I was working through Nate Silverâ€™s book The Signal and the Noise and got to his chapter about hypothesis testing. It is interesting reading and I thought I would expand on that by posing a couple of problems.

Problem one: suppose you knew that someone attempted some basketball free throws.

If they made 1 of 4 shots, what would the probability be that they were really, say, a 75 percent free throw shooter?

What if they made 2 of 8 shots instead?

Or, what if they made 5 of 20 shots?

Problem two: Suppose a woman aged 40-49 got a digital mammagram and got a â€śpositiveâ€ť reading. What is the probability that she indeed has breast cancer, given that the test catches 80 percent of the breast cancers (note: 20 percent is one estimate of the â€śfalse negativeâ€ť rate; and yes, the false positive rate is 7.8 percent. The actual answer, derived from data, might surprise you: it is : 16.3 percent.

Iâ€™ll talk about problem two first, as this will limber the mind for problem one.

So, you are a woman between 40-49 years of age and go into the doctor and get a mammogram. The result: positive.

So, what is the probability that you, in fact, have cancer?

Think of it this way: out of 10,000 women in that age bracket, about 143 have breast cancer and 9857 do not.

So, the number of false positives is 9857*.078 = 768.846; weâ€™ll keep the decimal for the sake of calculation;

The number of true positives is: 143*.8 = 114.4.

The total number of positives is therefore 883.246.

The proportion of true positives is So the false positive rate is 83.72 percent.

It turns out that, data has shown the 80-90 percent of positives in women in this age bracket are â€śfalse positivesâ€ť, and our calculation is in line with that.

I want to point out that this example is designed to warm the reader up to Bayesian thinking; the â€śreal lifeâ€ť science/medicine issues are a bit more complicated than this. That is why the recommendations for screening include criteria as to age, symptoms vs. asymptomatic, family histories, etc. All of these factors affect the calculations.

For example: using digital mammograms with this population of 10,000 women in this age bracket adds 2 more â€śtrueâ€ť detections and adds 170 more false positives. So now our calculation would be , so while the true detections go up, the false positives also goes up!

Our calculation, while specific to this case, generalizes. The formula comes from Bayes Theorem which states:

. Here: is the probability of event A occurring given that B occurs and P(A) is the probability of event A occurring. So in our case, we were answering the question: given a positive mammogram, what is the probability of actually having breast cancer? This is denoted by P(A|B) . We knew: P(B|A) which is the probability of having a positive reading given that one has breast cancer and P(B|not(A)) is the probability of getting a positive reading given that one does NOT have cancer. So for us: and .

The bottom line: If you are testing for a condition that is known to be rare, even a reasonably accurate test will deliver a LOT of false positives.

Here is a warm up (hypothetical) example. Suppose a drug test is 99 percent accurate in that it will detect that a certain drug is there 99 percent of the time (if it is really there) and only yield a false positive 1 percent of the time (gives a positive result even if the person being tested is free of this drug). Suppose the drug use in this population is â€śknownâ€ť to be, say 5 percent.

Given a positive test, what is the probability that the person is actually a user of this drug?

Answer: . So, in this population, about 16.1 percent of the positives will be â€śfalse positivesâ€ť, even though the test is 99 percent accurate!

Now that you are warmed up, letâ€™s proceed to the basketball question:

Question: suppose someone (that you donâ€™t actually see) shoots free throws.

Case a) the player makes 1 of 4 shots.

Case b) the player makes 2 of 8 shots.

Case c) the player makes 5 of 20 shots.

Now youâ€™d like to know: what is the probability that the player in question is really a 75 percent free throw shooter? (I picked 75 percent as the NBA average for last season is 75.3 percent).

Now suppose you knew NOTHING else about this situation; you know only that someone attempted free throws and you got the following data.

How this is traditionally handled

The traditional â€śhypothesis testâ€ť uses the â€śfrequentistâ€ť model: you would say: if the hypothesis that the person really is a 75 percent free throw shooter is true, what is the probability that weâ€™d see this data?

So one would use the formula for the binomial distribution and use n = 4 for case A, n = 8 for case B and n = 20 for case C and use p = .75 for all cases.

In case A, weâ€™d calculate the probability that the number of â€śsuccessesâ€ť (made free throws) is less than or equal to 1; 2 for case B and 5 for case C.

For you experts: the null hypothesis would be, say for the various cases would be respectively, where the probability mass function is adjusted for the different values of n .

