NATIONAL CENTER Series 2 

For HEALTH STATISTICS Number 5 

PROPERTYOF THE c 

PUBLICATIONSBRANCH 

EDITORIALLIBRARY 

VITAL and HEALTH STATISTICS 

DATA EVALUATION AND METHODS RESEARCH 

An Indexof Health: 

MathematicalModels 

Mathematical models of the distribution of i Ilness 

episodes, of il Iness duration, and of mortality in 

a population. An index of health reflecting both 

mortality and morbidity is proposed. 

Washington, D.C. May 1965 

U.S. DEPARTMENT OF 

HEALTH, EDUCATION , AND WELFARE Public Heolth Service 

Anthony J. Celebrezze Luther L. Terry 

Secretary Surgeon General 
Public Health Service Publication No. 1000-Series 2-No. 5 
NATIONAL CENTER FOR HEALTH STATISTICS 

FORREST E. LINDER, PH. D., Director 

THEODORE D. WOOLSEY, ~c$)q Di~ccto~ 

O. K. SAGEN, PH. D., Aktant D&ctor 

WALT R. SIMMONS, M.A., ~tat;~f;~az ~dv~~o~


ALICE M. WATERHOUSE, M.D., Medical Advisor


JAMES E. KELLY, D.D.S., Dental zfdu;$or


LOUIS R. STOLCIS, M.A., Executive Oficer


OFFICE OF HEALTH STATISTICS ANALYSIS 

IWAO M. MORXYAMA,PH. D., Chicj 

DIVISION OF VITAL STATISTICS 

ROBERT D. GROVE, PH. D., Chief 

DIVISION OF HEALTH INTERVIEW STATISTICS 

PHILIPS. LAWRENCE, Sc. D., Chief 

DIVISION	 OF HEALTH RECORDS STATISTICS 

MOXROE G. %RKEN, PH. D., Chief 

DIVISION OF HEALTH EXAMINATION STATISTICS 

ARTHURJ. MCDOWELL, Chief 

DIVISION OF DATA PROCESSING 

SIDNEY BINDER, Chitj 

Public Health service Publication No. 1000-Series 2-No. 5 

Library of Congress Catalog Card Number 65-60058 
PREFACE


This study is presented as a contribution to the methodology of measuring health status. Among the objectives of the 

National Center for Health Statistics is the development of new 

techniques in health measurement. 

Although it is generally conceded that mortality statistics 

no longer provide an adequate measure of the health status of 

a population, no generally accepted method of measuring health 

in terms of both mortality and morbidity has emerged. Dr. C. 

L. Chiang of the School of Public Health, University of California, was invited to develop mathematical models which might 

serve as the basis for a general index which reflects morbidity 

as well as mortality. The models which have been developed 

represent one of many possible approaches to the problem. 

It is hoped that the publication of his work will lead to more intensive investigation of both the conceptual and the mathematical problems involved in constructing such an index. 

As pointed out by the author, further testing is needed to 

determine whether the models presented provide a good description of observed data. 
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CONTENTS 

Preface 

I. htroduction


II. Probability Distribution of the Number of


III. Probability Distribution of the Durationof


Illnesses 

Illness 

Page 

... m 

IV. Time Lost Due to Death


V. Index of Health


VI. Remark-Adjusted Index of Health


References


Detailed Tables


1 

2 

5 

9 

10 

12 

12 

15 
AN INDEX OF HEALTH:


MATHEMATICAL MODELS


C. L. Chiang, 1%.D, School of Public Health, University of California, Be~keley 

1. INTRODUCTION 

The state of health of a nation is one of the 

most important aspects in the study of a human 

population; but the lack of quantitative measures 

to assess health has always been a problem in 

the field of public health and welfare activities. 

The purpose of this study is to suggest mathematical models for describing the state of health of 

a well-defined population over a given period of 

time, such as a calendar year. 

The health of a population is in part a function of such demographic variables as age, sex, 

and possibly race. People of different ages and 

sexes have different susceptibilities to diseases, 

and diseases may act differently upon them. To 

describe the health status adequately, the population should be divided into subpopulations according to these variables. For convenience of presentation, however, these subpopulations will be assumed homogeneous with respect to all demographic variables except age. From the public 

viewpoint, a simple and comprehensive index of 

the current state of health is most desirable. Because of the complexity of the problem, however, 

a satisfactory approach should begin with detailed 

investigations of the basic component variables. 

The state of health is best measured by the frequency and duration of illness, by the severity 

of illness, and by the number of deaths. These 

components taken together give a comprehensive 

picture of health; separately, each describes an 

aspect of the state of health. 

To measure the frequency of illnesses, we 

need to know the number of illnesses occurring in 

a calendar year to each individual of a given age 

group and the distribution of the subpopulation with 

respect to this variable. A mathematical model 

will be developed in section II. Although the model 

is not specifically developed for a particular type 

of illness, the general line of approach applies 

equally well for any specific disease. The derived 

probability distribution characterizes the pattern 

of proneness and susceptibility of a subpopulation 

to disease; it also provides an easy means of calculating incidence and prevalence rates. The mean 

number of illnesses and the corresponding standard deviation will serve to measure the average 

proneness and its variability for each subpopulation. Furthermore, all these measures can be used 

for comparing subpopulations or summarized for 

the entire nation. As a test the suggested model 

is fitted to actual data from a sickness survey. 

The severity of an illness varies with the disease and the individual concerned. It does not lend 

itself to quantitative measures except as’ it is related to duration or to termination in death. In 

section HI we present the derivation of a general 

model for the duration of illness, which again 

applies either for a particular disease or for all 

illnesses. Because of lack of data, no attempt is 

made to find a specific function; however, alternative approaches are described in detail. 

