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# Estimating Taxon-Environment Relationships: Measurement Error

## Measurement Error

Measurements of environmental variables that are used to develop taxon-environment relationships frequently exhibit high variability. This variability can be a manifestation of errors in the measuring instrument or method. For example, Wolman pebble count measurements of substrate composition can be subject to random errors introduced by the observer. Other measurements are highly variable simply because they are instantaneous measurements. For example, a grab stream temperature sample can be thought of as a highly variable measurement of average stream temperature over the sampling season.

In all of these cases, variability in the environmental measurements can have strong influences on the taxon-environment relationship that is estimated by regression. In simple models, regression coefficients estimated from highly variable measurements tend to be closer to zero than their true values. In multivariate models, the effects of measurement variability are more difficult to predict.

One approach for adjusting regression results for the effects of measurement error relies on introducing additional, known amounts of variability to the measured variables. After adding the known variability to the measured variables, regression models are refit, and the regression coefficient values computed. Then, the process is repeated with a different amount of added variability. Eventually, a relationship can be developed between the variability in the measured variables and the coefficient values. Then, we can extrapolate this relationship to predict coefficient values in the absence of measurement error. This simulation and extrapolation approach (SIMEX) was first proposed by Cook and Stefanski (1995), and has been used on several different types of ecological models.

Taxon-environment relationships provided in the CADDIS database have been adjusted for measurement error using the SIMEX procedure. More information on the implementation of this procedure for adjusting for variability of measurements can be found in Yuan (2007a). Examples of taxon-environment relationships for Heterlimnius and Malenka are shown in Figure 11. Biological inferences are more accurate when based upon taxon-environment relationships that have been corrected for measurement error.

Figure 11. Parametric relationship between probability of occurrence and temperature (degrees C) and percent sand/fines (SED) for Heterlimnius and Malenka using a parametric model. Contours represent mean probabilities of occurrence adjusted for measurement error.

### References

• Cook J, Stefanski LA (1995) A simulation extrapolation method for parametric measurement error models. Journal of the American Statistical Association 89:1314-1328.
• Yuan LL (2007) Effects of measurement error on inferences of environmental conditions. Journal of the North American Benthological Society 26:152-163.

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