mibiscreen ordination

General

ordination provides tools for multivariate statistics to calculate and visualize the interactions between any kind of data measured in the field, including contaminants, environmental factors, metabolite concentration and microbiota counts.

The general goal of ordination methods is to reduce the dimensionality of the data by arranging it along novel axes. Typically, two axis are used that represent the main gradients of the data. Then, the variables are evaluated by their correlation with these new axes. For each type of data correlation a different ordination method is defined.

Ordination methods can be subdivided in two types, unconstrained and constrained. Unconstrained ordination methods do not use any prior information about the data and treat each type of variable similarly. ordination provides the unconstrained method Principal Component Analysis (PCA). Constrained ordination uses prior knowledge of the data, differentiating between explanatory (or independent) variables and response (or dependent) variables. ordination provides two constrained ordination methods: Redundancy Analysis (RDA) and Canonical Correspondence Analysis (CCA). In the context of bioremediation, the explanatory variables are typically the environmental variables. The response variables are typically the species variables, e.g. microbiotic species or proxies for microbiotic species.

Ordination methods produce scores, called loadings, for the variables and scores for the measurement locations (referred to as sites). For constrained ordination methods, there are separate loadings for the dependent and the independent variables. In unconstrained ordination, there is no such separation in the loadings.

Principle of PCA

PCA determines the ordination axes by maximizing the amount of variance explained by each axis. In other words, it minimizes the total amount of residual variation per axis, by minimizing the amount of variation not explained by the particular axis. This results in a number of new axes equaling the number of variables.

The first two axes can then be used for plotting and represent the data in two uncorrelated directions that explain most of the variation in the data. The dissimilarity in the data is measured as Euclidean distance.

Principle of constrained ordination

Constrained ordination maximizes correlation between the independent and dependent variables. The implemented methods RDA and CCA are canonical ordination techniques, made to detect patterns in the dependent variables by the independent variables.

RDA bases its axes on the same principles as PCA, by maximizing the total variance for each axis. Like PCA, it is used when the assumed relationship between the independent and dependent variables is linear.

CCA bases its axes on a different principle. CCA determines the axes that maximizes the amount of dispersion/independence among variables, measured as chi-squared distance. It is used when the assumed relationship between the data is unimodal, i.e. the data having a probability distribution with a single peak.

Ordination plots

The two plot axis represent the two main axis identified by the ordination methods. The first ordination axis is oriented horizontally and the second vertically. The variable loadings are represented in the plot as arrows starting in the origin. The site scores shown as dots represent the coordinates of the sites in the new ordination axes.

While various axis scaling is possible, axes are generally between the minimal value of -1 and the maximum value of 1. Positive scores or loadings indicate positive correlation with the axis, where negative values indicate negative correlation. For example, a variable with negative loadings for the first two ordination axes is anticorrelated with the two largest trends in the data.

The direction of the arrow reflecting variable loading indicates to which ordination axis it correlates. The length of the vector is equivalent to the extent of that correlation. Thus arrows pointing in the same direction indicate that the variable are correlated. Arrows at an right angle to one another are uncorrelated. Arrows that point in opposite directions are anti-correlated. A vector very close to the origin shows little to no correlation with the axes. Proximity of the site scores in the plot indicate the similarity between the sample sites.

Ordination plots are biplots, when two different elements are displayed, this are e.g. variable loadings and site scores for unconstrained methods or dependent and independent variable loadings in constrained methods. When loadings and site scores are displayed in constrained methods, they are called a triplot.

Data Transformation

There are various ways to transform the data before ordination analysis: * centered: for each sample value of a variable \(x_i\) the mean of the variable over all samples \(\mu\) is subtracted: \(z_i = x_i − \mu\) * standardize: \(z_i = \frac{x_i - \mu}{\sigma}\) where \(\sigma\) is the standard deviation of the variable over all samples. * log transformed: \(z_i = \log( A x_i + B)\) where \(A\) and \(B\) scaling parameters (typically chosen \(A =1\) and \(B=1\))

Note that logarithmic transformation is performed before standardization or centering, since logarithms give no solution for negative values.

[not yet implemented] Samples or variables can be designated as supplementary. Then the values will not considered during ordnation analysis, but their scores and loadings relative to the axes will be determined for visualization. After performing the ordination analysis, data can be scaled or transformed again, for the purpose of plotting preferences. Scaling can be focused on either variable or sample distance.