**DISCUSSION**

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**December 30, 2009**

**A professor of medicine wrote: **

** "Well-known, well taught, still
neglected. In many applications the true positive rate is only 1 %, meaning that usually false positives will far outnumber
true positives. (......) The info content of criteria is best when the pre-criteria positivity estimate is close to **

**50 %. lf we survey a population, which only has 1 % RA
in it, like the full population, then we shall have a lot of false positives and only a few true positives. Google 'false positives' and look for Wikipedia refs!".**

** **

**Comment by B. Soric: **

** "False positive paradox" (from
Wikipedia) **

** http://en.wikipedia.org/wiki/False_positive_paradox **

** "The false positive paradox is
a situation where the incidence of a condition is lower than the false
positive rate of a test, and therefore, when the test indicates that a condition exists,
it is probable that the result is a false positive". **

** For example, if a subject has
the disease, a medical test may be, say, 99% likely to correctly indicate that she does, and if a subject does not
have the disease, it may be 99% likely to correctly indicate that she doesn't. Suppose that a disease occurs in
1 out of 10,000 people. The numbers from the Wikipedia example are here shown
in a contingency table:**

** p=0.99
; p'=0.99 ; 1-p'=0.01**

** **__True positive True
negative Sum _ __

**Test positive: a=
99 b= 9999 10,098 = S**

__Test negative: c= 1
d=989,901 989,902__

**
Sum: R=100 N= 999,900 1,000,000 = M**

** **

** 1. If we suppose that all the values are known, we calculate: **

**Positive predictive value = a/(a+b) = 1-P = 99/10,098
= 0.0098 **

**Proportion of false positives = b(a+b) = P = 0.9902**

** 2. **__But even if we know only____ p, p', M, and S,
we can still calculate__:

**R = [S-(1-p')M] /(p+p'-1) = 100 ;
N = M-R = 999,900**

**P = (1-p')N/S = 0.9902**

** So, the "false positive paradox"
may be well-known, but so far I have not found anywhere that the latter possibility of calculation (under 2.) has been published
by anybody. It is very simple, but it seems to be very important for researchers! **

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****December 31, 2009**

**Another professor wrote this: **

** The proportion (P)
of false positives cannot be exactly calculated, even for very large samples, because sensitivity and specificity are
known only approximately. **

** **

**Comment by B. Soric: **

** That
may be true, but unless we consider S/M or calculate P, we don't know at all whether P is too large or not. **

** Suppose, for example, that
we apply ACR criteria to 800 patients, whose diagnoses may be even totally unknown to us, and 417 of them are classified
as RA. So, we know: M= 800 ; S= 417 ; S/M= 0.521 ; p=
0.935 ; p'= 0.893 while
R, N and P are still unknown. Without
calculating P or taking into consideration S/M we have no idea of the percentage of wrong RA
diagnoses (false positives); but we can calculate as follows: **

** R= [S-(1-p')M] /(p+p'-1) = 400
; N= M-R = 400 ; and:**

** P= (1-p')N/S = 0.103 - though
this value of P is just an approximate
or expected value. (The same value of P
we find in Table 2. for S/M = 0.521 ; see
on page "Home" of this web-site!). ****So, we may expect or hope that a large part of those 417
patients indeed have rheumatoid arthritis. **

** On the contrary, if,
say, only 120 of the 800 patients are classified as RA, we find R=42 ; N=758 ; P=0.676 In the latter case we certainly cannot assume that most of those 120 patients
indeed have RA. **

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