Canonical Correspondence Analysis (CCA)
And Other techniques
Vamsi
Sundus
Shawnalee
What is CCA?
 “Commonly used by researchers trying to understand the
relationship between community composition and
environmental factors.”
 Or, more generally, comparing/testing one multivariate dataset
against a second one.
 Like DECORANA (the last presentation), it’s based off of
correspondence analysis (ordination technique).
CCA Purpose?
 To incorporate environmental data into the ordination so that
a better final ordination diagram can be created.
What’s needed (Part I)
Dependent matrix – contains data to be ordinated, usually
composed of population estimates for a bunch of species)
2. Environmental matrix – describes environmental
conditions. Must contain the same number of rows
(observations) as the species data, but must have fewer
columns than the number of observations.
1.
Problems
 Just like correspondence analysis, an arching effect may be
found resulting in the second ordination axis “being a
distortion of the first.”
 We eliminated this previously using a detrended technique.
DCCA
 In the same manner, CCA has detrended canonical
correspondence analysis (DCCA) that uses essentially the
same algorithm to terminate the second ordination axis and
eliminate the arch effect.
Complicated
 “Canonical correspondance analysis can be considered to be a
form of direct ordination, although it is so much more
complicated than conventional examples of direct
ordination…being a hybrid of direct and indirect ordination.”
What’s needed (Part II)
 Data must be collected from the same place at the same time.
 Autoregressive error?
 If not collected together  error of pseudoreplication.
Pseudoreplication (Reteaching)
 I forgot.
 Let’s say we want to observe the effects of a drug on estrus
(monthly period cycle).
 Let n=100. n1 = 50, n2 = 50, n = n1 + n2
 Trt A, Trt B
 Have all mice in same room.
Problems with this design
 Inherent in this design are problems:
 Chemical cues for setting cycle.
 One mice influences the next.
 Like in colleges.
 Pseudoreplication, apparently independent, but not really,
data.
Back to CCA
 End divergence.
Canonical
 Definition:
 Whenever used in this field (multivariate analysis), means
something is being optimized against some other constraint.
The Steps
 The only major difference between (regular) correspondance
analysis and canonical is the addition of two steps.
Step 1 - CA
 Start with a random weighting. It’s pretty kosher to start
from 0.0  100.0 in whatever increments are needed.
 In our case, we’ll do (0,50,100) for (A, B, C)
 Use this formula for nth species rank:
n 1
100 
| S  species
S 1
Step 2 - CA
 Use the starter weights (which are arbitrary essentially) and
compute a weighting for each of the years
Year Counts Counts Counts
1
100
0 0
2
90
10 0
3
80
20 5
4
60
35 10
5
50
50 20
6
40
60 30
7
20
30 40
8
5
20 60
9
0
10 75
10
0
0 90
Y1
--> 0.0
--> 5.0
--> 14.3
--> 26.2
--> 37.5
--> 46.2
--> 61.1
--> 82.4
--> 94.1
--> 100.0
0 100  50  0  100  0
 0.0 | Year1
100  0  0
Step 3
 We can now calculate a new weighting for each species using
these new year weightings.
0 100  5  90  14.3  80  ...  0  94.9  0 100
 19.1
100  90  ...  20  5
 Calculate similarly for B, C
Old weightings for
species
S10
S1a
0
19.1
50
43.9
100
78.5
New calculated
weightings for
species
Step 4
 These new weightings for each species though aren’t that
useful, so we need to rescale them back to 0  100, instead
of currently 19.1  78.5.
 So, to do this, simply use a logical rescaling method.
S1a
19.1
43.9
78.5
100  ( S1a  MIN )
S1b 
MAX  MIN
Step 4 cont.
 So, after computing the rescaled values, we find the
following:
S10
0
50
100
S1a 19.1
43.9
78.5
S1b 0.00 41.75 100.00
Step 5
 This is now one cycle of the CA completed.
 “Weightings for each year are recalculated using the new,
rescaled weightings for the species.”
 Eventually a stable patter will emerge.
 10-20 iterations.
CA vs. CCA
 Start with arbitrary but
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unequal site scores
Calculate species scores as
weighted average of site scores
Calculate new site scores as
weighted average of species
scores.
Standardize
Stop if acceptable; otherwise
iterate from step 2
Perform multiple regression of site
scores on environmental variables
Use multiple regression to derive
new predicted values.