Since death must be related to ill health in a 

population, a study of health is not complete without considering the mortality rate. Mortality is 

evaluated from the standpoint of health in section IV. 

While studies of the component variables 

give a more detailed picture of the state of health, 

1 
development of a single measure summarizing the 

information for the entire population is also essential. Basedon the ideas in the preceding sections, a heaith index is derived in section V. An 

adjusted index is suggested in section VI. 

II. PROBABILITY DISTRIBUTION 

OF THE NUMBER OF ILLNESSES 

Consider the time interval of 1 Year (O, 1), 

and for each t, o < ts 1, let the random variable 

N (t) be the number of illnesses that an individual has during the time interval (O, t), with 

N(O) -O. The purpose of this section is to derive 

the p~obability distribution of the random variable 

N(t), 

~(t)-Pr {N(t)-n I N(0)= 0~ . (1) 

This probability function is an idealization of the 

proportion of people in the population having n 

illnesses, for ZI-o, 1,... , during the interval 

(O,t).When an explicit form for the probability 

function is derived and computed for each value of 

n, we have a mathematical representation of the 

state of health of the population in terms of the 

number of illnesses. 

An assumption underlying the probability distribution (1) is that the probability of occurrence 

of an illness during the infinitesimal interval 

(t,t+h) equals Ath+0 (h), where At is afunction of time t and o(h) is a negligible quantity when 

h tends to zero. In essence, this means that the 

probability of an illness occurring within an infinitesimal time interval is a function of time and is 

independent of the number of previous illn$sses. 

This assumption leads to a system of differential 

difference equations for ~ (t). Consider two contiguous time intervals, (O, t) and (t,t+h). Exactly 

n illnesses can occur in the interval (O, t+h) 

in three mutually exclusive ways: (a) n illnesses 

will occur in (O, t) and none in [t, t+h) with a probability Pn(t) [1- A~h -0 (h)]; (h) n -1 illnesses 

will wcur in (O, t) and one in (t, t+h) with a prob1=0 in~=ate =WlicitlY that the models are developed for a sub 

population, say age group x, a subscript x should bs added in the 

appropriate places. For tht sake of simplicity of presentation, 

however, such a subscript will not be used in rhis secrion or in 

section III. 

ability F’o-l(t) [Xt h + o (h)];and (c) n -2 illnesses 

or less in (o, t)and two or more in (t,t+11), 

with a probability of o(h). Taking these possibilities together we have the formula: 

P“(t+zd= Pn(t)[l- )qh-o(lf)] 

(2) 

+ ~.l(t)[Ath + o(h)] + o(h). 

Transposing Pn(t), dividing through by h, and 

taking the limit as h tends to zero, yield a system of differential difference equations. 

(3) 

A P, (t) = -At Pn(t)+At Pn-l(t), n=l,2, . . . dt 

The first equation hqs the solution 

#t 

PO(t) = e 04 ; (4) 

the remaining equations are solved successively 

to give the probabilities 

+d’ [S.&’]n“) 

P“ (t)= e , n=l,2, . . . n! 

For a period of 1 year i.e. , for t = 1 the random 

variable N has the distribution 

-)’’,’q)’’,]n]n 

P{ N-n}= e n! * 

n= O,l, . . . (6) 

Within a period of 1 year, the instantaneous 

probability ht h + o(h) of occurring illness need 

not be dependent upon time t, and A~ may be 

2 
assumed to be constant. Under this assumption, 

we have the ordinary Poisson distribution 

~-AAn 

P{iV=n}= ~, , n= 0,1,2, . . . (7) 

The constant x in formula (7) signifies an individual’s susceptibility to diseases and, as such, 

is a measure of the degree of his health. In 

fact, A is the expected number of illnesses occurring to an individual during a period of 1 year. 

The larger the value of x , the more illnesses 

the individual may be expected to have. 

The value of Avaries from one individual to 

another. To describe mathematical y the health 

status of a population, we shall study the probability distribution of A. The distribution of Awill 

be denoted by g h) d A, the theoretical proportion 

of people having the specified value A. Since the 

sum of the proportions of individuals is unity, the 

function g satisfies the condition 

Jg(h)dA= 1, 

where the integral extends over all possible values 

of h. The probability distribution of illnesses 

will be a weighted average of the probability function (7) with the density function g (A)dx empIoyed as weights; that is, 

-Akn (8)

P {N=n}= ‘nl g(NdA , n= O,l, . . . / 

Roughly, formula (8) may be interpreted as 

follows: 

The expected proportion of individuals who will have n illnesses 

during the year is equal to the sum 

of the products of (a) the proportion 

of individuals having a specific value 

of A, and (b) the probability that an 

individual with the specific value 

of A will have n illnesses during the 

year, where the sum is taken over 

all possible values of A. 

Choice of function g (x) d h is dependent upon 

the health condition of the particular group of 

people in question. It appears, however, that the 

following function may describe the distribution 

in general: 

-1 e-i3A 

g(A)dA = ‘aAa 

r (a) (9) 

where the gamma function r(a) is defined by 

00 -1 r(a) = Ya e-y dy (lo)

J 

o 

The ranges of the constants for which(9) is defined 

area>O and /370. 

The function g(x) starts at A-0, increases 

as A increases at a rate of 

* L?(A)- L?(A)[(a- l) A-l -61 , (11) 

and reaches a maximum of 

(12) 

at 

a-1 

A=— B (13) 

After reaching the maximum value, g(A) decreases as A increases and assumes a value of 

zero as x tends to infinity. 

