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# lab8 - STAT 8200 Design of Experiments for Research Workers...

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STAT 8200 — Design of Experiments for Research Workers Lab 8 – Due: Friday, Dec. 3 Example: An experiment is designed to study pigment dispersion in paint. Four different methods of mixing a particular pigment are studied. The procedure consists of preparing a gallon can of paint using one of the four mixing methods and then applying each of the three one-third portions of that gallon to a wood panel using one of three application methods (brushing, rolling, and spraying). Because of the amount of time required to do the painting, the experiment was conducted over three consecutive days, with four gallons of paint (one with each of the four mixing methods) applied each day. The data are given below and are analyzed in pigment.sas which you should copy from the public directory and run in SAS. Application Mixing Method Day Method 1 2 3 4 brushing 64.5 66.3 74.1 66.5 1 rolling 68.3 69.5 73.8 70.0 spraying 70.3 73.1 78.0 72.3 brushing 65.2 65.0 73.8 64.8 2 rolling 69.2 70.3 74.5 68.3 spraying 71.2 72.8 79.1 71.5 brushing 66.2 66.5 72.3 67.7 3 rolling 69.0 69.0 75.4 68.6 spraying 70.8 74.2 80.1 72.4 This experiment has the same design as the alfalfa example discussed in class. The appropriate model for this design is y ijk = μ + α i + τ j + e ij + β k + ( αβ ) ik + ε ijk where y ijk is the percentage reflectance measured on the wood panel that was painted with application method k , on day j and with paint mixed with mixing method i . μ is a grand mean, α i is a (fixed) effect for mixing method i (the whole plot factor), τ j is a (random) effect for day j , e ij is the whole plot error term, β k is a (fixed) effect for the k th application method (the split-plot factor), ( αβ ) ik is a (fixed) effect for the i th mixing method when it is combined with application method k (it is the interaction effect), and ε ijk is the split-plot error term.

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Examine pigment.sas and its associated output. Here we illustrate how to fit a split plot model in PROC GLM and in PROC MIXED. In the first call to PROC GLM we fit the model described above to these data. One feature of this model is that it does not account for the possibility of a DAY*APPMETH interaction. That is, it implicitly assumes that differences in application method (the split-plot factor) do not change across days (the whole plot blocking factor). This assumption is very often reasonable a priori and I would say that it seems reasonable in the present context. However, if we want to avoid this assumption we can expand our model to include a ( τβ ) jk term. This is done in the second call to PROC GLM. The appropriate error term for this source of variation is the split-plot error term ε ijk . Therefore, the F statistic given in the SAS output on p.8 is formed correctly and gives some indication of the importance of including a DAY*APPMETH interaction in the model. Since F = 0 . 67 is not close to significance, there is not any evidence of an application method by day interaction, which means that our original model was suitable. The remainder of the analysis will be done based on the results from fitting the first model. (Here, I regard the first model as the model for this analysis. I’ve looked
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