The Lady Tasting Tea

Fisher’s Exact Test

Time: 1920s Place Cambridge, England Setting: An afternoon tea party at

the University Claim: “I can tell whether the milk is

poured first and the tea is added next, or whether the tea is poured first and the milk is added to the tea.”

Fisher was at the party!

Sir Ronald Aylmer Fisher was a British statistician

The most influential statistician so far

“Randomization, variance, the null hypothesis, maximum likelihoods – all these are tools used by every scientists today; we owe them all to Fisher. He almost single-handedly created the foundations of modern statistics.” Significance, 2013

Be careful what you claim if you are at a party with Dr. Fisher!

Brief digression:Was there really a lady tasting tea?

When I first heard this story, I thought it was apocryphal (made up). I thought Fisher was just using it as an example.

But I have come across some accounts that claim the story is real, although the accounts do not always agree on all the details.

In David Salsburg’s book: “The Lady Tasting Tea”

• The tea party took place in Cambridge, England.

• Late 1920s• “A group of university dons, their wives, and

some guests were sitting around an outdoor table for afternoon tea.”

• “Fisher does not describe the outcome of the experiment that sunny afternoon in Cambridge, but Professor Smith told me that the lady identified every single one of the cups correctly.”

In a recent article by Stephen Senn in “Significance”

• The tea party took place in Rothamsted research station, just north of London, England.

• “Three scientists were taking tea in the common room at an agricultural research station one afternoon in the early 1920s.”

• Fisher was there. Dr. Muriel Bristol, a researcher who studied algae, was “the lady.” William Roach, a biochemist, and the fiancé of Ms Bristol.

• “Our experiment consists in mixing eight cups of tea . . .”

Fisher considers how to test the claim

Try one cup?

But 50% chance of guessing correctly

Try two cups?

But then . . .

Zero 25%

One 50%

Two 25%

Plan a number of trials

Arrange a number of cups Fisher proposed 8 cups Half with tea first Half with milk first Present them in random

order

Then have the lady taste them and make her decisions

Taste the tea and make a decision

Tea first(Put four cups here)

Milk first(Put four cups here)

Taste the tea and make a decision

Tea first(Put four cups here)

Milk first(Put four cups here)

Now tally the results How many did she identify correctly?

3 1

1 3

Here is one possibility

“Milk”

“Tea”

The lady’s guess

Milk Tea

What really happened first

So she was mostly correct!

What are all the possibilities?

2 2

2 2

Evenly?

3

31

1 4 0

0 4

1

13

30

04

4

(a+b)! (c+d)! (a+c)! (b+d)!P (of an arrangement) = --------------------------------- N! a! b! c! d!

Fisher used the hypergeometric distribution to figure out the probabilities

Allincorrect Success!

Let’s try this for one possible outcome: What is the probability of getting six out of eight correct? (Just by chance)

3

31

1

(a+b)! (c+d)! (a+c)! (b+d)!P (of an arrangement) = --------------------------------- N! a! b! c! d!

a

dc

b

(4)! (4)! (4)! (4)!P (of an arrangement) = ------------------------ 8! 3! 1! 1! 3!

(4x3x2x1) (4x3x2x1) (4x3x2x1) (4x3x2x1)P = --------------------------------------------------- (8x7x6x5x4x3x2x1) (3x2x1) (1) (1) (3x2x1)

Notation for the cells

Here the results show a total of six correct

! = factorial notation

P = 0.229

The probability of getting this set of results just by chance.

0

04

4 1

13

3 2

22

2 3

31

1 4

40

0

p=.514 p=.229 p=.014p=.229p=.014

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Chances of success in the tea-tasting study

Number of successes

Pro

bab

ilit

y

And now . . .

We use Fisher’s Exact test in many situations beyond tea tasting!