Mixed Model: Split plot with two whole-plot factors, one split-plot factor, and CRD at the whole-plot level (e.g. fancier split-plot p.
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1 22s:173 Combining multiple factors into a single superfactor Mixed Model: Split plot with two whole-plot factors, one split-plot factor, and CRD at the whole-plot level (e.g. fancier split-plot p.422 Oehlert) Hamster example with three fixed factors of interest: DayLength (short/long), Climate (cold/warm), Tissue (heart/brain) Two hamsters randomly assigned to each combination of DayLength/Climate. Daylength long short Climate cold warm Then, two measurements are taken on each hamster, one from the brain and one from the heart. The response is the quantity of an enzyme (NI) in the blood. The main goal of the analysis is perform comparison tests of the three factors of interest. Each hamster is nested in one of the 4 cells shown above. In this split-plot setting, it s convenient to perceive the 4 cells (each a specific combination of DayLength(S/L) and Climate(C/W)) as coming from a single superfactor with 4 levels (SC/SW/LC/LW). In either perception, it takes 3 degrees of freedom to allow for 4 freely-fit means for the 4 cells. Source df superfactor 3 DayLength 1 Climate 1 DayLength*Climate 1 Hamster(superfactor) 4 random term Tissue 1 Tissue *superfactor 3 s.p. error 4 random term c.total 15 The benefit of using the original factor names is that you can potentially test for the main effects of DayLength and Climate (if there is no significant interaction between the terms). The benefit of perceiving the two original whole-plot factors as a superfactor is that you can very easily transfer your knowledge about a split-plot design with one whole plot factor and one split plot factor over to the present analysis (which has three factors of interest). 1
2 SAS statements for data input: proc import datafile="split_plot_hamsters.csv" out=ham dbms=csv replace; /*Create Superfactor column to be used for plotting:*/ data ham; set ham; if DayLength="short" & Climate="cold" then Superfactor="SC"; if DayLength="short" & Climate="warm" then Superfactor="SW"; if DayLength="long" & Climate="cold" then Superfactor="LC"; if DayLength="long" & Climate="warm" then Superfactor="LW"; ; proc print data=ham; Day Obs Hamster Length Climate Tissue NI Superfactor 1 1 short cold heart SC 2 1 short cold brain 11.3 SC 3 2 short cold heart SC 4 2 short cold brain SC 5 3 short warm heart SW 6 3 short warm brain SW 7 4 short warm heart SW 8 4 short warm brain SW 9 5 long cold heart 0.69 LC 10 5 long cold brain LC 11 6 long cold heart LC 12 6 long cold brain LC 13 7 long warm heart LW 14 7 long warm brain 9.9 LW 15 8 long warm heart 1.79 LW 16 8 long warm brain 8.8 LW symbol1 value=star i=std1mj color=black line=1; symbol2 value=circle i=std1mj color=blue line=2; proc gplot data=ham; plot NI*Superfactor=Tissue; 2
3 SAS statements for Proc GLM using original factor names: proc glm data=ham; class DayLength Climate Tissue Hamster; model NI=DayLength Climate Tissue Hamster(DayLength*Climate); random Hamster(DayLength*Climate)/test; The GLM Procedure Class Level Information Class Levels Values DayLength 2 long short Climate 2 cold warm Tissue 2 brain heart Hamster Dependent Variable: NI Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total
4 Source Type III Expected Mean Square DayLength Var(Error) + 2 Var(Hamst(DayLen*Climat)) + Q(DayLength,DayLength*Climate,DayLength*Tissue,DayLen* Climat*Tissue) Climate Var(Error) + 2 Var(Hamst(DayLen*Climat)) + Q(Climate,DayLength*Climate,Climate*Tissue,DayLen*Climat* Tissue) DayLength*Climate Var(Error) + 2 Var(Hamst(DayLen*Climat)) + Q(DayLength*Climate,DayLen*Climat*Tissue) Tissue Var(Error) + Q(Tissue,DayLength*Tissue,Climate*Tissue,DayLen*Climat* Tissue) DayLength*Tissue Climate*Tissue DayLen*Climat*Tissue Hamst(DayLen*Climat) Var(Error) + Q(DayLength*Tissue,DayLen*Climat*Tissue) Var(Error) + Q(Climate*Tissue,DayLen*Climat*Tissue) Var(Error) + Q(DayLen*Climat*Tissue) Var(Error) + 2 Var(Hamst(DayLen*Climat)) Above, you can see that SAS includes other fixed terms in the Q() value that contain the given term, but it is only testing for the given fixed term on the left. To match common notation, we could write... 4
5 Source DayLength Climate DayLength*Climate Tissue DayLength*Tissue Climate*Tissue DayLen*Climat*Tissue Hamst(DayLen*Climat) Type III Expected Mean Square Var(Error) + 2 Var(Hamst(DayLen*Climat)) + Q(DayLength) Var(Error) + 2 Var(Hamst(DayLen*Climat)) + Q(Climate) Var(Error) + 2 Var(Hamst(DayLen*Climat)) +Q(DayLength*Climate) Var(Error) + Q(Tissue) Var(Error) + Q(DayLength*Tissue) Var(Error) + Q(Climate*Tissue) Var(Error) + Q(DayLen*Climat*Tissue) Var(Error) + 2 Var(Hamst(DayLen*Climate)) You can see that the superfactor (DayLen, Climate, DayLen*Climate) should be tested at the whole-plot level using Hamster(superfactor) as the error term (which is listed above as Hamst(DayLen*Climat). Tissue and the Tissue*superfactor terms are tested at the split-plot level using the experimental error as the error term. 5
6 After choosing the test option in PROC GLM, the correct errors are used for the F-tests: Dependent Variable: NI The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Source DF Type III SS Mean Square F Value Pr > F DayLength Climate DayLength*Climate Error Error: MS(Hamst(DayLen*Climat)) Source DF Type III SS Mean Square F Value Pr > F Tissue DayLength*Tissue Climate*Tissue DayLen*Climat*Tissue Hamst(DayLen*Climat) Error: MS(Error)
7 SAS statements for Proc Mixed: proc mixed data=ham; class DayLength Climate Tissue Hamster; model NI=DayLength Climate Tissue/outp=predicted; /*save predicted & residuals*/ random Hamster(DayLength*Climate); The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Hamst(DayLen*Climat) 0 Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F DayLength Climate DayLength*Climate Tissue DayLength*Tissue Climate*Tissue DayLen*Climat*Tissue The Hamster-to-hamster variability was very small compared to the experimental error (splitplot error), and was estimated at zero by the REML method. 7
8 The 3-way interaction was not significant (which suggests that the pattern in any pair of 2-way interaction plots would be similar across all levels of the third factor). The only 2-way interaction that was significant was the DayLength*Tissue interaction, which seems apparent in the plot... The first two Superfactor levels are on the left and coincide with DayLength of long. The Tissue effect seems to be larger when DayLength is short and smaller when DayLength is long. Comparisons of means can be done as usual with a 3-way factorial. Diagnostics are still relevant, and we will consider the residuals now (at the level of the split-plot experimental error)... symbol1 value=plus i=; proc gplot data=predicted; plot resid*pred/vref=0; 8
9 Though there may be some non-constant variance, it is not a monotonic pattern, and no transformation will completely eliminate it (though one could argue it gets larger with the mean and try a log or square root transformation). The symmetric pattern around y=0 occurs due to the fact that we have only 2 observations in each cell. 9
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