Theory, Examples & Exercises
Constrained ordination includes multivariate regression analysis, and as in ordinary least squared regression, the effect size is measured by R2, the coefficient of determination. R2 quantifies the variation in species composition explained by a linear model of the environmental variable(s), and can be calculated (if no covariables are included) from the analysis output as the sum of eigenvalues of constrained axes divided by the total variation (sum of all eigenvalues). The value of R2 in constrained ordination suffers from the same drawback as in ordinary regression, namely that it increases with the number of explanatory variables and decreases with the number of samples in the dataset, making the values incomparable between analyses done on different datasets. The solution is to use adjusted R2. The value of explained variation itself is not too informative unless it is compared with the variation the same number of explanatory variables could possibly explain (which is usually far from 100%). Also, even if the explanatory variables are in fact randomly generated, the R2 is non-zero and positive (in contrast to adjusted R2, which may be zero or even negative), and to decide whether the results are interpretable, it is useful to test their significance by Monte Carlo permutation test.
R2 is known to depend on the number of samples in the dataset (sites in our case) and on the number of explanatory variables: with the number of samples R2 decreases, and with the number of predictors (even if these are randomly generated) it increases (Fig. 1). The relationship can be expressed numerically: p random predictors explain (in average) p/(n-1) of the variation (where n is the number of samples in the analysis).
The solution to this problem is to calculate adjusted R2. For linear ordination methods (as well for ordinary least squared multiple regression) the adjusted R2 can be calculated using Ezekiel’s formula:
where n is the number of samples and p is the number of predictors (explanatory variables). Resulting adjusted R2 are independent on the number of samples and predictors (Fig. 1).
In the case of unimodal ordination methods, however, the values returned by Ezekiel’s formula are overestimated (and the dependence of variation on the number of samples and/or explanatory variables is not removed), and the R2 needs to be adjusted using the permutational method proposed by Peres-Neto et al. (2006). The principle of this permutation adjustment is based on using modified Ezekiel's formula to compare observed variation explained by the variables (R2) with expected (mean) variation the same number of variables would explain if they are random (, Fig. 2).
In contrast to R2, the values of adjusted R2 can reach zero, which means that the explanatory variables do not explain any variation in species composition, or they can be even negative, which means that explanatory variables explain even less variation than would be explained (in average) by the same number of randomly generated ones. The negative values are usually ignored and not interpreted (this is important e.g. when interpreting fractions in variation partitioning).
The variation explained by constrained ordination may often seem as too low in the absolute terms. For example, in Example 1 in this section, the variation in species composition of vltava dataset explained by two explanatory variables, soil pH and soil depth, is less than 9%. Are the results of the analysis explaining less than 9% of variation worth to interpret/think of/publish?
To correctly interpret the value of explained variation, you need to consider that in the case of multiple linear regression (which constrained ordination is), two explanatory variables will be unable to explain 100% of the overall variation. In fact, the amount of variation explainable by a given number of explanatory variables in the case of the certain dataset can be exactly calculated, if we assume that the best explanatory variable for that dataset are sample scores on the ordination axes calculated by an unconstrained variant of the ordination on the same dataset. For example (Fig. 3), if you have one explanatory variable, and you want to know how much it could maximally explain in my constrained ordination analysis done on given dataset (soil pH used as explanatory in tb-RDA on log and Hellinger transformed species composition data from vltava dataset, explaining 4.8% of variation), you can do the following: 1) calculate unconstrained variant of given ordination method (here tb-PCA) on the same species composition dataset (with the same transformation of raw data if applicable), and 2) check the variation represented by the first ordination axis (calculated by dividing the eigenvalue of this axis by total variation in the dataset). This value (13.1% in our example) is the maximum variation any one predictor can explain in a given dataset. In our example, pH explained 4.8% of total variation, while it could maximally explain 13.1% - in fact, it explained more than third of what it could explain (4.8/13.1= 36.6%). This is perhaps not too bad.
The same applies for more than one explanatory variable – e.g. if you have two variables, you take the variation represented by the first two axes in the unconstrained variant of the ordination for comparison. In the case of more than one variables, however, the comparison starts to have the problem that ordination axes are by definition not correlated, while explanatory variables are (often) somewhat correlated. It means that the variation represented by n unconstrained ordination axes is the maximum variation which could be explained by n explanatory variables which are not intercorrelated.
Variation explained by explanatory variables in constrained ordination is routinely tested and often only variables (or sets of variables) which are significant are being considered for interpretation. The test is based on comparing observed variation explained by environmental variables with variation explained by (the same number of) randomized variables. The test statistic used is the so-called pseudo-F value, which is calculated from the axis eigenvalue, the residual sum of squares of the constrained model, numbers of explanatory variables, covariables and samples in the dataset. If not covariables are used in the analysis, pseudo-F value can be replaced by the values of raw R2.
Monte Carlo permutation test, in general, is a permutation test which compares the observed value of the test statistic with the expected distribution of the test statistic generated by permuting the original data under the assumption of the null hypothesis. In the case of constrained ordination, the test statistic is pseudo-F (but can be also R2 if no covariables are used); the observed value of the test statistic (Fdata) is from the analysis using the original explanatory variables, and the distribution of the null expectations for the test statistic (Fperm) are values from the analysis using the permuted explanatory variables (permutation removes the link of explanatory variables to species composition data, creating assumptions of the null hypothesis about not relationship between explanatory variables and species composition). The permutation needs to be repeated many times to build the null distribution to which the observed value of the test statistic is compared (Fig. 4). The resulting P-value is calculated as
where nx is the number of permutations where the expected value of the test statistic is higher than observed value (Fperm > Fdata), and N is the number of all permutations. The formula also explains why the numbers of permutations are usually set to end with 9 (e.g. 499 instead of 500): the observed value if the test statistic is considered to be also the part of the null distribution, and is therefore added together with expected values (and the number one is added to the formula). The lowest P-value you can reach depends on the number of all permutations (N), and can be calculated as Pmin = 1/(N+1). This means that for example with N = 199 permutations, the lowest P-value which can be obtained is 1/(199+1) = 0.005, and one cannot, therefore, hope to reject the null hypothesis at P < 0.001.
Several alternative tests are available. One can test the significance of only the first constrained axis or all constrained ordination axes (in the case of only one explanatory variable, a test of the first and all constrained ordination axes is identical). It is also possible to test each constrained ordination axis separately, e.g. to decide how many axes should be displayed in ordination diagram. If each axis is tested separately, consider correcting resulting P-values for the multiple testing issue (e.g. by Holm correction). Unconstrained ordination axes cannot be tested.