Use of the recursion formula of the Gompertz survival function to evaluate life-table data

and

Mechanisms of Ageing and Development

ELSEVIER

detdqnnent

89 (1996) 155-163

Use of the recursion formula of the Gompertz survival function to evaluate life-table data Ioannis Institute

of Medical

Radiation

D. Bassukas”

and Cell Resrawh (MSZ), Unicersity D-97078 Wiirzburg, Gcrman~~

of Wiirrburg.

Vrrsbackrrstr.

5,

Received 24 July 1995: revised 22 April 1996; accepted 24 April 1996

Abstract The recursion formula of the Gompertz function is an established method for the analysis of growth processes. In the present study the recursion formula of the Gompertz survival function In S(t + s) = a + h x In S(t) is introduced for the analysis of survival data, where S(t) is the survival fraction at age t. s is the constant age increment between two consecutive measurements of the survival fraction and a and h are parameters. With the help of this method-and provided stroboscopical measurements of rates of survival are available--the Gompertz survival function, instead of the corresponding mortality function, can be determined directly using linear regression analysis. The application of the present algorithm is demonstrated by analysing two sets of data taken from the literature (survival of Drosophila imagoes and of female centenarians) using linear regression analysis to fit survival or mortality rates to the corresponding models. In both cases the quality of fit was superior by using the algorithm presently introduced. Moreover, survival functions calculated from the fits to the mortality law only poorly predict the survival data. On the contrary, the results of the present method not only fit to the measurements, but, for both sets of data the mortality parameters calculated by the present method are essentially identical to those obtained by a corresponding application of a non-linear Marquardt-Levenberg algorithm to fit the same sets of data to the explicit form of the Gompertz survival function. Taking into consideration the advantages of using a linear fit (goodness-of-fit test and efficient statistical comparison of survival patterns) the method of the recursion formula of the Gompertz survival function is the most preferable method to fit survival data to the Gompertz function. * Present address: Fachklinik Hornheide. Univ. Mtinster. Germany. Tel.: + 49 251 3287465: fax.: + 49 251 3287299. 0047.6374/96/515.00

8 1996 Elsevier Science Ireland

PII SOO47-6374(96)01747-Z

Dorbaumstr.

Ltd. All rights

reserved

300, D-48157 Miinster.

156

I.D. Bassukus

Keywords: Mortality;

Gompertz

/ Mecllanism function;

Drosophila;

Female

of .4geing und Decelopnwnt

Recursion

formula;

89 (1996) 155-163

Difference

equation;

Survival;

centenarians

1. Introduction Since its first description [l], the Gompertz function has been extensively used in many biomedical research and application fields to fit to diverse sets of longitudinal data, both in the study of growth and regression phenomena. In the study of survival or mortality processes the so-called Gompertz mortality function (K,(t)) R,,,(t)=

R, x e”“’

(1)

is often applied to fit to mortality rate data, where R,(t) is the mortality rate at age t and R, and M are parameters (for a review of the different applications of Eq. (1) to fit mortality data see, e.g. [2,3]). Although the application of the mortality function is thought to be more directly interpreted in biological terms there are, however, some calculation problems inherent in the application of this equation, in practice. Among many different limitations (see Witten et al. [4,5]) probably the most serious one is related to the adequate definition of the mortality rates or hazards [3]. A way to overcome this difficulty is to use the Gompertz survival function, S(t), to fit to the corresponding survival data [4,6]: S(t)=exp

: (

x (1 -eXXt) >

where S(t) is the fraction of surviving individuals at age t, and R, and a are constants as in Eq. (1). In the first part of this paper an algorithm is introduced for the study of survival (mortality) kinetics, which is based on the recursion formula of the Gompertz function [7]. In the second part of this study the application of this algorithm is demonstrated by reanalysing two sets of survival data taken from the literature, one with uncensored, (Drosophila imagoes), and one with censored observations, (human female centenarians).

2. Materials and methods 2.1. Sourer of duta

Two sets of data compiled from the literature have been analyses according to the present method: (a) uncensored data on the survival of a cohort of male Drosophila imagoes [3,8]; and (b) censored data on the survival of female centenarians from 13 low-mortality countries [9, lo].

