# [ RadSafe ] AW: AW: Log-log plot for excess breast cancer incidence rate

Rainer.Facius at dlr.de Rainer.Facius at dlr.de
Fri Jul 22 08:34:29 CDT 2005

```George:

Thank you for insisting on explanations and I must apologise for not
being precise enough. As I read my remark again, that plea was meant to
apply only to questions regarding the fit statistics as copied from the
ORIGIN output. As far as the drawing is concerned I obviously erred in
considering it self explanatory.

Regarding the consistency in data analysis you call for, I am all with
you. Yet, I think I can demonstrate that you introduce inconsistencies

In a very formal sense, for linear curve fitting I have two "degrees of
freedom" corresponding to the two parameters I wish to determine. Once I
y-offset=background incidence rate) - guided by divine revelation, by
theoretical knowledge, by separate experiments, by hearsay, by visual
inspection, by formal parameter estimation or whatsoever, I have exactly
one degree of freedom left to improve the fit, in our case the slope=
absolute excess risk coefficient. So whether I subtract the maximum, or
the minimum or the average incidence, once I plot these reduced data
(reduced by this background), I have only the slope left to search for
further reduction of the variance. I am sure you have no problem to
concede this, if I did draw the reduced incidence in a linear-linear
plot, in which case the data pertaining to the reduced background would
scatter around 0. In this case I had to fix the fit line in the origin
and try to find the 'best' slope by rotating the line around the origin.
The same remains true if I try to find an 'optimum' slope in a log-log
plot. The counterpart of revolving the line around the origin is in
log-log the displacement of a 'slope=1' line. I only can shift around
this line until I am satisfied, but this straight line it must remain.
Otherwise I introduce/consume another degree of freedom! (Of course I
could go back and draw the reduced data with another trial value for the
background.) If this way I can't get no satisfaction, :-), this simply
means my trial function does not fit.

Initially I held that your unease regarding the "weird result"
(incidence above a LNT line at low doses) could simply be resolved by
the existence of a minimum. Yet actually it stems from a weird mixture
of this minimum and the actual scatter of the data. You can most easily
appreciate this by looking to the first of my attached graphs where by
eye only I drew LNT lines from the minimum, the average, and the maximum
respectively. Unless you take the maximum as background, every feasible
LNT line will you give the "weird result of an excess of low-dose
cancers - exactly the opposite of what we are claiming". The influence
of the scatter together with an initial negative trend of the data got
obscured in the log-log representation and hence the bottom line of the
otherwise for exploratory inspection useful log scale is that it proved
to be a nuisance which I better had avoided.

That brings me back to the graph whose explanation at first sight did
escape you, due to the negative slopes. You correctly observe that it is
"clearly not the conventional LNT". The problem however consists in the
conventional designation LNT as Linear-No Threshold being actually a
misnomer. In general, a LNT relation is given by

incidence - background = B * exposure.

Yet, this is exactly the relation which yields the lines with the
negative slopes when fitting to the ranges of low dose values indicated
by the extension of the lines (0 to 1000 and 0 to 2500 mSv). Of course I
did restrict the range to those exposures where I would be interested in
when being engaged in radiation protection. What you call the
conventional LNT postulate should actually be designated LNTPS, i.e.,
LNT plus the additional covert postulate Positive Slope, i.e., B>=0!

In the second attached graph I include a LNT line obtained with the
constraint B>=0 - LNTsub(2500, B>0), and you won't be surprised to see
that B=0 is the maximum likelihood estimate representing the data in
this range. If exposures up to 2500 mSv would really be of concern, of
course I would use neither the LNTsub(2500, B>0) line nor the
LNTsub(2500) line (without that additional constraint!) but the
LQsub(2500), a linear quadratic function fit to the data in the 0 to
2500 mSv range. Since lifetime exposures beyond 1000 mSv probably occur
only extremely rarely, the maximum likelihood estimate in this most
relevant range would be given by the LNTsub(1000).

You can arguably make a case for all of the approximations so far
mentioned. That someone would accept the line labelled LNTsub(EAR
Preston pooled) as a reasonable approximation in any dose interval does
defy my imagination. The pooling comprised the A-bomb data as well as
data from six other populations (medically) exposed also at
(fractionated) high dose rates. The only population truly representative
for chronic low dose exposures is the hemangioma population. It is your
guess to speculate why any one interested in breast cancer risk from
chronic low dose exposures would wish to intermingle these data and to
use the results from such a mix and claim: "The results support the
linearity of the radiation dose response for breast cancer.".

Finally, since we wasted so much time on log-log plots, I also attach a
third graph showing the 5 fit lines in log-log again.

Thank you once more for enticing me to carry on so far.

Kind regards, Rainer

Dr. Rainer Facius
German Aerospace Center
Institute of Aerospace Medicine
Linder Hoehe
51147 Koeln
GERMANY
Voice: +49 2203 601 3147 or 3150
FAX:   +49 2203 61970

________________________________

Von: George Stanford [mailto:gstanford at aya.yale.edu]
Gesendet: Mittwoch, 20. Juli 2005 20:56
An: Facius, Rainer
jjcohen at prodigy.net; jmarshall.reber at comcast.net; frantaj at aecl.ca;
jaro-10kbq at sympatico.ca; Jim_Hardeman at dnr.state.ga.us;
hflong at pacbell.net; maurysis at ev1.net; crispy_bird at yahoo.com;
merklejg at ornl.gov
Betreff: Re: AW: Log-log plot for excess breast cancer incidence rate

Rainer:

Thanks for the additional information, and the two new graphs.

I'm afraid I have to take issue with you on one point.

When fitting a curve to data, one has to use the observed values
consistently.  That is what I did with my fits.  The observed residual
background on the plot that I used is not zero.  It is approximately 25.
It's certainly legitimate to subtract a constant from the background,
which is what you did.  Subtracting the entire observed background leads
to negative data points -- no problem on a linear graph, but awkward in
a logarithmic representation (as you have pointed out).

If you were to repeat my process with the full background, the
result would be the same.  The LNT model** fits the hemangioma data
within the reported experimental error limits.

In your earlier graph, you selected the extreme (lowest) data
point from the low-dose region to use as the observed background.  That
led to the weird result that you depicted an excess of low-dose cancers
-- exactly the opposite of what we are claiming.   If I understand your
new loglog plot (PrestonHemExcess(1-1000)loglog.gif), you have corrected
that problem, and the result is the same as mine -- the LNT fit is
within the error limits.  That's something we have to live with --
accept it, and move on.

The interpretation of the other graph (PrestonHemLowFits.gif)
escapes me, since it has lines with negative slopes labeled "LNT" --
clearly not the conventional LNT -- but, as instructed, I will refrain
from asking for an explanation.  Maybe it will dawn on me sometime.

Best,

George

**  I can't think of a better term --"model" does not imply that it's a
good model.  The quadratic model is better.

```