator, called the window estimator,
f*(t )
' (lt6>
R*(t) = J fj(x)dx
"C
For a rectangular window
r l/2h , -h < w < h
K(w)
otherwise
1 o,
we have the rectangular window estimator
is a consistent estimator for Mt), where
f*(t) - / K(ln n(t. - t.))
n n i_j j i
and
p 00
(W)
-------
n
(t)
n
No. I
V*>
+ —
where Set A » (j ; t - n^ < t. < t+ 1^), and 1^ » h/(ln n).
The expression is derived in the following way. Let
D(W) • In nK(ln n - w) .
Then
D(w)
r
0 , otherwise
hence,
(50)
(51)
l*(t)
D(t - t) .
(52)
Since, D(t. - t) is 1/2^ when |t. - t| < h., the sum counts the number
of elements in the Set A « (j : t-h, < t. < t+h^l. The estimate of R(t)
given by /. f*(x)dx which, upon substituting w » t. - x,
« n j
r^"*
f*(x)dx - / D(w)dw
n J „
, t j - t > hx
, -l^ < t.j - t
, otherwise
(55)
55
-------
we obtain the denominator of Z (t).
A computational simplification of the window estimators was suggested
by Watson and Leadbetter [1964]. me denominator of the window estimator was
replaced by the estimator, Rn(*.i ) •" 1 - i/(n+ l). Using the density esti-
mator f*(t) suggested by Murthy [1965], we consider the window estimator
VV
n + 1
Since the denominator is not a random variable, we can write the expected
value of this as,
EZn(ti) - n n * * ± In n E ) k(ln n(T. - tj))
J-l
(55)
L L
(t)dt ,
where f . (t) is the density of the J order statistic
J
fj(t) - J( *-* ) p^Ct) R^^Ct) f(t) . (56)
This is applied for rectangular and triangular windows to the three sets of
model data.
A triangular window estimator is obtained when
56
-------
0 , w < -h
JL + 1
J. + h
-h < w <0
K(w)
+ = , 0 < w < h
,2 T h
h < w
Let D(W) » In nK(v In n), then,
D(w)
0 , w <
-^ + vT > -hn < w < 0
v,^ h. i — -
hl 1
J*
-•%+&• 0
-------
The reliability estimator is
- x)dx
^ rV*
£ ) / D(u)du ,
11 £_! J_ «>
(61)
where
r*^"*
/ D(u)du
w _ oo
o , tj-t<-h1
(t4-t)2 (t4-t) ,
V^
2h,
bl 2
<*-t<0 » (62)
0 < t.-t < 0
J ~
< t. - t
J
and, the hazard rate estimator is
(63)
_i_ y
n h. A
1 -
1
n
JcA,
vhere SGN(x) » x/ |x|, (i.e., the sign of x).
-------
The simplified triangular-window estimator is
__
nhi
ZA:
Test Results
The empirical estimator, given in (36), the simple rectangular-window
estimator of (5^)> and the simple triangular-window estimator of (6k) were
used on a set of randomly generated data from a known truncated exponential
distribution and on a set of 200 times-to-death of people from the Los
Angeles area* The time-to-death data of people WBJS obtained from the Los
Angeles Public Health Cooperative Data File. We wish to thank Dr. Anne
Coulson of the School of Public Health, UCLA, for her assistance in obtaining
a random sample of 200 people from that file. The date of birth and month
and day of death in 1970 was received from cards; if the year of birth was
not given, the age at death was subtracted from 1970 and used for the year of
birth. The birth date and death date were converted to days and subtracted to
give time-to-death in days. A FORTRAN subroutine, CALL DATE (I YEAR, J
MONTH, K YEAR, N DAYS), supplied by Steven Chasen of the Health Sciences
Computing Facility was used to make the conversion and account for leap years.
The 200 exponential, random numbers were generated by a FORTRAN sub-
routine, CALL R GAMMA (l.,15000.,200,v), developed by Frane and Murthy
59
-------
[197?]* The mean was 15,000 days and the variables were truncated at
U3,800 days. The hazard is constant at A = 6.7.10 , with a spike at
120 years.
A sort routine, CALL SORT (A,l,200) developed by Steven Chasen was
used to put the data in ascending order.