I.D. BUSSU~YZS / Mechanisms of’ Ageing ad

2.2. Statistical

157

Deoelopment 89 (1996) 15% 163

evaluation

For both data sets (drosophila imagoes and female centenarians) the survival probability at a given age was calculated according to the life-tables method [l 11. In order to compare the results of the present fitting method to the classical use of the mortality function (Eq. (1)) mortality rates were derived from actuarial analysis, i.e. as probability of dying in a given age interval, conditional on having survived to the youngest age of this age interval. Unweighted actuarial survival probabilities were further used in the regression analysis in order to fit to survival or mortality functions [lo]. It is worthwhile to note, that in the case of the uncensored Drosophila data the absolute number of individuals, which survived a given age, can also be directly applied to calculate the corresponding best fit Gompertzian recursion formula. In fitting the Gompertz mortality equation (Eq. (1)) the age, which corresponds to each mortality rate, was adjusted to be the midpoint of the corresponding interval of ages. Gavrilov and Gavrilova [3] and Wilson [6] discuss the practical limitations of the different definitions of the mortality rate. Standard least-square linear regression analysis, provided on the commercial PC-MEDAS software for statistical analysis (Grund, Wtirzburg, Germany), was applied to calculate best-fit regression lines. This analysis provided best-fit values for both the slope and the axis-intercept. Goodness-of-fit was tested on the basis of the iteration or run test [12]. The Marquardt-Levenberg algorithm was applied to directly fit the explicit form of the survival function (Eq. (2)) to the same data using the commercial SigmaPlot software (DOS-Version, Jandel Scientific, Erkrath, Germany). Further details of the statistical methods used here have been discussed elsewhere in relation to the analysis of growth curves by the method of the recursion formula of the Gompertz function [13,14].

3. Results and discussion 3. I. The recursion formula of the Gompertr the corresponding mortality law The recursion

formula

or difference

In s(t + r) = a + b x In S(t)

survirul jiuxtion

equation

and its relationship

of the Gompertz

growth

to

function, (3)

where s(t) is the value of a Gompertz function at time t, u and b are constants and r is the constant time interval between stroboscopic s(t) measurements, has been previously used to study growth phenomena and it was shown that it may, potentially, be also used in the study of regression phenomena obeying similar kinetics [7]. Applying linear regression analysis and provided stroboscopic measurements are available (quite a common situation in reports of demographic data), Eq. (3) can be applied to study numerous dynamic processes. The main advantage of this method is that it unifies a wide spectrum of alternative patterns of alteration. This permits comparisons of growth processes on the basis of a single model [6,14].

I.D. Bassukas / Mechanisms of’ Ageing and Deoelopment 89 (1996) 155-163

158

Thus, depending on the exact value of the parameter b, i.e. the slope of the regression line, diverse growth or involution patterns emerge, which range from increase (decrease) with decreasing growth/regression rate for 0 < b < 1 (decelerating alteration rate, i.e. classical Gompertz function) to increase (decrease) with increasing growth/regression rate for b > 1 (accelerating alteration rate, ‘hyperexponential’ function). For b = 1 a special solution exists, the exponential model, i.e. growth or regression with no alteration of the specific growth or regression rate. Meanwhile, myself and others have used this algorithm to analyze and compare growth processes in vivo and in vitro [13- 181. It is worth noting, that essentially the same model has been previously explored for use primarily in marine biology [19] and insect ecology [20] but not, however, for the description and comparison of mortality kinetics as a function of ageing. In the following it will be demonstrated that essentially the same algorithm can be applied to the analysis of life-table data. For the use of Eq. (3) to analyze survival or mortality data the equivalence of the parameters a (intercept) and b (slope) of the recursion formula of the Gompertz function with the parameters R, and CIof Eqs. (1) and (2) emerges, if one compares Eqs. (2) and (3) as follows, In b R=-=>b=enxl

(da) (4b)

From Eq. (4a) it follows that the parameter CIof the mortality function positively correlates to the parameter b of Eq. (3). They both express the acceleration constant of the mortality rate as a function of aging. In terms of the Gompertzian mortality law b is that factor by which the mortality rate increases after the constant stroboscopical time interval, S, has elapsed, i.e. between two successive measurements or ages. As can further be deduced from Eq. (4a) the parameter c( of Eqs. (1) and (2) equals the natural logarithm of the normalized b-value of the difference equation, i.e. the b-value for r = 1. It is worth noting, that the survival data of two populations with the same b-value, i.e. with parallel Gompertz survival difference equations, are only shifted on the time scale, i.e. at any time or age the surviving fraction of one population is proportional to the surviving fraction of the other. The parameter u of the Gompertzian recursion formula is a measure of the survival fraction after completing the first iteration, i.e. at time r, and thus proportional to parameter R, of the mortality function. Note that these two parameters (n and R,) are arbitrarily defined on the basis of the corresponding parameter of the alteration of the mortality rate (b or a) and time (age). Eq. (4b) also makes obvious the strong interrelationship of the two parameters of the mortality function and the corresponding Gompertzian difference equation (Eqs. (1) and (3)) to each other, & -=x

a l-b

(5)

