Don’t panic over a positive medical test…use Bayes’ theorem to put it in perspective

My baby’s newborn hearing screening came back positive – that is, they referred us to an audiologist for further testing. Like all parents, I’m worried. But understanding statistics and probability makes a me a little less worried, so I’m writing this up to help me relax about it until we see the audiologist in a few weeks (one possible cause is simply some residual fluid in the baby’s ear, so they wait a few weeks before testing again).
The basic intuition behind the statistics in reacting to a positive medical test is that it maters how rare the condition is, and how likely the test is to get it wrong. When 1,000 newborns are screened for hearing loss about 22 will test positive. Two of those 22 will be the 2 in 1,000 who actually have hearing loss (the actual rate of hearing loss in newborns). But of the 998 newborns who don’t have hearing loss, about 20 of them will get false positives (a 2 percent false positive rate). That means of the 22 positive test results – the category my daughter falls in – only 2 out of the 22, or about 9 percent actually have hearing loss.
To calculate more precisely the probability that my daughter really does have some hearing difficulty given she had a positive hearing screening, we need to know three numbers: the overall percentage of newborns with hearing loss(between .001 and .003); the rate of false positives on the test (.02); and the rate of false negatives on the test (between .02 and .09). (Note: I found these numbers by reading abstracts in medical journals for the ABR hearing test).
Combining these three numbers using Bayes’ theorem, the odds of hearing loss given a positive screening result is somewhere between 4 percent and 13 percent. That’s enough to make me worry and see an audiologists – but it’s a lot less worrying than if I only knew she had a positive screening on a test with a 2 percent false positive rate. (I’ve posted the code for combining the three numbers below for those who are interested in the details of how this works).
A few more things that aren’t in the math: 1) most importantly, she can clearly hear us;  2) this is one of many screenings for newborns, which increases the odds that at least one of them will be false positive; 3) my daughter is demographically in a low-risk group (e.g. young mother etc.);  4) the test depends at least as much on the newborn being relatively calm during testing as it does on hearing ability, and she was being a little fussy; and 5) she passed in one ear but not the other. Combine all those things and it makes me think we should lean towards the 4 percent end of the range of estimates.
Code is below:
#Using Bayes rule with medical testing info to see how worried a test should make you
#A is hearing loss
#B is a postive test
bayes_med<-function(base_rate, false_positive, false_negative){
hit<-1-false_negative
prob_B<-((hit*base_rate)+false_positive*(1-base_rate))#P(B)=P(B|A)*P(A)+P(B|not_A)*P(not_A)
posterior<-(hit*base_rate)/prob_B#P(A|B)=(P(B|A)*P(A))/P(B), Baye's Theorem
posterior
}
bayes_med(.002, .02, .055)#medium estimate
bayes_med(.001, .02, .09)#low estimate
bayes_med(.003, .02, .02)#high estimate
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