Principal Response Curves for Treatments with Repeated Observations

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

prc(response, treatment, time, ...)
# S3 method for class 'prc'
summary(object, axis = 1, scaling = "sites", const,
        digits = 4, correlation = FALSE, ...)
# S3 method for class 'prc'
plot(x, species = TRUE, select, scaling = "symmetric",
     axis = 1, correlation = FALSE, const, 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 prc result object.

axis

Axis shown (only one axis can be selected).

scaling

Scaling of species scores, identical to the scaling in scores.rda.

The type of scores can also be specified as one of "none", "sites", "species", or "symmetric", which correspond to the values 0, 1, 2, and 3 respectively. Argument correlation can be used in combination with these character descriptions to get the corresponding negative value.

const

General scaling constant for species scores (see scores.rda for details). Lower values will reduce the range of species scores, but will not influence the regression coefficients.

digits

Number of significant digits displayed.

correlation

logical; if scaling is a character description of the scaling type, correlation can be used to select correlation-like scores for PCA. See argument scaling for details.

species

Display species scores.

select

Vector to select displayed species. This can be a vector of indices or a logical vector which is TRUE for the selected species

type

Type of plot: "l" for lines, "p" for points or "b" for both.

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 legend. A guess is made if this is not supplied, and NA will suppress legend.

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.

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.

Author

Jari Oksanen and Cajo ter Braak

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.

