Normal Suite: One Sample, Two Sided, Known Variance

Description

Generates the suite of functions related to the one sample normal experiment with a two-sided alternative hypothesis of interest when the variance is known.

Usage

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  norm1KV.2sided(sigma, theta0, prob, mu, tau)

Arguments

sigma

Scalar. The known standard deviation of the population of interest.

theta0

Scalar. The critical value of the mean under the null hypothesis: theta==theta0.

prob

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

mu

Scalar. The mean of the normal prior density on theta under the alternative hypothesis. See documentation for dnorm.

tau

Scalar. The standard deviation for the normal prior density on theta under the alternative hypothesis. See documentation for dnorm.

Details

norm1KV.2sided is used to generate a suite of functions for a one-sample normal experiment with a two-sided alternative hypothesis when the variance is known. That is, when

X ~ Normal(theta,sigma2)

H0: theta == theta0 vs. H1: theta != theta0

using the following prior on theta

pi(theta) = u*I(theta==theta0) + (1-u)*I(theta!=theta0)Normal(mu,tau2),

where Normal(mu,tau2) is Normal density with mean mu and variance tau2 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 theta, as well as constructing the Bayes Factor and determining the sample size via an average error based approach.

The arguments of norm1KV.2sided are passed to each of the additional functions upon their creation as default values. That is, if mu is set to 1 in the call to norm1KV.2sided, each of the functions returned will have the defaualt value of 1 for mu. If an argument is not specified in the call to norm1KV.2sided, then it remains a required parameter in all functions created.

Value

norm1KV.2sided returns a list of 5 functions:

logm

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. The function takes the following usage:

logm(xbar, n, sigma, theta0, prob, mu, tau)
  • xbar: Vector. Observed sample mean from the experiment.

  • n: Scalar. Sample Size.

  • Remaining parameters described above for norm1KV.2sided.

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(xbar, n, sigma, theta0, prob, mu, tau)

See logm above for details on the parameters of the function.

prior

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

prior(theta, theta0, prob, mu, tau)
  • theta: Vector. The quantiles at which to evaluate the prior.

  • Remaining parameters described above for norm1KV.2sided.

post

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

post(theta, xbar, n, sigma, theta0, prob, mu, tau)
  • theta: Vector. The quantiles at which to evaluate the prior.

  • xbar: Vector. Observed sample mean from the experiment.

  • n: Scalar. Sample Size.

  • Remaining parameters described above for norm1KV.2sided.

ssd.norm1KV.2sided

Sample size calculations for this particular set-up. The function has the following usage:

ssd.norm1KV.2sided(alpha, w, sigma, theta0, prob, mu, tau, 
    minn = 2, maxn = 1000, all = FALSE)

See ssd for more details. The suite-specific parameters are described above for norm1KV.2sided.

See Also

binom1.1sided,binom1.2sided, binom2.1sided,binom2.2sided, norm1KV.1sided,norm2KV.2sided norm1UV.2sided,ssd,BAEssd

Examples

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############################################################
# Generate the suite of functions for a one-sample normal
# with a two-sided test. Consider the hypothesis
#      H0: theta==0  vs.  H1: theta!=0
#
# with a normal prior for theta with prior mean 2 and 
# prior standard deviation 1. The known standard 
# deviation for the population is 5. The prior proability
# of the null hypothesis is set to 0.5.

# generate suite
f6 <- norm1KV.2sided(sigma=5,theta0=0,prob=0.5,mu=2,tau=1)

# attach suite
attach(f6)

# plot the prior and posterior density when the observed data is
#  n = 30, xbar = 1.
mus <- seq(-5,5,0.01)
mu1 <- prior(mus)
mu2 <- post(mus,xbar=1,n=30)

plot(c(mu1,mu2)~rep(mus,2),type="n",ylab="Density",xlab="mu",main="")
lines(mu1~mus,lty=1,lwd=2)
lines(mu2~mus,lty=2,lwd=2)

# perform sample size calculation with TE bound of 0.25 and weight 0.5
#   using both the example-specific function as well as the general.
ssd.norm1KV(alpha=0.25,w=0.5,logm=logm)
ssd.norm1KV.2sided(alpha=0.25,w=0.5)

# detain suite
detach(f6)