# Statistics Resembling Deviance and AIC for Constrained Ordination

### Description

The functions extract statistics that resemble deviance and AIC from the
result of constrained correspondence analysis `cca`

or
redundancy analysis `rda`

. These functions are rarely
needed directly, but they are called by `step`

in
automatic model building. Actually, `cca`

and
`rda`

do not have `AIC`

and these functions
are certainly wrong.

### Usage

1 2 3 4 5 | ```
## S3 method for class 'cca'
deviance(object, ...)
## S3 method for class 'cca'
extractAIC(fit, scale = 0, k = 2, ...)
``` |

### Arguments

`object` |
the result of a constrained ordination
( |

`fit` |
fitted model from constrained ordination. |

`scale` |
optional numeric specifying the scale parameter of the model,
see |

`k` |
numeric specifying the "weight" of the |

`...` |
further arguments. |

### Details

The functions find statistics that
resemble `deviance`

and `AIC`

in constrained
ordination. Actually, constrained ordination methods do not have a
log-Likelihood, which means that they cannot have AIC and deviance.
Therefore you should not use these functions, and if you use them, you
should not trust them. If you use these functions, it remains as your
responsibility to check the adequacy of the result.

The deviance of `cca`

is equal to the Chi-square of
the residual data matrix after fitting the constraints. The deviance
of `rda`

is defined as the residual sum of squares. The
deviance function of `rda`

is also used for
`capscale`

. Function `extractAIC`

mimics
`extractAIC.lm`

in translating deviance to AIC.

There is little need to call these functions directly. However, they
are called implicitly in `step`

function used in automatic
selection of constraining variables. You should check the resulting
model with some other criteria, because the statistics used here are
unfounded. In particular, the penalty `k`

is not properly
defined, and the default `k = 2`

is not justified
theoretically. If you have only continuous covariates, the `step`

function will base the model building on magnitude of eigenvalues, and
the value of `k`

only influences the stopping point (but the
variables with the highest eigenvalues are not necessarily the most
significant in permutation tests in `anova.cca`

). If you
also have multi-class factors, the value of `k`

will have a
capricious effect in model building. The `step`

function
will pass arguments to `add1.cca`

and
`drop1.cca`

, and setting `test = "permutation"`

will provide permutation tests of each deletion and addition which
can help in judging the validity of the model building.

### Value

The `deviance`

functions return “deviance”, and
`extractAIC`

returns effective degrees of freedom and “AIC”.

### Note

These functions are unfounded and untested and they should not be used
directly or implicitly. Moreover, usual caveats in using
`step`

are very valid.

### Author(s)

Jari Oksanen

### References

Godínez-Domínguez, E. & Freire, J. (2003)
Information-theoretic approach for selection of spatial and temporal
models of community organization. *Marine Ecology Progress
Series* **253**, 17–24.

### See Also

`cca`

, `rda`

, `anova.cca`

,
`step`

, `extractAIC`

,
`add1.cca`

, `drop1.cca`

.

### Examples

1 2 3 4 5 6 7 8 |