Examples

## 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)

## IGNORE_RDIFF_BEGIN
## PRC
mod <- prc(pyrifos, dose, week)
mod            # RDA
#> Call: prc(response = pyrifos, treatment = dose, time = week)
#> 
#> -- Model Summary --
#> 
#>                Inertia Proportion Rank
#> Total         288.9920     1.0000     
#> Conditional    63.3493     0.2192   10
#> Constrained    96.6837     0.3346   44
#> Unconstrained 128.9589     0.4462   77
#> 
#> Inertia is variance
#> 
#> -- Eigenvalues --
#> 
#> Eigenvalues for constrained axes:
#>   RDA1   RDA2   RDA3   RDA4   RDA5   RDA6   RDA7   RDA8   RDA9  RDA10  RDA11 
#> 25.282  8.297  6.044  4.766  4.148  3.857  3.587  3.334  3.087  2.551  2.466 
#>  RDA12  RDA13  RDA14  RDA15  RDA16  RDA17  RDA18  RDA19  RDA20  RDA21  RDA22 
#>  2.209  2.129  1.941  1.799  1.622  1.579  1.440  1.398  1.284  1.211  1.133 
#>  RDA23  RDA24  RDA25  RDA26  RDA27  RDA28  RDA29  RDA30  RDA31  RDA32  RDA33 
#>  1.001  0.923  0.862  0.788  0.750  0.712  0.685  0.611  0.584  0.537  0.516 
#>  RDA34  RDA35  RDA36  RDA37  RDA38  RDA39  RDA40  RDA41  RDA42  RDA43  RDA44 
#>  0.442  0.417  0.404  0.368  0.340  0.339  0.306  0.279  0.271  0.205  0.179 
#> 
#> Eigenvalues for unconstrained axes:
#>    PC1    PC2    PC3    PC4    PC5    PC6    PC7    PC8 
#> 17.156  9.189  7.585  6.064  5.730  4.843  4.518  4.105 
#> (Showing 8 of 77 unconstrained eigenvalues)
#> 
summary(mod)   # PRC
#> 
#> Call:
#> prc(response = pyrifos, treatment = dose, time = week) 
#> Species scores:
#>     Simve     Daplo     Cerpu     Alogu     Aloco     Alore     Aloaf     Copsp 
#>  2.688099  1.464566  0.542739  0.280040  0.177019  0.315038  0.426524  1.169368 
#>     Ostsp     Slyla     Acrha     Aloex     Chysp     Alona     Plead     Oxyte 
#>  2.312186 -0.556899  0.105535  0.228092  0.095042  0.063689  0.138397  0.025401 
#>     Grate     Copdi     NauLa     CilHa     Strvi     amosp     Ascmo     Synsp 
#>  0.096840  1.428854  4.847070  0.895241  3.069709 -1.357663  0.069736 -0.026494 
#>     Squro     Squmu     Polar     Kerqu     Anufi     Mytve     Mytvi     Mytmu 
#>  0.264390 -0.452667  0.461989  0.495348  0.432767  0.074372  0.090928  0.105891 
#>     Lepsp     Leppa     Colob     Colbi     Colun     Lecsp     Lecqu     Lecco 
#>  0.998286  0.084809 -0.723051 -0.139569 -0.828338  0.472866 -0.088860 -0.290531 
#>     Leclu     Lecfl     Tripo     Cepsp     Monlo     Monae     Scalo     Trilo 
#>  0.049788 -0.408035  0.215234 -0.809597 -0.527913 -0.089948 -0.077192 -0.039086 
#>   Tripo.1     Tricy     Trisp     Tepat     Rotne     Notla     Filsp     Lopox 
#>  0.246435  0.335400  0.078423  0.007719  0.143730 -0.114301 -0.168356 -0.030990 
#>  hydrspec  bothrosp  olchaeta  erpoocto  glsicomp  alglhete  hebdstag   sphidae 
#> -0.048698  0.398665 -1.165154 -0.901030 -0.144389 -0.073049  0.902223  1.463655 
#>  ansuvote  armicris  bathcont  binitent  gyraalbu  hippcomp  lymnstag  lymnaes7 
#>  0.140685  1.680010  0.073282 -1.950500  0.033051  0.404473 -0.263679  0.135150 
#>  physfont  plbacorn  popyanti  radiovat  radipere  valvcris  valvpisc  hycarina 
#> -0.026383  0.084761  1.272223 -0.019815 -0.625468  0.010579 -0.267577  1.044034 
#>  gammpule  aselaqua  proameri  collembo  caenhora  caenluct  caenrobu  cloedipt 
#>  1.526450  1.578743  0.116577  0.029906  5.767844  2.376188  0.126181  4.734035 
#>  cloesimi  aeshniae  libellae  conagrae  corident  coripanz  coripunc  cymabons 
#>  1.242207  0.212699 -0.081867  1.630574  0.013761  0.120439 -0.176746  0.046360 
#>  hesplinn  hespsahl  notoglau  notomacu  notoobli  notoviri  pacoconc  pleaminu 
#> -0.069465  0.033240  0.555201  0.050060  0.082356  0.215863  0.016709  0.071168 
#>  sigadist  sigafall  sigastri  sigarasp  colyfusc  donacis6  gyrimari  haliconf 
#>  0.076479 -0.018364  0.060206 -0.277312  0.036493 -0.078608  0.010579  0.446911 
#>  haliflav  haligruf  haliobli  herubrev  hya_herm  hyglpusi  hyhyovat  hypoplan 
#>  0.044538  0.450146  0.131472 -0.128486 -0.322379 -0.011775  0.024196  0.014976 
#>  hyporusp  hytuinae  hytuvers  laphminu  noteclav  rhantusp  sialluta  ablalong 
#>  0.232042  2.316179  1.772351  0.632897  0.007912  0.067618  1.109341  0.014976 
#>  ablaphmo  cltanerv  malopisp  mopetenu  prdiussp  pstavari  chironsp  crchirsp 
#>  2.992695  0.075631  0.047658  0.008716  0.554911  0.082842  1.889916  0.016709 
#>  crclglat  ditendsp  mitegchl  pachgarc  pachgvit  popegnub  popedisp  acriluce 
#>  0.028953  0.083481  0.230629 -0.012187 -0.029907  0.224272 -0.069649 -0.007950 
#>  chclpige  conescut  cricotsp  liesspec  psclbarb  psclgsli  psclobvi  psclplat 
#> -0.008744  0.821036  0.121530  0.107387 -0.028639  0.601568 -0.362378 -0.052054 
#>  psclpsil  pscladsp  cladotsp  laa_spec  patanysp  tatarssp  zaa_spec  anopmacu 
#>  0.007339  0.005674  0.539894  0.034105  0.146807  0.669430  0.049949 -0.163731 
#>  cepogoae  chaoobsc  cucidae4  tabanusp  agdasphr  athrater  cyrncren  holodubi 
#>  2.555403  2.442310  0.033240 -0.011601  0.271815  0.067618  0.071168  0.094754 
#>  holopici  leceriae  lilurhom  monaangu  mystazur  mystloni  oecefurv  oecelacu 
#>  0.611618  0.298633  0.009063  0.644295  0.033240  2.998460  0.536628  0.259064 
#>  triabico  paponysp 
#>  0.088915  0.097788 
#> 
#> Coefficients for dose + week:dose interaction
#> which are contrasts to dose 0 
#> rows are dose, columns are week
#>           -4      -1     0.1        1       2       4        8      12      15
#> 0.1 -0.07218 -0.1375 -0.1020 -0.04068 -0.2101 -0.1364 -0.08077 -0.1536 -0.1123
#> 0.9 -0.08106 -0.1935 -0.1936 -0.47699 -0.4977 -0.4306 -0.13532 -0.3548 -0.2408
#> 6   -0.16616 -0.1232 -0.4539 -1.15638 -1.0835 -1.1511 -0.56112 -0.4698 -0.3078
#> 44  -0.13979 -0.1958 -0.7308 -1.26088 -1.2978 -1.4627 -1.29139 -1.0081 -0.7819
#>          19       24
#> 0.1 -0.2163 -0.07835
#> 0.9 -0.1756 -0.15442
#> 6   -0.3293 -0.18227
#> 44  -0.5768 -0.31022
logabu <- colSums(pyrifos)
plot(mod, select = logabu > 100)

## IGNORE_RDIFF_END
## 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)
#> Permutation test for rda under reduced model
#> Plots: ditch, plot permutation: free
#> Permutation: series
#> Number of permutations: 99
#> 
#> Model: prc(response = pyrifos, treatment = dose, time = week)
#>          Df Variance      F Pr(>F)   
#> RDA1      1   25.282 15.096   0.01 **
#> Residual 77  128.959                 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1