How to get the most out of null results using Bayes

Zoltán Dienes

The problem:

Does a non-significant result count as evidence for the null hypothesis or as no evidence either way?

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Difference in verbal ability

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Geoff Cummin: http://www.latrobe.edu.au/psy/esci/index.html

The solutions:

1. Power2. Interval estimates3. Bayes Factors

Problems with Power

I) Power depends on specifying the minimal effect of interest (which may be poorly specified by the theory)

II) Power cannot make use of your actual data to determine the sensitivity of those data

Confidence intervals solve the second problem

Bayes Factors can solve both problems

By making use of the full range of predictions of the theory, it makes maximal use of the data in assessing the sensitivity of the data in distinguishing your theory from the null

A Bayes Factor can show strong evidence for the null hypothesis over your theory, when it is impossible to say anything using power or confidence intervals

Difference between means ->

Minimal interesting value

If the 95% confidence/ credibility/ likelihood interval is completely contained in this region, conclude there is good evidence that population value lies in null region – accept the null region hypothesis

If the interval is completely outside this region, conclude there is good evidence that population value lies outside null region – reject the null region hypothesis

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Null region

If the upper limit of the interval is below the minimal interesting value, conclude there is evidence against a theory postulating a positive difference

If the interval includes both null and theoretically interesting values, the data are insensitive

The four principles of inference by intervals:

The Bayes Factor

Cat hypothesisDevil hypothesis

If a cat, you lose finger only 1/10 of time

If a devil, you will lose finger 9/10 of time

Evidence supports the theory that most strongly predicted it

Evidence supports the theory that most strongly predicted it

John puts his hand in the box and loses a finger.

Which hypothesis is most strongly supported, the cat hypothesis or the devil hypothesis?

Evidence supports a theory that most strongly predicted it

John puts his hand in the box and loses a finger.

Which hypothesis is most strongly supported, the cat hypothesis or the devil hypothesis?

Cat hypothesis predicts this result with probability = 1/10Devil hypothesis predicts this result with probability = 9/10

Evidence supports a theory that most strongly predicted it

John puts his hand in the box and loses a finger.

Which hypothesis is most strongly supported, the cat hypothesis or the devil hypothesis?

Cat hypothesis predicts this result with probability = 1/10Devil hypothesis predicts this result with probability = 9/10

Strength of evidence for devil over cat hypothesis = 9/10 divided by 1/10 = 9

The evidence is nine times as strong for the devil over the cat hypothesis

OR

Bayes Factor (B) = 9

Consider:

John does not lose a finger

Consider:

John does not lose a finger

Now evidence strongly supports cat over devil hypothesis (BF = 9 for cat over devil hypothesis or 1/9 for devil over cat hypothesis)

Probability of losing finger given cat = 4/10Probability of losing finger given devil = 6/10

Now if John loses finger strength of evidence for devil over cat = 6/4 = 1.5

Not very strong

We can distinguish:

Evidence for cat hypothesis over devil

Evidence for devil hypothesis over cat

Not much evidence either way.

Bayes factor tells you how strongly the data are predicted by the different theories (e.g. your pet theory versus null hypothesis):

B =

Probability of your data given your pet theory

divided by

probability of data given null hypothesis

If B is greater than 1 then the data supported your theory over the null

If B is less than 1, then the data supported the null over your theory

If B = about 1, experiment was not sensitive.

(Automatically get a notion of sensitivity;

contrast: just relying on p values in significance testing.)

Jeffreys, 1961: Bayes factors more than 3 or less than a 1/3 are substantial

To know which theory data support need to know what the theories predict

The null is normally the prediction of e.g. no difference

Population difference between conditions

Plausibility

0 2-2 4

On the null hypothesis only this value is plausible

To know which theory data support need to know what the theories predict

The null is normally the prediction of e.g. no difference

Need to decide what difference or range of differences are consistent with one’s theory

Difficult - but forces one to think clearly about one’s theory.

To calculate a Bayes factor must decide what range of differences are predicted by the theory

1) Uniform distribution

2) Half normal

3) Normal

Plausibility

Population difference in means between conditions0 2 4 8

Example: The theory predicts a difference will be in one direction.

Subjects give 0-8 ratings in two conditions

Maximum difference allowed

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Seems more plausible to think the larger effects are less likely than the smaller ones:

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Plausibility

Population difference in means between conditions

But how to scale the rate of drop?

0 4

Implies: Smaller effects more likely than bigger ones; effects bigger than 8 very unlikely

Plausibility

Population difference in means between conditions

Similar sorts of effects as those predicted in the past have been on the order of a 5% difference between conditions

0 5

Implies: Smaller effects more likely than bigger ones; effects bigger than 10% very unlikely

Plausibility

Population difference in means between conditions

Plausibility

Difference between conditions

0 5 10

To calculate Bayes factor in a t-test situationNeed same information from the data as for a t-test:

Mean difference, Mdiff

SE of difference, SEdiff

To calculate Bayes factor in a t-test situationNeed same information from the data as for a t-test:

Mean difference, Mdiff

SE of difference, SEdiff

Note: t = Mdiff / SEdiff

=> SEdiff = Mdiff/t

To calculate a Bayes factor:

1) Google “Zoltan Dienes”

2) First site to come up is the right one:http://www.lifesci.sussex.ac.uk/home/Zoltan_Dienes/

3) Click on link to book

4) Click on link to Chapter Four

5) Scroll down and click on “Click here to calculate your Bayes factor!”