The expectation and variance of A maybe directly computed from (9): 

-~a As-l ~-@A dA- ~ E(X)= JA— 

r (a) 6 (14) 

o 

0 

(15) 

3 
Thus the ratio a/P measures theaveragehealth 

of a population, and the reciprocalof aistherelative variance, 

(16) 

which is a measure of variation of health among 

individuals in the subpopulation relative to the 

mean health. 

Assuming (9) as the function underlying the 

distribution of the population with respect to health 

condition, we have from (8) the probability function of the number of illnesses during the year: 

(17) 

- r (n+d ~ 

n! r (a) f3 (1+/9)-(n+a) , n= O,l, . . . 

This probability is the expected proportion of individuals in the population having n illnesses during the year, taking into account the variability 

among individuals in the population as described 

by formula (9). The expected number of illnesses 

occurring to an individual in the subpopulation is 

given by 

E(N)- —;’ (18) 

and the variance by 

~z _ a(l+~) (19) N $2 “ 

Formula (17) represents a family of infinitely 

many probability distributions, depending upon the 

constants a and 13. The health status of a subpopulation may best be described as a member of 

the probability distribution family for which 

a and p assume particular values, In order to 

estimate these values, it is necessary to know the 

observed frequency distribution of the number of 

illnesses occurring to the individuals of the subpopulation from which the mean N and variance 

s; of the number of illnesses are computed. 

Substituting N and S ~ for E (N) and u ~ , respectively, in (18) and (19) and solving the resulting equations for ~ and B give the estimates 

-2
 A (20) a - (s;%


and 

(21) ‘ 

Using the estimated values ~and $in (17), we 

have 

(1+6)- (n+@) , 

n= O,l, . . . (22) 

The probability (22) multiplied by the total number of individuals in the subpopulation is the expected number of individuals having n illnesses 

during the year, for n = 0,1,2,. . . . This expected frequency distribution may be compared 

with the observed frequent y distribution by means 

of the chi-square test to determine whether the 

model described by (17) is an adequate measure 

of the state of health. Material collected by the 

Canadian Sickness Survey, 1950-1951, is used for 

this purpose (see references 3 and 4). 

The data in the Survey were based on a sample 

of approximately 10,000 households 2 inflated to 

give the national figures as appeared in the publications. Thus the published figures are much 

greater than the actual counts in the sample. 

Not knowing the exact number, we take 13,538 

as the sample size and each thousand in the 

published data as a single count (the total population size is 13,538,000, see table 3). Since the 

actual sample size is probably larger than 10,000 

the exact chi-square values in our test should be 

somewhat greater. 

Two indirect measures of illness were used— 

th; number of doctors’ calls and clinic visits 

and the number of complaint periods that an 

individual had during the year. For the number 

of doctors’ calls and clinic visits the model is 

2see page 17 of Ref exence 3. 

4

fitted forsixage groups—under 15,15-24,25-44, 

45-64, 65 and over, and all ages. The results for 

the first two age groups and for all ages are presented in tables 1, 2, and 3, and figures 1, 2, and 

3, respectively. In each of the first two cases, the 

fit is quite good. For all ages, however, the chisquare value exceeds the critical value at the 1 

percent level of significance. 

Data on the number of complaint periods were 

divided into only four age groups—under 15, 15-64, 

65 and over, and all ages. Only the age group under 

15 is well described by the present model as 

shown in table 4 and figure 4. 

Although neither of the underlying random 

variables is that of our model, the chi-square tests 

show promising prospects when the age intervals 

are not too large. It is hoped that more appropriate 

material will be made available for furt)ier testing. 

Ill. PROBABILITY DISTRIBUTION 

OF THE DURATION OF ILLNESS 

Let random variable T be the duration of an 

illness so that 

Pr(T<t) (23) 

is the probability that a person will recover from 

an illness within a period of time t. We are interested in an explicit function of the probability 

in (23) and of the corresponding probability density 

function of T. 

Consider an infinitesimal time interval 

(t, t+ At) for t> O. The conditional probability 

that an individual who is ill for the period of time 

t will recover from the illness during the interval (t, t + At) is certainly a function of time t 

and length At. Let this function be denoted by 

#(t, At), We shall assume that the function 

@ is continuous with respect to A t and has the 

first derivative, say Vt, at At = O for each possible value of t. It follows from Taylor’s theorem that 

*(t, At)=vt At+o(Af) , (24) 

where v, is a function of t and o(A t) is a quantity of a smaller order of magnitude than At. 

To derive an explicit function for (23), consider 

Q(t)= l- F’r(T<t)=Pr(T>t) (25) 

the probability that an individual will be ill 

for a period longer than t, and the time interval (t,t+ A f). In order for an individual to be 

ill for a period longer than (t+ A t,)he must be ill 

for a period longer than t and not recover within 

the period (t,t+ At). According y, the corresponding probabilities have the following relationship: 

Q(t+At)= Q(t) [1-v, At-o(At)] . (26) 

Subtracting ~(t) from both sides of (26), dividing 

(24) through by A t, and letting A t tend to zero, 

we have the differential equation 

+ Q(t)=–!2 (f)vt . (27) 

The general solution of (27) is easily found to be 

t 

-J vrdr+c 

Q (t)= e“ 1 (28) 

where c is the constant of integration. Since 

Q(O)= Pr(T > O)= 1 for t=O the constant of integration must be O, and we have the desired solution 

-Y 

t 

VTdr 

Q (t)= e“ 9 (29) 

and the probability 

— t

Vr dr J 

Pr(T<t)=l-eO (30) 

