This client arrived with two asks: to verify analysis of data she has already collected and to design an experiment for the next stage in her research. She was interested in the prescence of phosphorus in retention ponds caused by agricultural runoff.
To study this phenomena she measured phosphorus at different locations of each pond: inflow, center, outflow; and at different depths: surface, a few meters down, and on the pond bottom. For each measurement she also recorded a few covariates such as disolved oxygen.
Because the client was primarily interested in the effect of depth, location, and other covariates on phosphorus concentration and was not interested in each individual pond, her advisor reccomended (and we concurred) that she should apply a linear mixed model for this problem, in particular: by including a random intercept for each pond.
Including a random intercept (random average) in the model allows us to account for differences between ponds and better isolate the effect of the variables of interest. The client sampled a set of ponds from the space of all possible ponds (not that these ponds were of particular interest, just the fact that they were available ponds) so we have strong theoretical justification for allowing this intercept/average parameter to be random. A final advantage is that by specifying a random intercept we can pool information across ponds to provide better estimates for small ponds with fewer observations (some ponds were small and shallow).
According to the client, research in this area is somewhat divided between two competeting hypotheses and the results of this analysis support the newer, less accepted hypotheses.
Now that the client had demonstrated the prescence of agricultural runoff in local ponds she was intent on developing an inexpensive filtration system to remove these pollutants before they reached the water system.
She was interested in comparing between a variety of filter media of various quantities to understand which materials filtered phosphorus best but also important was understanding how the performance of these filters degraded over time as the filter reached capacity.
Additionally she was severely limited by budget and time constraints as she was going to have to build each filter as well as the large tank simulating the pond by hand. If she ever wanted to graduate we were going to have to keep the experiment small.
The result of this story is that the client and her advisor had already developed a design and desired for us to simply sign off on it. We experessed our reservations but couldn’t develop a better design under the resource constraints provided. We were able to correct some notions, namely that you couldn’t “gain more power by observing the filters more frequently” and that true replication would require building a copy of each filter and testing the copy. Needless to say, that didn’t go over well. But sometimes you just have to do the best with what you have