whose outcomes are monitored along time and compared with control units. Such
experimental results can be analysed with standard RDA, but the resulting plots and
numerical results are complex and the relevant patterns difficult to identify. In
particular, the differences between control and treatment units at each time step do
not stand out; they should, because they are the most important features of such
experiments.
Principal response curves address the problems related to the analysis of multivariate results of designed experiments that involve repeated measurements over
time by means of a modified form of RDA. The PRC method focuses on the
differences between control and treatments; it provides a clear graphical illustration
of treatment effects at the community as well as the species level.
Let D be the total number of treatment levels and T the total number of time
points. With a balanced design with R replicates per level, a standard RDA would
involve the response data with n ¼ D Â T Â R observations and m species; the
explanatory variables would consist in an interaction factor of D Â T levels indicating to which combination of treatment level and time point each observation belongs
(Van den Brink and ter Braak 1999) plus, if needed, the main effects D and T. The
PRC method, in contrast, specifically focuses on the difference between control and
treatment level at each time point. To achieve that, one must remove the overall
effect of time, i.e., use the time factor as a covariable. This makes sure that any
overall time effect is removed. The explanatory factor, on the other hand, is the one
with D Â T levels as above, but with the levels corresponding to the control removed
“so as to ensure that the treatment effects are expressed as deviations from the
control” (Van den Brink and ter Braak 1999). The canonical coefficients resulting
from the RDA are plotted against time; curves representing these scores for each
treatment along time are called the principal response curves of the community. The
species weights can be assessed by means of their regression coefficients against the
site scores. They represent “the multiple by which the principal curves must be
multiplied to obtain the fitted response curves of [each] species” (Van den Brink and
ter Braak 1999). A high positive weight indicates a high likelihood that the species
follows the overall pattern of the PRC. A negative weight shows a tendency of the
species to follow an opposite pattern. A weight close to 0 indicates either no pattern
or a pattern of a shape differing from the overall one. Note that, in this case, a small
weight would not indicate a lack of response of the species to the treatment, but a
response pattern that may be strong but different to the overall (communitylevel) one.
Principal response curves can be computed with the function prc() of package
vegan. The example available in the documentation file of the function is the one
used by Van den Brink and ter Braak (1999) in their paper and consists in the study
of the effects of insecticide treatments on aquatic invertebrate communities. We will
present it here. It is based on observations on the abundances of 178 invertebrate
species (macroinvertebrates and zooplankton) subjected to insecticide treatments in
aquatic mesocosms (“ditches”). The species data are log-transformed abundances,
y tr ¼ ln(10y + 1). The experiment involved twelve mesocosms, which were surveyed
on eleven occasions, so n ¼ 12 Â 11 ¼ 132. Four mesocosms served as controls
6.6 Other Asymmetric Analyses
269
experimental results can be analysed with standard RDA, but the resulting plots and
numerical results are complex and the relevant patterns difficult to identify. In
particular, the differences between control and treatment units at each time step do
not stand out; they should, because they are the most important features of such
experiments.
Principal response curves address the problems related to the analysis of multivariate results of designed experiments that involve repeated measurements over
time by means of a modified form of RDA. The PRC method focuses on the
differences between control and treatments; it provides a clear graphical illustration
of treatment effects at the community as well as the species level.
Let D be the total number of treatment levels and T the total number of time
points. With a balanced design with R replicates per level, a standard RDA would
involve the response data with n ¼ D Â T Â R observations and m species; the
explanatory variables would consist in an interaction factor of D Â T levels indicating to which combination of treatment level and time point each observation belongs
(Van den Brink and ter Braak 1999) plus, if needed, the main effects D and T. The
PRC method, in contrast, specifically focuses on the difference between control and
treatment level at each time point. To achieve that, one must remove the overall
effect of time, i.e., use the time factor as a covariable. This makes sure that any
overall time effect is removed. The explanatory factor, on the other hand, is the one
with D Â T levels as above, but with the levels corresponding to the control removed
“so as to ensure that the treatment effects are expressed as deviations from the
control” (Van den Brink and ter Braak 1999). The canonical coefficients resulting
from the RDA are plotted against time; curves representing these scores for each
treatment along time are called the principal response curves of the community. The
species weights can be assessed by means of their regression coefficients against the
site scores. They represent “the multiple by which the principal curves must be
multiplied to obtain the fitted response curves of [each] species” (Van den Brink and
ter Braak 1999). A high positive weight indicates a high likelihood that the species
follows the overall pattern of the PRC. A negative weight shows a tendency of the
species to follow an opposite pattern. A weight close to 0 indicates either no pattern
or a pattern of a shape differing from the overall one. Note that, in this case, a small
weight would not indicate a lack of response of the species to the treatment, but a
response pattern that may be strong but different to the overall (communitylevel) one.
Principal response curves can be computed with the function prc() of package
vegan. The example available in the documentation file of the function is the one
used by Van den Brink and ter Braak (1999) in their paper and consists in the study
of the effects of insecticide treatments on aquatic invertebrate communities. We will
present it here. It is based on observations on the abundances of 178 invertebrate
species (macroinvertebrates and zooplankton) subjected to insecticide treatments in
aquatic mesocosms (“ditches”). The species data are log-transformed abundances,
y tr ¼ ln(10y + 1). The experiment involved twelve mesocosms, which were surveyed
on eleven occasions, so n ¼ 12 Â 11 ¼ 132. Four mesocosms served as controls
6.6 Other Asymmetric Analyses
269