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Re: Type II Poisson Errors (REPOSTED)
James,
Chris is correct in his statements. In particular, subtracting even a
constant from a Poisson distribution changes the distribution. See the work
by Potter, William E.
Neyman-Pearson confidence intervals for extreme low-level, paired
counting, Health Physics 76:186:1999
and
Confidence Intervals for low-level, paired counting, Operational
Radiation Safety, 77, No. 5, November 1999, S111.
There is no formal method I have found for an LLD concept for a Poisson
distribution. I am preparing such a publication. A simple way to determine
the Lc is obtain an average background, BG, for the counting period (round
up to the nearest integer). In Excel use the formula =CHIINV(0.05,
2*(BG+1))/2. This will equal Lc (95% confidence interval of BG). To find
the LLD, substitute for LLD in =CHIINV(0.95, 2*(LLD+1))/2 (5% confidence
interval of LLD) until you obtain the smallest value greater than Lc. This
LLD is total counts. The difference LLD-BG is the number you are
interested in, but remember, as soon as you subtract you change the
distribution and increase the uncertainty. You need Potter's method to
determine the uncertainty. More when the paper is finished.
Joe
----- Original Message -----
From: Christoph Hofmeyr <chofmeyr@nnr.co.za>
To: <james.g.barnes@ATT.NET>
Cc: <radsafe@list.vanderbilt.edu>
Sent: Monday, September 24, 2001 10:03 AM
Subject: RE: Type II Poisson Errors (REPOSTED)
> James,Radsafers,
> I think you are posing a very interesting question. Poisson statistics
> are applicable when we are dealing with a counting problem. When one
> mentions 'background' to any HP person, or, for that matter to most
> physicists, their immediate urge is to want to subtract it somewhere.
> Now, the Poisson distribution has the fairly unique and valuable
> property that the variance is equal to the average. Futher, the sum of
> two Poisson distributions (say, background and say, source) is again a
> Poisson distribution. But, and this is a big BUT, the difference is in
> general NOT a Poisson distribution. So, having lovingly subtracted the
> background, you have destroyed the Poisson distribution of your data.
> Makes one think, doesn't it.
> One redeeming feature is that the equivalent Normal gaussian
> distribution is very similar for anything above about 20 'events' (with
> SD = square root of 'mean') in a given time, but obviously the main
> difference lies in the tails (Poisson stops at 0), if one is willing to
> overlook the continuous nature of the Gaussian. But here one must also
> remember that variances ADD, so subtraction should be avoided, if
> possible. Hope these pointers are useful. Own thoughts.
> Chris Hofmeyr
> chofmeyr@nnr.co.za
>
> -----Original Message-----
> From: james.g.barnes@att.net [mailto:james.g.barnes@att.net]
> Sent: Friday, September 21, 2001 7:45 PM
> To: RadSafe Bulletin Board
> Cc: james.g.barnes@boeing.com
> Subject: Type II Poisson Errors (REPOSTED)
>
>
> PLease forgive me if this is a repost. I am in the
> process of shifting email accounts, and the first posting
> may have been lost in the shuffle. . . .
>
> Folks,
>
> We have been working with determining detection levels in
> some low background count situations, and have been
> examining the counter behavior using Poisson statistics.
> Poisson statistics will basically allow one to calculate
> a cumulative probability of the number of counts a person
> would see for a given count with an expected background
> for that count period. This approach appears to
> calculate a value that could be considered (I suppose)
> equivalent to a "Type 1" scenario (i.e., roughly
> equivalent to the Decision Level; prone to "false
> positive" detections).
>
> Is there an approach using a poisson distribution that
> calculates something akin to the Type II error (i.e, the
> LLD value that anticipates and corrects for false
> positive events). I have looked at all the usual
> references, but haven't seen any treatment of this. Is
> "LLD" even a meaningful concept in a Poisson
> distribution?
>
> Thanks,
>
> Jim Barnes, CHP
> Radiation Safety Officer
> Rocketdyne/Boeing
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