Programs for the estimators were tested for numerical correctness by
using uniformly generated data with t, » l82(i - l), 1 < i < 200; the
hazard is l/(c - t), 0 < t < c, with c - 36,218 days and *(o) •
2.75.10"5.
The results of the three estimators are shown in Figures 6-12. The
time-to-death in days was divided by 365 to give time-to-death in years and
the hazard was multiplied by 10 to provide clarity in the presentation.
Various window widths, h, were used; the effect of h on the estimate is
shown in Figure 5»
60
-------
. '93 *
800 »
Model I
TOO *
*.
600 *•
H
A
I
A
R 500 «•
0
400 »
*
300 »
200 +
100 *
. ! PPP P
o +. ooo n
0
-5 5
' tml •
0 »
SIMPLE TRIANGULAR WINDOW
h= 10,000 p - estimated point *
o ~ data point 0 .
* - both *
'• :
0 . .
' • V *
• 4-
o • .
r 00 I
0 .
0
• o
0 »
0
• . o . .1
oo • .1
00 • P
000 PP P P »l
ppp»»ppppp .
ppppppp»*pppp*noo .1
PPPPP PP PPPPPPPPP 000000
PP p p p PPPP PPP PP p p oonoonnooooono .
oo o oo ooou onn oo o ocarina do nn »
10 20 30 40 50 60 7O 80 90 ^(J0
15 . 25 35 45 55 65 /5 85 9^
Figure 2
-------
1125 »
- . .
. 1000 +
*
875 »
Model II
750 »
• '
625 *
A
Z 500 +
A
R
D,75 ;
•
250 *
125 I
0 > *** •
•
-125 »
0
-5 5
P
0 »
:
SIMPLE TRIANGULAR WINDOW 0 I
h=10,000 p - estimated point *
•*. •
o - data point ' .
0
* - both p *
;
* .
"*
p , ..
0 . • *
• ' •
00 .
p
* ' .
• »
* .
PPO
P 0 .
POO
**o »
**o .
p**
0****0
0000********* • .
** * * * **** *** ** •* ****** ** *****«ppppp «
'•
10 20 30 40 50 60 70 8C 90 1JO
15 25 35 *B S5 65 ~ " '/!> Bi '" 75 ~
AGE(YRS)
Figure
-------
o\
H
A
Z
A
R
D
• ' ^ • • • • •
'1000 +
•
•
P7-5 <•
Model lU SIMPLE TRIANGULAR WINDOW
•
. • H = 10,000 p - estimated point
I o - data point
• • •
625 I * ' both
•
•
•
•
•
. 373 *
•
•
•
253 »
•
125 »:
•
. a * *•• * ** « * * •«»* *•• »* * ««•*»»
•
•
•
0 10 20 30 40 50
•
f
. . m
o .;''
• . •
•
- . •
*
a . •
•
*
•
*
0
•. • »
p
•
•
»
* '
* •
• •
0 . +
00 I
— •
. *
* *
PO
PPO
•P*0
• •0 »
• »p I
0*****
00***********? .
»* *»»»*»pp ^
•
'•
•
60 70 80 90 100
15
25
5
55
65
75
85
9»
AG6IVPSI
Figure
-------
10-
8-
Zn(t). 6-
I xlO"5-
T r i n r
40 60 80
YEARS
TIME-TO-DEATH IN LOS ANGELES
I SI
Zn(tj)s2htn JEAj
where Aj = |j: /tj-ti/
-------
2n (t) EMPIRICAL Z EXPONENTIAL DATA
100-i
80-
60-
40-
' 1 h= 1000
i i i i i i i
Theoretical
T I
80 100
YEARS
TIME-TO-DEATH EXPONENTIAL
Figure 6
65
-------
2n (t) SIMPLE TRIANGULAR WINDOW EXPONENTIAL DATA
= 10,000
Theoretical
100
TIME-TO-DEATH EXPONENTIAL
Figure 7
66
-------
2n ft) SIMPLE RECTANGULAR WINDOW
120-n
100-
80-
Zn(t)
60-
40-
20-
lOxlO'5 -
!_•-
h = 10,000
h= 1,000
i i I i I IIT i \
20 40 60 80 100
YCARS
TIME-TO-DEATH IN LOS ANGELES
Figure 8
67
-------
Zn tt) SIMPLE TRIANGULAR WINDOW
100-
80-
Zn(t)
60-
40-
.h = 10,000
YEARS
TIME-TO-DEATH IN LOS ANGELES, n=200
Figure 9
68
-------
Z't) EMPIRICAL
n
Zn(t)
100-1
80-j
60-1
40-1
0
20
Zn (tj)— In
n i n
n|
h= 1,000 days
•40
YEARS
TIME-TO-DEATH DATA
Figure 10
-------
Zn ft) SIMPLE TRIANGULAR WINDOW
10-
6-
4-
2-
I xlO'5
0
*«r
1
}
20
40
60
i i
80
100
YEARS
TIME-TO-DEATH IN LOS ANGELES
Figure 11
70
-------
Zn (t) SIMPLE RECTANGULAR EXPONENTIAL
40n
20H
lOxlO"5
h=IO,000
i heoretical
0-h
0
\\l\\\ \.\ \
20 40 60 80
YEARS
TIME-TO-DEATH EXPONENTIAL
100
j?*ig\irt 12.
