Repeated measures mixed model with AR(1) correlation structure in nlme vs SAS Proc Mixed

Simon Bate <simontbate at ...> writes:

 Hi all, I don't know if anyone has any thoughts on this, I suspect
it is a local convergence issue but am not sure. I have been trying
to move from SAS Proc Mixed to R nlme and have an unusual result.  I
have several subjects measured at four time points. I want to model
the within-subject correlation using an autoregressive
structure. I've attached the R and SAS code I'm using along with the
results from SAS.  With R lme I get an estimate of the
autoregressive parameter phi = 0.2782601, whereas SAS gives me an
estimate of 0.3389 Intriguingly if I include a between subject
factor, a covariate or delete one of the observations, then the
results appear to agree.  I'm surprised the seemingly simpler model
if different between the two implementations whereas the more
complex models agree.  Any ideas would be most welcome!  Regards,
Simon

Hmmm.  I do agree it could be a convergence problem of some sort.
  I agree that it's worth trying to figure out what the difference
is, but I think it's going to be hard.  I went through some basic
sanity checking with R (see below) and don't see an obvious problem.

library(nlme)
sasdata <- data.frame(Response=c(0.55,0.86,0.21,0.36,0.46,
	    0.32,0.11,0.24,0.36,0.29,0.48,0.93,
            0.56,0.67,0.36,0.55,0.51,0.4,0.34,0.51,
            1,0.61,0.65,0.41,0.99,0.86,0.64,0.86,0.31,
            0.19,0.21,0.36,0.41,0.47,0.16,0.81,0.9,0.72,0.87,0.02),
        Subject=c(1,1,1,1,2,2,2,2,3,3,3,3,
                  4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,
                  8,8,8,8,9,9,9,9,10,10,10,10),
    ## or: rep(1:10,each=4)
         Day=c(1,2,4,6,1,2,4,6,1,2,4,6,1,2,4,6,
               1,2,4,6,1,2,4,6,1,2,4,6,1,2,4,6,1,2,4,6,1,2,4,6))
    ## or: rep(c(1,2,4,6),4)

sasdata$Time<-factor(sasdata$Day)

fit0<-lme(Response~Time, random=~1|Subject,
          data=sasdata, na.action = (na.omit), method = "REML")

plot(ACF(fit0),alpha=0.05,grid=TRUE)
## it actually looks like AR1 is not a very good model here
##   anyway?
AR1<- update(fit0,
             correlation=corAR1(form=~as.numeric(Time)|Subject,
               fixed =FALSE))

plot(ACF(AR1),alpha=0.05,grid=TRUE)
## ACF(AR1) shows that the raw rho(1) is about 0.2

## identical (based on order of observations within groups)
AR1B<- update(fit0,
             correlation=corAR1(form=~1|Subject,
               fixed =FALSE))


## construct likelihood profile on rho
rhovec <- seq(-0.99,0.99,by=0.01)
Lvec <- sapply(rhovec,
               function(r) {
                 logLik(update(fit0,
                      correlation=corAR1(form=~1|Subject,
                        value=r,
                        fixed =TRUE)))
               })

plot(rhovec,-2*Lvec,type="b")
abline(h=-2*logLik(AR1)+3.84,lty=2) ## LRT cutoff
abline(v=c(-0.53,0.82),lty=2)   
## CIs on rho from intervals() -- reasonable ballpark


## zoom in
plot(rhovec,-2*Lvec,type="b",ylim=c(11.5,15),xlim=c(-0.5,0.7))
abline(v=c(0.278,0.339)) ## SAS/R answers

## zoom in further
plot(rhovec,-2*Lvec,type="b",ylim=c(11.6,11.8),xlim=c(0.2,0.4),
     las=1,bty="l")
abline(v=c(0.278,0.339),lty=2)
text(c(0.278,0.339),rep(par("usr")[4],2),c("lme","SAS"),
         pos=3,xpd=NA)
abline(h=11.63168,lty=2)
text(par("usr")[2],11.63168,"SAS",pos=1,xpd=NA)

  So ...  I don't see anything obviously wrong or weird about
what R is doing.  You can try

update(AR1,control=lmeControl(msVerbose=TRUE))

to see more detail on the optimization, but you will have
to dig into Pinheiro & Bates 2000 and/or the code and
documentation for details on the internal parameterizations
of rho in order to interpret the results ...

###
R Code: 

SAS Code: 

proc mixed; 
class Subject Day; 
model Response = Day / outp=pout; 
repeated Day / subject = Subject type=AR(1); 
run; 


SAS Results: 


Model Information 

Data Set WORK.ALLDATA 
Dependent Variable Response 
Covariance Structure Autoregressive 
Subject Effect Subject 
Estimation Method REML 
Residual Variance Method Profile 
Fixed Effects SE Method Model-Based 
Degrees of Freedom Method Between-Within 


Class Level Information 

Class Levels Values 
Subject 10 1 10 2 3 4 5 6 7 8 9 
Day 4 1 2 3 4 


Dimensions 

Covariance Parameters 2 
Columns in X 5 
Columns in Z 0 
Subjects 10 
Max Obs Per Subject 4 


Number of Observations 

Number of Observations Read 40 
Number of Observations Used 40 
Number of Observations Not Used 0 


Iteration History 

Iteration Evaluations -2 Res Log Like Criterion 
0 1 14.67045653 
1 2 11.63168913 0.00000018 
2 1 11.63168429 0.00000000 


Convergence criteria met. 



Covariance Parameter Estimates 

Cov Parm Subject Estimate 
AR(1) Animal1 0.3389 
Residual 0.06862