dfba_binomial {DFBA} R Documentation

## Bayesian Binomial Rate Parameter Inference

### Description

Given binomial frequency data, provides a Bayesian analysis for the population binomial rate parameter.

### Usage

dfba_binomial(n1, n2, a0 = 1, b0 = 1, prob_interval = 0.95)


### Arguments

 n1 Integer number of binomial observations for a category 1 response (e.g., the number of successes) n2 Integer number of binomial observations for a category 2 response (e.g., the number of failures) a0 The first shape parameter for the prior beta distribution that corresponds to the population binomial parameter (default is 1). Must be positive and finite. b0 The second shape parameter for the prior beta distribution for the population binomial rate parameter (default is 1). Must be positive and finite. prob_interval Probability within interval estimates for the population binomial rate parameter (default is .95)

### Details

The binomial distribution with size = n and probability = \phi has discrete probabilities

p(x) = \frac{n!}{z!(n - x!)}\phi^{x}(1-\phi)^{n-x}

where x is an integer from 0 to n in steps of 1. The binomial model assumes a Bernoulli process of independent trials where there are binary outcomes that have the same probability (say, \phi) for a response in one of the two categories and a probability of 1-\phi for the other category. Before any data are collected, there are n + 1 possible values for x number of outcomes in category 1 and n - x number of outcomes in category 2. The binomial distribution is a likelihood distribution. A likelihood is the probability of an outcome given a specific value for the population rate parameter. Yet for real applications, the population parameter is not known. All that is known are the outcomes observed from a set of binomial trials. The binomial inference problem is to estimate the population \phi parameter based on the sample data.

The frequentist approach to statistics is based on the relative frequency method of assigning probability values (Ellis, 1842). From this framework, there are no probabilities for anything that does not have a relative frequency (von Mises, 1957). In frequency theory, the \phi parameter does not have a relative frequency, so it cannot have a probability distribution. From a frequentist framework, a value for the binomial rate parameter is assumed, and there is a discrete distribution for the n + 1 outcomes for x from 0 to n. The discrete likelihood distribution has relative frequency over repeated experiments. Thus, for the frequentist approach, x is a random variable, and \phi is an unknown fixed constant. Frequency theory thus delibrately eschews the idea of the binomial rate parameter having a probability distribution. Laplace (1774) had previously employed a Bayesian approach of treating the \phi parameter as a random variable. Yet Ellis and other researchers within the frequentist tradition delibrately rejected the Bayes/Laplace approach. For tests of a null hypothesis of an assumed \phi value, the frequentist approach either continues to assume the null hypothesis or it rejects the null hypothesis depending on the likelihood of the observed data plus the likelihood of more extreme unobserved outcomes. The confidence interval is the range of \phi values where the null hypothesis of specific \phi values would be retained given the observed data (Clopper & Pearson, 1934). However, the frequentist confidence interval is not a probability interval since population parameters cannot have a probability distribution with frequentist methods. Frequentist statisticians were well aware (e.g., Pearson, 1920) that if the \phi parameter had a distribution, then the Bayes/Laplace approach would be correct.

Bayesian statistics rejects the frequentist theoretical decisions as to what are the fixed constants and what is the random variable that can take on a range of values. From a Bayesian framework, probability is anything that satisfies the Kolmogorov (1933) axioms, so probabilities need not be limited to processes that have a relative frequency. Importantly, probability can be a measure of information or knowledge provided that the probability representation meets the Kolmogorov axioms (De Finetti, 1974). Given binomial data, the population binomial rate parameter \phi is unknown, so it is represented with a probability distribution for its possible values. This assumed distribution is the prior distribution. Furthermore, the quantity x for the likelihood distribution above is not a random variable once the experiment has been conducted. If there are n_1 outcomes for category 1 and n_2 = n-n_1 outcomes in category 2, then these are fixed values. While frequentist methods compute both the likelihood of the observed outcome and the likelihood for unobserved outcomes that are more extreme, in Bayesian inference it is only the likelihood of the observed outcome that is computed. From the Bayesian perspective, the inclusion of unobserved outcomes in the analysis violates the likelihood principle (Berger & Wolpert, 1988). A number of investigators have found paradoxes with frequentist procedures when the likelihood principle is not used (e.g., Lindley & Phillips, 1976; Chechile, 2020). The Bayesian practice of strictly computing only the likelihood of the observed data produces the result that the likelihood for the binomial is proportional to \phi^{n_1}(1 - \phi)^{n_2}. In Bayesian statistics, the proportionality constant is not needed because it appears in both the numerator and the denominator of Bayes theorem and thus cancels. See Chechile (2020) for more extensive comparisons between frequentist and Bayesian approaches with a particular focus on the binomial model.

