Bayesian One Sample Inference: Binomial

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

The Bayesian One Sample Inference: Binomial procedure provides options for executing Bayesian one-sample inference on Binomial distribution. The parameter of interest is π, which denotes the probability of success in a fixed number of trials that may lead to either success or failure. Note that each trial is independent of each other, and the probability π remains the same in each trial. A binomial random variable can be seen as the sum of a fixed number of independent Bernoulli trials.

Although it is not necessary, a prior from the Beta distribution family is normally chosen when estimating a binomial parameter. The Beta family is conjugate for the binomial family, and as such leads to the posterior distribution with a closed form still in the Beta distribution family.

1. From the menus choose:

   Analyze > Bayesian Statistics > One Sample Binomial

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 Success Categories 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, the table automatically adds or removes the same variables from its variable pair columns.
   • When Characterize Posterior Distribution is selected as the Bayesian Analysis, the Success Categories column is enabled.
   • When Estimate Bayes Factor or Use Both Methods are selected as the Bayesian Analysis, all editable columns are enabled.

   Null Prior Shape

   Specifies the shape parameter a₀ under the null hypothesis of Binomial inference.

   Null Prior Scale

   Specifies the scale parameter b₀ under the null hypothesis of Binomial inference.

   Alternate Prior Shape

   A required parameter to specify a₀ under the alternative hypothesis of Binomial inference if Bayes factor is to be estimated.

   Alternate Prior Scale

   A required parameter to specify b₀ under the alternative hypothesis of Binomial inference if Bayes factor is to be estimated.

   Success Categories

   Provides options for defining conjugate prior distributions. The provided options specify how success is defined, for numerical and string variables, when the data value(s) are tested against the test value.

   Last Category

   The default setting that performs the binomial test using the last numerical value found in the category after it is sorted in an ascending order.

   First Category

   Performs the binomial test using the first numerical value found in the category after it is sorted in an ascending order.

   Midpoint

   Uses the numerical values ≥ the midpoint as cases. A midpoint value is the average of the minimum and maximum sample data.

   Cutpoint

   Uses the numerical values ≥ a specified cutoff value as cases. The setting must be a single numeric value.

   Level

   Treats user specified string values (can be more than 1) as cases. Use commas to separate the different values.

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: Binomial/Poisson Priors settings (conjugate or custom prior distributions).