Bayesian regression analysis using JASP

Comments

• EJ

  July 2019

  Dear Thalia,

  First, if you set the *model prior* to Uniform this assigns equal prior plausibility to each model (i.e., each unique combination of predictors). This is standard practice, but the problem is that, implicitly, this setting leads to a preference for models with about half of the predictors included. Scott & Berger (2006, 2010) have argued for a different approach that is more sophisticated.

  Within a given model (i.e., a set of predictors) you need to assign a prior distribution to the regression coefficient. This can be done in different ways, and JASP offers plenty of options. I personally prefer the default that we provide (which is the same as the default in the BayesFactor package).

  As far as the non-continuous predictors are concerned, yes, this is a problem. You could close your eyes and pretend everything is OK, but that is dangerous. Let me include Don van den Bergh and Alexander Ly in this conversation, maybe they have some words of wisdom...

  Cheers,

  E.J.
• AlexanderLy

  July 2019

  Dear Thalia, 

  To perform a Bayesian linear regression with nominal predictors we recommend using a Bayesian ANCOVA. Most likely you’re interested in the effects table. The effects table provides the evidence for the inclusion of a predictor across models. 

  In these linear model there are basically two types of priors: (1) the priors on the models, and (2) the priors on the parameters within a model. 

  
  

  1. Priors on the models

  With p number of predictors there are in principle 2^p models. For instance, if p = 8, then there will be 2^8=256 models. These models can be represented by an indicator variable that tells you which of the variables are active. For instance, the first model that only includes the intercept, thus, none of the predictors, can be represented by 

  0, 0, 0, 0, 0, 0, 0, 0

  and the last model that, on top of the intercept, includes all 8 predictors can be represented by 

  1, 1, 1, 1, 1, 1, 1, 1

  In between we have models such as 

  0, 0, 0, 0, 0, 0, 1, 1

  which has two active predictors, namely, the last two. When you choose a uniform prior on the models, then each of these 256 models gets a prior model probability of 1/256. After data observation, these prior model probabilities are updated to posterior model probabilities. If all predictors are relevant then the last model represented by 

  1, 1, 1, 1, 1, 1, 1, 1

  gets a relative high posterior model probability compared to the first model where each predictor is inactive. Similarly, the model 

  0, 0, 0, 0, 0, 0, 1, 1

  should then also get a higher posterior model probability than the first model. 

  When p=8, the effects table summarises the importance of each predictor across the 256 models by weighting with respect to the posterior model probabilities. Note that the prior and posterior model probabilities are discrete. In Bayesian linear regression in JASP you can change this prior on the models to, for instance, a beta-binomial. In Bayesian ANCOVA a uniform prior on the models is used. 

  
  

  2. Priors on the parameters within a model

  For a Bayes factor we also require priors on the active parameters within each model. These priors are continuous and as a default we recommend using a multivariate Cauchy prior with scale parameter r for this. This set-up is referred to as JZS in Bayesian linear regression, and in Bayesian ANCOVA the UI doesn’t mention JZS, but it allows users to tune the scale parameter r. 

  For more on these two types of priors and the roles they play, see for instance

  https://psyarxiv.com/dhb7x

  I hope that this helps. 

  Cheers,

  Alexander
• eniseg2

  July 2019

  Dear both,

  thank you for this elaboration. I'm facing a similar issue to Thalia. One of my predictors is likely going to be categorical but it will have four levels. Is this acceptable? Also, in the ANCOVA, random factors do not produce post hoc tests, is that correct?

  Thank you very much!

  Anna