The ordinations in the previous section can be described as unconstrained. While individual methods may assume a certain underlying structure within the set of dependent variables (e.g., PCA derives an ordination space from a linear combination of species values and CA maximizes the center of a unimodal distribution of sample values), no structure of the samples itself is assumed. Any inferences about potential causal effects such as multiple environmental covariates or group effects (e.g., site effects, experimental treatments) are made in an ad hoc fashion. Unconstrained ordinations are a type of 'indirect gradient analysis' - any interpretation of potential effects of other factors that generated the patterns can only be made indirectly because those factors were not explicitly included in the analysis.
In contrast, constrained ordination or 'direct gradient analysis' explicitly includes two or more different sets of ecological information into a single analysis and hence directly examines relationships between sets of variables. For example, if the abundance of a set of organisms and a set of environmental variables were recorded in the same sample, the data set could be thought of as containing two discrete sets of variables that could have an ecological relationship. For example, organism distributions are considered a function of different environmental conditions. Common constrained ordinations in ecology are redundancy analysis, canonical correlation, canonical correspondence analysis, canonical discriminant (or canonical variate) analysis, and more recently canonical analysis of principal coordinates. All methods can be lumped under the general term 'canonical analysis' in that they are direct comparisons of more than one data matrix.
The basic idea behind these constrained ordination methods is that an unconstrained ordination of one set of variables (usually the set that represents the 'dependent' variables; e.g., organism abundance) is carried out with an additional proviso - the ordination ofthe 'dependent' variables is further constrained to be a function also of the other set of variables (usually considered 'independent' variables; e.g., environmental variables). Thus constrained ordination is usually considered an extension of unconstrained ordination methods, and their development closely parallels that ofunconstrained methods.
In a data set with both dependent and independent variables, one way of constraining an ordination of dependent variables is by calculating an ordination not of the dependent variables themselves, but their fitted values from a series of multiple regressions of each dependent variable on the set of independent variables. In this way, an ordination space of the dependent variables is constructed, with an additional constraint that the ordination space also be an optimized linear combination of a set of 'independent' variables. The simplest approach to do this is redundancy analysis, which is an extension of PCA. A simple linear multiple regression of each dependent variable on the set of independent variables is calculated and a PCA run on the fitted values. The eigenvectors from the PCA define the ordination space. As with PCA, it is assumed that the responses ofthe dependent variables to the independent variables are essentially linear along any gradients. Because the regressions are calculated independently for each dependent variable, it also follows that redundancy analysis maximizes the predictability of the dependent variables, given the set of independent variables, and so is analogous to a multivariate multiple regression. Typical data presentation from a redundancy analysis would be a biplot either of the original dependent variables or the fitted values projected onto the ordination (the sample scores), and the eigenvectors. An additional set of information can also be added: the correlations ofeach independent variable with the sample scores, to indicate how the independent variables are associated with the ordination space. Another feature of interest is the redundancy statistics, which indicate how much variation in the dependent variables is explained by the independent data set.
In many ecological settings, there are no clear 'dependent' and 'independent' variables, or researchers are not prepared to assume a directional, causal relationship. In univariate or bivariate analyses, these two different situations are characterized as regression and correlation problems, respectively. Redundancy analysis is clearly analogous to regression, maximizing the ability of independent variables to predict values of dependent variables. The constrained ordination equivalent to correlation is canonical correlation analysis, in which there is a symmetric relationship between different sets of variables. Canonical correlation is similar to redundancy analysis, except that instead of maximizing the ability of one set of variables to predict another, the joint correlation between the two sets is maximized. Instead of calculating a PCA on the predicted values, as in redundancy analysis, canonical correlation conceptually calculates a PCA on the 'dependent' variables, conditional on the axes being maximally correlated with the axes of a PCA on the 'independent' variables. This relationship is symmetric because either set ofvariables could be treated as 'dependent' or 'independent' without changing the result, unlike a regression approach.