5 
The derivative of (30) gives the probability density function of the general model of the duration 

of an illness 

— 

t

VTdr 

f(t)dt=d [1J 

l-e’ 

t 

-r Vr dr 

J 

= Vt e“ dt, t>o. (31) 

It should be noted that the random variable 

T in the above model is the complete duration 

of an illness rather than the duration of an illness observed within a calendar year as needed 

in our formulation. In the study of the health of a 

current population, we are considering a truncated 

case, where the duration of an illness is defined 

as the interval extending from the beginning of the 

year (if it is an illness continuing from the preceding year) or from the date of onset to the date of 

recovery, death, or the end of the year, whichever 

comes first. Therefore, we shall consider a truncated distribution of T. For illnesses occurring 

during the calendar year, truncation could be made 

at random. But this would result in truncation at 

a point approximately 6 months after onset and 

would misrepresent those illnesses having durations of more than 6 months within a calendar year. 

Random truncation would further distort the picture of the entire distribution which includes 

chronic illnesses carried over from the preceding 

year. To avoid distortions, we use one year as the 

point of truncation. 

Let the random variable T* be the truncated 

duration of illness expressed in unit of years. The 

probability distribution, say f* (t) dt of T*, is 

equal to the truncated distribution of the original 

random variable T, for T< 1; symbolically, 

for O<t<l, 

. 0, elsewhere. (32) 

Substituting (30) and (31 ) in (32) gives the required 

model 

t —	 VT dr 

f 

0 

f*(t)dt= ‘te ~ dt. (33) 

VT dr 

f 

1-4


The sum of f “(f)dt over all possible values of 

t for O<t<l is unity or 

t l_ vrdr J 

Vt‘=0 dt=l, (34)

1

/

a -/ 

Vr dr 

l–e” 

as can be proven by direct integration. Both the 

probability distribution (X3) and the expectation of 

T*, 

1 

E(T*) = tf*(t)df (35) ~ 

0 

t 

VTdr 

1 Y 

Vte“ t dt 1 

“1 V7 dr o f 

l–e” 

t 1 

VTdr VT dr 

1– f -J

e“ dt – e“ 

“r 

0 

1 

VT dr 

Y 

l–e” 

are	 dependent upon the function vt. 

The general model (33) may be applied to a 

practical situation by specifying the function 

6 
, 

VI, Without appropriate data for testing aspecif- typical statistical approach is to use the method 

ic model, we consider only the following simple of maximum likelihood. Consider a total of M illcase to illustrate a few points. nesses and let ti* be the truncated duration of the 

Supposing that Vt= v is independent of f, we i-th illness, for i = 1,2 ,. . ., M. The joint denhave from (33) a truncated negative exponential sity function of the M random variables is 

distribution 

Ve– vt*, v M –v i$, t; 

1=1 1- e–” () e-v 

e , (41) f“ (t) Cft - —	

v e-vt ~t 

for OSL’S 1, 4=1 —= 

l-e-v ‘ l-

- 0, elsewhere. (36) Maximizing the likelihood function (41) with reUnder this model the function f*(t) assumes a spect to v leads to an equation


tT121XhN.12n value of v (1 – e ‘“)-1 at t =’0 and de- e -0


creases as tincreases at a rate of F“-Q–l – l–e–o (42)


( 

V2 

-“t (37) where ~_e–v e 

The expected value and the second moment of (43) 

T* can be obtained from (36). 

1 is the mean truncated duration of an illness and 

Ve–vt 

E(T”) - t dt @ is the likelihood estimate of v. Formula (42) 

/ 1 -~–v can be solved for ~ by a reiterative procedure. o By comparing (42) with (38) we see that the maximum likelihood estimate O may be obtained by 

estimating the expectation E(T*) with the corre­

-v -1 _ e —v 

(38) sponding observed mean duration ~-. l–e–v An alternative method of estimating v is to 

and consider the probability distribution f (t)dt in (31) 

of the complete duration of an illness. When 

1 Vt = v is independent of t, 

E (T”z) = tz ‘e-”t dt 

/ 1- e-v (39) o 

f (t)dt - ve-vt dt, for t2 O, (44) 

= 2 V-2 + (2 v-1+1) [l-(l-e-v)-l]. and we have the expected duration of an illness 

Therefore the variance of T* is (45) 

2 u,. - E (T*2)- [E(T*)]2 Now we take a sample of M illnesses and record 

the complete duration of each illness. The obe— v served mean duration, say ~, so determined is 

= V-2 - (40) an estimate of E(T), and its inverse is an estimate 

(l–e-v)z of v, 

A 1 

In applying the model in formula (33) to actual v.= 

T 

(46) 

data, we need to estimate the parameter v. A 
For application later, we also compute the second Since rrj is known and Vj can be estimated for 

and the third moments about the origin, each disease, the estimate of E (T*) can be computed from (50). 00 

2 Illnesses are often classified as acute or 

E(T2]= Jt2ve-Vtdt= _ 

“2 chronic. A mathematical model to describe illness 

o	 from this viewpoint is essential but difficult to 

formulate, because the demarcation line between 

and	 (47) acute and chronic illness is not always well de-

. fined, and the exact proportions of illnesses in the

6 E (T3) = 1t3 ve -Vt dt. — 

V3 two categories are never known. In the discussion 

to follow, we shall use the duration of the illness o 

as a criterion of classification and make an attempt 

Using the estimated value @from either to solve the problem. 

method, the expected relative frequency of ill- According to our formulation, a general mathnesses with a truncated duration between, say tl ematical model in this case may be represented 

and t*may be cpmputed from by the probability density function -

J t2 *

ve-+t ~t= e 

-Otl _e-&2 

(48) 

f*(t) lit = (51) 

