95% Credibility Intervals for Bayesian ANCOVA

eniseg2

Hello Team JASP,

Just curious, if I want to have the 95%CI for my best model's posterior distribution, how would I go about that? I saw that we can have the posteriors and 95%CI for the individual coefficients but what about the overall model's 95% CI?

Best,

Anna

JohnnyB

Hi Eniseg,

In order to get the posterior estimates per model, you can specify your desired model under the section "single model inference". There, you can specify the model (e.g., the best model according to the model comparison table), and then you get a bunch of statistics for that model specifically, such as posterior distributions and credible intervals for each of the coefficients in the model.

This is contrast to the coefficients table that is in the default output, which are the metrics per coefficient, averaged over all the models included in the analysis. My colleague Don van den Bergh wrote an awesome tutorial paper on Bayesian ANOVA's in JASP, which you can find here: https://psyarxiv.com/spreb/ in case you want to read more about this in greater detail.

Does that answer your question?

Kind regards,

Johnny

eniseg2

Hi Johnny!

Yes, that helped thank you very much. :) I just have a curious interpretation question. For my most recent analysis, I'm seeing some reasonably large Bayes Factors (30 and above) with an R2 of .30, give or take. But my coefficients are teeny tiny. For some of my most important variables, the coefficient is something like 0.05! My null model contains all known risk factors for a certain outcome, so the H1 model contains some protective factors, the coefficient of which is around 0.05 but with Bayes factors above 30 and R2 of .30.

So I can conclude the hypothesis that my new factor is protective is strongly supported because of the large Bayes factor, it explained a small amount of variance and had a very weak influence? That sounds counterintuitive. Is this correct?

Kind regards

Anna

JohnnyB

Hi Anna,

Great to hear! It's certainly possible to obtain evidence in favor of an effect, even though it's small, most commonly if you have a large data set - is that the case for you?

I like playing around in the summary statistics module to get some intuition for this. If you want, you can try to enable the summary statistics module (you can enable modules by clicking the "+" sign to the right of the screen), go to the Summary statistics -> Bayesian correlation, specify a certain correlation (something small, for instance 0.1), and start with a small sample size (like 10) and see how the bayes factor changes when you increase the sample size. For instance, you can go from 10 to 100, to 1000, to 10,000. At first, you will see evidence in favor of the null, but as you increase n, you will start seeing evidence in favor of the alternative hypothesis. Here it also helps to have the prior/posterior plot enabled.

Kind regards,

Johnny

eniseg2

Hi Johnny!

Thanks for the fast response. So our data set only has 60 people in it. I study a rare disease and 60 is quite sizeable for the field of this disease (i.e. if every single possible patient that had a clinic appointment with our team over the course of 12 months agreed to participate in our study, we would get 60 patients, max.). But obviously that doesn't make much of a difference numerically.

In order to detect an effect of 0.10 (Pearson's rho), we would have needed a larger sample set of around 750 people. That would have got us a BF=4 in favour of our H1. So in order to detect larger effects we would need a much much larger sample? This seems counterintuitive, I thought large samples were for very small effects. I'm also not sure if a sample size of 750 patients with our disease exists anywhere in the world, I've certainly never seen a paper with that many.

So my conclusion saying "strong evidence in favour of a small effect explaining a moderate amount of variance, larger effects may appear in larger samples" is appropriate?

All the best!

Anna