and I think that this would be the 'long-hand' Bayesian proof (I have assumed a high sensitivity as an example):
if,
• Sensitivity 98% (i.e. a FN rate of 2%)
• Specificity 95% (i.e. a FP rate of 5%)
• Prevalence = 1:1000
and let,
• A be the event that a patient truly has disease.
• ¬A be the event they truly don’t have the disease
• B be the event that they test positive.
Our question is:
“What is the chance the patient testing positive actually has the diseaseI?”
Expressed in Bayesian form, this is p(A/B) i.e.
“what is the probability of the patient truly having the disease given the event that they test positive?”
So,
“98% sensitivity” means that p(B/A) = .98
“95% specificity” means that p(B/ ¬A) = .05
Using Bayes theorem:
p(A/B) = _____p(B/A)p(A)______
p(B/A)p(A) + p(B/¬A )p(¬A)
thus:
p(A/B) = _____.98x.001_______ = 0.0192 (i.e. 1.9%)
.98x.001 + .05x.999
I think this is correct. Apologies if the formula de-formats.
Best
Roger
Roger Kerry FMACP MSc MCSP
Associate Professor
Division of Physiotherapy Education
University of Nottingham
United Kingdom
________________________________________
From: Evidence based health (EBH) [[log in to unmask]] On Behalf Of Pasquale Urbano [[log in to unmask]]
Sent: Thursday, February 10, 2011 4:31 PM
To: [log in to unmask]
Subject: R: Question about screening and denominators
Click on http://araw.mede.uic.edu/cgi-alansz/testcalc.pl?DT=95&Dt=5&dT=4995&dt=94905&2x2=Compute and see what happens with your numbers: the pre-test probability [prevalence, if you have no other info: 1/1000], increases 19fold to the post test probability of about 2/100; a negative result will rule out the desease with confidence.
To screen for rare diseases you better use the most sensitive test available [>99% ?], followed by the most specific, to rule out the false positives.
___________________________________________________________________
Prof. Pasquale Urbano
Ordinario di Microbiologia, F.R.
Dipartimento di Sanità Pubblica,
Università di Firenze
Da: Evidence based health (EBH) [mailto:[log in to unmask]] Per conto di Simon Hatcher
Inviato: giovedì 10 febbraio 2011 11.34
A: [log in to unmask]
Oggetto: Re: Question about screening and denominators
After another night this time on the red wine my friend has provided me with the source of the problem - it is John Lanchester's book "Whoops" where Lanchester writes "if a test for some disease is 95% accurate, and the disease affects one person in a thousand, and you go for a test and it comes back positive, what's the probability that you have the disease? Most respondents say, well the test is accurate, so the probability is 95%. The correct answer is 2%, because if you test 1000 people, the test will give fifty positives, whereas only one of the population has the illness". The reference for this is Nassim Nicholas Taleb's "Fooled by Randomness". Is he right?
________________________________
From: Evidence based health (EBH) on behalf of Miranda Cumpston (Med)
Sent: Thu 10/02/2011 13:08
To: [log in to unmask]
Subject: Re: Question about screening and denominators
Hi all,
Just for the sake of it, here are some numbers, assuming that you mean 95% for both specificity and sensitivity (which is a pretty unlikely test) – that is, the test will detect 95% of true positives and true negatives (using a population of 100,000 to keep everything in whole numbers):
True +
True -
Total
Test +
95
4,995
5,090
Test -
5
94,905
94,910
Total
100
99,900
100,000
So, assuming you have a positive test (i.e. you’re in the top row), then the probability that you actually have the disease is only 95/5090, or about 2%, even with a pretty good test. The reason is that the sheer number of people without the disease is so much greater that even a low % of false positives translates into large absolute numbers.
For rare diseases, tests are often much better at ruling out disease than ruling it in. If you had tested negative, your probability of having the disease would be 5/94,910, or 0.0001%.
Hope that’s right – would be a shame to get it wrong in front of such an audience : ). And of course, if you mean something else by “correctly detects the disease”, then you’d have to adjust the numbers accordingly.
cheers,
Miranda
From: Evidence based health (EBH) [mailto:[log in to unmask]<mailto:[log in to unmask]>] On Behalf Of Paul Glasziou
Sent: Thursday, 10 February 2011 10:55 AM
To: [log in to unmask]<mailto:[log in to unmask]>
Subject: Re: Question about screening and denominators
Hi Simon,
Brian is right that you need to know more to answer the question.
If the "test which correctly detects the disease 95%of the time" and we interpret that as the sensitivity.
If the specificity is 5% then the test is useless - a Youden index (sensitivity + specicity - 100%)
and so after a positive test the chance of disease is still 1:1000
Paul Glasziou
PS If instead the 95% is the overall accuracy, then just call everyone negative and the test is 99.9% accurate ;-)
On 2/10/2011 9:31 AM, Brian Alper MD wrote:
Here’s my quick thoughts without doing any math or providing formulas.
More than one answer is possible depending on how one interprets “correctly detects the disease 95% of the time”
If you know the sensitivity (% correct in patients with known disease), specificity (% correct in patients known to not have disease) and disease prevalence (1:1000 in this example) you can answer the question.
But overall measures of diagnostic accuracy (% correct overall based on combining sensitivity and specificity) could reach 95% through different combinations of sensitivity and specificity. The different combinations could lead to different results for your question (a measure of positive predictive value)
Brian S. Alper, MD, MSPH
Editor-in-Chief, DynaMed (www.ebscohost.com/dynamed<http://www.ebscohost.com/dynamed>)
From: Evidence based health (EBH) [mailto:[log in to unmask]] On Behalf Of Simon Hatcher
Sent: Wednesday, February 09, 2011 6:22 PM
To: [log in to unmask]<mailto:[log in to unmask]>
Subject: Question about screening and denominators
I had this discussion with a friend over a beer last night and we couldn't agree on the answer. Here's the scenario:
The incidence of a disease in a population is 1:1000
There is a test which correctly detects the disease 95%of the time
If I test positive with the test what is my risk of having the disease?
Be interested in any thoughts on the "correct" answer.
Cheers
Simon
--
Paul Glasziou
Bond University
Qld, Australia 4229
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