norm1UV.2sided {BAEssd} R Documentation

## Normal Suite: One Sample, Two Sided, Unknown 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 unknown.

### Usage

```  norm1UV.2sided(theta0, prob, mu, scale, shape, rate)
```

### Arguments

 `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`. `scale` Scalar. Used to determine the standard deviation for the normal prior density on theta under the alternative hypothesis. The standard deviation is equal to `scale*sigma`. See documentation for `dnorm`. `shape` Scalar. The shape parameter for the gamma prior on the inverse of the unknown standard deviation `sigma2`. See documenation for `dgamma`. `rate` Scalar. The rate parameter for the gamma prior on the inverse of the unknown standard deviation `sigma2`. See documentation for `dgamma`.

### Details

`norm1UV.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 unknown. That is, when

X ~ Normal(theta,sigma2)

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

using the following prior on theta and sigma2

pi(theta|sigma2) = u*I(theta==theta0) + (1-u)*I(theta!=theta0)Normal(mu,(scale*sigma)^2),

pi(sigma2) = InverseGamma(shape,rate),

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 parameters `theta` and `sigma2`, as well as constructing the Bayes Factor and determining the sample size via an average error based approach.

The arguments of `norm1UV.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 `norm1UV.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 `norm1UV.2sided`, then it remains a required parameter in all functions created.

### Value

`norm1UV.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, s2, n, theta0, prob, mu, scale, shape, rate)` `xbar`: Vector. Observed sample mean from the experiment. `s2`: Vector. Observed sample standard deviation from the experiment. `n`: Scalar. Sample size. Remaining parameters described above for `norm1UV.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, s2, n, theta0, prob, mu, scale, shape, rate)` For details on the arguments, see `logm` above. `prior` Returns a vector. The value of the prior density. The function has the following usage: `prior(theta, sigma2, theta0, prob, mu, scale, shape, rate)` `theta`: Vector. The quantiles of the mean at which to evaluate the prior. `sigma2`: Vector. The quantiles of the standard deviation at which to evaluate the prior. Remaining parameters described above for `norm1UV.2sided` `post` Returns a vector. The value of the posterior density. The function has the following usage: ```post(theta, sigma2, xbar, s2, n, theta0, prob, mu, scale, shape, rate)``` `theta`: Vector. The quantiles of the mean at which to evaluate the posterior. `sigma2`: Vector. The quantiles of the standard deviation at which to evaluate the psterior. `xbar`: Vector. Observed sample mean from the experiment. `s2`: Vector. Observed sample standard deviation from the experiment. `n`: Scalar. Sample size. Remaining parameters described above for `norm1UV.2sided` `ssd.norm1UV.2sided` Sample size calculations for this particular set-up. The function has the following usage: ```ssd.norm1UV.2sided(alpha, w, theta0, prob, mu, scale, shape, rate, m = 2500, minn = 3, maxn = 1000, all = FALSE)``` See `ssd` for more details. The suite-specific parameters are described above for `norm1UV.2sided`. Note that this example will not work with `ssd.norm1KV`.

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

### Examples

```############################################################
# 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
# scale of 1/3 for the standard deviation. The prior proability
# of the null hypothesis is set to 0.5. The prior density
# on sigma2 is taken to be InverseGamma with parameters
# 11 and 30 for the shape and rate.

# generate suite
f8 <- norm1UV.2sided(theta0=0,prob=0.5,mu=2,scale=(1/3),shape=11,rate=30)

# attach suite
attach(f8)

# calculate the Bayes Factor for the following observed data
#   n = 30, xbar = 1, s2 = 2
logbf(xbar=1,s2=2,n=30)

# perform sample size calculation with TE bound of 0.25 and weight 0.5
ssd.norm1UV.2sided(alpha=0.25,w=0.5)

# detain suite
detach(f8)
```

[Package BAEssd version 1.0.1 Index]