binom1.2sided: Binomial Suite: One Sample, Two Sided

Description

Generates the suite of functions related to the one sample binomial experiment with a two-sided alternative hypothesis of interest.

Usage

binom1.2sided(p0, prob, a, b)

Arguments

Scalar. The value of p under null hypothesis Ho: p==p0. Must be a value between 0 and 1.

prob

Scalar. The prior probability that the null hypothesis is true. Must be a value between 0 and 1.

Scalar. Shape1 parameter for prior Beta distribution. See documentation for dbeta.

Scalar. Shape2 parameter for prior Beta distribution. See documentation for dbeta.

Value

logm

binom1.2sided Returns a list of three vectors: the log marginal density under the null hypothesis (logm0), the log marginal density under the alternative hypothesis (logm1), the log marginal density (logm). Each are evaluated at the observed data provided. This function is passed to ssd.binom to calculate required sample sizes. The function has the following usage:

logm(x, n, p0, prob, a, b)

x: Vector. Number of successes observed, out of n independent Bernoulli trials.
n: Scalar. Sample size, the number of independent Bernoulli trials.
Remaining parameters specified above for binom1.1sided.

logbf

Returns a vector: the value of the log Bayes Factor given the observed data provided and the prior parameters specified. The function has the following usage:

logbf(x, n, p0, prob, a, b)

For details on the parameters, see above function logm

prior

Returns a vector: the value of the prior density. The function has the following usage:

prior(p, p0, prob, a, b)

p: Scalar. Quantiles for the prior distribution.
Remaining parameters specified above for binom1.1sided.

post

Returns a vector: the value of the posterior density. The function has the following usage:

post(p, x, n, p0, prob, a, b)

p: Scalar. Quantiles for the posterior distribution.
x: Scalar. Number of successes observed, out of n independent Bernoulli trials.
Remaining parameters specified above for binom1.1sided.

Details

binom1.2sided is used to generate a suite of functions for a one-sample binomial experiment with a two-sided alternative hypothesis. That is, when $$X \sim Binomial(n,p)$$ $$H_0: p = p_0 \quad vs. \quad H_1: p \neq p_0$$ using the following prior on p $$\pi(p) = u*I(p=p_0) + (1-u)*I(p\neq p_0)Beta(a,b)$$ where Beta(a,b) is Beta density with parameters a and b and u is the prior probability of the null hypothesis (prob). The functions that are generated are useful in examining the prior and posterior densities of the parameter p, as well as constructing the Bayes Factor and determining the sample size via an average error based approach. The arguments of binom1.2sided are passed to each of the additional functions upon their creation as default values. That is, if p0 is set to 0.5 in the call to binom1.2sided, each of the functions returned will have the defaualt value of 0.5 for p0. If an argument is not specified in the call to binom1.2sided, then it remains a required parameter in all functions created.

Examples

Run this code

############################################################
# Generate the suite of functions for a one-sample binomial
# with a two-sided test. Consider the hypothesis
#      H0: p==0.5   vs.  H1: p!=0.5
#
# with a uniform prior on p under the alternative and a
# prior probability of the null hypothesis equal to 0.5.

# generate suite
f2 <- binom1.2sided(p0=0.5,prob=0.5,a=1,b=1)

# attach suite
attach(f2)

# plot prior and posterior given x = 25, n = 30
#  - don't forget that point mass is not shown on plot
ps <- seq(0.01,0.99,0.01)
p1 <- prior(ps)
p2 <- post(ps,x=25,n=30)

plot(c(p1,p2)~rep(ps,2),type="n",ylab="Density",xlab="p",main="")
lines(p1~ps,lty=1,lwd=2)
lines(p2~ps,lty=2,lwd=2)

# perform sample size calculation with TE bound of 0.25 and weight 0.5
ssd.binom(alpha=0.25,w=0.5,logm=logm)

# detain suite
detach(f2)

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