I.D. Bassukas

/ Mechanisms qf’ Ageing and Deueiopment 89 (1996) 155-163

I59

3.2. Mortulity vs. survival formulation of the Gompertz function Both sets of analyses survival data were each fitted to three different functions: (i) Gompertz mortality function (Eq. (1)); (ii) recursion formula of the Gompertz survival (Eq. (3)) using linear regression analysis; and (iii) Gompertz survival function (Eq. (2)), using the Marquardt-Levenberg algorithm. Table 1 and Fig. 1 summarize the results of all these fittings. The application of the recursion formula of the survival law to both sets of data resulted in very good fits (correlation coefficients of 0.9908 and 0.9942 for Drosophila imagoes and female centenarians, respectively; P > 0.05 for the lack-of-fit-test for both data sets). The slope values (parameter b) were in both cases > 1.O, which implies an exponentially accelerating regression process. On the contrary the quality of the fit to the corresponding mortality function was disappointing (P < 0.01 for the lack-of-fit-test in both cases) in spite of comparatively high correlation coefficients (0.9694 and 0.9763 for Drosophila and centenarians, respectively; compare with Fig. 1). This comparison is mandatory to avoid the possibility that a good model can be misused by applying an inappropriate fitting procedure. Recently Eakin et al. [4] have extensively reviewed this potential pitfall in estimating parametric survival model parameters in gerontological aging studies. In Fig. 2 a comparison of the results from the three different fitting procedures applied to the survival data of Drosophila imagoes (Fig. 2a) and female centenarians (Fig. 2b) are presented. Either method (recursion formula or non-linear) gives Table I Survival kinetics of a cohort of Drosophila imagoes and of human female centenarians: comparison of the estimated values of the parameters ( i standard errors) of the Gompertz mortality function (R,, and x. Eq. (1)). Gompertz survival function (R, and r, Eq. (2)) and recursion formula of the Gompertz survival function (a and b, Eq. (3)).

Drosophila” R,, 5L (2 h

Mortality”

SurvivaP

Recursion

0.0121 f 0.0034 0.0589 f 0.0056

0.0012 * 0.0003 0.0687 f 0.0024

(0.0013) (0.0687) -0.0096~0.1119 1.5096 k 0.0727

0.3474 * 0.02 13 0.0683 ) 0.0096

0.3782 * 0.0239 0.1170~0.0103

(0.3794) (0. I 172) ~ 0.4026 i 0.1666 1.1243 k 0.0385

survival’

Centenarians R,, r ‘1 h I’ By fitting data to Eq. h By fitting data to Eq. L By fitting data to Eq. ’ Values refer to days. ’ Values in parenthesis

(1). (2). (3). are calculated

on the basts of the recursion

formula

estimates.

I.D. Bas.cukas

I Mechunisms of Age&g and Decelopment 89 (1996) lS5- I63

lo-'

Age

10-J

1

o-2

10-l

Survtval fractron at

[dl

age

100 X

[d]

BI 110

105

Age

[dl

Survwal

probability

at

age

X

[yr]

Fig. I. Fit of the Drosophila survival data (A) and the human female centenarians survival data (B) to the Gompertz mortality function (Eq. (I): Al and BI) and to the recursion formula of the Gompertz survival function (Eq. (3); A2 and B2). Solid lines represent best-fit regression lines and dashed lines bound the 95% confidence space of the latter. For the source of the retrieved data and the corresponding equations see text. Note the bad fit in both cases using the Gompertz mortality formulation (Al and Bl).

better fits for both data sets as compared with those tits which are calculated by applying the parameters obtained by fitting the mortality function (for the calculation of the explicit form of the survival function on the basis of the results of the fit to the recursion formula see Appendix A). The ‘error’ made by the application of the mortality function, i.e. the deviation from the measured data, is also systematic in both cases: it increases with decreasing survival fraction. Additionally, in the case of the Drosophila data the application of the mortality function leads to an underestimation, whereas the same function overestimates the survival rates of the female centenarians. This result agrees with the conclusions of recent papers by Wilson [6] and Eakin et al. [4], who also discuss some reasons why non-linear analysis of survival functions gives better fitting results compared with the appliedtion of the corresponding Gompertz mortality function. Probably, the most impor-

I.D.

