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PHILOSOPHY OF SCIENCE:
Neyman-Pearson approach
Jerzy Neyman
Egon Pearson
April 16, 1894August 5, 1981
11 August 1895 12 June 1980
Zoltán Dienes, Philosophy of Psychology
'The statistician cannot excuse himself from the duty of getting his
head clear on the principles of scientific inference, but equally no
other thinking person can avoid a like obligation'
Fisher 1951
Prior to 1930s:
There were many statistical procedures
But no coherent account of what they achieved or of how to choose
the right test.
Neyman and Pearson put the field of statistics on a firm logical
footing
It is now orthodoxy
(but note: there are passionate attacks on just how firm their logical
footing is!)
What is probability?
Relative frequency interpretation
Need to specify a collective of elements – like throws of a dice.
In the long run – as number of observations goes to infinity – the
proportion of throws of a dice showing a 3 is 1/6
The probability of a ‘3’ is 1/6 because that is the long run frequency of
‘3’s relative to all throws
One cannot talk about the probability of a hypothesis e.g.
“this cancer drug is more effective than placebo” being true
“genes are coded by DNA” is not true 2/3 of the time in the long
run – it is just true. There is no relevant long run.
A hypothesis is just true or false.
When we say what the probability of a hypothesis is, we are
referring to a subjective probability
Neyman-Pearson (defined the philosophy underlying standard
statistics):
Probabilities are strictly long-run relative frequencies – not subjective!
Statistics do not tell us the probability of your theory or the null
hypothesis being true.
So what relative frequencies are we talking about?
If D = some data and H = a hypothesis
For example, H = this drug is just a placebo cure for depression
D = 17 out of 20 people felt happier on the drug than the placebo
One can talk about p(D|H)
The probability of the data given the hypothesis
e.g. p(‘17 or more people happier out of 20’|’drug is a placebo’)
A collective or reference class we can use:
the elements are
‘measuring for each of 20 people whether they are happier on the drug
day or the placebo day’ given the drug operates just as a placebo.
Consider a hypothetical collective of an infinite number of such
experiments.
That is a meaningful probability we can calculate.
One can NOT talk about p(H|D)
The probability of our hypothesis given the data
e.g. p(‘my drug is a placebo’| ‘obtaining 17 out of 20 people happier’)
What is the reference class??
The hypothesis is simply true or false.
P(H|D) is the inverse of the conditional probability p(D|H)
Inverting conditional probabilities makes a big difference
e.g.
P(‘dying within two years’|’head bitten off by shark’) = 1
P(‘head was bitten off by shark’|’died in the last two years’) ~ 0
P(A|B) can have a very different value from P(B|A)
Statistics cannot tell us how much to believe a certain hypothesis.
All we can do is set up decision rules for certain behaviours
– accepting or rejecting hypotheses –
such that in following those rules in the long run we will not often be
wrong.
E.g. Decision procedure:
Run 40 subjects and reject null hypothesis if t-value larger than a
critical value
Our procedure tells us our long term error rates BUT it does not
tell us which particular hypotheses are true or false or assign any
of the hypotheses a probability.
All we know is our long run error rates.
State of World:
Decision:
Ho true
Accept Ho
Reject Ho
Ho false
Type II error
Type I error
Need to control both types of error:
α = p(rejecting Ho|Ho)
β = p(accepting Ho|Ho false)
Consider a year in which of the null hypotheses we test, 4000 are
actually true and 1000 actually false.
State of World
___________________________
Decision
H0 true
H0 false
___________________________________________________
Accept H0
3800
500
Reject H0
200
500
___________________________
4000
α =?
β=?
1000
State of World:
Decision:
Ho true
Ho false
Accept Ho
Reject Ho
Type II error
Type I error
Need to control both types of error:
α = p(rejecting Ho/Ho)
β = p(accepting Ho/Ho false)
power:
P(‘getting t as extreme or more extreme than critical’/Ho false)
Probability of detecting an effect given an effect really exists in the
population. ( = 1 – β)
Decide on allowable α and β BEFORE you run the experiment.
e.g. set α = .05 as per normal convention
Ideally also set β = .05.
α is just the significance level you will be testing at.
But how to control β?
Controlling β:
Need to
1) Estimate the size of effect you think is plausible or interesting
given your theory is true
2) Power tables or online programs tell you how many subjects you
need to run to keep β to .05 (equivalently, to keep power at 0.95)
Most studies do not do this!
But they should. Strict application of the Neyman-Pearson logic means
setting the risks of both Type I and Type II errors in advance (α and β).
Most researchers are extremely worried about Type I errors (false
positives) i.e. whether p < .05
but allow Type II errors (false negatives) to go uncontrolled.
Leads to inappropriate judgments about what results mean and what
research should be done next.
Read handout for details!
You read a review of studies looking at whether meditation reduces
depression. 100 studies have been run and 50 are significant in the
right direction and the remainder are non-significant. What should
you conclude?
You read a review of studies looking at whether meditation reduces
depression. 100 studies have been run and 50 are significant in the
right direction and the remainder are non-significant. What should
you conclude?
If the null hypothesis were true, how many would be significant?
How many significant in the right direction?
"The continued very extensive use of significance tests is alarming."
(Cox 1986)
"After four decades of severe criticism, the ritual of null hypothesis
significance testing---mechanical dichotomous decisions around a
sacred .05 criterion---still persist. “
“[significance testing] does not tell us what we want to know, and ..
out of desperation, we nevertheless believe that it does!"
(Cohen 1994)
“statistical significance testing retards the growth of scientific
knowledge; it never makes a positive contribution”
(Schmidt & Hunter, 1997, p. 37).
“The almost universal reliance on merely refuting the null
hypothesis is a terrible mistake, is basically unsound, poor
scientific strategy, and one of the worst things that ever happened
in the history of psychology”
(Meehl, 1978, p. 817).
A lot of criticism arises because most researchers do not follow
the Neyman and Pearson demands in a sensible way
e.g. habitually ignoring power
BUT
The (orthodox) logic of Neyman and Pearson is also
controversial
To summarise:
You are allowed to draw a back and white conclusion when the
decision procedure has known low error rates
Anything that affects the error rates of your decision procedure affects
what decisions you can draw
In general: The more opportunities you give yourself to make an
error the higher the probability of an error becomes. So you must
correct for this.
E.g.
Multiple tests: If you perform two t-tests the overall probability of
an error is increased
Multiple tests:
Testing the elderly vs the middle aged
AND the middle aged vs the young
That’s two t-tests so for the overall Type I rate to be controlled at .05
could conduct each test at the .025 level.
If one test is .04, would reject the null if just doing that one test but
accept the null if doing two tests.
Cannot test your data once at .05 level
Then run some more subjects
And test again at .05 level
Type I error rate is no longer .05 because you gave yourself two
chances at declaring significance.
Each test must be conducted at a lower p-value for the overall error
rate to be kept at .05.
Does that make sense?
Should our inferences depend on what else we might have done or
just on what the data actually is?
If when they stopped collecting data depends on who has the better
kung fu
Then the mathematically correct result depends on whose kung fu is
better!
The mathematically correct answer depends on whose unconscious wish
to please the other is strongest!!
The Bayesian (and likelihood) approaches do not depend on
when you planned to stop running subjects,
whether you conduct other tests,
or whether the test is planned or post hoc!