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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Ashlie Johnson
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1 STAT:5201 Applied Statistic II 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 OLRT) Hamster example with three fixed factors of interest: DayLength (short/long), Climate (cold/warm), Tissue (heart/brain) (see HW 2 for the full set-up.) Two hamsters randomly assigned to each combination of DayLength/Climate. Then, two measurements are taken on each hamster in the brain and heart. The main goal of the analysis is perform comparison tests of the three factors of interest. SAS statements for data input: proc import datafile="split_plot_hamsters.csv" out=ham dbms=csv replace; /*Create Superfactor column 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 2.32 SC 2 1 short cold brain SC 3 2 short cold heart 0.29 SC 4 2 short cold brain 9.93 SC 5 3 short warm heart 2.26 SW 6 3 short warm brain SW 7 4 short warm heart 1.79 SW 8 4 short warm brain SW 9 5 long cold heart 0.49 LC 10 5 long cold brain 5.18 LC 11 6 long cold heart 0.54 LC 12 6 long cold brain 9.28 LC 13 7 long warm heart 2.31 LW 14 7 long warm brain LW 15 8 long warm heart 0.79 LW 16 8 long warm brain 7.80 LW 1

2 symbol1 value=star i=std1mj color=black line=1 height=2; symbol2 value=circle i=std1mj color=blue line=2 height=2; proc gplot data=ham; plot NI*Superfactor=Tissue; SAS statements for Proc GLM and expected mean squares: proc glm data=ham; class DayLength Climate Tissue Hamster; model NI=DayLength Climate Tissue Hamster(DayLength*Climate); random Hamster(DayLength*Climate)/test; NOTE: I recommend using residual plots from PROC MIXED for diagnostics rather then PROC GLM. 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

3 The GLM Procedure 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 essentially only testing for the given fixed term on the left. To match our previous notation, we could write... The GLM Procedure 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)) 3

4 After choosing the test option, 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) SAS statements for Proc Mixed: proc mixed data=ham plots(only)=residualpanel(conditional); class DayLength Climate Tissue Hamster; model NI=DayLength Climate Tissue/ddfm=satterth outp=preddata residual; /* preddata contains*/ random Hamster(DayLength*Climate); /* conditional resids and predicted values*/ The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Hamst(DayLen*Climat) Residual

5 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 3-way interaction was not significant. 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 (from the split-plot experimental error)... 5

6 Constant variance plot has an issue with non-constant variance. It does look like a pattern of non-constance variance (monotonically increasing) that may be helped by a transformation (the square root transformation looks pretty good). Normality looks OK. SAS Documentation on available residuals: viewer.htm#statug_mixed_sect027.htm#statug.mixed.mixeddetailres 6

7 Fitting this model in R (extra information) ## Use the same dummy coding as SAS: > options(contrasts=c("contr.sas","contr.poly")) ## Load the linear mixed-effects package lme4. > library(lme4) # Get the data > hamsterdata <- read.csv("split_plot_hamsters.csv") > head(hamsterdata) Hamster DayLength Climate Tissue NI 1 1 short cold heart short cold brain short cold heart short cold brain short warm heart short warm brain ## Coerce the hamster variable to be a factor: > hamsterdata$hamster <- as.factor(hamsterdata$hamster) ## Fit the model... ## Include all main effects and interactions for fixed effects. ## Include a random hamster effect for each hamster nested in ## the treatment combination of Climate:DayLength. > model <- lmer(ni ~ 1 + DayLength + Climate + DayLength:Climate + Tissue + Tissue:DayLength + Tissue:Climate + Tissue:DayLength:Climate + (1 (Climate:DayLength):Hamster), data = hamsterdata) ## The first part of the model statement is for the fixed effects. ## The second part in parentheses is for the random effects. ## The (1 (Climate:DayLength):Hamster) tells lmer to fit a different ## random intercept for all Climate-by-DayLength-by-hamster combinations. SIDENOTE: The traditional split-plot example with secretaries (where the whole-plot looked like a CRD) can be fit in R using the code below. model <- lmer(numsorted ~ periods + (1 periods:secretary) + timeday + timeday:periods, data = secretaries) Diagnostics: > plot(model) > qqnorm(residuals(model)) > qqline(residuals(model)) 7