Bassukas

/ Mechanisms

of Ageing

and Developmcm

89 (1996)

155-163

161

tant reason for this discrepancy is the increasing uncertainty in the calculation of hazards, either by the actuarial or by the Kaplan-Meier method, as the population diminishes. On the contrary, the application of the difference equation of the survival function gives fits indistinguishable to that gained by using the non-linear Marquardt-Levenberg algorithm to directly fit the explicit form of the survival function to the same data (Fig. 2). Finally, the application of the present method seems to provide better fits than a corresponding log-likelihood based fitting procedure previously applied by Zelterman [lo] to fit the Gompertz survival function to the same survival data of the presently analyzed female centenarians cohort. Notably this is true not only for the use of the difference equation. but also for the direct non-linear fit using the Marquardt-Levenberg algorithm. Wilson [6] also noticed good fits to survival functions by using the Marquardt-Levenberg algorithm. Since the quality of fit to survival data, which is provided by the method of the recursion formula of the Gompertz survival function, is indistinguishable from that obtained by the use of the non-linear Marquardt-Levenberg algorithm, its use is preferable over the use of the latter procedure for at least two reasons. Firstly, the method of the recursion formula enables the application of a goodness-of-fit test to clarify the overall applicability of the model used. i.e. the Gompertz equation (see [13] for details of the statistical evaluation). Secondly, it is the simplest model still adequate to fit the Gompertz survival function, which is a two-parameter model. It is worth noting, that although the fitting procedure of the recursion formula of the survival function is mathematically equivalent to that fitting procedure of the mortality function, fitting to the former algorithm clearly supplies better fitting results as compared to the Gompertz mortality function. Furthermore. Eq. (3) can be extended to be used for the comparison of the patterns of mortality processes.

A.ee IdI

Age aI last hirlllclay

[)rI

Fig. 2. Comparison of the fits of the Drosophila survival data (A) and the human female centenarians survival data (B) to the Gompertz mortality function (Eq. (I ): dashed line), the recursion formula of the Gompertz survival function (Eq. (3): dotted line) and to the explicit form of the Gompertz survival function by the Marquardt-Levenberg algorithm (Eq. (2): solid line). Note the unsatisfactory fit of both data sets to the Gompertz mortality function,

162

I.D. Bassukas

/ Meclzanisms of Ageing and Development 89 (1996) 155-163

I have recently described a corresponding method of comparison of patterns of growth processes on the basis of the difference equation of the Gompertz function by applying a covariance algorithm and Scheffe’s F-projection calculations [14]. Finally simple non-linear extensions of the present basic model may be used in the analysis and better understanding of survival data with survival kinetics deviating from the simple Gompertzian mode. A common form of such a deviation is the case of decreasing mortality rates as a function of age, the so-called tailing of the survival curves [21]. It is worthwhile to see if models derived from the present model, by allowing the b parameter to decrease as a function of the iteration steps, can sufficiently fit to survival data showing this latter deviation.

Acknowledgements

The valuable criticism of Professor Dr. B. Maurer-Schultze during the preparation of this manuscript is gratefully acknowledged. I am indepted to a referee of this paper for very instructive suggestions.

Appendix A: Construction Gompertzian function

of survival curves using the recursive form of the

Using the values of the parameters a and b of a best-fit Gompertz difference survival function (Eq. (5)) and setting a starting value (SO) the survival fraction after n cycles of recursion (S, ) can generally be calculated according to In S,, =

a x (1 -b”) l-b+b”xlnS,,

(Al)

Eq. (Al) is further simplified for use in the case of survival studies by setting S,, = 1, which leads to

(A21 Eq. (A2) can be applied to construct survival curves using the parameters of the fit of the Gompertzian recursion formula to corresponding data.

References [I] B. Gompertz. On the nature of the function expressive of the law of human mortality. and on a new mode of determining the value of life contingencies. Philos. Trans. R. Sot. London, 115 (1825) 513-585. [2] C.E. Finch, Longevity, Senescence, and tlze Genomr, University Chicago Press, Chicago, IL, 1990. [3] L.A. Gavrilov and N.S. Gavrilova, The Biology cf L@ Span, A Quantitatil;e Approach, Harwood, London, 1991.

I.D.

[4] T. Eakin,

[5] [6] [7]

[8] [9] [IO] [I l] [I21 [I31 [I41 [I51 [I61

[I71

[I@

(191 [20] [2l]

Bass&as

/ Mrchanisms

yf’ A,@ng

and Dtwlopmet~

89 (1996)

155- I63

163

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