Dose response, or rate response tests, form the basis for product usage instructions. Theophrastus' advice ‘the dose makes the poison’ rings true across product types, from medicine to agricultural inputs.
Today apply a mixed model to a quantitative explanatory variable within the hypothetical dataset ‘field yield’. The dataset includes four maize trial sites in the central Mexico highlands (attached). To simplify the discussion, today's analysis is restricted to a single site, San Felipe.
The response variable is yield in metric tons per hectare, and the variable of interest is MBoost, a mythical microbial product that may improve yields at small application rates (between 0.5 to 1 kg per hectare).
Each site had a consistent design: three treatments: 0, 0.5 and 1kg/Ha Mboost. Treatments were blocked in a randomized complete block design. If we were interested running the dose response analysis, it might look like:
*The field yield data set was restricted to the San Felipe site; PROC Mixed DATA=WORK.RATETEST plots=all; CLASS Block Mboost; MODEL Yield_tonnes_ha = Rate_kg_ha Mboost/ Solution; RANDOM Block; RUN;
We model the quantitative rate variable (deliberately omitted from the class statement) and the Mboost variable against yield. Random factor is block.
The residual plots seem OK for a small field data set. The output shows the rate was significant, however a lack of effect of Mboost. This is expected since rate is completely orthogonal to Mboost and consumed almost all the variance otherwise that could be explained by the binary (yes-no) inclusion of the ingredient. Next time, we will omit Mboost.
Another thing to note what the Type 3 fixed effects for rate do not test: they do not test the slope including whether the response was linear or quadratic. All we can determine from the test is that the differences of at least one of the means are significantly different from 0. Chapter 3.4 of SAS® for Mixed Models: Introduction and Basic Applications outlines several ways to go fitting the response trajectory.
From this analysis we’ve determined the rate is a significant effect. For our combined site analysis, we may want to test the same factors, also including location interactions and random effects.
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