Bayesian One Sample Inference: Normal

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

The Bayesian One Sample Inference: Normal procedure provides options for making Bayesian inference on one-sample and two-sample paired t-test by characterizing posterior distributions. When you have normal data, you can use a normal prior to obtain a normal posterior.

1. From the menus choose:

   Analyze > Bayesian Statistics > One Sample Normal

2. Select the appropriate Test Variables from the source variables list. At least one source variable must be selected.
   Note: The source variables list provides all available variables except for Date and String variables.
3. 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.
4. Select and/or enter the appropriate Data Variance and Hypothesis Values settings. The table reflects the variables that are currently in the Test Variables list. As variables are added or removed from the Test Variables list, the table automatically adds or removes the same variables from its variable columns.
   • When one or more variables are in the Test Variables list, the Variable Known, and Variance Value columns are enabled.

     Variance Known

     Select this option for each variable when the variance is known.

     Variance Value

     An optional parameter that specifies the variance value, if known, for observed data.

   • When one or more variables are in the Test Variables list, and Characterize Posterior Distribution is not selected, the Null Test Value and g Value columns are enabled.

     Null Test Value

     A required parameter that specifies the null value in the Bayes factor estimation. Only one value is allowed, and 0 is the default value.

     g Value

     Specifies the value to define ψ² = gσ²_x in the Bayes factor estimation. When the Variance Value is specified, the g Value defaults to 1. When the Variance Value is not specified, you can specify a fixed g or omit the value to integrate it out.

5. You can optionally click Criteria to specify Bayesian One Sample Inference: Criteria settings (credible interval percentage, missing values options, and numerical method settings), or click Priors to specify Bayesian One Sample Inference: Normal Priors settings (type of priors, such as inference parameters, mean given variance, or precision).