CADDIS Volume 4: Data Analysis

## Predicting Environmental Conditions from Biological Observations (PECBO) Appendix

- Introduction

- Using Existing Taxon-Environment

Relationships - Estimating Taxon-Environment

Relationships - Computing Inferences

- R Scripts

##### Topics in Computing Inferences

- Overview
- Weighted Average Inferences
- Maximum Likelihood Inferences
- Categorical Tolerance Data

### Weighted Average Inferences

A weighted average inference is computed as the average of the single-value descriptors of the taxa observed at the site. A single-value descriptor of a taxon-environment relationship can quantify the central tendencies of a taxon with respect to an environmental gradient or the environmental limits of a taxon.

The computation of weighted average inference can be expressed as follows:

where *u _{m,i}* is the weighted average inference at
site

*i*,

*N*is the number of taxa, and

*u*is the single-value descriptor for taxon

_{j}*j*.

*Yij*is defined as previously: for presence/absence data,

*Y*= 1 when a taxon

_{ij}*j*is present, and

*Y*= 0 when taxon

_{ij}*j*is absent from site

*i*; for abundance data

*Y*is the abundance of taxon

_{ij}*j*at site

*i*.

An example of a weighted average inference is shown in Figure 14. Central tendencies for different genera were computed with regard to stream temperature using EMAP-West data. Then, these values were used to compute the weighted average inferences at sites in an independent dataset, collected from western Oregon. The weighted average inference at each site was plotted versus stream temperature measured at the same site.

The relationship between the weighted average inferences and observed temperature is relatively strong. Note though that the range of possible values of the weighted average inference is much narrower than the observed range. This "shrinkage" in the range of values is a consequence of the averaging operations that have been applied to compute the central tendencies of each taxon and to compute the inference.