Medical tests, a first example of Bayesian calculations
Sep 11, 2014
Note:
2015, May 26 I have a post that runs through all of these calculations using Python and a package named Lea-- check it out: Probabilistic programming with Python and Lea
In this post I will discuss a first example of a Bayesian calculation using a well-known example of testing for breast cancer. I choose this example for a few reasons:
- The answer often surprises people and supposedly many doctors get this wrong.
- The problem provides a nice example of updating a prior (information before) to a posterior (information after). This is really essential to understanding Bayesian thinking and I will emphasize this idea, making a point of connecting the concept to mathematical notation.
- The calculations are relatively simple (at least compared with other examples). If you understood my post on Joint, conditional and marginal probabilities , you should have no problems.
Before we get started, I will point out cancer test example 1 and cancer test example 2 , which both provide discussions of this exact same example. You should also check those posts out for another view of the problem. Also, there is a wikipedia drug test example that covers a different but very similar problem.
Now, on to the problem of what a positive medical test means. I'll use a direct quote from cancer test example 1
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
Parsing the problem
The first, and maybe most confusing, part of answering the above question is parsing the provided information. First, what is being asked? I would reword the last sentence(s) and say:
What is the probability that a woman (age 40) has breast cancer given that she had a postive mammogram?
Notice that I'm not adding any new information, just restating what is given above. Let's start by connecting the given information to mathematical notation. First, let's define the two items (variables) that we are interested in:
- \( C \) -- cancer: This variable represents the presence or absence of cancer and can be equal to yes: \( C=\textrm{yes} \), or no: \( C=\textrm{no} \)
- \( M \) -- mammogram: This variable represents the outcome of the mammogram and be positive: \( M=\textrm{pos} \), or negative: \( M=\textrm{neg} \)
So, in mathematical notation, the problem is asking for us to calculate the following conditional probability:
\[ P(C=\textrm{yes} \vert M=\textrm{pos}) = \, ? \]In English, this is the probability that cancer is yes given that a mammogram is pos (positive). Notice the phrase given that, this is a key indicator that we are considering a conditional probability.
Givens in the problem statement
Prior probability of cancer
There is a lot of information in the problem statement, let's properly assign probabilities to their given values. First, we have the statement:
1% of women at age forty who participate in routine screening have breast cancer.
Mathematically, this is giving us the value for the prior probability that a woman (age 40) has breast cancer:
\[ P(C=\textrm{yes}) = 0.01 \]Also note that the probability of no breast cancer in the same group of women must be 99% because the sum of probabilities for all outcomes must be 100% (or 1):
\[ \begin{array}{ll} P(C=\textrm{no}) & = & 1 - P(C=\textrm{yes}) \\ & = & 0.99 \end{array} \]This is our information about the probability of breast cancer (or no breast cancer) before the mammogram.
Effectiveness of tests
Our next information gives probabilities relevant to the accuracy of the mammograms. First, we have:
80% of women with breast cancer will get positive mammographies.
Mathematically, this turns into:
\[ P(M=\textrm{pos} \vert C=\textrm{yes}) = 0.80 \]Again, in simple English, this is the probability that a mammogram is pos given that cancer is yes-- you should agree that this is the same as the statement above. We also get another probability for free:
\[ P(M=\textrm{neg} \vert C=\textrm{yes}) = 0.20 \]This is because the probabilities of a mammogram being positive or negative must sum to one (100%), even with the conditioning on \( C=\textrm{yes} \):
\[ P(M=\textrm{pos} \vert C=\textrm{yes}) + P(M=\textrm{neg} \vert C=\textrm{yes}) = 1 \]Finally (at least for the givens), we have the specificity of the test. This provides the probability of getting a positive mammogram even when there is no breast cancer. The relevant part of the problem is:
9.6% of women without breast cancer will also get positive mammographies.
Mathematically, this translates to the following probability:
\[ P(M=\textrm{pos} \vert C=\textrm{no}) = 0.096 \]Again, in simple English, this is the probability that a mammogram is pos (positive) given that cancer is no (there is no cancer).
