# Principal Response Curves for Treatments with Repeated Observations

### Description

Principal Response Curves (PRC) are a special case of
Redundancy Analysis (`rda`

) for multivariate responses in
repeated observation design. They were originally suggested for
ecological communities. They should be easier to interpret than
traditional constrained ordination. They can also be used to study how
the effects of a factor `A`

depend on the levels of a factor
`B`

, that is `A + A:B`

, in a multivariate response
experiment.

### Usage

1 2 3 4 5 6 7 8 | ```
prc(response, treatment, time, ...)
## S3 method for class 'prc'
summary(object, axis = 1, scaling = "symmetric",
digits = 4, correlation = FALSE, ...)
## S3 method for class 'prc'
plot(x, species = TRUE, select, scaling = "symmetric",
axis = 1, correlation = FALSE, type = "l", xlab, ylab, ylim,
lty = 1:5, col = 1:6, pch, legpos, cex = 0.8, ...)
``` |

### Arguments

`response` |
Multivariate response data. Typically these are community (species) data. If the data are counts, they probably should be log transformed prior to the analysis. |

`treatment` |
A factor for treatments. |

`time` |
An unordered factor defining the observations times in the repeated design. |

`object, x` |
An |

`axis` |
Axis shown (only one axis can be selected). |

`scaling` |
Scaling of species scores, identical to the
The type of scores can also be specified as one of |

`digits` |
Number of significant digits displayed. |

`correlation` |
logical; if |

`species` |
Display species scores. |

`select` |
Vector to select displayed species. This can be a vector
of indices or a logical vector which is |

`type` |
Type of plot: |

`xlab, ylab` |
Text to replace default axis labels. |

`ylim` |
Limits for the vertical axis. |

`lty, col, pch` |
Line type, colour and plotting characters (defaults supplied). |

`legpos` |
The position of the |

`cex` |
Character expansion for symbols and species labels. |

`...` |
Other parameters passed to functions. |

### Details

PRC is a special case of `rda`

with a single
factor for `treatment`

and a single factor for `time`

points
in repeated observations. In vegan, the corresponding
`rda`

model is defined as ```
rda(response ~ treatment *
time + Condition(time))
```

. Since the `time`

appears twice in the
model formula, its main effects will be aliased, and only the main
effect of treatment and interaction terms are available, and will be
used in PRC. Instead of usual multivariate ordination diagrams, PRC
uses canonical (regression) coefficients and species scores for a
single axis. All that the current functions do is to provide a special
`summary`

and `plot`

methods that display the
`rda`

results in the PRC fashion. The current version only
works with default contrasts (`contr.treatment`

) in which
the coefficients are contrasts against the first level, and the levels
must be arranged so that the first level is the control (or a
baseline). If necessary, you must change the baseline level with
function `relevel`

.

Function `summary`

prints the species scores and the
coefficients. Function `plot`

plots coefficients against
`time`

using `matplot`

, and has similar defaults.
The graph (and PRC) is meaningful only if the first `treatment`

level is the control, as the results are contrasts to the first level
when unordered factors are used. The plot also displays species scores
on the right vertical axis using function
`linestack`

. Typically the number of species is so high
that not all can be displayed with the default settings, but users can
reduce character size or padding (`air`

) in
`linestack`

, or `select`

only a subset of the
species. A legend will be displayed unless suppressed with
`legpos = NA`

, and the functions tries to guess where to put the
legend if `legpos`

is not supplied.

### Value

The function is a special case of `rda`

and returns its
result object (see `cca.object`

). However, a special
`summary`

and `plot`

methods display returns differently
than in `rda`

.

### Warning

The first level of `treatment`

must be the
control: use function `relevel`

to guarantee the correct
reference level. The current version will ignore user setting of
`contrasts`

and always use treatment contrasts
(`contr.treatment`

). The `time`

must be an unordered
factor.

### Author(s)

Jari Oksanen and Cajo ter Braak

### References

van den Brink, P.J. & ter Braak, C.J.F. (1999). Principal response curves: Analysis of time-dependent multivariate responses of biological community to stress. Environmental Toxicology and Chemistry, 18, 138–148.

### See Also

`rda`

, `anova.cca`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
## Chlorpyrifos experiment and experimental design: Pesticide
## treatment in ditches (replicated) and followed over from 4 weeks
## before to 24 weeks after exposure
data(pyrifos)
week <- gl(11, 12, labels=c(-4, -1, 0.1, 1, 2, 4, 8, 12, 15, 19, 24))
dose <- factor(rep(c(0.1, 0, 0, 0.9, 0, 44, 6, 0.1, 44, 0.9, 0, 6), 11))
ditch <- gl(12, 1, length=132)
# PRC
mod <- prc(pyrifos, dose, week)
mod # RDA
summary(mod) # PRC
logabu <- colSums(pyrifos)
plot(mod, select = logabu > 100)
## Ditches are randomized, we have a time series, and are only
## interested in the first axis
ctrl <- how(plots = Plots(strata = ditch,type = "free"),
within = Within(type = "series"), nperm = 99)
anova(mod, permutations = ctrl, first=TRUE)
``` |

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