-------
CHAPTER k
Applications of the Models and an Example
Real Life Examples to Which the Model Can Be Applied
Die mortality rates corresponding to many real life situations fall
within the domain of our models. From the practical point of view, we dis-
tinguish among three categories of mortality rates. A, B and C, depending
on the shape. Category A is marked by a decreasing risk function, followed
by an increasing one. Category B is purely decreasing, while category C is
increasing.
In the biological field, the classical example of a bath-tub-hazard
curve — category A — is the survival data of humans (as well as other
animals). It is characterized by: high infant mortality during the first
year of life with slowly decreasing risk from age 1 through k, and the
lowest mortality rate starting from age 5 to around 35 » increasing risk
is seen as age increases above the 35 year level (see Figure 1 of Chapter 3).
This category can be used for survival data pertaining to many var-
ied diseases, as long as they have this general pattern. For example, Gehan's
[1969] study was an application of category A, in which F(t) is called the
distribution function associated with the mortality rate A(t) and
B(t) - l-F(t) . ey-y (1)
is called the survivorship function. The density function corresponding to
(l) is given by
72
-------
f(t) -
Ul.
It is clear from Equations (l) and (2) that there is a one-to-one
correspondence between the mortality rate on the one hand and the distribution
and survivorship functions on the other. If we can discover mortality func-
tions Mt) with the bath tub property and obtain the integral y(t) -
/* Mt)dt in a closed form, Equations (l) and (2) will give the correspond-
ing distribution and survivorship functions, respectively, in closed form. In
what follows, we obtain families of distributions whose corresponding mortal-
ity rates have the above desired properties.
Statistical Models with Bath-Tab Property
MODEL I; Hiis class of distributions is characterized by the hazard
rate
Mt) - IT^bt + c dfcd"1» * - ° (5)
where the parameters a,b,c > 0, d > 1. Some special cases of this model
are: when c » 0,
a decreasing function of time.
When
73
-------
a • 0, d • 1, X(t) » c
c - 0, b • 0, Mt) •* a
Here, the mortality rate reduces to a constant and we have an exponential dis-
tribution. When b « 0, d » 2, we have the form described by Kbdlin [1967]:
that form the observed data on survival of males with angina pectoris, the
death rate per yeat is highest in the first year after diagnosis. For the
next nine years, the hazard function remains fairly constant and increases
somewhat later.
Category B curves characterize situations where there is an initial
high risk which decreases slowly as time progresses. Survival data on
patients after surgery, transplant operation, or other medical treatments that
would Improve the chances of survival, belong to this category.
Category C is marked by an increasing risk function. Experiments that
introduce harm to initially healthy animals, or sick control groups that are
unattended are good examples for this. Garg, et al [1970] and Kodlin [1967]
gave numerical examples of this type in their papers.
Graphics Program
We shall now demonstrate the versatility of the models considered in
this chapter, with respect to their ability to describe the various mortality
rates that one encounters in real life. To this end, an interactive graphics
program was written in such a way that a person at the console can select one
of the models at a time, and type in different values of the parameters
-------
through a keyboard. The corresponding hazard function will then be displayed
on the screen. If desired, different increments of the parameters can also
be typed in and plots of these curves will be displayed one after the other,
as the parameters increase or decrease by the specified amount each time.
