This is lesson 4.2.1 Bayes' rule. In the previous few lessons when we calculated icers we assumed we had all the information necessary to perform the calculations. There are instances however, when all the information is not necessary in that it's necessary to use the information you have in a different way to calculate information that's needed for the technology assessment. In other words, you need to generate or incorporate new information into the model. The information you want is not always the information you have. The question then is how should one rationally incorporate new information into their beliefs or the assessment? For example, suppose one gets a positive test result where the test is imperfect, what is the probability that one has the condition? In this case, you have the information of a positive test result. But what you really want to know is what's the probability that you have the condition? The approach to dealing with this situation is Bayes' Rule. This is particularly useful for the analysis of screening, like the example I just described. But it applies more broadly to incorporating new information into a technology assessment model. Before using Bayes' Rule, let's discuss some simple notation, in the context of the screening example, I just described. The probability of event A, or the probability of having a condition is just written, the probability of A or P, (A). The probability of information being available, or P(B) is the probability of the positive test. Then we would say, that the probability of a positive test given that you have the condition or the probability of B occurring if A, Is what we know. And that's the situation we are in right now. What we actually want to know however is the probability of A given B. In other words what's the probability of having the condition. Given that we know we had a positive test. The calculation for Bayes' Rule is then the probability of B given A times the probability of A divided by the probability of B. This results in the probability of A given B. Now lets consider a more formalized example. What is the probability that a women has breast cancer? What we know is that 1% of women age 40 who participate in routine screenings have breast cancer. We also know that 80% of women with breast cancer will get a positive mammogram result. 9.6% of women without breast cancer will also get a positive mammogram result. That means some women with a positive result do not have breast cancer. So the question that we need to solve using Bayes' rule is for a woman in this age group who had a positive result in a routine screening, what's the probability that she actually has breast cancer? Recall the notation that we just described, but now in the context of this example the probability A is the probability of woman has breast cancer. The probability of B is the probability of a positive mammogram result. The probability of B given A is the probability a woman will have a positive mammogram given she has breast cancer. The probability of B given not A, is the probability a woman will have a positive mammogram result given she does not have breast cancer. And the probability that we're interested in, which is the probability of A given B, is the probability a woman has breast cancer given that she had a positive mammogram result. We can think about this graphically as well and apply the information that we know from the example. In this case, U is the circle, is the large circle in the universe of all woman age 40 who have routine mammograms. Then there's a 1% chance that a randomly selected woman age 40 who participates in routine screening will have breast cancer. This probability A is equal to .01 is a small circle in the universe. The probability of a positive mammogram, the probability B is also in the universe. And it's equal to the sum of the probabilities of positive mammograms for women with and without breast cancer. So the percent of women with breast cancer that have a positive mammogram would be the area where A and B overlap. We can calculate the total probability of B by multiplying the probability of B given A, times the probability of A, and the probability of B, given not A, times the probability of not A. This then can be rewritten in the values that we know which is the probability A and one minus the probability of A. When we apply the numbers from the example we get 0.8 times 0.01 plus 0.096 times 0.99, thus the probability of B is just over 10%. So now we can apply all of these values to calculate the probability of A given B or, a forty year old woman who had a positive mammogram. We can calculate the probability that she actually has breast cancer. The probability of A given B, using Bayes' rule, is the probability of B given A, times the probability of A, divided by the probability of B. If we refer back to some of the previous slides, we know the probability of A. The probability of B given A, and the total probability of B that we just calculated. And then we can calculate the probability of A given B. Or the probability that a woman has breast cancer given a positive test result, is just over 7.7%. So there is a 7.8% chance a woman has breast cancer given a positive mammogram which we calculated using Bayes' rule. This was a simple example, but it demonstrates how Bayes' rule can be used to incorporate information into a model. In particular using information that you have but not information that you need.