While canonical ordinations, as with unconstrained ordinations, can be used to generate both a reduced space plot and a plot of the variables in the reduced space, in general the reduced space plot of the samples is of less interest than the relationship between the different sets of variables. A canonical correlation of the triplefin data set with broad-scale habitat types indicates that there is a characteristic suite ofspecies associated with Ecklonia radiata macroalgal forests - F. malcolmi, N. segmentatus, F. flavonigrum, and F. varium. Forsterygion lapillum is characteristic of rock flats. Ruanoho whero is associated with E. radiata forest, and not generally found on the rock flats (Figure 3). These patterns of course are also influenced by the between-site availability of the different habitat types. Not all habitat types are found at all sites. This provides a strong caution, generalizable across any regression problem,
Figure 3 Canonical correlation of triplefin abundance with broad-scale habitat classification. The reduced space plot of samples is generally not of much interest as the samples are usually considered as tools to measure correlations between different sets of variables. Consequently, the vectors of the correlation coefficients between the variables and the position of the samples in ordination space are usually presented (circle represents the 0.2 correlation coefficient between the variables and the axes). The variation-explained reported on the axes is the proportions of the total eigenvalues. The standardized variation values explained in each data set are less optimistic - the redundancy statistics for the triplefin abundances are 19.43% explained in two habitat axes, whereas 42.1% of the standardized variation in habitat values are explained by their ordination axes. These low values are not uncommon in canonical analyses of ecological data sets. There is a characteristic suite of species associated with Ecklonia radiata macroalgal forests-F. malcolmi, N. segmentatus, F. flavonigrum, and F. varium. Forsterygion lapillum is characteristic of rock flats. Ruanoho whero is associated with E. radiata forest, and not generally found on the rock flats. Canonical analysis plots can be difficult to interpret because they attempt to represent multidimensional patterns from two (or more) different data sets onto a single graph. The results from other commonly used canonical ordinations, redundancy analysis and canonical CA, were qualitatively similar to canonical correlation. While this is not always guaranteed to be the case (particularly if data are sampled across a large ecological or spatial range), in many ecological data sets these different analyses will yield similar results. Any differences in qualitative answers indicate that the underlying response model (linear, unimodal) is inappropriate. In this particular example, most species occupy a wide range of broad-scale habitat types, so there is no clear signal in the data. Most of the strong patterns occur at the site level, as indicated in the PCA, CA, and MDS analyses.
about comparing factors that are potentially confounded. In canonical analyses, the samples themselves are usually viewed as tools to measure the relationship between variables, so their patterns are generally not of interest.
The constraining variables in a canonical analysis need not be continuous. A common ordination, canonical discriminant analysis (CDA, also known as canonical variate analysis), uses categorical variables as constraining variables. Typically CDA is used as an ordination to interpret the results of multivariate analysis of variance (MANOVA). It can be thought of as a canonical correlation on a set of dummy binary variables corresponding to the level of the classification variable (e.g., site, experimental treatment). Within a MANOVA framework, it is usually thought of as a PCA-type ordination on a rescaled and partitioned variance-covariance matrix, HE-1; where H is the variance-covariance matrix of the hypothesis matrix, and E the variance-covariance matrix of the error matrix. Conceptually, this is the equivalent of a PCA on the multivariate equivalent of an f-statistic in ANOVA. If samples are explicitly grouped (e.g., replicated transects from different sites), then CDA will usually provide a better ordination of differences between sites than PCA or CA (Figure 4).
N. caerulepunctu N. yaldwyni K ste
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Canonical discriminant 1 (82.6%)
Canonical discriminant 1 (82.6%)
Figure 4 Canonical discriminant analysis of triplefin abundance at sites of varying exposure (circles, sheltered mainland; diamonds, semiexposed mainland; squares, exposed mainland; inverted triangles, exposed offshore). Data were transformed by yij = y,j/(yi+^y+j) to preserve the x2distanceand enable comparison with the unconstrained ordinations in Figure 2. Species vectors are the structure coefficients and have been multiplied by 8 for clarity. The length of the vector indicates the strength of the correlation; the circle represents a correlation of 111. As with the unconstrained ordinations, sheltered sites are distinct from all others. Exposed offshore sites are distinct from mainland sites, with a smaller difference between exposed and semiexposed mainland sites. These patterns are driven by the abundance of F. lapillum and R. whero at sheltered sites, the absence of F. varium on exposed offshore sites and dominance of N. segmentatus, and increased abundance of N. caerulepunctus, N. yaldwyni, K. stewarti, and F. flavonigrum at exposed offshore sites.