Joint probabilities
Before moving onto the final calculation, it is worth putting together the joint probabilities for all possible outcomes. We can do this using the following relationships:
\[ P(C=c, M=m) = P(M=m \vert C=c) P(C=c) \]where I use \( c \) and \( m \) to represent any valid values for the presence of cancer and the mammogram result. This allows us to substitute the desired values for \( c \) and \( m \) and use the above equation over and over again. For example, let's start with \( c = \textrm{yes} \) and \( m = \textrm{pos} \):
\[ \begin{array}{ll} P(C=\textrm{yes}, M=\textrm{pos}) & = & P(M=\textrm{pos} \vert C=\textrm{yes}) P(C=\textrm{yes}) \\ & = & 0.80 \times 0.01 \\ & = & 0.008 \end{array} \]This says the probability that cancer is yes and a mammogram is pos is only 0.8% -- I'm sure that will surprise many people. Why is it so small? Well, this is driven by the fact that only 1% of women at this age have breast cancer at all-- see the 0.01 in the calculation. Only 80% of those women get a positive mammogram, resulting in the very low 0.8%.
Wait a second, isn't that the answer we want??? Actually, no. We want to know the probability cancer is yes given that a mammagram is pos, not the probability cancer is yes and a mammogram is pos. That is, we want a probability conditioned on the fact that we know for certain that we have a positive mammogram. We'll get to the answer in a bit.
Let's try another joint probability:
\[ \begin{array}{ll} P(C=\textrm{no}, M=\textrm{pos}) & = & P(M=\textrm{pos} \vert C=\textrm{no}) P(C=\textrm{no}) \\ & = & 0.096 \times 0.99 \\ & = & 0.09504 \end{array} \]This says the probability that cancer is no and a mammogram is pos is 9.5%, fairly high. Again, this is driven by the fact that most women at this age do not have breast cancer-- the 0.99 in the calculation-- and the probability of false positives is fairly high at 9.6%.
Let's calculate the probability of the other two joint outcomes:
\[ \begin{array}{ll} P(C=\textrm{no}, M=\textrm{neg}) & = & P(M=\textrm{neg} \vert C=\textrm{no}) P(C=\textrm{no}) \\ & = & 0.904 \times 0.99 \\ & = & 0.89496 \end{array} \]and
\[ \begin{array}{ll} P(C=\textrm{yes}, M=\textrm{neg}) & = & P(M=\textrm{neg} \vert C=\textrm{yes}) P(C=\textrm{yes}) \\ & = & 0.20 \times 0.01 \\ & = & 0.002 \end{array} \]From the above, we can see that it is most common to have no cancer and a negative mammogram, at 89.5%.
Joint probability table
Let's put together a joint probability table to see everything at once:
\[ \newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{c|c|c|c} \T & C=\textrm{yes} & C=\textrm{no} & \\ \hline M=\textrm{pos} \T & \color{blue}{0.008} & \color{blue}{0.09504} & 0.10304 \\ M=\textrm{neg} \T & \color{blue}{0.002} & \color{blue}{0.89496} & 0.89696 \\ \hline \T & 0.01 & 0.99 & 1 \end{array} \]The central part of the table (blue values) provides the joint probabilities calculated above. Notice that the sum of these values is equal to one because they cover all possible combinations of \( C \) and \( M \). The bottom row provides the sum of probabilities in the column-- these sums are also known as marginal probabilities. For example, we have
\[ \begin{array}{ll} P(C=\textrm{yes}) & = & P(M=\textrm{neg}, C=\textrm{yes}) \\ & + & P(M=\textrm{pos}, C=\textrm{yes}) \\ & = & 0.008 + 0.0002 \\ & = & 0.01 \end{array} \]This result just reconstructs a probability that we already knew and were given in the problem statement.
Also note that the last column provides the sum of the probabilities in each row. Here, we can find something new:
\[ \begin{array}{ll} P(M=\textrm{pos}) & = & P(M=\textrm{pos}, C=\textrm{yes}) \\ & + & P(M=\textrm{pos}, C=\textrm{no}) \\ & = & 0.008 + 0.09504 \\ & = & 0.10304 \end{array} \]This says \( P(M = \textrm{pos}) \) is 10.3% and gives the probability of a positive mammogram in all testing, including women that have, and do not have, breast cancer.
