### Constrained analyses

The main idea...

The analyses that this wizard will guide you through are known as constrained analyses. Constrained analysis is a form of direct gradient analysis, which attempts to explain variation in a data table directly through the variation in a set of explanatory variables  (e.g. environmental factors) stored in a corresponding table or tables (Figure 1). When direct gradient analysis is used with ordination techniques, the axes that are built to represent high-dimensional data in a low-dimensional space are constrained to be functions of the explanatory factors. The explanatory factors can often be visualised in an ordiation bi- or triplot (Figure 2). Any variation that cannot be 'explained' by the explanatory variables is treated as residual variation.

Read on for some more general information on constrained analyses or click here to start the wizard...

#### b

Figure 1: Most constrained analyses require two tables or matrices as input: a) a table of objects (such as sites or samples) by a set of response variables (here, species) and b) a table of the same objects by a set of explanatory variables (such as environmental factors). Many analyses can accept data types in addition to numerical data (such as categorical data), although recoding and the creation of 'dummy variables' may be needed.

 Figure 2: Schematic representation of a biplot generated by constrained ordination. Points may represent objects or response variables. Vectors represent explanatory variables. Colours represent grouping.

## Warnings

• The choice of which explanatory variables you measure and feed into these analyses is crucial. Simply because a set of explanatory variables explain variation in your response variables mathematically is no guarantee that they have true explanatory power. There may always be a covariate that is the real causal influence on the ecology of the system. As always, subject the results of these analyses to further scrutiny and knowledge-guided criticism.
• If the number of explanatory variables approaches or exceeds the number of sampling objects (sites, samples, observations), it is possible that the analysis will no longer be constrained. Increasing the number of explanatory variables typically increases the ability for constrained analyses to fit the observed data to some combination of these variables. This is rarely useful, as it prevents any ecologically meaningful interpretation of the response data's relationship to explanatory variables.