Given a beta distribution prior for the binomial \phi parameter, it has been shown that the resulting posterior distribution from Bayes theorem is another member of the beta family of distributions (Lindley & Phillips, 1976). This property of the prior and posterior being in the same distributional family is called conjugacy. The beta distribution is a natural Bayesian conjugate function for all Bernoulli processes where the likelihood is proportional to \phi^{n_1}(1 - \phi)^{n_2} (Chechile, 2020). The density function for a beta variate is

f(x) = \begin{cases} Kx^{a-1}(1-x)^{b-1} & \quad \textrm{if } 0 \le x \le 1, \\0 & \quad \textrm{otherwise} \end{cases}

where

K = \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)}

(Johnson, Kotz, & Balakrishnan, 1995). The two shape parameters a and b must be positive values. If the beta prior shape parameters are a0 and b0, then the posterior beta shape parameters are a_{post} = a_0 + n_1 and b_{post} = b_0 + n_2. The default prior for the dfba_binomial() function is a0 = b0 = 1, which corresponds to the uniform prior.

Thus, the Bayesian inference for the unknown binomial rate parameter phi is the posterior beta distribution with shape parameters of a_post and b_post. The dfba_binomial() function calls the dfba_beta_descriptive() function to find the centrality point estimates (i.e., the mean, median, and mode) and to find two interval estimates that contain the probability specified in the prob_interval argument. One interval has equal-tail probabilities and the other interval is the highest-density interval. Users can use the dfba_beta_bayes_factor() function to test hypotheses about the \phi parameter.

### Value

A list containing the following components:

 n1 Observed number of category 1 responses n2 Observed number of category 2 responses a0 First shape parameter for the prior beta distribution of the binomial rate parameter b0 Second shape parameter for the prior beta distribution of the binomial rate parameter prob_interval Probability within interval estimates for the population binomial rate parameter a_post First shape parameter for the posterior beta distribution for the binomial rate parameter b_post Second shape parameter for the posterior beta distribution for the binomial rate parameter phimean Mean of the posterior beta distribution for the binomial rate parameter phimedian Median of the posterior beta distribution for the binomial rate parameter phimode Mode of the posterior beta distribution for the binomial rate parameter eti_lower Lower limit for the posterior equal-tail interval that has the probability stipulated in the prob_interval argument eti_upper Upper limit for the posterior equal-tail interval that has the probability stipulated in the prob_interval argument hdi_lower Lower limit for the posterior highest-density interval that has the probability stipulated in the prob_interval argument hdi_upper Upper limit for the posterior highest-density interval that has the probability stipulated in the prob_interval argument

### References

Berger, J. O., & Wolpert, R. L. (1988). The Likelihood Principle (2nd ed.) Hayward, CA: Institute of Mathematical Statistics.

Chechile, R. A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Statistics. Cambridge: MIT Press.

Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26, 404-413.

De Finetti, B. (1974). Bayesianism: Its unifying role for both the foundations and applications of statistics. International Statistical Review/ Revue Internationale de Statistique, 117-130.

Ellis, R. L. (1842). On the foundations of the theory of probability. Transactions of the Cambridge Philosophical Society, 8, 1-6.

Johnson, N. L., Kotz S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 1, New York: Wiley.

Kolmogorov, A. N. (1933/1959). Grundbegriffe der Wahrcheinlichkeitsrechnung. Berlin: Springer. English translation in 1959 as Foundations of the Theory of Probability. New York: Chelsea.

Laplace, P. S. (1774). Memoire sr la probabilite des causes par les evenements. Oeuvres complete, 8,5-24.

Lindley, D. V., & Phillips, L. D. (1976). Inference for a Bernoulli process (a Bayesian view). The American Statistician, 30, 112-119.

Pearson, K. (1920). The fundamental problem of practical statistics. Biometrika, 13(1), 1-16.

von Mises, R. (1957). Probability, Statistics, and Truth. New York: Dover.

Distributions for details on the functions included in the stats regarding the beta and the binomial distributions.

dfba_beta_bayes_factor for further documentation about the Bayes factor and its interpretation.

dfba_beta_descriptive for advanced Bayesian descriptive methods for beta distributions

### Examples

# Example using defaults of a uniform prior and 95% interval estimates
dfba_binomial(n1 = 16,
n2 = 2)

# Example with the Jeffreys prior and 99% interval estimates
dfba_binomial(n1 = 16,
n2 = 2,
a0 = .5,
b0 = .5,
prob_interval = .99)



[Package DFBA version 0.1.0 Index]