Thus, the characteristics of each model can be fully illustrated easily,
Maximum Likelihood Procedure for Estimation
The method of maximum likelihood is discussed here. Since similar
techniques are applied to all three models, more detailed explanations will
be given for Model I, while only brief summaries of equations will be given
for Models II and III.
MODEL I
Let t-,tg,...,t be a random sample of sine N from the distribu-
tion F(t) given by 00. The likelihood of the observed sample is
— e
1-1 (l+bti)a/b
f r
a. \ ••_/•. . ^^j \ _ \ ^d
an i - - 5 / m(i + bti) - c ^ ti(
1-1
(7)
75
-------
The following equations can be solved, vising a,£,c and & that will maxi-
mize (7); the second partial derivatives given by:
a2 in L V2
s— ° ~ b
U2(b-l)
In L
-"
N
I
In L
ad
In L
« - a
ZN
In
- ab
f1 *2i°
1 + b In t
J
ctt)
ctt)
-1 In tj,(2 + b In t±)
- a
InL
ob oc
ad
b In
(1
76
-------
*1.
InL
l 2d
ct
. d
Ct . ^ HI- —
1 c
-H 2d
[X(t1)](i+
f
+ d >
-------
and
In L
3d
- c
tJ in t,
+ d In
where
Mt±)
(8)
With the aid of a computer, these equations can be solved iteratively. Init-
•
ial estimates can be obtained from nonlinear least-squares fit to the empiri-
cal hazard function.
Estimates of the asymptotic variances and covariances of the maximum
M
estimates may be
_
-
-------
MODEL II
The equations Involved in solving for Model II are, briefly:
In L - ) lir(ab
ctj (10)
with
-bt.
In L
-bt,
(1 -
-bt.
0
1 ab e
din L
f e'*tl(l " bti} V
1 ab e + cdt1" 1
and
din L
dc
Ld-l
Li -bt.
1 ab e + cdt
"1
4
0
t 1(1 + d In- t.)
8d
- c > t" In t^ - 0
1 ab e + cdt.
79
-------
MODEL III
InL » > In
J
1 + bt,
+ cd e
- c
In 1 + bt,
(11)
vith
la L
~aT~
& " J
a dti
at,
dt,
1 ( , .\t + cd e '* ]( 1
0 ;
? I Ki+bti)
b L> H-bt,
0 ;
dt,
In L
de
dt,
bt,
cd e
and
In L
ZN c e i(H- dt.) ^
— ^r - °t
1 , .a. . •»• c de ^^ 1
1 +• Dt;.
dt.
80
-------
A Numerical Example
A hypothetical cohort of 1,000 people born in the United States in
the year 1900 was made up, using information recorded in Vital Statistics of
the United States. (The probability of dying for ages > 70 was approximated
using the values in the latest 1967 of the aforementioned reference.) The
survival data are presented below and the empirical hazard function was cal-
culated using
Mt J . '«! .
mi h.(l + p.)
where
t . • midpoint of the ith interval;
mi
h± - width of the ith interval;
q. = conditional proportion dying in the i
interval; and
p. = conditional proportion surviving in the
ith interval;
as described by Kimball [I960].
Nonlinear least-squares fits to the empirical hazard function were
attempted for all three models. (Computations were performed using BMIK8^,
Nonlinear Least Squares, a package program written at the Health Sciences
Computing Facility, UCLA.) The estimated parameters and hazard functions
are shown in Table 2.
It was found that Model I seems to fit the data best, so the method
81
-------
of.maximum likelihood was performed, giving estimates as follows:
a - 1.59^ E 01
£ » 2.612 E 02
c - 8.635 E-ll
£ • 6.299 E oo
The value of the log likelihood (Eq. 7), using maximum likelihood estimates
is -0.4378E04, compared to -0.4413E04 using the least-squares estimates.
(12)
TABLE 1
U.S. SURVIVAL DATA
Mt) ° 2q/(h(l+p)) where h « year interval, according to Kimball's method.