We could do the calculations by hand, or rely on this handy calculator.

The answers are:

Case A: .0508

Case B: .0042

Case C: .0000 ()

By traditional standards: Case A: we would be on the verge of â€śrejecting the null hypothesis that p = .75 and weâ€™d easily reject the null hypothesis in cases B and C. The usual standard (for life science and political science) is p = .05).

(for a refresher, go here)

So that is that, right?

Well, what if I told you more of the story?

Suppose now, that in each case, the shooter was me? I am not a good athlete and I played one season in junior high, and rarely, some pickup basketball. I am a terrible player. Most anyone would happily reject the null hypothesis without a second thought.

But now: suppose I tell you that I took these performances from NBA box scores? (the first one was taken from one of the Spurs-Heat finals games; the other two are made up for demonstration).

Now, you might not be so quick to reject the null hypothesis. You might reason: â€śwell, he is an NBA player and were he always as bad as the cases show, he wouldnâ€™t be an NBA player. This is probably just a bad game.â€ť In other words, youâ€™d be more open to the possibility that this is a false positive.

Now you donâ€™t know this for sure; this could be an exceptionally bad free throw shooter (Ben Wallace shot 41.5 percent, Shaquille Oâ€™Neal shot 52.7 percent) but unless you knew that, youâ€™d be at least reasonably sure that this person, being an NBA player, is probably a 70-75 shooter, at worst.

So â€śhowâ€ť sure might you be? You might look at NBA statistics and surmise that, say (I am just making this up), 68 percent of NBA players shoot between 72-78 percent from the line. So, you might say that, prior to this guy shooting at all, the probability of the hypothesis being true is about 70 percent (say). Yes, this is a prior judgement but it is a reasonable one. Now youâ€™d use Bayes law:

Here: A represents the â€ś75 percent shooterâ€ť being actually true, and B is the is the probability that we actually get the data. Note the difference in outlook: in the first case (the â€śfrequentistâ€ť method), we wondered â€śif the hypothesis is true, how likely is it that weâ€™d see data like thisâ€ť. In this case, called the Bayesian method, we are wondering: â€śif we have this data, what is the probability that the null hypothesis is trueâ€ť. It is a reverse statement, of sorts.

Of course, we have and weâ€™ve already calculated P(B|A) for the various cases. We need to make a SECOND assumption: what does event not(A) mean? Given what Iâ€™ve said, one might say not(A) is someone who shoots, say, 40 percent (to make him among the worst possible in the NBA). Then for the various cases, we calculate respectively.

So, we now calculate using the Bayesian method:

Case A, the shooter made 1 of 4: .1996. The frequentist p-value was .0508

Case B, the shooter made 2 of 8: .0301. The frequentist p-value was .0042

Case C, the shooter made 5 of 20: 7.08 x 10^-5 The frequentist p-value was 3.81 x 10^-6

We see the following:

1. The Bayesian method is less likely to produce a â€śfalse positiveâ€ť.

2. As n, the number of data points, grows, the Bayesian conclusion and the frequentist conclusions tend toward â€śthe truthâ€ť; that is, if the shooter shoots enough foul shots and continues to make 25 percent of them, then the shooter really becomes a 25 percent free throw shooter.

So to sum it up:

1. The frequentist approach relies on fewer prior assumptions and is computationally simpler. But it doesnâ€™t include extra information that might make it easier to distinguish false positives from genuine positives.

2. The Bayesian approach takes in more available information. But it is a bit more prone to the userâ€™s preconceived notions and is harder to calculate.

How does this apply to science?

Well, suppose you wanted to do an experiment that tried to find out which human gene alleles correspond so a certain human ailment. So a brute force experiment in which every human gene is examined and is statistically tested for correlation with the given ailment with null hypothesis of â€śno correlationâ€ť would be a LOT of statistical tests; tens of thousands, at least. And at a p-value threshold of .05 (we are willing to risk a false positive rate of 5 percent), we will get a LOT of false positives. On the other hand, if we applied bit of science prior to the experiment and were able to assign higher prior probabilities (called â€śposterior probabilityâ€ť) to the genes â€śmore likelyâ€ť to be influential and lower posterior probability to those unlikely to have much influence, our false positive rates will go down.

Of course, none of this eliminates the need for replication, but Bayesian techniques might cut down the number of experiments we need to replicate.