In this post, I mentioned that it can be easier to think in terms of conditional probability sensitivity as p(test+|dz+), than to remember the formula sensitivity = tp/(tp+fn). I will try to show why p(test+|dz+) is intuitive.

Before talking about the conditional prob, p(test+|dz+), how to estimate unconditional prob, p(test+)? To do so, we ask: what proportion of a sample test positive? So, if I have a sample of 100 people, and 10 have a positive test, then p(test+) can be estimated by 10/100.

Now, how do I estimate a conditional prob, p(test+|dz+)? I start with the idea that p(test+|dz+) is the prob of testing positive in the subset of the sample with the disease. We then compute the proportion of that subset who test positive, as w/ unconditional prob.

Example: to estimate p(test+|dz+), we can ask, for all those who have the disease, what proportion test positive? Recall the 2×2 table.

	test +	test –
dz+	tp	fn
dz-	fp	tn

To estimate p(test+|dz+), we are considering the subset of the sample for which dz+ is true. That is the first row of the table. We can ignore the rest.

	test +	test –
dz+	tp	fn

In this subset, what is the probability of testing positive? It’s the number of people with positive tests divided by the number of people in the sample. tp / (tp+fn). That’s the formula for sensitivity.

BTW, p(dz+|test+) is ppv. Thinking in terms of conditional probabilities reveals certain relationships. There is a close relationship between ppv and risk models.