Age
N
Dead
q
P
pp
Mt)*
0- 0.99
1- 4.99
5-1^.99
15-24.99
25-3^.99
35-^.99
45-5^.99
55-64.99
65-74.99
75-84.99
85-9^.99
1000
838
799
776
738
703
666
609
503
314
66
162
39
23
38
35
37
57
106
189
248
66
0.1620
0.0465
0.0288
0.0490
0.0474
0.0526
0.0856
0.1740'
0.3757
0.7898
1.0000
0.8380
0.9535
0.9712
0.0951
0.9526
0.9474
0.9144
0.8260
0.6243
0.2102
0.0000
0.83800
0.79903
0.77602
0.73800
0.70302
0.66604
0.60903
0.50306
0.31406
0.06602
0.00000
0.176270
0.011900
0.002922
0.005023
0.004855
0.005402
0.008942
0.019058
0.046259
0.130534
0.200000
82
-------
TABLE 2
U.S. SURVIVAL DATA
(Hazard Function)
Midpoint
Year
0.5
3.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80,0
90.0
Observed
0.17630
0.01190
0.00292
0.00502
0.00485
0.0051*0
0.00894
0.01906
0.01*626
0.13052
0.20000
Model I
a - 1.9872E 01
b « 2.2764E 02
c - 2.4398E-12
d - 6.1851E 00
Model I
0.17307
0.02906
0.00873
0.00445
0.00360
0.00524
0.01147
0.02649
0.05693
0.11237
0.20591
Model II
a • 2 .8035E-01
b - 1.0781E 00
c » 3.9204E-12
d • 6.0841E 00
Model II
0.17630
0.01191
0.00001
0.00010
0.00077
0.00333
0.01036
0.02617
0.05730
0.11299
0.20563
Model III
a - 3.2761E 01
b • 3.7340E 02
c • 1.153UE-20
d - 4.9992E-01
Model III
0.17453
0.02922
0.00877
0.00439
0.00292
0.00219
0.00175
0.00146
0.00126
0.00244
0.20087
Least-squares estimation on Mt).
-------
CHAPTER 5
Computer Program for Graphics
The computer program described in this section was used to display, on
a single plot, both the mortality rate and the mean residual lifetime for
three realistic models of the mortality rate. This program was written in
FORTRAN IV and was run on an IBM 360/91 at the Health Sciences Computing
Facility, UCLA.
The mean residual lifetime, v(t), may be defined as
r ffe
Jt *(t
where •R(t)»l-F(t) is the survivorship function and F(t) the cumulative
distribution function. For Model I, the mortality rate, Mt) and the sur-
vivorship function R(t) are
t", t > 0, a,b,c > 0, d > 1
~~ — —
and
R(t) -
. + bt)a/D
For Model II, the equations are
Mt) - abe"bt + cdt*"1, t > 0, d > 1, a,b,c > 0
and
R(t)
84
-------
Finally, for Model III the equations are
Mt) « * + cdedt, t > 0, a,b,c,d>0
and
_/dt
R(t)
_
(1+ bt)a/b
The mortality functions for the three models are computed in the sub-
routine HAZ listed below. The value of LM may be 1, 2 or 3, and refers
to the appropriate model number with HAZ being the mortality rate value at a
given time.
FUNCTION HAZ(T)
DOUBLE PRECISION A,B,C,D,Z,X
COMMON LM
COMMON/PARS/A, B, C , D
X-DBLE(T)
GO TO (l,2,e),LM
1 CONTINUE
Z»A/(L.+B*X)
HAZ-&4-C*D*X**(D-l. )
RETURN
2 CONTINUE
Z-A*B*DEXP(-B«X)
RETURN
3 CONTINUE
Z=A/(l.+B*X)
RETURN
END
The survivorship functions for the three models are computed in the double
precision subroutine FUN listed below. The value of LM may be 1,2 or e
and refers to the appropriate model number with FUN being the survivorship
-------
function value at a given time.
DOUBLE PRECISION FUNCTION FUN(X)
DOUBLE PRECISION X,A,B,C,D,EX,BX
COMMON LM
COMMON/PARS/A,B,C,D
GO TO (l,2,3),LM
1 CONTINUE
EX»C*X**D
IF(EX.GT.IOO.) EX-IOO.
EX»DEXP(-EX)/(l.OIX>fB*X)**(A/B)
FUN-EX
RETURN
2 CONTINUE
BX»B*X
IF(BX.GT.170.) BX-170.
