# Function for Bayesian analysis of one- and two-sample designs

### Description

This function computes Bayes factors for contingency tables.

### Usage

1 2 3 | ```
contingencyTableBF(x, sampleType, fixedMargin = NULL,
priorConcentration = 1, posterior = FALSE, callback = function(...)
as.integer(0), ...)
``` |

### Arguments

`x` |
an m by n matrix of counts (integers m,n > 1) |

`sampleType` |
the sampling plan (see details) |

`fixedMargin` |
for the independent multinomial sampling plan, which margin is fixed ("rows" or "cols") |

`priorConcentration` |
prior concentration parameter, set to 1 by default (see details) |

`posterior` |
if |

`callback` |
callback function for third-party interfaces |

`...` |
further arguments to be passed to or from methods. |

### Details

The Bayes factor provided by `contingencyTableBF`

tests the independence assumption in
contingency tables under various sampling plans, each of which is described below.
See Gunel and Dickey (1974) for more details.

For `sampleType="poisson"`

, the sampling plan is assumed to be
one in which observations occur as a poisson process with an overall
rate, and then assignment to particular factor levels occurs with
fixed probability. Under the null hypothesis, the assignments to the
two factors are independent. Importantly, the total N is not fixed.

For `sampleType="jointMulti"`

(joint multinomial), the sampling
plan is assumed to be one in which the total N is fixed, and observations
are assigned to cells with fixed probability. Under the null hypothesis, the
assignments to the two factors are independent.

For `sampleType="indepMulti"`

(independent multinomial), the
sampling plan is assumed to be one in which row or column totals are fixed,
and the each row or column is assumed to be multinomially distributed.
Under the null hypothesis, each row or column is assumed to have the
same multinomial probabilities. The fixed margin must be given by
the `fixedMargin`

argument.

For `sampleType="hypergeom"`

(hypergeometric), the sampling
plan is assumed to be one in which both the row and column totals are fixed.
Under the null hypothesis, the cell counts are assumed to be governed by the
hypergeometric distribution.

For all models, the argument `priorConcentration`

indexes
the expected deviation from the null hypothesis under the alternative,
and corresponds to Gunel and Dickey's (1974) "a" parameter.

### Value

If `posterior`

is `FALSE`

, an object of class
`BFBayesFactor`

containing the computed model comparisons is
returned.

If `posterior`

is `TRUE`

, an object of class `BFmcmc`

,
containing MCMC samples from the posterior is returned.

### Note

Posterior sampling for the hypergeometric model under the alternative has not yet been implemented.

### Author(s)

Richard D. Morey (richarddmorey@gmail.com)

Tahira Jamil (tahjamil@gmail.com)

### References

Gunel, E. and Dickey, J., (1974) Bayes Factors for Independence in Contingency Tables. Biometrika, 61, 545-557

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
## Hraba and Grant (1970) doll race data
data(raceDolls)
## Compute Bayes factor for independent binomial design, with
## columns as the fixed margin
bf = contingencyTableBF(raceDolls, sampleType = "indepMulti", fixedMargin = "cols")
bf
## Posterior distribution of difference in probabilities, under alternative
chains = posterior(bf, iterations = 10000)
sameRaceGivenWhite = chains[,"pi[1,1]"] / chains[,"pi[*,1]"]
sameRaceGivenBlack = chains[,"pi[1,2]"] / chains[,"pi[*,2]"]
hist(sameRaceGivenWhite - sameRaceGivenBlack, xlab = "Probability increase",
main = "Increase in probability of child picking\nsame race doll (white - black)",
freq=FALSE, yaxt='n')
box()
``` |