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http://www.latrobe.edu.au/psy/esci/index.htmlBayes p

2.96 .0814.88 .0340.52 .744.88 .0342.70 .090.46 .8174.40 .0281024.6 .0013.33 .0564.88 .0311.73 .2794.28 .0242.96 .08349.86 .0022.16 .1672.12 .1721.01 .3870.65 .6140.75 .47628.00 .0064.28 .02849.86 .0025.60 .0242.36 .1441.73 .23

The tai chi of the Bayes factors

The dance of the p values

A Bayes Factor requires establishing predicted effect sizes. How?

Do digit-colour synesthetes show a Stroop effect on digits?

You display: 3 … 4 … 5 … 6

What they see: 3 … 4 … 5 … 6

You get a null effect (incongruent minus congruent RTs) . . . What size effect would be predicted if there were one?

A Bayes Factor requires establishing predicted effect sizes.

Do digit-colour synesthetes show a Stroop effect on digits?

You display: 3 … 4 … 5 … 6

What they see: 3 … 4 … 5 … 6

You get a null effect (incongruent minus congruent RTs) . . . What size effect would be predicted if there were one?

Run normals on a condition in which digits are coloured in the way synesthetes say they are. The Stroop effect is presumably the maximum one could expect synesthetes to show.

Use a uniform:

0 Effect for normals with real colours

Plausibility

Possible population Stroop effects

Another condition in your experiment might help settle expectations:

Jiang et al 2012

Obtained significant amount of unconscious knowledge (5%)

Conscious knowledge was 6% with a SE of 7%

Another condition in your experiment might help settle expectations:

Jiang et al 2012

Obtained significant amount of unconscious knowledge (5%)

Conscious knowledge was 6% with a SE of 7% (non-significant)

To assess meaning of non-significant result, used a half-normal with SD = 5%

BF = 1.25

Another group in your experiment might help settle expectations:

Jiang et al 2012

Obtained significant amount of unconscious knowledge (5%)

Conscious knowledge was 6% with a SE of 7%

Used a half-normal with SD = 5%

BF = 1.25

Nothing follows about whether subjects had conscious knowledge or not

If you have a manipulation meant to reduce an effect, effect of manipulation unlikely to be larger than the basic effect

e.g. Dienes, Baddeley & Jansari (2012)Predicted sad mood would reduce learning compared to neutral mood

So e.g. if on 2-alternative forced choice test, in neutral condition people get 70% correct

If you have a manipulation meant to reduce an effect, effect of manipulation unlikely to be larger than the basic effect

e.g. Dienes, Baddeley & Jansari (2012)Predicted sad mood would reduce learning compared to neutral mood

So e.g. if on 2-alternative forced choice test, in neutral condition people get 70% correct

Sad condition expected to be somewhere between 50 and 70%

So effect of mood must be?

My typical practice:

If think of way of determining an approximate expected size of effectÞUse half normal with SD = to that typical size

If think of way of determining an approximate upper limit of effect=> Use uniform from 0 to that limit

Moral and inferential paradoxes of orthodoxy:

1. On the orthodox approach, standardly you should plan in advance how many subjects you will run.

If you just miss out on a significant result you are not allowed to just run 10 more subjects and test again.

You are not allowed to run until you get a significant result.

Bayes: It does not matter when you decide to stop running subjects. You can always run more subjects if you think it will help.

Moral paradox:If p = .07 after running planned number of subjectsi) If you run more and report significant at 5% you have cheatedii) If you don’t run more and bin the results you have wasted tax payer’s money and

your time, and wasted relevant dataYou are morally damned either way

Inferential paradoxTwo people with the same data and theories could draw opposite conclusions

Moral and inferential paradoxes of orthodoxy:

2. On the orthodox approach, it matters whether you formulated your hypothesis before or after looking at the data.

Post hoc vs planned comparisons

Predictions made in advance of rather than before looking at the data are treated differently

Bayesian inference: It does not matter what day of the week you thought of your theory

The evidence for your theory is just as strong regardless of its timing

Moral and inferential paradoxes of orthodoxy:

3. On the orthodox approach, you must correct for how many tests you conduct in total.

For example, if you ran 100 correlations and 4 were just significant, researchers would not try to interpret those significant results.

On Bayes, it does not matter how many other statistical hypotheses you investigated (or your RA without telling you). All that matters is the data relevant to each hypothesis under investigation.

For orthodoxy but not Bayes:

Different people with the same data and theories can come to different conclusions

You can thus be tempted to make false (albeit inferentially irrelevant claims), like when you thought of your theory

What is the aim of statistics?

1) Control the proportion of errors you make in the long run in accepting and rejecting hypotheses

(conventional statistics)

2) Indicate how strong the evidence is for one hypothesis rather than another / how much you should change your confidence in one hypothesis rather than another

(Bayesian statistics)

Dienes 2011 Perspectives on Psychological Science