# Refit Constrained Ordination as a Multiple Response Linear Model

### Description

Functions refit results of constrained ordination (`cca`

,
`rda`

, `capscale`

) as a multiple response
linear model (`lm`

). This allows finding influence
statistics (`influence.measures`

). This also allows
deriving several other statistics, but most of these are biased and
misleading, since refitting ignores a major component of variation in
constrained ordination.

### Usage

1 | ```
as.mlm(x)
``` |

### Arguments

`x` |
Constrained ordination result. |

### Details

Popular algorithm for constrained ordination is based on iteration
with regression where weighted averages of sites are used as dependent
variables and constraints as independent variables.
Statistics of linear regression
are a natural by-product in this algorithm. Constrained ordination in
vegan uses different algorithm, but to obtain linear regression
statistics you can refit an ordination result as a multiple response
linear model (`lm`

). This regression ignores residual
unconstrained variation in the data, and therefore estimates of
standard error are strongly biased and much too low. You can get
statistics like *t*-values of coefficients, but you should not use
these because of this bias. Some useful information you can get with
refitted models are statistics for detecting influential observations
(`influence.measures`

including
`cooks.distance`

, `hatvalues`

).

### Value

Function returns an object of multiple response linear model of class
`"mlm"`

documented with `lm`

.

### Note

You can use these functions to find *t*-values of coefficients
using `summary.mlm`

, but you should not do this because the
method ignores unconstrained residual variation. You also can find
several other statistics for (multiple response) linear models with
similar bias. This bias is not a unique feature in vegan
implementation, but also applies to implementations in other
software.

Some statistics of linear models can be found without using
these functions: `coef.cca`

gives the regression
coefficients, `spenvcor`

the species-environment
correlation, `intersetcor`

the interset correlation,
`vif.cca`

the variance inflation factors.

### Author(s)

Jari Oksanen

### See Also

`cca`

, `rda`

, `capscale`

,
`cca.object`

, `lm`

, `summary.mlm`

,
`influence.measures`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 | ```
data(varespec)
data(varechem)
mod <- cca(varespec ~ Al + P + K, data=varechem)
lmod <- as.mlm(mod)
## Coefficients
lmod
coef(mod)
## Influential observations
influence.measures(lmod)
plot(mod, type = "n")
points(mod, cex = 10*hatvalues(lmod), pch=16, xpd = TRUE)
text(mod, display = "bp", col = "blue")
``` |

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