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

George Stanford gstanford at aya.yale.edu
Sat Jul 23 00:46:44 CDT 2005


         Thanks for your further explanations.  I
really do appreciate your patience.

         The observed cancer rate consists of two
components: (1) a background rate; (2) an "excess"
rate (positive or negative) due to radiation.  The
LNT hypothesis is that excess cancers can be
predicted by extrapolating linearly from
moderately high doses back to the origin (linear
plot).  In fitting the LNT model to observed
cancer-rate data (CR), then, there are two
parameters (as you say), and only two.  One of
them is the y-value (bkg) of a horizontal line
representing the background.

         The other parameter is the slope of the LNT
representation of the excess, where "excess" is
defined as (CR - bkg).  Call it CRx.  The slope of
that straight line is determined by two
requirements: (a) the line must pass through the
origin (zero excess at zero dose), and (b) it must
pass through the values of CRx at dose rates high
enough that the excess is clearly evident.

         The fitting process is very simple, and can
usefully be done by eyeball, without so much as
thinking about least-squares minimization of
residuals.  Please note that the term "polynomial
fit" immediately rules out the possibility of
fitting to the LNT model, and therefore cannot be
used to refute that model.

         The value of the parameter bkg MUST be the
the y-intercept of the horizontal line that best
fits the observed low-dose rates.  It would not be
representative to use any other guess, because the
resulting slope of the LNT line would be wrong.  I
respectfully suggest that you do not have freedom
to make some other choice.  You would be leaving
yourself open to the charge that you selected the
"background" arbitrarily to suit your purposes.

         Also, you seem to using "LNT" as though it
were a synonym for "linear," but it's not.  It's
the term used to refer to the particular linear
model for radiation damage defined above.  The LNT
line MUST be the best one-parameter (slope) fit to
the high-dose excess.  Any other usage of “LNT”
will not be understood, and will only confuse the
reader. Thus, on the linear and loglog plots you
just sent, none of the curves labeled "LNT"
qualifies for that designation.  The rose-colored
curve labeled "LNT EAR Preston pooled" is a
particular mystery, since it does not fit any of
the data at all, although it does come fairly
close at very low dose rates.

         Your LQ fit, on the other hand, looks
excellent, with an appropriate y-intercept and
good match to the data.  It's clearly better than
a properly done LNT fit -- although, as shown in
the picture that I sent you earlier, had you used
the comparable procedure for your "LNT" fit, you
would find that the curve falls within the error
bands (except, perhaps, for the low point at 750
mGy) -- as you, too showed in one of your graphs:
To repeat from my preceding transmission, "If I
understand your new loglog plot
you have corrected [the background] 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."

         I hope this helps.

                 Best wishes,


At 08:34 AM 7/22/2005, Rainer.Facius at dlr.de wrote:

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 into your 

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 
have made a decision about one parameter (in our case the 
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 

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
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
Cc: radsafe at radlab.nl; Zack.Clayton at epa.state.oh.us; jimm at WPI.EDU; 
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


         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.



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


More information about the RadSafe mailing list