Note on Fitting the Multi-Hit Survival CurveAuthor(s): Joan M. GurianSource: Biometrics, Vol. 12, No. 2 (Jun., 1956), pp. 123-126Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/3001756 .

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NOTE ON FITTING THE MULTI-HIT SURVIVAL CURVE*

JOAN M. GURIAN

Division of Biological and Medical Research

Argonne National Laboratory, Lemont, Ill.

The expected proportions (S) of a population of micro-organisms surviving irradiation dose x has been described by

S = 1 - (1 - ekx)n.

Kimball [1] proposes a graphical method for evaluating the constants k and n in this expression. He defines his dependent variable

ui = In (I1-Si) i = 1 , 2, .. **p

where Si is the observed proportion surviving, and minimizes the quantity

2p

V = E [u - n In (1 e-kzx)]2 t =1

with respect to k, by a graphical procedure. This method has the advantages of being simple, requiring no arbitrary approximations, and permitting the use of all data. However, application of the transforma- tion used in this procedure, without making corresponding adjustments of the weights, will give the larger Si unduly heavy weight. This results in a poor fit at the lower values of Si , if Si varies by several orders of magnitude.

Survival experiments are usually designed to give equal percentage plating error for all Si or constant variance C_ for all log Si . If u= In (1 - SJ) is used as the dependent variable the sampling variance of the dependent variable is no longer constant, but is approximately equal to o_2[Si/(1 - Si)]2 [2]. The theoretical weight for ui is then [(1 - Si)/S]2. Errors in dose, dilution, etc., are often propagated as Si decreases [3], hence these effects must also be taken into consideration in the choice of the weighting function. Ideally the appropriate weights should be derived empirically from a properly designed experiment; but prior knowledge may be used to give a rough estimate of the weights, which may prove adequate for the evaluation of the parameters.

*This work was performed under the auspices of the U. S. Atomic Energy Commission.

123

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124 BIOMETRICS, JUNE 1956

TABLE 1

Formulae for Approximating n and k

Quantity Kimball's formulae Weighted Kimball's formulae

k Graphical estimate Graphical estimate

n 2 2

v2 WiV2 i =1 i =1

vi = In (1 e-kxi) v= In (1 -e-kx)

P P P P

V E 2- E UiVi 2 _ z2n 1: wjuivi V -~~U n z wiui

t=1 j=1 =1=1

2 S2B S2B Sk AB-C2 AB-C2

2 s2A s2A S~n .AB - C2 AB - C2

where:

82 v v s ~ ~~~ p -2 p - 2

P P A Vi2 WiV2

2 =1 i =1

B ( -e i r

22 _W(j _ )

C E (l ~~e Ii, ( iie $

TABLE 2

n, k and Their Errors Computed

Quantity Kimball's Weighted Kimball's Method Method

n 2.220 2.654 k .525 .600 Sn .140 .199 Sk .033 .028

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CURVE FITTING 125

Table 1 contains in Column 1 Kimball's formulae used in the approxi- mations of n and k, and the sampling errors of their final values, and in Column 2 the corresponding modified formulae. The parameters k and n are determined by the same method of iterative approximations as in the original procedure. The only increase in labor and complexity is that due to the presence of differing weights.

Table 2 contains the constants and their errors as computed by the two procedures, using the data of Dr. S. Pomper on the survival of

1.0

0.I

z

Z 0.01

o

0~

0~

0.001 - -Curve fitted by Kimball's Method -Curve fitted by Weighted Kimball

Method

* Experimental Points

0.0001 1

0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

X-RAY DOSE IN 104 ROENTGENS

FIGURE 1

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126 BIOMETRICS, JUNE 1956

Saccharomyces cerevisiae, which were employed by Kimball. The weighting function used is wi = (1-S)2. It is based on the assump- tion that in this experiment errors in dilution, etc., increase with de- creasing Si in such a manner, that they offset the decreasing theoretical sampling error, so that the variance of Si is constant.

Figure 1 shows the data as fitted by the two methods.

ACKNOWLEDGEMENTS

The author wishes to express her gratitude to Mr. George A. Sacher and Mr. Sylvanus A. Tyler for many helpful discussions.

REFERENCES

(1) Kimball, A. W. The fitting of multi-hit survival curves. Biometrics, 9: 201-211. 1953.

(2) Deming, W. Edwards. Statistical Adjustment of Data. John Wiley and Sons, Inc., New York, 1948. p. 39.

(3) Jennison, M. W. and G. P. Wadsworth. Evaluation of the errors involved in estimating bacterial numbers by the plating method. J. Bact., 39: 389-397. 1940.

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