Hypothesis Testing: Frequentist vs. Bayesian
I was working through Nate Silver’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 4049 got a digital mammagram and got a “positive” 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 “false negative” 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.
I’ll talk about problem two first, as this will limber the mind for problem one.
So, you are a woman between 4049 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; we’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 8090 percent of positives in women in this age bracket are “false positives”, 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 “real life” 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 bracketadds 2 more “true” 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 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 . We knew: which is the probability of having a positive reading given that one has breast cancer and 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 “known” 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 “false positives”, even though the test is 99 percent accurate!
Now that you are warmed up, let’s proceed to the basketball question:
Question: suppose someone (that you don’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 you’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 “hypothesis test” uses the “frequentist” 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 we’d see this data?
So one would use the formula for the binomial distribution and use for case A, for case B and for case C and use for all cases.
In case A, we’d calculate the probability that the number of “successes” (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 .
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 “rejecting the null hypothesis that and we’d easily reject the null hypothesis in cases B and C. The usual standard (for life science and political science) is p = .05).
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 SpursHeat 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: “well, he is an NBA player and were he always as bad as the cases show, he wouldn’t be an NBA player. This is probably just a bad game.” In other words, you’d be more open to the possibility that this is a false positive.
Now you don’t know this for sure; this could be an exceptionally bad free throw shooter (Ben Wallace shot 41.5 percent, Shaquille O’Neal shot 52.7 percent) but unless you knew that, you’d be at least reasonably sure that this person, being an NBA player, is probably a 7075 shooter, at worst.
So “how” 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 7278 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 you’d use Bayes law:
Here: A represents the “75 percent shooter” 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 “frequentist” method), we wondered “if the hypothesis is true, how likely is it that we’d see data like this”. In this case, called the Bayesian method, we are wondering: “if we have this data, what is the probability that the null hypothesis is true”. It is a reverse statement, of sorts.
Of course, we have and we’ve already calculated for the various cases. We need to make a SECOND assumption: what does event mean? Given what I’ve said, one might say 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 pvalue was .0508
Case B, the shooter made 2 of 8: .0301. The frequentist pvalue was .0042
Case C, the shooter made 5 of 20: 7.08 x 10^5 The frequentist pvalue was 3.81 x 10^6
We see the following:
1. The Bayesian method is less likely to produce a “false positive”.
2. As n, the number of data points, grows, the Bayesian conclusion and the frequentist conclusions tend toward “the truth”; 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 doesn’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 user’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 “no correlation” would be a LOT of statistical tests; tens of thousands, at least. And at a pvalue 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 “posterior probability”) to the genes “more likely” 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.
5 Comments »
Leave a Reply

Archives
 August 2016 (31)
 July 2016 (51)
 June 2016 (48)
 May 2016 (63)
 April 2016 (59)
 March 2016 (75)
 February 2016 (76)
 January 2016 (60)
 December 2015 (59)
 November 2015 (60)
 October 2015 (54)
 September 2015 (50)

