Marine Biological Laboratory

WOODS HOLE, MASSACHUSETTS a .
~) v' } &
_jund-3b, 1949
Dear Josh:

Thanks for your lettey, which I take very seriously.
I violently disagree with assertion that g is difficult to
compute; a Pickett & Eckel model 4 slide rule will convert
1 datum/5 sec. At low S, ¢ and S are the same, so you don't
have to deal with anything off scale.

It is too bad that finished figs. were not ready to
send. The superiority of the @ plot is evident when shown
eraphically and compared with S plots of the same data.
Your vcoint that ¢ should he defined as an experimental
statistic I think ts well taken and I plan to add something
to emphasise this point. This leads me tc an imvasse in
doing something about your suspicion that the ¢ plot may
not vive a linear relation to dose when thers is a Poisson
distribution of units regardless of whether the 5 plot has
approximated linearity, since the whole purpose of the
derivation (13)-(14) is to show under what conditions the
experimental statistic, ¢, is equal to np. Perhaps this
could be shown as follows: Where p is the probability that
& unit survives a dose D:

(1) p= e kD

Where there is an average of n units per organism
np is the average number of surviving units per organism
and,

(2) np = ne7*D

If the units have a Poisson distribution and Py is
the proportion of organisms in which m units survive out of
a total of n units;

i
(3) Py = e-np tip)
and,
(4) P,= ef?

Where S is the experimental survival of organisms,
we essume according to the multi-unit hypothesis that:

(5) (1-S)=Po

Now assume that in a given experiment we do not
know whether equation (4) holds, 1.e. we do not know the
type of distribution of our hypothetical units at D=0
nor do we know the type of distribution of assumed surviving
units at any dose. How much can we find out about this
from the data, and what is the easiest way to do it?
Marine Bi ological Laboratory

WOODS HOLE, MASSACHUSETTS

One way is to just look at the data, or perhaps the S plot.
I daresay that with three months of intensive training an
intelligent individual could be taught to make fairly good
guesses this way. You will admit that it is easier to
temporarily assume that equation (4) holds, set 1-5 equal
to e~"P, compute np, then see where in the experiment lognp
has a linear relation to dose. If we were able somehow to
know the values of np in an experiment, it is evident from
eqyation (2) above that log np would be found to have a
linear relation to dose. Therefore if auch a linear
relation is found for np computed from (4) above, we are
justified in thinking that our method of computing np

was correct. Also, where this relation is non-linear we
know that hthe theoretical np 1s an upper limit.

It may help to point out p is a free variable in
(3) above, and is the only one which 1s a function of dose.
Obviously the substitution of any of the possible values
of p (i.e. the delivery of any dose) in (3) cannot change
the form of the distribution, and hence the validity of (4),
but merely shifts the expected (mean) value. This can be
clearly seen in your derivation of the Delbruck equation
where shifting q within its boundaries can have no effect
on the derivation. Lable

With constant initial n the error in deriving (1§) \ oe US
from (1%) depends on 1-psse~P, and not on the value of n. ¢
For a given p, S is a function of n. Therefore the survival

at which (14) holds is higher for higher n. It is evident on
plotting "ge" curves for various assumed distributions of

initial n, that the approximation becomes good more rapidly

for distributed n than for constant n, but I don't see the
necessity for attempting a rigorous proof of this. In the

light of your criticism I will try to further clarify the
meaning of "g". Also, I would like your permiission to use

your truncation correction and derivation of the Delbruck
equation. ,

We wish you could come to Woods Hole, its cool
Here.

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