| dfba_beta_contrast {DFBA} | R Documentation |
Bayesian Contrasts
Description
This function implements a Bayesian analysis of a linear contrast of conditions when there are 2 or more independent conditions and where the variate for each condition is a binomial.
Usage
dfba_beta_contrast(
n1_vec,
n2_vec,
contrast_vec,
a0_vec = rep(1, length(n1_vec)),
b0_vec = rep(1, length(n1_vec)),
prob_interval = 0.95,
samples = 10000
)
Arguments
n1_vec |
A vector of length K that consists of the observed number of successes for the categorical variable in each of the K separate conditions |
n2_vec |
A vector of length K that consists of the observed number of failures for the categorical variable in each of the K separate conditions |
contrast_vec |
A vector of coefficients of a linear comparison among the conditions where the sum of all the coefficients must be 0 and the sum of the positive coefficients must be 1 and the sum of the negative coefficients must be -1 |
a0_vec |
A vector of length K that consists of the prior |
b0_vec |
A vector of length K that consists of the prior |
prob_interval |
Desired probability for equal-tail interval estimate on the contrast (default is 0.95) |
samples |
The desired number of Monte Carlo samples taken from each posterior beta variate (default is 10000) |
Details
Since the Bayesian analysis for each separate condition has a posterior beta distribution with known shape parameters, the program approximates, via Monte Carlo sampling, a linear contrast among the set of independent beta distributions because the contrast of beta distributions is not a known probability model.
Given a binomial categorical variate for each of K independent
conditions with K \ge 2, the standard frequentist nonparametric
analysis is to do a \chi^2 test with K - 1 degrees of freedom
(Siegel & Castellan, 1988). Hypothesis testing for the frequentist \chi^2
test assesses the sharp-null hypothesis that the binomial success rate is
exactly equal in all the conditions. But this point-null hypothesis is not an
interesting question about the population success rate from a Bayesian
viewpoint because the probability of any single point hypothesis has a
probability measure value of zero (Chechile, 2020). Although it is possible
that the frequentist null hypothesis can be retained for small-n studies,
the hypothesis itself is about the population in the case of unlimited sample
size, and surely for this limiting case it is almost certain that the
hypothesis is not exactly true. Thus, from the Bayesian framework, the point-
null hypothesis is not a good use of scientific effort and resources, and it
is more scientifically meaningful to assess a linear comparison of the
conditions, such as to assess if the population success rate in one condition
is greater than the success rate in another condition. An interval hypothesis
such as this has a meaningful probability value, as does the complimentary
hypothesis. If \phi_1 and \phi_2 are, respectively, the population
success rates for the binomials in conditions 1 and 2, then a meaningful
comparison might be to assess the probability distribution for \Delta =
\phi_2 - \phi_1. This example is a simple linear contrast with contrast
coefficient weights of -1 and 1, which are the multipliers for the two
population success rates. If the posterior interval estimate for the contrast
contains 0, then the hypothesis of \Delta = 0 has some credibility in light
of the given sample size. Thus, by estimating the distribution of \Delta,
the user learns important information about condition differences. As another
example of a contrast, suppose there are three conditions where the first
condition is a standard control and the other two conditions are different
alternative conditions. In this case, a user might want to compare the
mean of the control data against the average of the two experimental-
condition means, i.e., the contrast of
\Delta = -1\phi_1 +.5\phi_2 + .5\phi_3.
In this second example, the coefficients of the contrast are [-1, +.5, +.5].
As a third example, the user might also be interested in a comparison where
the two experimental conditions are compared, i.e., the contrast of
\Delta = 0\phi_1 + 1\phi_2 - 1\phi_3.
For the dfba_beta_contrast() function, the user is required to
stipulate the coefficients of a contrast such that the sum of all the
coefficients is 0, the sum of the positive coefficients is 1, and the sum of
the negative coefficients is -1. This constraint on the coefficients forces
\Delta to be on the [-1, +1] interval.