1- e-+ 1- e-+ t t 

VI‘r dr 9 V27 dr 

-f 

* Vlt e“ v2te“ 

for any interval (tl,t2). Multiplying (48) by M 1 —— + (I-T) 1 dt . 

gives the expected number of illnesses. The dis- Vlr d~ V2T dr -~ -~ crepancy between the observed and expected numI–e” l-e” ber of illnesses for each interval (tl, tz) may be


evaluated by the chi- square test to determine the


sufficiency of the model in formula (36). When Vlt = v ~ and V2t= V2 are assumed to be in-


Equation (36) may be applied either to all dependent of time t, the probability function bediseases as a whole or separately to individual comes 

diseases. When all diseases are taken as a single 

group,the computations involved are quite simple. vle -Vit V2e _v2t 1 

(52) 

When individual diseases are studied separately, f*(r) dt = r + (1-~) dt.


they can be summarized by the compound distri- [ I–e-vl l–e-v2€

bution


Here ir is an unknown proportion and may be inf*(t) dt= y m,fj* (t)dt (49) terpreted as the probability that an illness will be 

acute with the severity signified by VI. A similar 

interpretation holds for (I - r ) and V2, Conwhere Xj is the known proportion of illnesses sequent to our arbitrary classification of illof the j th disease and fi*(t) is the corresponding nesses as acute or chronic, a graphic represenprobability density function as given in (36). In tation of the model will show a bimodal curve. 

this case, the overall expected duration of illness The statistical problem is to fit such a model 

is given by to empirical data and to estimate the parameters 

r, VI, and V2 in the formula. In this case the 

(50) simplest approach is to consider the complete 

duration of a sample of M illnesses and to use the 

8 
method of moments, which allows the parameters and formulate a quadratic equation and 

to be estimated from the first three sample moments U1, LIZ, and U3 (see equations ~45) and 

(47) ) 

Y2-a1Y+a2-0. (60) 

+ ]+ u*.	 x+ —..._ The two roots of (60) are 41 and 02. Substiy 

tuting these values into the first equation in 

(53), we obtain the estimate of r: 

u2” 

2+ 2(1-+) 

Aii+ A2 

‘1 ‘2 (53) 

A (1– u*@2) 

+. ‘lA ~ (61) 

‘1 — V2 

Using the estimated values of T, VI , and V2 , 

we can compute the expected relative frequency 

64? + 6(1-@) from 

‘3= AT A3 

‘1 V2 t2 

VI 92 

f (t) dt = (62) 

tlJ 

These equations may now be solved for + , ~1 

and +2. Eliminating @ from the first two equa­


tions in (53) gives t2 ~~ e -$lt ~ .+2t


tld r 1- ~-$ + (1–+) ‘2e dt 

l-e-$ -0 (54) 1 

and from all three equations gives for each interval (t~,t2). The expected number 

of illnesses of durations between tl and t2 is ob6u1- 3u2(~ +02) + u3q~= o (55) tained by multiplying (62) by the total number of 

illnesses. 

Solving (54) and (55) simultaneously, we have 

IV. TIME LOST DUE TO DEATH 

AA 6U1U2- 2U3 

V1+V2= 3@ - 2U1U3 

(56) h general, a high frequency of mortality 

indicates ill health within a population and, consequently, the death rate is one basic measure of 

and the state of health. When the amount of illness in 

a” calendar year is studied, not only the length of 

AA 12u; -6u2 the, illness before death must be considered, but 

‘1 ‘2 - 31.4- 2U1U3 

(57)	 also the period from the time of death to the end 

of the year. This period will be referred to as 

“the time lost due to death” and will be denoted 

These can be computed 

ments. Now let 

from the sample mo- by t. We want to determine the probability distribution of f, and the expected value of ~ 

L among the deaths occurring during the year. 

~++2 - a, (58) The time lost due to death is determined by 

the time of death. Although deaths may be subject 

and to seasonal variation, as an approximation we assume that the y take place uniformly throughout AAv, V2=~z, (59) the year. Consequently the random variable ~ 
also has a uniform distribution within the interval (O, 1), and its distribution function is given by 

P{t<e}-~ed7-e , for O<e<l . (63) 0 

Direct computation gives the expected value and 

variance of f, 

m=+- (64) 

and 

.<=OJ’ [,-m]’ d,=+ (65) 

This formulation indicates that the average time 

lost for each death is one-half year. 

Let m, denote the age-specific death rate, 

which is the approximate proportion of people of 

age x who die during the year. Since, on the average, one-half year is lost by each death, the 

average time lost to each individual in the entire 

subpopulation is ~ m ~. 

The problem now is to determine how much 

weight to apply to the time lost due to death as 

compared with the duration of illness. In general, 

we may say that 1 day lost due to death is equivalent to w days of illness. The problem is to determine w. From the point of view of health, 

both illness and death are states of ill health, and 

time lost due to death and time lost due to illness 

should be weighed equally. It may then be assumed 

that w is unity. Thus ~ mX, the average time 

lost due to death for an individual in the entire 

subpopulation, is directly comparable to the average duration of illness and will be used in the 

formulation of a measure of the state of health in 

the following section. 

