covariance structure for multilevel modelling - r

Question

I have a multilevel repeated measures dataset of around 300 patients each with up to 10 repeated measures predicting troponin rise. There are other variables in the dataset, but I haven't included them here.
I am trying to use nlme to create a random slope, random intercept model where effects vary between patients, and effect of time is different in different patients. When I try to introduce a first-order covariance structure to allow for the correlation of measurements due to time I get the following error message.
Error in `coef<-.corARMA`(`*tmp*`, value = value[parMap[, i]]) : Coefficient matrix not invertible
I have included my code and a sample of the dataset, and I would be very grateful for any words of wisdom.
#baseline model includes only the intercept. Random slopes - intercept varies across patients
randomintercept <- lme(troponin ~ 1,
data = df, random = ~1|record_id, method = "ML",
na.action = na.exclude,
control = list(opt="optim"))
#random intercept and time as fixed effect
timeri <- update(randomintercept,.~. + day)
#random slopes and intercept: effect of time is different in different people
timers <- update(timeri, random = ~ day|record_id)
#model covariance structure. corAR1() first order autoregressive covariance structure, timepoints equally spaced
armodel <- update(timers, correlation = corAR1(0, form = ~day|record_id))
Error in `coef<-.corARMA`(`*tmp*`, value = value[parMap[, i]]) : Coefficient matrix not invertible
Data:
record_id day troponin
1 1 32
2 0 NA
2 1 NA
2 2 NA
2 3 8
2 4 6
2 5 7
2 6 7
2 7 7
2 8 NA
2 9 9
3 0 14
3 1 1167
3 2 1935
4 0 19
4 1 16
4 2 29
5 0 NA
5 1 17
5 2 47
5 3 684
6 0 46
6 1 45440
6 2 47085
7 0 48
7 1 87
7 2 44
7 3 20
7 4 15
7 5 11
7 6 10
7 7 11
7 8 197
8 0 28
8 1 31
9 0 NA
9 1 204
10 0 NA
10 1 19

Answer 1

You can fit this if you change your optimizer to "nlminb" (or at least it works with the reduced data set you posted).
armodel <- update(timers,
correlation = corAR1(0, form = ~day|record_id),
control=list(opt="nlminb"))
However, if you look at the fitted model, you'll see you have problems - the estimated AR1 parameter is -1 and the random intercept and slope terms are correlated with r=0.998.
I think the problem is with the nature of the data. Most of the data seem to be in the range 10-50, but there are excursions by one or two orders of magnitude (e.g. individual 6, up to about 45000). It might be hard to fit a model to data this spiky. I would strongly suggest log-transforming your data; the standard diagnostic plot (plot(randomintercept)) looks like this:
whereas fitting on the log scale
rlog <- update(randomintercept,log10(troponin) ~ .)
plot(rlog)
is somewhat more reasonable, although there is still some evidence of heteroscedasticity.
The AR+random-slopes model fits OK:
ar.rlog <- update(rlog,
random = ~day|record_id,
correlation = corAR1(0, form = ~day|record_id))
## Linear mixed-effects model fit by maximum likelihood
## ...
## Random effects:
## Formula: ~day | record_id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 0.1772409 (Intr)
## day 0.6045765 0.992
## Residual 0.4771523
##
## Correlation Structure: ARMA(1,0)
## Formula: ~day | record_id
## Parameter estimate(s):
## Phi1
## 0.09181557
## ...
A quick glance at intervals(ar.rlog) shows that the confidence intervals on the autoregressive parameter are (-0.52,0.65), so it may not be worth keeping ...
With the random slopes in the model the heteroscedasticity no longer seems problematic ...
plot(rlog,sqrt(abs(resid(.)))~fitted(.),type=c("p","smooth"))