8 Normal Q-Q Plot resid(., type = "pearson") Sample Quantiles fitted(.) Theoretical Quantiles > summary(model) Linear mixed model fit by REML [ lmermod ] Formula: NI ~ 1 + DayLength + Climate + DayLength:Climate + Tissue + Tissue:DayLength + Tissue:Climate + Tissue:DayLength:Climate + (1 (DayLength:Climate):Hamster) Data: hamsterdata REML criterion at convergence: 40 Random effects: Groups Name Variance Std.Dev. (Climate:DayLength):Hamster (Intercept) Residual Number of obs: 16, groups: (Climate:DayLength):Hamster, 8 Compare these estimates to the SAS covariance parameter estimates below (they match): The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Hamst(DayLen*Climat) Residual Fixed effects: Estimate Std. Error t value (Intercept) Tissuebrain Climatecold DayLengthlong Tissuebrain:Climatecold Tissuebrain:DayLengthlong Climatecold:DayLengthlong Tissuebrain:Climatecold:DayLengthlong

9 Compare the above estimates to the SAS fixed effects estimates below (they match): One of the issues with the lme4 package is that it does not directly provide p-values for the fixed effects (a common request from clients). Presently, I am using the package called lmertest to get p-values and lsmean values for mixed models in R. > library(lmertest) The lmertest package has a function called anova() which will mask the base package anova() function with one that gives you Type III tests (I find it annoying that they used the exact same function name... causes confusion, but it will get you what you want). You MUST RE-FIT THE MODEL after loading lmertest (it s actually the lmertest::lmer() function now) > model <- lmer(ni ~ 1 + DayLength + Climate + DayLength:Climate + Tissue + Tissue:DayLength + Tissue:Climate + Tissue:DayLength:Climate + (1 (Climate:DayLength):Hamster), data = hamsterdata) > lmertest::anova(model) Analysis of Variance Table of type III with Satterthwaite approximation for degrees of freedom Sum Sq Mean Sq NumDF DenDF F.value Pr(>F) DayLength Climate Tissue *** DayLength:Climate DayLength:Tissue * Climate:Tissue DayLength:Climate:Tissue

10 Compare the above F-statistics to the SAS output below (they match): 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 lmertest package also has a function called lsmeanslt to provide SAS-type LSMEANS: > lsmeanslt(model,test.effs="daylength:tissue",ddf="satterthwaite") Least Squares Means table: DayLength Climate Tissue Estimate Standard Error DF t-value DayLength:Tissue long brain 1 NA DayLength:Tissue short brain 2 NA DayLength:Tissue long heart 1 NA DayLength:Tissue short heart 2 NA Lower CI Upper CI p-value DayLength:Tissue long brain e-04 *** DayLength:Tissue short brain <2e-16 *** DayLength:Tissue long heart DayLength:Tissue short heart Signif. codes: 0 *** ** 0.01 * And to give estimated differences of LSMEANS: > difflsmeans(model,test.effs="daylength:tissue",ddf="satterthwaite") Differences of LSMEANS: Estimate Standard Error DF t-value Lower CI DayLength:Tissue long brain - short brain DayLength:Tissue long brain - long heart DayLength:Tissue long brain - short heart DayLength:Tissue short brain - long heart DayLength:Tissue short brain - short heart DayLength:Tissue long heart - short heart Upper CI p-value DayLength:Tissue long brain - short brain ** DayLength:Tissue long brain - long heart ** DayLength:Tissue long brain - short heart ** DayLength:Tissue short brain - long heart e-04 *** DayLength:Tissue short brain - short heart e-04 *** DayLength:Tissue long heart - short heart Signif. codes: 0 *** ** 0.01 *

11 But it looks like you can not apply any multiple comparison adjustment using this difflsmeans. You can always apply the Bonferroni adjustment, but if you want a Tukey adjustment, you could use the lsmeans package by Russ Lenth. > library(lsmeans) > lsmeans(model, list(pairwise ~ DayLength:Tissue), adjust = "tukey") NOTE: Results may be misleading due to involvement in interactions $ lsmeans of DayLength, Tissue DayLength Tissue lsmean SE df lower.cl upper.cl long brain short brain long heart short heart Results are averaged over the levels of: Climate Degrees-of-freedom method: satterthwaite Confidence level used: 0.95 $ pairwise differences of contrast contrast estimate SE df t.ratio p.value long,brain - short,brain long,brain - long,heart long,brain - short,heart short,brain - long,heart short,brain - short,heart long,heart - short,heart Results are averaged over the levels of: Climate P value adjustment: tukey method for comparing a family of 4 estimates Below is a quick plot of the means: > with(hamsterdata,interaction.plot(factor(daylength:climate), Tissue, NI)) mean of NI Tissue brain heart long:cold long:warm short:cold short:warm factor(daylength:climate) 11

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.

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)