V. INDEX OF HEALTH 

The term “illness” discussed in the preceding 

sections needs clarification before we proceed 

further. An illness may be defined as a continuous 

state of ill health over a period of time regardless of the number of diagnoses; it may also be defined as a continuous state of ill health for each 

disease. Consider, for example, a child who had 

the first symptom of chickenpox on February 1 and 

came down with a cold on February 11; he recovered from the chickenpox on February 14 and 

from the cold on February 18. According to the 

first definition, the child had one illness with a 

duration of 18 days; but according to the second, 

he had two illnesses with durations of 14 and 8 

days, respectively. Each of the two definitions has 

its merit in describing health. The models suggested in the preceding sections should apply 

equally well in both cases, although the constants 

involved in the models will take on different values. 

For the purpose of deriving an index of health, 

however, we shall use the first definition and consider” that the child enjoyed good health 365- 18= 

347 days during the year, providing, of course, 

there were no other illnesses for the rest of the 

year. 

In the study of the state of health of a population, tfie paramount question would seem to be: 

What is the average fraction of the year in which 

an individual is healthy? This fraction will be referred to as the mean duration of health and will 

be used as an index of the health of a population. 

The index of health so defined is closely related to the three component variables presented 

in the preceding sections. From the distribution 

of the number of illnesses, we can calculate the 

expected number of illnesses occurring to an individual in a calendar year; and from the study of 

the duration of illness, the expected duration of an 

illness. The product of the two quantities is the expected total duration of illness during the year. For 

age group x let RXbe the observed average number of illnesses per person and TX*be the average 

duration of an illness in a year. The product 

~X TX� is an estimate of the expected duration 

that an individual is ill. From the standpoint of 

the index of health, RXTX*is simply the average 

duration of illness per person per year and can 

be estimated directly from a single sample. Suppose a sample is taken from age group x. For 

each individual in the sample the fraction (IX) of 

the year that he is ill is determined; the average 

of this fraction, denoted by 7X, is equal to the 

product, 

(66) 

10 
-----

In the discussion of mortality, wedeterminedthat 

the average time lostdue todeathfor an individual 

of age group x is one-half the age-specific death 

rate. Since the age-specific death rate is usually 

available in vital statistics publications, it need 

not be computed again from a sample. The average duration of ill health is the sum of the average length of time that an individual is ill and 

the time lost due to death 

~. ++-xnx . (67)

Let Hx denote the mean duration of health, or the 

fraction of the year in which an individual in age 

group x is living and free from illness. Obvi­

&sl~, Hx is the complement of the aver%e duration of ill health, 

Hx=l-(7..++--mx). (68) 

In formula (68) the unit of 1X is years; the average duration of ill health cannot exceed one; and 

Flx is between zero and unity. 

To derive the corresponding measure for the 

entire population, only a weighted average of 

Hx need be computed. Several principles maybe 

used to determine the weights. The simplest is 

to use the population proportion in each age group 

as weights. Let Px be the age-specific population and 

P-? P, (69) 

be the total population. Then the weighted average 

H =+: PXHX (70) 

is the mean duration of health, or the index of 

health, for the entire population. Since Px is the 

actual population having experienced the duration 

of health indicated by the value of HX, in formula 

(70) we have a meaningful measure of the state of 

health. 

The values of the index of health H are obviously between zero and one; the healthier a population is, the larger will be the value of H, and 

vice versa. If no illnesses and no deaths occur to 

the people of a current population, then ~= O and 

mx= O for each x; hence Hx=l and H=l. At the 

other extreme, if every individual in a population 

were ill during the entire year, then 7X= 1, (m ~= O 

in this case), and both H, and the index of health 

would assume the value of zero. 

The quantities H, and H are, of course, random variables, and they are estimates of the corresponding true unknown expected values. In terms 

of the models discussed in the preceding sections, 

the expected values may be derived as follows: 

E(H) = ~ Z ~,E (Hx), (71) 

E(HX)= 1– [E(~)+~E (m,)] (72) 

and 

E(~x)=E(@x~*) (73) 

Assuming independence between fiX and ~,*, 

we have 

E(fix TX*) = E(flx) E(~x*) 

= E(NX) E(~*) ; (74) 

where, in light of’-mrrnulas “ (18) “”” ancl (35),

‘(NX)= (75) ? 

and 

E(~*)=vx-] -e -“’ [pe-”x]-l (76) 

The expected value of mx for the age group of 

nX years is approximately equal to 

n, [1 -’71 -ax) 9X] (77) 

where 9X is the probability that an individual of 

exact age x will die before reaching exact age 

x+n X, and aX is the average fraction of the age 

11

interval (x, x+nX)lived by individuals who die at 

the age covered by the interval. It follows that 

ax -l_e -v% (l-e -~x)-l 

‘(HX)= ’-K [ ‘x 1 

.- 1 qx 

2 nx ~– (l–ax)q,] (78) 

and the expectation of the index of health can be 

obtained upon substitution of (78) in (71). 

For making statistical inferences and for 

comparing the state of health in different calendar 

years, it is necessary to know the variance of the 

index of health. It is not necessary hereto derive 

the complicated formula for the true unknown variance; for practical purposes, it suffices to have 

the formula for the sample variance, which can 

be computed directly from the observed data without referring to the models discussed in the preceding sections. The sample variance of H can 

be written as 

9 (79) 

and for each age group x 

S;x
= .+ + +:x * (80) x 

where the sample variance of the average duration 

of illness S; can be computed directly from the x 

sample. The sample variance of the age-specific 

death rate is given in reference 2. 

