# as.mlm: Refit Constrained Ordination as a Multiple Response Linear... In vegan: Community Ecology Package

## 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

`cca`, `rda`, `capscale`, `cca.object`, `lm`, `summary.mlm`, `influence.measures`.
 ``` 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") ```