## Test a one sided hypothesis from a numerically specified posterior CDF

### Description

Calculates the probability of a one sided null hypothesis from a numerically calculated posterior CDF.

### Usage

pnullNum(theta0, theta, cdf, type = 'upper')


### Arguments

 theta0 the hypothesized value, i.e. H0: theta <= theta0 theta the values over which the the posterior CDF is specified cdf the values of the CDF, F(θ) = \int_{-∞}^{θ}f(t).df where f(t) is the PDF. type the type of probability to return, 'lower' = Pr(theta <= theta0) or 'upper' = Pr(theta >= theta0). It is sufficient to use 'l' or 'u'

### Details

This function uses linear interpolation to calculate bounds for points that may not be specified by CDF

### Value

a list containing the element prob which will be the upper or lower tail probability depending on type

### Examples

## commands for calculating a numerical posterior CDF.
## In this example, the likelihood is proportional to
## \eqn{\theta^{3/2}\times \exp(-\theta/4)} and a N(6, 9) prior is used.
theta <- seq(from = 0.001, to = 40, by = 0.001)
prior <- dnorm(theta,6,3)
ppnLike <- theta^1.5*exp(-theta/4)
ppnPost <- prior*ppnLike
scaleFactor <- sintegral(theta, ppnPost)$int posterior <- ppnPost/scaleFactor cdf <- sintegral(theta, posterior)$y
pnullNum(1, theta, cdf)