mcmc.norm.hier {asbio} | R Documentation |

## Gibbs sampling of normal hierarchical models

### Description

These functions are designed for Gibbs sampling comparison of groups with normal hierarchical models (see Gelman 2003), and for providing appropriate summaries.

### Usage

```
mcmc.norm.hier(data, length = 1000, n.chains = 5)
norm.hier.summary(M, burn.in = 0.5, cred = 0.95, conv.log = TRUE)
```

### Arguments

`data` |
A numerical matrix with groups in columns and observations in rows. |

`length` |
An integer specifying the length of MCMC chains. |

`n.chains` |
The number of chains to be computed for each parameter |

`M` |
An output array from |

`burn.in` |
The burn in period for the chains. The default value, 0.5, indicates that only the latter half of chains should be used for calculating summaries. |

`cred` |
Credibility interval width. |

`conv.log` |
A logical argument indicating whether convergence for |

### Details

An important Bayesian application is the comparison of groups within a normal hierarchical model.
We assume that the data from each group are independent and from normal populations with means
`\theta[j]`

, `j = (1,...,J)`

, and a common variance, `\sigma^2`

. We also assume that group means,
are normally distributed with an unknown mean, `\mu`

, and an unknown variance , `\tau^2`

.
A uniform prior distribution is assumed for `\mu, log\sigma`

and `\tau`

; `\sigma`

is
logged to facilitate conjugacy. The function `mcmc.norm.hier`

provides posterior distributions
of `\theta[j]`

's, `\mu, \sigma`

and `\tau`

. The distributions are derived from univariate
conditional distributions from the multivariate likelihood function. These conditional distributions
provide a situation conducive to MCMC Gibbs sampling. Gelman et al. (2003) provide excellent summaries of these sorts of models.

The function `mcmc.summary`

provides statistical summaries for the output array from `mcmc.norm.hier`

including credible intervals (empirically derived directly from chains) and the Gelman/Rubin convergence criterion, `\hat{R}`

.

### Value

The function `mcmc.norm.hier`

returns a three dimensional (step x variable x chain) array. The function `mcmc.summary`

returns a summary table containing credible intervals and the Gelman/Rubin convergence criterion, `\hat{R}`

.

### Author(s)

Ken Aho

### References

Gelman, A., Carlin, J. B., Stern, H. S., and D. B. Rubin (2003) *Bayesian Data Analysis, 2nd edition*. Chapman and Hall/CRC.

### See Also

### Examples

```
## Not run:
data(cuckoo)
mcmc.norm.hier(cuckoo,10,2)
## End(Not run)
```

*asbio*version 1.9-7 Index]