Fixed and mixed-effects models for multi-watershed experiments

The usual analysis of paired-watershed studies is simple and conclusions are consequently very limited. Linear regression models are computed, relating a response in two similar watersheds, before and after a land treatment is applied to one of them. If there is a significant difference between the regression lines, then the treatment is deemed to have had an effect, and the size of the effect is estimated as the average change in response. Sometimes the magnitude of the effect is estimated for different event size classes after testing subsets of events, but the sample sizes are often too small to detect real effects, especially for large, infrequent events. Descriptors of the treatment are not explicitly included in the model, so it is impossible to use the model for prediction unless the new treatments can be considered identical to the original. Data collected during application of the treatment are typically lumped with post-treatment data, diluting the overall effect. Serial and spatial autocorrelation in the data are typically ignored, resulting in underestimation of variance and rates of Type I error (i.e. rejection of true hypotheses). If more than one watershed is treated, the analysis is repeated for each, instead of applying a single model to all the data. Splitting the data reduces statistical power and increases the probability of Type I errors unless adjustments are made for the testing of multiple hypotheses. 

Fixed and mixed-effects regression modeling tools available in many statistical software packages provide solutions to these problems, but have not been widely applied in hydrological research. The discussion begins with some general concepts, followed by an example showing the development of a model for storm flows.