There are a few different measures of the contribution of variables to the pattern. In canonical correlation, for example, two different ordination spaces are generated - that defined by the left set of variables (e.g., species) and that defined by the right set (e.g., habitat). Each of these spaces is a linear product of either the species (constrained by habitat) or habitat (constrained by species). Hence there are two potential multivariate spaces that samples could be projected into, each defined either by the species or the habitat variables. The coefficients of the species/habitat variables that define the ordination space are analogous to multiple regression coefficients and termed 'canonical coefficients'. They represent the contribution of each variable to defining the ordination space, after correcting for other variables in the model. An important consequence of this is that their values will change with the exclusion or inclusion of other dependent variables in the analysis. This instability of the importance of variables in the analysis is also found in multiple regression. An alternative set of coefficients that describe the importance of a variable in the analysis are the 'structure coefficients'. These are the simple correlations between a variable's value and the position of the sample in ordination space and these are the coefficients plotted in Figure 3. The main distinction between the two types of coefficients is that if a structure coefficient lies in the same direction on an ordination axis as a sample value, then that sample will have a higher value of that variable. Structure coefficients will generally have a more ecologically intuitive interpretation. In contrast, canonical coefficients are sensitive to other variables in the model. It is possible for a variable to have a canonical coefficient on a variable that is strongly correlated with the pattern in the samples that is either weak or even negatively associated with the pattern. This may be due to its covariance with other variables in the analysis. Further complexity is introduced into analytical interpretation by the dual space calculated in the ordination. The left set of variables (e.g., species) has a correlation with its own constrained ordination (the ordination defined by the species, constrained by habitat) and a correlation with the ordination of the right set of variables (e.g., habitat - the ordination defined by habitat, constrained by species). Canonical ordinations should be applied with extreme care to interpret the coefficients that are relevant to the question at hand.
Redundancy analysis, being an extension of PCA, is a method that is best used to describe linear patterns of organisms along gradients due to its preservation of Euclidean distance. As with PCA, its use has been questioned when dependent variables occur unimodally along environmental gradients. One modification to redundancy analysis is canonical CA, which replaces a
PCA of the multiple regression fitted values with a CA of the weighted multiple regression values. This ordination (as with CA) preserves the x distance and hence can represent unimodal patterns in ordination space. Canonical CA is simply a weighted redundancy analysis on x2-transformed data. Consequently, the ordination presentation is similar to that of redundancy analysis, except the projection of the dependent variables into the ordination has a somewhat different interpretation. Instead of the dependent variables being presented as vectors, indicating that the value of the variable continues to increase linearly in that direction, the projection of dependent variables should be interpreted as the center of the mode of the dependent variables distribution. This parallels the distinction between PCA biplots and CA joint plots.
The similarities between each canonical method results from their common mathematical approach. If the data are in sample*variable form, then canonical analyses, in simplified form, partition a covariance (equal to a correlation, if data are standardized first - common in constrained ordination) matrix:
where Syy is a submatrix representing the covariances between the dependent variables, SYX and SXY are submatrices representing the covariances between the dependent-independent and independent-dependent variables, respectively, and SXX is a submatrix representing the covariance between the independent variables. In redundancy analysis, SYY is defined as the identity matrix so the predictive ability of independent variables is maximized. In canonical correlation, SYY is defined as the covariance matrix between the dependent variables, so that the joint covariance between the dependent and independent variables is maximized in the ordination. In canonical CA, SYY is defined as a matrix with the species abundance on the diagonal and 0 on off-diagonals (note relationship to redundancy analysis, which has 1's on the diagonal), and SXX is the covariance of the independent variables weighted by the sample size (which enables the weighted-multiple regression). Canonical discriminant analysis is simply a canonical correlation in which the SXX submatrix is calculated from a set of dummy binary variables corresponding to the grouping variables. It is clear that these different constrained ordination approaches are not so much different analyses, but rather the same analysis with different standardizations, weightings, and rescalings. It should be clear also, that there is considerable flexibility in the types of independent variables that can be analyzed. Although all except canonical discriminant analysis have continuous independent variables, it is clear that combinations of continuous and categorical variables can be incorporated as dependent variables. In this way, the multivariate equivalent of analysis of covar-iance can be presented in a constrained ordination space, using either the redundancy analysis, canonical correlation, or canonical CA approaches. The effect of different data weightings will have an effect on what data structures the analyses can detect, and also subtly alter the interpretation of biplots. These decisions should be used to guide the selection of an analysis for any data set.
As with unconstrained ordination, constrained ordinations can also be used to partial the effect of different sets of variables to enable a 'corrected' ordination to be calculated, similar in principle to a covariate approach. For example, if fish abundance and habitat variables were measured at different spatial locations, it might be of interest to partition independent fractions of the relationship between fish and habitat after correcting for common spatial pattern, fish-space association after correcting for habitat differences, and the relationship between fish and the interaction of space and habitat. This can be achieved by regressing both fish and habitat variables on space, then ordinating their residuals to generate ordinations of 'space-independent' habitat association. The difference between the sum of the 'independent' variance fractions and the variance fraction of all variables included indicates whether there is an interaction effect. The significance of these fractions are usually tested by permutation methods.
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