EX-A (l.ODO-DEXP(-BX))fC*X**D
IF(EX.GT.100.) EX»100.
FUN-DEXP(-EX)
RETURN .
3 CONTINUE
EX-D*X
IF(EX.GT.IOO.) EX-IOO.
EX-C (DEXP(EX)-I.ODO))
IF(EX.GT.IOO.) EX-IOO.
EX»DEXP(-EX)
FUN«EX/(l. ODO*-B»X V* (A/B )
RETURN
END
Within the main program, the appropriate model number is read, the
values for the parameters a,b,c and d of the model are read and values
for the mean residual lifetime are computed.
Finally, the specified information is rpinted along with the computed
time values, mean residual lifetime values, v, and the mortality rate values
Z; the main program calls a subroutine GPLOT which plots Z and v at
various times. The routines used in GPLOT were developed at the Health
86
-------
Sciences Computing Facility, U.C.L.A.
Sample output from this program is included after the listings of the
main program and subroutine GPLOT.
S ION V ( 4 IHJ-H Tl-ME-H 00)- » -G <
N CMC 3)
CUUBLE PRECISION D X ,K , L , T , E , A RLA
DIMENSION Z(40G», X(40G» t Yf4004
UOU8LE PRECISION A, Bf C, 0
DOUBLE PRECISION DIES
Cu-^MGN /PARS/ AtBfC fD
DATA Ch/« V1, «Z','* '/
COMMON LM
10 KtAO(fflOC) LM
IFtLM .Lt . 0) GO TO 9G
17 KEAOl 5iUl) AtBtCtO
111 FORMAK^FIO. 5)
WKlTE(ttl20) LMtAtBtCtO
IF (A .LE. -i.O) GO TO 1C
TMIN = C.
TMAX=AC.
STEP=2.
AREA = C.UDO
MSTtP=100
NNUM-={ IMA X-TMIN) /STEP
TU'=TMA)(
DU 11 1=1,1000
1=DBLE(TU)
1F(FUN(T) .LE. l.OD-6) GO TO 20
11 rU=TU+STcP
v^lR I TF ( 6f 1 10 I TU
110 FORMA !(' TAIL TOG BIG AT TU't F10.5)
20 CONTINUE
XX= STEP AJ STEP
OX=OBLE( XX)
R=FUN( 1 )
IENU=lCOv>00
UiJ I 1=1, IcND
TEST=TN'AX-S1EP*NUM
UTES=C-BLE (TEST )-
IF( OABSU-OTESI .GT. l.CD-3) CO TC 2
J=iMiMUM + l -NUM
IF(R .tEr--l.-€t-6h 6G TC 12 • •- •••
V< J)
87
-------
12
13
GO TO 13
vm=c.
CJNTIMt.
L( J)=hA£( TEST)*1.0t2
•TI-MEt J»»TfcST ------
NUM=NlM+l
1F(NUM .GT. NNUM) GO TC 69
T=T-DX
L=FUN( T)
E = ( L+K )/
-------
OOEL
I
1
2
3
4
5
6
7
8
9 -
10
11
12
13
14
15
16
17
ia
19
20
21
1 A, 6, C,
T IME
-U.C
2.GOOOO
4 . CO-JuiJ
6.COOOO
8.COOOO
10.00000
12.00000
14.COOOC
16. GOO 00
13.00030
20.00000
22.00000
24.COOOO
26.COOOO
2tfiOOOOO -
30.00000
32.00000
34.00000
36.00000
38.COOOO
40.00000
D= 0.0
V
IS; 375C2
17. 50595
it>. 88o43
14. 40230
13. 26U82
12. 19372
11. 257^1
10.43222
9. -70 t 74
9. C5235
8.47264
7.95308
7, 4356*
7. 06355
6nr6 81-10
6. 33339
6. C162d
5. 72620
5. 460U8
5. 2152)3
4,^6S5U
10.GCOOC
I
c.o
C. 74533
1.55460
2.39003
3.24256
-4.10819
4.98442
5.86955
—6. -76 23 8
7.6619S
8.56768
9.47886
10.39506
11.31589
1^-2-4101
13.17013
14.10301
15.03943
15.97916
16. 9220o
17.86797
2.06040
-------
MODEL
A, 6, C,
D =
L. 00000
0. 1COOO
0.0002U
2.00000
I
1
2
3
4
6
6
7
8
9
10
11
12
13
14
-15
16
17
13
19
20
2-1
T iv, t
-O.G
2.COOOO
-4.GOOUO
6.COOOO
8.00000
1-0.00000
12.CCOOO
14.00000
-1-6.CGGGO
Id. 00000
20.COOOO
22.COOGG
24.CGGGG
26.COOGO
2-8.GG4KHJ
30eCOQQO
32.00000
34.00000
36.00000
38.00000
--- 4G.COOOG
V
2 7 . 7 5 S s S,
31. 11 boo
34.C2692
36.45024
3d. 38336
39. 65265
40. 9G19fc
41. 58397
41. S5340
42. C63Cb
41.96121
41.69G55
41. 28772
40. 78357
-4^,203o 7
29. 56L86
38. 89590
38. 19H52
. 37. 48705
36. 70991
- —-3-6.05354,
I
1C. 00000
6.26731
6.06319
5.7281^
4.81329
4.07879
3.4919^
3.02597
2.65896
2.37299
2.15335
1.98803
1.8671b
1.78273
1.72810
i.697b/
1.68762
1.69373
1.71324
i. 74371
1.78-316
-------
Z,V VERSUS T
45 +
V V V V V
v V V
«G « V V V
V
V V
V V
v . • • • - v
.V
20
10 t?
0. «•
/! i -
^ / 111 lit II! I.
i. •!. 15 ii <; /
0. 6. I? 1>1 t'r 30
92
-------