S;x = mx (l-ax n%m%) 

~ [1+ (l-ax) nxmx] (81) 1 

W. REMARK—ADJUSTED INDEX 

oF tiEALTH 

The index of health His evidently a meaningful and useful measure of the state of health 

in a single &pulation. Since it is a weighted 

average of HXwith the current population proportion Px\P as the weight, however, the value of 

H is affected by the current population composition. Such an effect will produce a distortion when 1“ 

two populations with different age compositions I 

are compared. To adjust for the difference, we 

may use a standard population and compute the 

weighted average of Hx, 

H*=+ 2 “P%Hx , (82) s 

where P3X/P~ is the population proportion of age 

group x in the standard population. This weighted 

average may be called the age-adjusted index of 

health. The sample variance of H* is given by 

(83) 

where the ,sample variance of Hx may be computed from formula (80). 

REFERENCES


1. Feller, W.: An Introduction to Probability Theory and lts App[i. 

cations. 2d edition. New York. John Wiley and Sons, Inc., 1957. 

2. National Vital Statistics Division: Standard error of the age-adjusted death rate, by C. L. Chiang. Vital Statistics--Special Reports, Vol. 47, No. 9. Public Health Service. Washington, D. C., 

1961. 

3. The Deparrmeat of Narional Health and Welfare and The Dominion 

Bureau of Statistics: Illness and Health Care in Canada, Canadian 

—ooo 

Sickness Survey, 1950.51. Ottawa. The Queen’s Printer and Con. 

troller of Stationery, 1960. 

4. The Dominion Bureau of Statistics and Tire Deparnnent of Na- ( 

tional Health snd Welfare; Canadian Sickness Survey, 1950-S1. No. 

8. VoIume of HeaIth Care (National Estimates); D. B. S. Reference 

Paper No. >1. Ottawa The Qrreen’s Printer and Controller of Stationery, 1955. 

— I 

12 
I Ml

Observed Expected 

836 840 

~ 4 5 6 7 8 9 10-19 

NUMBER OF DOCTORS, CALLS OR CLINIC VIsns 

NOTE E!!5,d o“ do!a Show” In fable 1. 

Figure 1. Observed and expected number	 of persons under 15 years of age by number of doctors’ calls or


clinic visits in a yepr.


135 

1326 1327 

30 

observed Expected 

28 

L 

> 

; 20 

, 

I 

; 15 

10 

58 

akzz- 0 I z 3 4 5 6 7 8 9 10-20 

NUMBER OF DOCTORS, CALLS OR CLINIC VISITS 

NOTE: Bused on aloha shown [n fable 2. 

Figure 2. Observed and expected number of persons 15-24 years of age by number of doctors’ calls or clinic


visits in a year.


13

514 

10-20 

NUMBER OF DOCTORS’CALLS OR CLINIC VISITS 

NOT= Based m do!o $hmm In table 3, 

Figure 3. Observed and expected number ofpersons of all ages by number ofdoctorsr calls or clinic visits


in a year.


900~ . . . 

ma -

U!l/ 700 - Obs,rvedEw.cttd 

~ 600 - . 

125 127 

78 77 

H Biwz_ 

7 8 9 

NUMBER OF cOMPLAINT PERIODS 

NOTE: Bas8d m data shown!!! Iobls 4. 

Figure 4. Observed and expected number of persons under 15 years of age by number of complaint periods


in a year.


14

------------------------------------------

------------------------------------------------

----------------------------------- ------------------

--------------------------------------------------------

DETAILED TABLES 

Page


Table 1.	 Observed and expected number of persons under 15 years of age, by number of doc-


tors’ calls or clinic visits in a year 16


2.	 Observed and expected number of persons 15-24 years of age,by number of doctors’


cslls or clinic visits in s year 17


3. Observed and expected number of persons of all ages, by number of doctors’ calls 

or clinic visits in a year -
 18


4.	 Observed and expected number of persons under 15 years of age, by number of com­


plaint periods in a year 19


15

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--------------------

--------------------

--------------------

--------------------

--------------------

--------------------

--------------------

--------------------

-------------------

-------------------

------------------

Table 1. Observed andd~;~~ec~~d number of persons under 15 years of age, by number of


calls or clinic visits in a year I


Number of doctors’ calls or clinic Observed Expected Difference (fn-q)z I 

visits, n f, F, fn -F, Fn 

Number of persons in thou;andsl 

Total 4,116 4,116 0 13.899 

O call or visit 2,367 2,379 -12 0.060


1 call or visit 749 715 +34 1.617


2 calls or visits 350 372 -22 1.301


3 calls or visits 222 221 -1-1 0.004


4 calls or visits 136 140 -4 0.114


Subtotal, 5-9 calls or visits-- 239 242 -3 0.037


5 calls or visits 95 91 +4 

6 calls or visits 64 61 -I-3 

7 calls or visits 41 41 o 

8 calls or visits 25 29 -4 

9 calls or visits 14 20 -6 

Subtotal, 10+ calls or visits-- 53 47 +6 0.766


10 calls or visits 12 14 -2 

11 calls or visits 11 10 +1 

12 calls or visits------------------- 9 7 +2 

13 calls or visits------------------- 8 5 +3 

14 calls or visits------------------- 5 3 +2 

15+ calls or visits 8 8 o 

R= l_.163 Q= 0.4(35 X2= 3.899 

S2= 4.500 @=0.348 d.f.= 4 

Source: Observed f“ were calculated from percent distributions shown in table 7-c, 

page 27 of Reference 4,and population totals shown in table 114, page 193 of Reference 

3. To estimate the parameters involved in the model, subtotals shown were distributed 

by the number of calls or viaits to obtain fn for each n in the respective groups. 

lFor justification in using thousand as a single count in computing the X2, see text


on page 4.