Other Techniques
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There are many other techniques that are available for multivariate analysis.
COR
CVA
FA
MDS
MRPP
MANCOVA
MANOVA
NMS
NMDS
Procustes Rotation
RDA
PRC
COR
 Canonical Correlation Analysis
 Similar to CCA.
 Continuation of the progression from bivariate to multiple
linear regression.
 Bivariate = 1 independent to explain 1 dependent
 Multivariate = n independent to explain 1 dependent
 Canonical = n independent to explain m dependent
COR (cont.)
 Major difference in limitations:
 (Number of species + environmental variables) < number of
sites. //COR
 Weaker requirement for CCA
 (Number of environmental variables alone < number of
observations. //CCA
 Both result in similar outputs. CCA is preferred. (easier
limitations to meet on allowable number of variables).
CVA
 Canonical Variates Analysis
 Purpose: generate a score for each inidvidual, which, using a
1 way anova by category would return the highest possible F
value
 Maximize variance within dataset  hence canonical.
 Limitations: multivariate normality, categories need to be
known a priori.
FA
 Factor Analysis is used as a synonym for PCA (Principal
component analysis) in the US
 How it began:
 School students – scores in Classics, French, English, Math,
Discrimination of Pitch, and Music
 Abilities in each due to smaller number of fundamental skills
(factors).
 Derive absolute parameter estimates.
FA (cont.)
j p
X j   ( F1   j1 ...  Fm   jm )   j
j 1
Fn = value of nth factor
Lamdajn= loading variable j on factor n
ej = residual for variable j
P = number of variables
M = number of factors
FA (cont)
 FA becomes an eigenvector problem hence Similar to PCA
(eigenanalysis of correlation matrix).
 “…the results are…difficult to interpret and based on
assumptions that are probably invalid.”
 “FA is not worth the time necessary to understand and
perform it.” (Hills 1977)
MDS
 Multidimensional Scaling
 Takes square matrix of distances between individuals and
recreates maps
 Discussed previously
MRPP
 Multiresponse Permutation Procedure
 Assesses the probability that two or more groups consisting
of multivariate data differ
 Different from normal mulivariate ANOVA in that it’s nonparametric  can be used on biological data without
worrying about multivariate normality
MANCOVA
 Multivariate Analysis of Covariance
 Multivariate equivlent of ANOVA
 Assumption of normality
 Lacks non-parametric test though
MANOVA
 Multivariate ANOVA
 Analagous to univariate ANOVA  provides estimate of the
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probability that the observed patter arises from random data.
Each mean is treated as a coordinate in multivariate space.
Used specifically in assessing whether “an overall response has
occurred, but will not identify which variables contributed to
treatments if significance is found.”
Requires normality, or else.
Or else use MRPP
NMS, NMDS
 Non-metric multidimensional scaling
 Ordinal scaling
 Square distance matrix  map reconstructed
 Differs from other multivariate techniques
NMS, NMDS (cont)
 Differs from other multivariate techniques
 Uses only one distance measure derived from ranked
differences between individuals.
 So, can be used with non-normal, discontinuous or questionable
distributions.
 Ordinations axes will differ according to how many axes are
requested.
 Where two or more ordination axes are requested, the first axis
need not be more important than the second or higher axes. 
axis numbering is arbitrary.
 A lot of subjectivity in the technique in choice of axis, hence
not used that often.
Procrustes Rotation
 Compares two different ordinations applied to the same data.
 Has m2 statistic (residual sum of squares) to assess after
Procrustes operations have been applied.
 No significance test
 No clear guildelines to interpret m2 values
Procrustes Rotation
 Named is derived from Greek mythology.
 Inn keeper who ensured al his customers fittyed perfectly to
his bed by stretching them or chopping their feet off.
RDA
 Redundancy Analysis
 Derivative or PCA with bonus feature
 Values entered into analysis aren’t original data but the best-fit
values estimated from a multiple linear regression between each
variable and second matrix of environmental data.
 Thus, this is a canonical version of PCA
 Constrained to optimally correlate with another dataset.
 Interpretation is by biplot
 Collinearity, which is likely in biological data, makes
canonical coefficients unreliable.
 RDA = technique that underlies PRC
PRC
 Principal response curves
 1999, New technique
 Derived from RDA and specfically intended to help interpret
planned experiements on biological communities.
 Two treatments, one is a control
 Reapeated sampling
 <not enough details>
END