There is a standard Bayesian posterior for each condition, which is a beta
distribution (see Chechile (2020) for a detailed discussion of this
literature). In short, it is well known that the beta distribution is a
natural Bayesian conjugate function for Bernoulli random processes. Thus, a
prior beta distribution with shape parameters a_0 and b_0 results
(via Bayes's theorem) in a posterior beta with shape parameters a
and b where a = a_0 + n_1 and b = b_0 + n_2, where n_1
and n_2 are the respective successes and failures of the categorical
variable. While the Bayesian analysis of each beta distribution for the
separate conditions are known, a comparison among 2 or more separate beta
distributions is not distributed as a beta. The posterior mean of a linear
contrast of separate beta variates has a known mean regardless of the
correlations among the variates, but the distributional form of the contrast
of independent betas is not known in closed form. The distributional form is
important for ascertaining issues such as determining the probability that
the contrast is positive or specifying a probability interval for the
contrast. But, with the dfba_beta_contrast() function, these important
aspects of the Bayesian analysis are approximated via Monte Carlo simulation.
The samples argument stipulates the number of random values to be
drawn from each of the K posterior conditions. The default value for
samples is 10000. The default value of 10000 is also the minimum value
that can be selected (increased values of samples provide increased
precision). Posterior interval estimation and the Bayes factor for the
contrast are provided on the basis of the Monte Carlo sampling. If samples
is equal to N and if \phi_1, \ldots, \phi_K are the parameters
for the population success rates, then there are N random values drawn
from each of \phi_i parameters for i = 1, \ldots , K. Given the
contrast coefficients stipulated in the arguments, there are N delta random
posterior values where \Delta_j = \Psi_1\phi_{1j}+ \ldots +\Psi_i\phi_{Kj} for
j = 1, \ldots, N, where \Psi_i are the contrast coefficients specified
in the contrast_vec argument. The Monte Carlo sampling from each posterior
beta with known shape parameters uses the rbeta() function. Thus, unlike
Bayesian procedures that employ Markov chain Monte Carlo algorithms, the
Monte Carlo sampling in the dfba_beta_contrast() function does not depend
on a burn-in process or a starting estimate. Thus, all the N sampled
values are valid random samples. Repeated use of the dfba_beta_contrast()
function for the same input will naturally exhibit some random variation in
the interval estimate and in the Bayes factor for a contrast greater than 0.
However, the point estimate for the contrast does not depend on the Monte
Carlo sampling, and it is constant given the vectors for n1_vec and
n2_vec and given the same prior.
Value
A list containing the following components:
mean |
Exact posterior mean estimate for the contrast |
eti_lower |
The lower equal-tail limit for the contrast for the probability interval value specified by |
eti_upper |
The upper equal-tail limit for the contrast for the probability interval value specified by |
prob_positive_delta |
Posterior probability that the contrast is positive |
prior_positive_delta |
Prior probability that the contrast is positive |
bayes_factor |
The Bayes factor for the posterior-to-prior odds for a positive contrast to a non-positive contrast |
delta_quantiles |
Quantile values ( |
a_vec |
A vector of length K that consists of the posterior |
b_vec |
A vector of length K that consists of the posterior |
a0_vec |
A vector of length K that consists of the prior |
b0_vec |
A vector of length K that consists of the prior |
contrast_vec |
A vector for the contrast coefficients for a linear comparison of posterior beta variates |
prob_interval |
The probability for the equal-tail estimate for the contrast (default is 0.95) |
samples |
The number of Monte Carlo samples from the K separate posterior beta distributions |
References
Chechile, R. A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods. Cambridge: MIT Press.
Siegel, S. & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. New York: McGraw Hill.
Examples
## Suppose there are four conditions from a factorial design
# where the conditions labels are A1B1, A2B1, A1B2, and A2B2
# where the frequencies for success for the binomial variate are:
n1_vec <- c(22, 15, 13, 21)
# and the frequencies for failures per condition are:
n2_vec <- c(18, 25, 27, 19)
# Let us test the following three orthogonal contrasts
contrast.B1vsB2 <- c(.5, .5, -.5, -.5)
contrast.A1vsA2 <- c(.5, -.5, .5, -.5)
contrast.ABinter <- c(.5, -.5, -.5, .5)
dfba_beta_contrast(n1_vec = n1_vec,
n2_vec = n2_vec,
contrast_vec = contrast.B1vsB2)
dfba_beta_contrast(n1_vec,
n2_vec,
contrast_vec = contrast.A1vsA2)
dfba_beta_contrast(n1_vec,
n2_vec,
contrast_vec = contrast.ABinter)
# Plot the cumulative distribution for AB interaction
testABinteraction<-dfba_beta_contrast(n1_vec,
n2_vec,
contrast_vec = contrast.ABinter)
plot(testABinteraction)