Bayesian Inference about Linear Regression Models

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics option.

Regression is a statistical method that is broadly used in quantitative modeling. Linear regression is a basic and standard approach in which researchers use the values of several variables to explain or predict values of a scale outcome. Bayesian univariate linear regression is an approach to Linear Regression where the statistical analysis is undertaken within the context of Bayesian inference.

You can invoke the regression procedure and define a full model.

1. From the menus choose:

   Analyze > Bayesian Statistics > Linear Regression

2. Select a single, non-string, dependent variable from the Variables list. You must select one non-string variable.
3. Select one or more categorical factor variables for the model from the Variables list.
4. Select one or more, non-string, covariate scale variables from the Variables list.
   Note: Both of the Factors and Covariate lists cannot be empty. You must select at least one Factors or Covariate variable.
5. Select a single, non-string, variable to serve as the regression weight from the Variables list. The Weight variable field can be empty.
6. Select the desired Bayesian Analysis:
   • Characterize Posterior Distribution: When selected, the Bayesian inference is made from a perspective that is approached by characterizing posterior distributions. You can investigate the marginal posterior distribution of the parameter(s) of interest by integrating out the other nuisance parameters, and further construct Bayesian confidence intervals to draw direct inference. This is the default setting.
   • Estimate Bayes Factor: When selected, estimating Bayes factors (one of the notable methodologies in Bayesian inference) constitutes a natural ratio to compare the marginal likelihoods between a null and an alternative hypothesis.

     Table 1. Commonly used thresholds to define significance of evidence

     Bayes Factor	Evidence Category	Bayes Factor	Evidence Category	Bayes Factor	Evidence Category
     >100	Extreme Evidence for H0	1-3	Anecdotal Evidence for H0	1/30-1/10	Strong Evidence for H1
     30-100	Very Strong Evidence for H0	1	No Evidence	1/100-1/30	Very Strong Evidence for H1
     10-30	Strong Evidence for H0	1/3-1	Anecdotal Evidence for H1	1/100	Extreme Evidence for H1
     3-10	Moderate Evidence for H0	1/10-1/3	Moderate Evidence for H1

     H0: Null Hypothesis

     H1: Alternative Hypothesis

     ¹

     ²

   • Use Both Methods: When selected, both the Characterize Posterior Distribution and Estimate Bayes Factor inference methods as used.