Z,V VERSUS T
i i .'.••« • • -
I
v z
11..G \ - - - -- : - - Z •• --.
I
I
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CHAPTER 6
CONCLUDING REMARKS
During the next twelve months, the marginal and combined effects of
the basic pollutants on morbidity for the City of Los Angeles and the mile-
high city of Denver, Colorado, will be assessed using the methods developed
in this report. The various ways for modeling for the competing risk situa-
tion will be explored. Both parametric and nonparametric methods will be
considered. We will show how prior knowledge of empirical distributions of
the concentrations of the various pollutants can be used to sharpen the
statistical procedures of estimation and testing for goodness-of-fit as a
follow-on effort, including applications of paramount importance to the
Environmental Protection Agency of the United States.
94
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CHAPTER 7
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"The Comparison of Survival
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98
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
W-W/t-76,015
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
REALISTIC MODELS FOR MORTALITY RATES AND THEIR
ESTIMATION
5. REPORT DATE
peb>uarv 1P76
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
V. K. Murthy
8. PERFORMING ORGANIZATION REPORT NO.
I9. PERFORMING ORGANIZATION NAME AND ADDRESS
University of California at Los Angeles and
University of Southern California
Los Anqeles, California
10. PROGRAM ELEMENT NO.
1AA601
11. CONTRACT/GRANT NO.
800Z30
1:2. SPONSORING AGENCY NAME AND ADDRESS
Health Effects Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, N.C. 27711
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
EPA-ORD
1!>. SUPPLEMENTARY NOTES
Hi. ABSTRACT
The objective of a medical follow-up study is generally to determine
the effectiveness of each of several treatments by analyzing the responses
of the patients. Frequently the response data coming out of these
investigations is time to death of patients who are riot otherwise lost
to the follow-up of our investigation. The statistical nature of this
data are characterized in this report.
By definition the, "Force of mortality or mortality rate function",
is the rate associated with the probability of the patients' death in
a specified short interval of time, given that the patient has survived
to this instant in time.
Mathematical models are presented which, as special cases, represent
constant, increasing and decreasing mortality rates, along with combinations
of these properties. Usually, these mortality rate curves are "U" shape.
The first part of the curve corresponds to infantile mortality, the second
part corresponds to useful life, and finally, the last part corresponds
to decay, aging, etc., culminating in death. Their corresponding probability
distributions and survivorship functions are obtained in closed form.
Methods of estimating the parameters are developed and procedures dealing
with computational details and statistical properties are disrusspd.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Mortality
Mathematical Models
12A
05B
06G
8. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
21. NO. OF PAGES
104
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
99
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