Re: Chi-Square Test and Counting Statistics

[webber@nist.gov]

>Date: Thu, 06 Nov 1997 17:42:33 -0800
>From: Judd Sills <sillsj@GAT.COM>
>To: radsafe@romulus.ehs.uiuc.edu
>Subject: Re: Chi-Square Test and Counting Statistics
>Message-ID: <3.0.4.16.19971106174233.389f7524@vaxd.gat.com>
>At 06:25 PM 11/6/97 -0600, you wrote:
>>All,
>>
>>I am looking for a good definition and examples of Chi-square tests and 
>>its applications.  The math is easy, but I'm having a hard time 
>>understanding what the Chi-square number is telling me and what is a 
>>good Chi-square number and what is a bad Chi-square number.  Good 
>>references would be a appreciated.
>>
>>Also, if I do a chi-square determination with a Tc99 source for betas 
>>and a Th230 source for alphas, should I do daily reliability checks 
>>with a Pb210 source?
>>
>>Thanks in advance.
>>
>>Denny
>>
>>----------------------
>>Denis J. Rinkacs, Jr.
>>Radiation Safety Officer
>>Carnegie Mellon University
>>412-268-3221
>>412-268-6976 Fax
>>dr4i@MAIL1.andrew.cmu.edu
>>
>>
>>
>The Chi-square test is a means to test the variability
>of a counter-scaler system to ensure that the results are
>consistent with the random nature of radioactive decay which can be
>predicted according to a probability.  Simply
>stated, when multiple counts of a radioactive sample
>(or natural background) are made, the results should
>approximate a Poisson distribution.  The Chi-square test
>is a method to compare the variability of the results to
>the predicted variability from a true Poisson distribution.
>If the system results are too consistent (little variability) then it
>probably isn't performing properly, and similarly, if the variability is
>excessive, then it also probably isn't performing properly.  The
>Chi-square test should be performed for both background and a test
>source on a regular basis (e.g., monthly) and whenever the system has
>been altered by maintenance (repair or adjustments) or calibration.  It
>is performed by collecting a  series of counts (source or background,
>according to which Chi-square is being performed).  The more
>observations that are collected, the smaller the standard deviation of
>the observations would be.  I recommend that you perform 20 observations
>(n).  The Chi-square is calculated as the sum of the squares of the
>differences of the observed count rates from the mean count rates
>divided by the mean count rate itself.
>                     2
>                    X  = {Sum(i to n) [xi - xbar]}/xbar
>where xbar is the sample mean for the n counts.
>For 20 observations, following the 99th percentile, the Chi-square range
>should fall between 8.91 and 32.9.   Other acceptance criteria are
>needed if a different number of observations (counts) are taken.
>After collecting this data and meeting the acceptable Chi-square result,
>control charts (graphs) should be developed to test daily background and
>source checks for acceptability.  Determine the standard deviation of
>the observations:
>     SD = Square Root(sum(i to n)[xi - xbar]/{n - 1})
>where n is the number of observations (e.g., 20 for my recommendation)
>On the Control Chart, draw a line representing the mean (xbar), two SD
>(xbar +/- 2SD), and three SD (xbar +/- 3SD).  On a daily basis plot the
>observation (background or source) and compare it to these bounds.  If
>the daily observations are between the two 2SD lines, the result  is
>acceptable.  If the daily observation is between the 2SD and 3SD line,
>then a recount should be performed (if it is between the 2 SD lines,
>then this is acceptable).  If the daily observation is beyond the 3SD
>lines, the system should be taken out of service for repair/evaluation.
>I recommend that you use the same sources for daily testing that you
>used for the Chi-square testing.  Your source should be representative
>of your typical sample decay energy.  It is also recommended that you
>compare your empirical counter efficiency with what your calibration lab
>has reported for the isotope that you are using for your test source as
>another check of system performance.
>This was a long winded answer, but hopefully explained the reasons for
>the test, and how to perform it and use it properly.
>Best regards.
>Judd M. Sills, CHP           |   Office: (619)455-2049
>General Atomics, Room 01-166C|      Fax: (619)455-3181
>3550 General Atomics Court   |   E-Mail:  sillsj@gat.com
>San Diego, CA  92121         |

My knowledge of statistics is rusty but:
I question the usefullness of performing a Chi-square test for background when the background rate is less than 1 count per min.

1. Twenty measurements of ten minutes each (200 min total count time) would result in less than 10 counts per measurement. At this level the power of the Chi-squared test would be very weak, i.e. only able to detect the grossest of differences from expected variation.

2. I believe that the Chi-squared test assumes a normal distribution not a Poisson distribution. The Poisson distribution can be approximated by a normal distribution for large (50-100) gross counts, but is not a good model when the gross count is less than 10.

Someone please correct me if I am wrong.

=====================================================
William R. Webber, Health Physicist (Warp Factor OS/2, connect)
The opinions expressed above are those of the author alone
and do not represent those of the National Institute of
Standards and Technology nor the US Department of Commerce.
E-mail: wwebber@nist.gov
=====================================================