Prior to posterior
Finally, let's calculate the posterior to get the desired quantity. Again, the idea is that we are updating the prior to the posterior:
- prior: \( P(C = \textrm{yes}) = 0.01 \) -- information before mammogram
- posterior: \( P(C = \textrm{yes} \vert M = \textrm{pos}) \)-- information after the mammogram (conditioned on a positive result)
We've already calculated every relevant probability, so let's construct Bayes' rule for this problem. We can relate the joint, conditional and marginal probabilities of interest in two ways (both correct):
\[ P(C=\textrm{yes}, M=\textrm{pos}) = P(M=\textrm{pos} \vert C=\textrm{yes}) P(C=\textrm{yes}) \]and
\[ P(C=\textrm{yes}, M=\textrm{pos}) = P(C=\textrm{yes} \vert M=\textrm{pos}) P(M=\textrm{pos}) \]Note that \( P(C=\textrm{yes}, M=\textrm{pos}) \) is on the left side of both equations. So, we can set the right side of each equation equal and re-arrange to get:
\[ P(C=\textrm{yes} \vert M=\textrm{pos}) = \frac{P(M=\textrm{pos} \vert C=\textrm{yes})P(C=\textrm{yes})}{P(M=\textrm{pos})} \]Notice that the left side of the equation has the (unknown) quantity that we want and the right side of the equation has only known quantities that we were given or calculated above. So, we can find our answer:
\[ P(C=\textrm{yes} \vert M=\textrm{pos}) = \frac{0.8 \times 0.01}{0.01034} = 0.077 \]So, the probability of breast cancer given a positive mammogram is just 7.7%. Most people are shocked at how low this value is. However, remember that before the mammogram the probability of cancer was just 1%. The positive test increased the probability of cancer by a factor of roughly eight times: \( 0.01 \rightarrow 0.077 \).
- Why isn't it 100%, or at least really high probability?
Many people find it helpful to think of this problem using 1,000 imaginary women and think in terms of number of people instead of probabilities. We can do that by multiplying all of the probabilities in our joint probability table by 1000 to get the number of women in each status:
\[ \newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{c|c|c|c} \T & C=\textrm{yes} & C=\textrm{no} & \\ \hline M=\textrm{pos} \T & \color{blue}{8} & \color{blue}{95} & 103 \\ M=\textrm{neg} \T & \color{blue}{2} & \color{blue}{895} & 897 \\ \hline \T & 10 & 990 & 1,000 \end{array} \]Looking at this table, the problem should be very apparent: of 103 positive mammograms, only 8 really have breast cancer. The other 95 women have false positives. Also, 2 women with breast cancer get negative mammograms. As a result, a positive mammogram increases the probability of cancer, from 1% to 7.7%, but does not make it certain.
Bayes' theorem
While digging into the details of the above calculations it would be easy to loose site of where Bayes' theorem appeared and what it looks like. A more typical presentation would look something like:
\[ \color{blue}{P(C=c \vert M=m)} = \color{black}{ \frac{P(M=m \vert C=c) \color{red}{P(C=c)} }{P(M=m)} } \]In the above we colored the posterior blue and the prior red. As always, we think of updating a prior to a posterior given some information or data. Another form of Bayes' theorem that is often used, if the term in the denominator is expanded, is:
\[ P(C=c \vert M=m) = \frac{P(M=m \vert C=c)P(C=c) }{ \sum_{\hat{c}=\textrm{yes},\textrm{no}} P(M=m \vert C=\hat{c})P(C=\hat{c}) } \]where we use the relation between joint and marginal probabilities (see my post on Joint, conditional and marginal probabilities if this doesn't make sense to you):
\[ \begin{array}{ll} P(M=m) & = & \sum_{\hat{c}=\textrm{yes},\textrm{no}} P(M=m \vert C=\hat{c})P(C=\hat{c}) \\ & = & P(M=m \vert C=\textrm{yes})P(C=\textrm{yes}) \\ & + & P(M=m \vert C=\textrm{no})P(C=\textrm{no}) \end{array} \]Let's apply this last form of Bayes' theorem to do the calculation (again) for \( c = \textrm{yes} \) and \( m = \textrm{pos} \):
\[ \begin{array}{ll} P(C=\textrm{yes} \vert M=\textrm{pos}) & = & \frac{ P(M=\textrm{pos} \vert C=\textrm{yes})P(C=\textrm{yes}) }{ \sum_{\hat{c}=\textrm{yes},\textrm{no}} P(M=\textrm{pos} \vert C=\hat{c})P(C=\hat{c}) } \\[0.5em] & = & \frac{ 0.80 \times 0.01}{0.80 \times 0.01 + 0.096 \times 0.99} \\[0.5em] & = & 0.077 \end{array} \]The same answer, whew! Seriously, make sure that all of the substitutions make sense and that you can relate this calculation back to the more incremental calculation done above.
Summing up
So, that's it, a first example of a Bayesian calculation done a couple of ways. I hope the level of detail will encourage you replicate the calculations and understand how all of the probabilities are related. As always leave comments, questions, and corrections below.