Categories
 2008 Election
 2010
 2010 election
 2012 election
 2014 midterm
 2016
 Aaron Schock
 Ad
 affirmative action
 Agricultural Commisioner
 aircraft
 Alabama
 alternative energy
 america
 April 1
 arizona
 astronomy
 atheism
 Barack Obama
 barback obama
 Barbara Boxer
 baseball
 basketball
 bicycling
 Biden
 big butts
 bikinis
 bill maher on mosque
 bill richardson
 biology
 blog humor
 Blogroll
 blogs
 blood donation
 Bobby Jindal
 books
 boxing
 brain
 bushera
 business & economy
 butt
 Cheri Bustos
 civil liberties
 Claire McCaskill
 climate change
 college football
 comedy
 cop
 cosmology
 creationism
 d k hirner
 dark energy
 dave koehler
 deadline
 Democrats
 Dick Durbin
 Dick Morris
 disease
 dk hirner
 draw Mohammad day
 draw Muhammad day
 economics
 economy
 education
 edwards
 energy
 entertainment
 environment
 evolution
 extension
 family
 flu
 football
 Fox News Lies Again
 free speech
 Friends
 frogs
 geese
 glenn beck
 glenn hubbard
 green news
 ground zero mosque
 gwen ifill
 haunting songs
 health
 health care
 Herman Cain
 High Speed Rail
 hiking
 hillary clinton
 history
 hsr
 huckabee
 human sexuality
 humor
 if rich people have to pay taxes
 IL17
 IL18
 Illinois
 illness
 immigration. racial profiling
 injury
 internet issues
 interviews
 Intrade Prediction
 islamophobia
 jan brewer
 jim lehrer
 job
 Joe Biden
 John McCain
 jon stewart
 Judicial nominations
 knee rehabilitation
 lahood
 laughing at myself
 liars
 marathons
 mathematics
 matter
 mccain
 media
 michelle bachmann
 Mid Life Crisis
 Middle East
 Mike Huckabee
 mike's blog round up
 mind
 Mitt Romney
 money
 moron
 morons
 movies
 music
 nanotechnology
 national disgrace
 nature
 Navel Staring
 NBA
 neuroscience
 newshour
 Newt Gingrich
 NFL
 north america
 north carolina
 NSFW humor
 obama
 obesity
 Olympic Spandex
 Olympics
 Peoria
 Peoria/local
 Personal Issues
 photos
 physics
 Political Ad
 political humor
 political/social
 politics
 politics/social
 poll
 poor
 poverty
 public policy and discussion from NPR public radio program Science Friday with host Ira Flatow. Science Videos
 pwnd
 quackery
 racewalking
 racism
 ranting
 rebulican party
 recession
 relationships
 religion
 Republican
 republican party
 republican senate minority leader
 republicans
 republicans political/social
 republicans politics
 restaurants
 resume
 rich
 rick perry
 rick santorum
 running
 Rush Limbaugh
 sarah palin
 sb1070
 science
 Science Friday teachers
 Science Friday teens.
 SCOTUS
 shinkansen
 shoulder rehabilitation
 sickness
 social/political
 space
 spandex
 Spineless Democrats
 sports
 statistics
 stem cells
 stephen colbert
 story
 summer
 superstition
 swimming
 tax cuts
 taxes
 technology
 the colbert report
 Tim Pawlenty
 time trial/ race
 training
 trains
 Transportation
 travel
 ultra
 Uncategorized
 walking
 war on drugs
 wealth
 weight training
 whining
 wise cracks
 workouts
 world events
 WTF
 yoga

RSS
Entries RSS
Comments RSS
[…] So right now, for women between 4049, traditional mammograms are about 80 percent accurate in the sense that, if a woman who really has breast cancer gets a mammogram, the test will catch it about 80 percent of the time. The false positive rate is about 8 percent in that: if 100 women who do NOT have breast cancer get a mammogram, 8 of the mammograms will register a “positive”. Since the breast cancer rate for women in this age group is about 1.4 percent, there will be many more false positives than true positives; in fact a woman in this age group who gets a “positive” first mammogram has about a 16 percent chance of actually having breast cancer. I talk about these issues here. […]
Pingback by What do you mean by “more accurate”? « blueollie  July 12, 2013 
[…] So right now, for women between 4049, traditional mammograms are about 80 percent accurate in the sense that, if a woman who really has breast cancer gets a mammogram, the test will catch it about 80 percent of the time. The false positive rate is about 8 percent in that: if 100 women who do NOT have breast cancer get a mammogram, 8 of the mammograms will register a “positive”. Since the breast cancer rate for women in this age group is about 1.4 percent, there will be many more false positives than true positives; in fact a woman in this age group who gets a “positive” first mammogram has about a 16 percent chance of actually having breast cancer. I talk about these issues here. […]
Pingback by An example to apply Bayes’ Theorem and multivariable calculus  College Math Teaching  July 12, 2013 
[…] However the best part of the book (and it is all pretty good, IMHO), is his discussion of Bayes law and Bayesian versus frequentist statistics. I’ve talked about this. […]
Pingback by Review of Nate Silver’s book: The signal and the noise: why so may predictions fail, but some don’t « blueollie  July 22, 2013 
[…] However the best part of the book (and it is all pretty good, IMHO), is his discussion of Bayes law and Bayesian versus frequentist statistics. I’ve talked about this. […]
Pingback by Nate Silver’s Book: The signal and the noise: why so many predictions fail but some don’t  College Math Teaching  July 23, 2013 
[…] I really wonder: “what is the difference in the various metrics, what are the pvalues, and is this one of those famous “false positives” that we are constantly warned about? My guess: “probably”; such announced results usually are. But this is such a fun […]
Pingback by Maybe that is why I like “the big butt”? :) « blueollie  October 31, 2013 