16

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--------------------

--------------------

--------------------

-------------- ------

--------------------

--------------------

--------------------

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-------------------

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-------------------

-------------------

-------------------

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-------------------

-------------------

-------------------

------------------

� 

Table 2. Observed and expected number of persons 15-24 years of age, by number of


doctors’ calls or clinic visits in a year


Number of doctors’ calls or clinic Observed Expected Difference (f”- F“)2 

visits, n f“ F, fn-F, F, 

Total


o	 call or visit 

1 call or visit 

2 calls or visits 

3 calls or visits--------------------


4 calls or visits


Subtotal, 5-9 calls or visits--


5 calls or visits


6 calls or visits


7 calls or visits


8 calls or visits


9 calls or visits


Subtotal, 10+ calls or visits--


10 calls or visits 

11 calls or visits 

12 calls or visits 

13 calls or visits 

14 calls or visits 

15 calls or visits 

16 calls or visits 

17 calls or visits 

18 calls or visits 

19 calls or visits


20+ calls or visits


R= 1,347 &=O.223


S2= 9.4,78 6= 0.166


Number of persons in thousands


2,050 2,050 0 16.955 

1,326 1,327 -1 0.001 

248 254 -6 0.142 

141 136 +5 0.184 

88 86 +2 0.047 

68 59 +9 1.373 

113 133 -20 3.008 

47 43 +4 

28 32 -4 

17 24 -7 

12 19 -7 

9 15 -6 

66 55 +11 2.200 

8 11 -3 

7 9 -2 

7 7 0 

6 6 0 

5 5 0 

4 4 0 

4 3 +1 

3 3 o 

3 2 +1 

2, 2 o 

17 I 3 +14 

X2= 6.955


d.f.= 4 .


Source: Observed fn were calculated from percent distributions shown in table 7-c, 

$age 27 of Reference 4,and population totals shown in table 114, page 193 of Reference . To estimate the parameters involved in the model, subtotals shown were distributed 

by the number of calls or visits to obtain fn for each n in the respective groups. 

lFor justificationin using thousand as a single count in computing the X2, see text on page 4. 

17

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--------------------

--------------------

--------------------

--------------------

--------------------

-------------------

-------------------

-------------------

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------------------

Table 3. Observed and expected number of persons Of all ages, by number of doctors’


calls or clinic visits in a year


Number of doctors’ calls or clinic Observed Expected Difference (f,- F’,)z 

visits,n f, F, f=- Fn F, 

Number of persons in thousands


Total 13,538 13,538 0 124.072 

O call or visit---------------------- 7,690 7,692 -2 0.001 

1 call or visit---------------------- 2,044 1,996 +48 1.154 

2 calls or visits 1,097 1,099 -2 0.004 

3 calls or visits-------------------- 718 714 +4 0.022 

4 calls or visits-------------------- 474 492 -18 0.658 

Subtotal, 5-9 calls or visits-- 1,001 1,105 -104 9.788 

5 calls or visits 310 357 -47 

6 calls or visits 250 267 -17 

7 calls or visits 190 202 -12 

8 calls or visits 140 157’ -17 

9 calls or visits 111 122 -11 

Subtotal, 10+ calls or visits-- 514 440 +74 12.445 

10 calls or visits------------------- 97 96 +1 

11 calls or visits------------------- 79 77 +2 

12 calls or visits 57 61 -4 

13 calls or visits 45 49 -4 

14 calls or visits------------------- 33 39 -6 

15 calls or visits------------------- 29 31 -2 

16 calls or visits 24 26 -2 

17 calls or visits 18 20 -2 

18 calls or visits 13 16 -3 

19 calls or visits------------------- 11 14 -3 

20+ calls or visits 108 11 -1-97 

~= 1.631 G =0.307 X2= 24.072 

S2= 10.286 f) =0.188 d.f.= 4 

Source: Observed fn were calculated from percent distributions shown in table 7-c, 

page 27 of Reference 4,and population totals shown in table 114, page 193 of Reference 

3. To estimate the parameters involved in the model, subtotals shown were distributed 

by the number of calls or visits to obtain fn for each n in the respective groups. I 

lFor justification in using thousand as a single count in computing the X2,see text


on page 4.


18

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------------------

------------------

------------------

------------------

------------------

------------------

------------------

------------------

-----------------------

-----------------

-----------------

-----------------

-----------------

-----------------

----------------

Table 4. Observed and expected number of persons under 15 years of age, by number of


complaint periods in a year


Number of complaint periods, n Observed Expected Difference (f,- FII)2


Total


O complaint period


1 complaint period


2 complaint periods


3 complaint periods


4 complaint periods


5 complaint periods 

6 complaint periods 

7 complaint periods 

8 complaint periods 

9 complaint periods 

Subtotal, 10+ complaint


periods


10 complaint periods


11 complaint periods


1.2complaint periods


13 complaint periods


14 complaint periods


15+ complaint periods


iv= 2.826 G= 3.113 

S2= 5.392 P= 1*102 

f“ F, f, -Fn F, 

Number of persons in thousands


4,116 4,116 17.871


522 551 -29 1.526 

875 817 +58 4.118 

787 799 -12 0.180 

637 648 -11 0.187 

458 471 -13 0.359 

316 318 ‘2 0.013 

206 205 +1 0.005 

125 127 -2 0.031 

78 77 +1 0.013 

53 45 +8 1.422 

59 58 0.017 

31 26 +5 

14 15 -1 

5 8 -3 

4 5 -1 

2 3 -1 

3 1 -I-2 

X2= 7.871 

d.f.= 8 

Source: Table 31, page 122 of Reference 3. To estimate the parameters involved in


the model, the subtotal shown was distributed by the number of complaint periods to


obtain fn for each n in the group. 

lFor justification in using thousand as a single count in computing the X2,see text


on page 4.


19

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