Bayesian Independent - Sample Inference

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

The Bayesian Independent - Sample Inference procedure provides options for using a group variable to define two unrelated groups, and make Bayesian inference on the difference of the two group means. You can estimate the Bayes factors by using different approaches, and also characterize the desired posterior distribution either assuming the variances are known or unknown.

1. From the menus choose:

   Analyze > Bayesian Statistics > Independent Samples Normal

2. Select the appropriate Test Variables from the source variables list. At least one source variable must be selected.
3. Select the appropriate Grouping Variable from the Available Variables list. A grouping variable defines two groups for the unpaired t-test. The selected grouping variable can be either a numeric or a string variable.
4. 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 credible 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 H1	1-3	Anecdotal Evidence for H1	1/30-1/10	Strong Evidence for H0
     30-100	Very Strong Evidence for H1	1	No Evidence	1/100-1/30	Very Strong Evidence for H0
     10-30	Strong Evidence for H1	1/3-1	Anecdotal Evidence for H0	1/100	Extreme Evidence for H0
     3-10	Moderate Evidence for H1	1/10-1/3	Moderate Evidence for H0

     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.
5. Use the Define Groups options to define two groups for the t test by specifying two values (for string variables), or two values, a midpoint, or a cut point (for numeric variables).
   Note: The specified values must exist in the variable, otherwise an error message displays to indicate that at least one of the groups is empty.

   For numeric variables:

   • Use specified values. Enter a value for Group 1 and another value for Group 2. Cases with any other values are excluded from the analysis. Numbers need not be integers (for example, 6.25 and 12.5 are valid).
   • Use midpoint value. When selected, the groups are separated into < and ≥ midpoint values.
   • Use cut point.
     • Cutpoint. Enter a number that splits the values of the grouping variable into two sets. All cases with values that are less than the cutpoint form one group, and cases with values that are greater than or equal to the cutpoint form the other group.

   For string grouping variables, enter a string for Group 1 and another value for Group 2, such as yes and no. Cases with other strings are excluded from the analysis.

6. You can optionally click Criteria to specify Bayesian Independent-Sample Inference: Criteria settings (credible interval percentage, missing values options, and adaptive quadrature method settings), click Priors to specify Bayesian Independent-Sample Inference: Prior Distribution settings (data variance, prior on variance, and prior on mean conditional on variance), or click Estimate Bayes Factor to specify Bayesian Independent - Sample Inference: Estimate Bayes Factor settings.