BANOVA-package {BANOVA} | R Documentation |

## BANOVA: Hierarchical Bayesian ANOVA Models

### Description

This package includes several hierarchical Bayes Analysis of Variance models. These models are suited for the analysis of experimental designs in which both within- and between- subjects factors are manipulated, and account for a wide variety of distributions of the dependent variable. Floodlight analysis and mediation analysis basaed on these models are also provided. The package uses 'Stan' and 'JAGS' as the computational platform.

### Details

Package: | BANOVA |

Type: | Package |

Version: | 1.2.1 |

Date: | 2022-06-18 |

License: | GPL (>= 2) |

Model:

`E(y_i) = g^{-1}(\eta_i)`

where `\eta_i = \sum_{p = 0}^{P}\sum_{j=1}^{J_p}X_{i,j}^p\beta_{j,s_i}^p`

, `s_i`

is the subject id of data response `i`

. Missing values (NAs) of `y_i`

are allowed. The within-subjects factors and their interactions are indexed by `p (p = 1,2,.,P)`

. Each index `p`

represents a batch of `J_p`

coefficients: `\beta_{j,s}^p, j = 1,.,J_p`

;`s = 1,.,S`

indexes subjects. Note that if the subject-level covariate is continuous, `J_p=1`

, so that ANCOVA models are also accommodated (relaxing their "constant slope" assumption).

The population-level model allows for heterogeneity among subjects, because the subject-level coefficients `\beta_{j,s}^p`

are assumed to follow a multivariate normal distribution.The between-subjects factors and their interactions are indexed by `q,(q = 1,2,.,Q)`

, `q = 0`

denotes the constant term. The population-level ANOVA can be written as:

`\beta_{j,s}^p = \sum_{q = 0}^Q \theta_{j,k_s^q}^{pq} + \delta_{j,s}^p`

The population-level ANCOVA model can be expressed as a linear model with a design matrix `Z`

that contains all between-subjects factors and their interactions and a constant term:

`\beta_{j,s}^p = \sum_{k = 1}^Q Z_{s,k}\theta_{j,k}^{p} + \delta_{j,s}^p`

where `Z_{s,k} `

is an element of `Z`

, a `S \times Q`

matrix of covariates. `\theta_{j,k}^p`

is a hyperparameter which captures the effects of between-subjects factor `q`

on the parameter `\beta_{j,s}^p`

of within-subjects factor p. The error `\delta_{j,s}^p`

is assumed to be normal: `\delta_{j,s}^p`

~ `N(0,\sigma_p^{-2} )`

. Proper, but diffuse priors are assumed: `\theta_{j,k}^p`

~ `N(0,\gamma)`

, and `\sigma_p^{-2}`

~ `Gamma(a,b)`

, where `\gamma,a,b`

are hyper-parameters. The default setting is `\gamma = 10^{-4}, a = 1, b = 1`

.

Note that missing values of independent variables are currently not allowed in the package.

### Author(s)

Chen Dong; Michel Wedel

Maintainer: Chen Dong <cdong@math.umd.edu>

### References

Dong, C. and Wedel, M. (2017)
*BANOVA: An R Package for Hierarchical Bayesian ANOVA*, Journal of Statistical Software, Vol. 81, No.9, pp. 1-46.

McCullagh, P., Nelder, JA. (1989)
*Generalized linear models*, New York, NY: Chapman and Hall.

Gelman, A. (2005)
*Analysis of variance-why it is more important than ever*, Ann. Statist., Vol. 33, No. 1, pp. 1-53.

Rossi, P., Allenby,G., McCulloch, R. (2005)
*Bayesian Statistics and Marketing*, John Wiley and Sons.

Gill, J. (2007)
*Bayesian Methods for the Social and Behavioral Sciences*, Chapman and Hall, Second Edition.

Gelman, A., Carlin, J., Stern, H. and Dunson, D. (2013)
*Bayesian Data Analysis*, London: Chapman and Hall.

Wedel, M. and Dong, C. (2016) *BANOVA: Bayesian Analysis of Variance for Consumer Research*. Submitted.

*BANOVA*version 1.2.1 Index]