WOODS HOLE, MASSACHUSETTS

1949

Dear Josh:

     Thenks for your letter, which I take very seriously.
I violently disagree with assertion that g Is difficult to
compute; a Pick&t 8c Eckel model 4 slide rule will convert
1 datum/5 sec.  At low S, g and3 are the same, 80 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
graphically and compared with S plots of the same data.
Your Taint that g should be defined as an experimen%al
statistic I think is well taken and T plan to add something
to emphasise this point.  This leads me tc an impasse in
doinp someth.inP about your suspicion that the g plot may
not aive a linear relation to dose when the*= is a Poisson
distribution of units regardless of whether the S plot has
approximated linearity, since the whole purpose of $he
derivation (l-J)-( 14) Is to show under what conditions the
experimental statistic, g, is equal to np.  Perhaps this
could be shown as follows:  Were p is the probability that
a unit survives a dose D:

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

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

and,

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

(5)      (I-S)"Pg

     Now assume that in a given experiment we do not

know whether equation (4) holds, i.e. we do not know the
type of distribution of our hypothetical unite at D=O
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?


WOODS HOLE, MASSACHUSETTS

&e 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
t,zrnEo;;rily assume that equation (4) holds, set 1-S equal
     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
eqjlation (2) above that log np would be found to have a
linear relation to dose.  Therefore if such 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 la an upper limit.

    It may help to point out p Is a free variable In
(3) above, and is the only one which Is a function of doae.
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.

    With constant initial n the error In deriving (18)
from (15) depends on l-p#e'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 "g" 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 8;; ;l-~z
necessity for attempting a rigorous proof of this.
light of your criticism I will try to further clarify the
meaning of '&I".  Also, I would like your permllssion to use
your truncation correction and derivation of the Delbruck
equation.

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


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