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# BF and Cred.Interval seem to conflict ?

edited March 2016

OK all, n00b question again (but don't worry, I'm a fast learner, especially with some help from you all. And yes, I did try, very hard, to solve it myself)

QUESTION 1: HOW SHOULD I UNDERSTAND THE CREDIBLE REGION NUMBERS?

My data is a range of scores (0 ... 10), called "rapportcijfer".

Descriptive Statistics
rapportcijfer
Valid 175
Missing 0
Mean 6.594
Std. Deviation 1.535
Minimum 2.000
Maximum 10.00

I now want to do a Bayesian one sample t-test. Question: is the population mean > 6.4 ? In frequentist stats this would be:
H0: score =< 6.4
H1: score > 6.4

OK, let's do the Bayesian test. These are the JASP-results: [IMG]http://img.cogsci.nl/uploads/56e031605b471.jpg[/IMG]

Test: > 6.4
BF10: 0.630
BF01: 1.58
95% CI: 0.015 .. 0.27

OK, BF10 is 0.63, which means that these data are less likely under H1 than under H0. Right? BF01 is 1.58, which means that these data are 1.58 times more likely under H0.

The 95% Credible Interval is 0.015 ....0.27. I read this as: the value is between (6.4 + 0.015) and (6.4 + 0.27) = 6.415 ...6.67

Or should I read this as: Cred. Int. = (sample mean + 0.015) ... (sample mean + 0.27) ??

QUESTION 2: WHAT IS THE H0 and H1 BEING TESTED?

The Bayes Factor compares the probability of data under two hypotheses. Where (in JASP) can I see these hypotheses being made explicit?

QUESTION 3: CONFLICTING BF AND CREDIBLE REGION NUMBERS?

• The Bayes Factor suggests that the data are less likely under (some?) H1, with a mean > 6.4
• the CI ranges from 6.415 ... 6.67

This does not go well together.

Please tell me what is wrong in my interpretations. THANKS AGAIN !

• edited March 2016

Here we go:
1. "H0: score =< 6.4 H1: score > 6.4" This is the classical test. I have always considered it illogical, for the following reason. We have the two-sided test that compares H0: score = 6.4 against H1: score \neq 6.4. Now we have the expectation that the effect, if it exists, is really > 6.4. In response, the classical test then goes and changes....the null hypothesis! Weird. In the Bayesian version, what changes is the prior parameter distribution under H1, so that all mass is assigned to values greater than 6.4. This is also apparent from the prior and posterior plot.

You say: "The 95% Credible Interval is 0.015 ....0.27. I read this as: the value is between (6.4 + 0.015) and (6.4 + 0.27) = 6.415 ...6.67". Yes, this is correct. Note that this is a central credible interval, not a highest posterior density interval. Also note that this is the interval conditional on H+ being true.

1. H0 is always the absence of an effect. H1 is specified in the input panel; it is also visualized in the prior-posterior plot. The one-sided specification simply "folds" the prior around 0, assigning all mass to values consistent with the proposed direction of the effect.

2. Yes there can be an apparent conflict between the credible interval and the BF. Note that the credible interval is conditional on H+ being true. The data suggest that H+ may not be true (although the data are ambiguous). In general, the credible interval does not provide information about H0. It is tempting to think it does, but that's wrong. For enlightenment, see for instance:

http://www.ejwagenmakers.com/inpress/MarsmanWagenmakersOneSidedPValue.pdf

https://osf.io/m6bi8/

Cheers,
E.J.

• edited March 2016

Super again. Will read those papers too.

• edited 4:36AM

I am confused. Wouldn't it be clear to show

• TWO priors (corresponding to H0 and H+)
• the data
• TWO posteriors

What I now see in JASP, is:

• a single prior, which I assume belongs to H0
• a single prior, which (as EJ tells) depends on H+

My question: what exactly is depicted in the Bayesian T-Test plots?

• edited 4:36AM

Hi Pieter,
1. For the parameter under test (such as effect size), H0 specifies a point (e.g., delta=0), not a distribution. So under H0 we have a spike at 0, both as a prior and as a posterior.
2. Therefore, the prior and the posterior you see in the plot belong to H1 (or H+, for a one-sided test). The dotted line is the prior, the solid line is the posterior.
E.J.

• edited 4:36AM

EJ, what I am ultimately interested in, are posterior odds. That is, how likely are H0 and H+, given the data.

Of course I'd need to enter prior odds. Also, I'd like to define a more realistic H0, that is, not a spike but a distribution.

How would you do this in JASP? Or would I have to go to R ?

• edited 4:36AM

Hi Pieter,

Entering prior odds is something you can do easily yourself; multiplying them with the Bayes factor gives the posterior odds. If you want H0 to be a distribution, then you'll have to look at some of Richard's work on interval Bayes factors. We might implement this in a future JASP release, but before we do so we'll probably allow the user to be more flexible with respect to the expectations under H1 instead.

Cheers,
E.J.