1. ## Re: Help with Bayesian question please

Silly, silly, silly me! I never checked your guys numbers. I made the mistake of trusting your arithmetic over mine. Probably because you usually seem correct and there were two of you. I was just getting ready to type out a huge reply with how I used the Simple Bayes Theorem based on odds. Then I went back to write out the results every way you could calculate them and was going to reference your calculations. That is when I saw you both had your numbers mixed up. I will show you below. If you are interested in my approach, Google search: Bayes Likelihood Ratio, you will get hundreds of thousands of hits. Also, below I once again included the calculation for repeated testing.

Originally Posted by Dason
I hate to keep this going but I'm going to just flat out say that your number for P(D+ | T+) is wrong.

Rogojel and I both come to that solution.

The formula is correct you just switched numbers around. It should be:

Original Information:

Prevalence or Pre-Test Probability or Prior: 0.0001

Given the increased sample size of 1,000,000

Sensitivity (P(T+|D+): 90/100 = 0.90

Specificity (P(T+|D-): 979902/999900 = .98

Positive Likelihood Ratio (+LR): SEN/(1-SPEC): 0.90/0.02 = 45

Simple Bayes Theorem based on Odds:

P(D+|T+) = Pre-Test Odds * +LR = Post-Test Odds

Post-Test Odds: (0.0001/(1+0.0001)) * 45 = 0.0045

Post-Test Probability: 0.0045 / (1-0.0045) = 0.0045

Formula For Serial Tests:

Post-Test Odds = Pre-test Odds * +LR1 * +LR2

+LR due to repeat test will always by 45 in this case, so:

0.0001 * 45 * 45

~ 0.167, after two positive tests.

_____________________________________________________________________

Lastly, a very very simple way to get at the Post-Test Probability after a test is just to report the Positive Predictive Value, which is just the first cell in the 2x2 classification table divided by the first row total.

2. ## Re: Help with Bayesian question please

I disagree that I got the numbers mixed up. OP says sensitivity is .98 which is the true positive rate which is P(T+ | D+).

Also in your last post you say
Specificity (P(T+|D-)
but specificity is P(T- | D-)

3. ## Re: Help with Bayesian question please

Well there is the issue, I was switching the two numbers around in my head. You typically present the SEN before the SPEC in results and I was erroneously switching them in my calculations. I will repost #46

4. ## Re: Help with Bayesian question please

Yup, writing that out definitely helped me figure out where my issue was. You two were totally right!

Original Information:

Prevalence or Pre-Test Probability or Prior: 0.0001

Given the increased sample size of 1,000,000

Sensitivity (P(T+|D+): 98/100 = 0.98

Specificity (P(T-|D-): 899910/899912 = .90

Positive Likelihood Ratio (+LR): SEN/(1-SPEC): 0.98/0.10 = 9.8

Simple Bayes Theorem based on Odds:

P(D+|T+) = Pre-Test Odds * +LR = Post-Test Odds

Post-Test Odds: (0.0001/(1+0.0001)) * 9.8 = 0.00098

Post-Test Probability: 0.00098 / (1-0.00098) = 0.00098

Formula For Serial Tests:

Post-Test Odds = Pre-test Odds * +LR1 * +LR2

+LR due to repeat test will always by 9.8 in this case, so:

0.0001 * 9.8 * 9.8

~ 0.0096, after two positive tests.