## Bayesian inference for simple linear regression

### Description

This function is used to find the posterior distribution of the simple linear regression slope variable \beta when we have a random sample of ordered pairs (x_{i}, y_{i}) from the simple linear regression model:



 y_{i} = \alpha_{\bar{x}} + \beta x_{i}+\epsilon_{i} 



where the observation errors are, \epsilon_i, independent normal(0,\sigma^{2}) with known variance.

### Usage

bayes.lin.reg(
y,
x,
slope.prior = c("flat", "normal"),
intcpt.prior = c("flat", "normal"),
mb0 = 0,
sb0 = 0,
ma0 = 0,
sa0 = 0,
sigma = NULL,
alpha = 0.05,
plot.data = FALSE,
pred.x = NULL,
...
)


### Arguments

 y the vector of responses. x the value of the explantory variable associated with each response. slope.prior use a “flat” prior or a “normal” prior. for \beta intcpt.prior use a “flat” prior or a “normal” prior. for \alpha_[\bar{x}] mb0 the prior mean of the simple linear regression slope variable \beta. This argument is ignored for a flat prior. sb0 the prior std. deviation of the simple linear regression slope variable \beta - must be greater than zero. This argument is ignored for a flat prior. ma0 the prior mean of the simple linear regression intercept variable \alpha_{\bar{x}}. This argument is ignored for a flat prior. sa0 the prior std. deviation of the simple linear regression variable \alpha_{\bar{x}} - must be greater than zero. This argument is ignored for a flat prior. sigma the value of the std. deviation of the residuals. By default, this is assumed to be unknown and the sample value is used instead. This affects the prediction intervals. alpha controls the width of the credible interval. plot.data if true the data are plotted, and the posterior regression line superimposed on the data. pred.x a vector of x values for which the predicted y values are obtained and the std. errors of prediction ... additional arguments that are passed to Bolstad.control

### Value

A list will be returned with the following components:

 post.coef the posterior mean of the intecept and the slope post.coef the posterior standard deviation of the intercept the slope pred.x the vector of values for which predictions have been requested. If pred.x is NULL then this is not returned pred.y the vector predicted values corresponding to pred.x. If pred.x is NULL then this is not returned pred.se The standard errors of the predicted values in pred.y. If pred.x is NULL then this is not returned

### Examples


## generate some data from a known model, where the true value of the
## intercept alpha is 2, the true value of the slope beta is 3, and the
## errors come from a normal(0,1) distribution
set.seed(123)
x = rnorm(50)
y = 2 + 3*x + rnorm(50)

## use the function with a flat prior for the slope beta and a
## flat prior for the intercept, alpha_xbar.

bayes.lin.reg(y,x)

## use the function with a normal(0,3) prior for the slope beta and a
## normal(30,10) prior for the intercept, alpha_xbar.

bayes.lin.reg(y,x,"n","n",0,3,30,10)

## use the same data but plot it and the credible interval

bayes.lin.reg(y,x,"n","n",0,3,30,10, plot.data = TRUE)

## The heart rate vs. O2 uptake example 14.1
O2 = c(0.47,0.75,0.83,0.98,1.18,1.29,1.40,1.60,1.75,1.90,2.23)
HR = c(94,96,94,95,104,106,108,113,115,121,131)
plot(HR,O2,xlab="Heart Rate",ylab="Oxygen uptake (Percent)")

bayes.lin.reg(O2,HR,"n","f",0,1,sigma=0.13)

## Repeat the example but obtain predictions for HR = 100 and 110

bayes.lin.reg(O2,HR,"n","f",0,1,sigma=0.13,pred.x=c(100,110))