I used the
nlme package to implement a mixed effects model with an AR(1) correlation structure on the residuals.
nlme, the grouping structure of the AR(1) process must be the same as the random effects structure, as far as I understand. While my AR(1) process was in fact discrete-time, because I had missing data within my random effects groups, I had to instead implement this as a continuous-time AR(1) model (
?corCAR1). To my knowledge, I cannot simply specify that there are multiple AR(1) groups nested within my random effect groupings.
I've calculated some confidence intervals, and I wanted to make a short parametric bootstrap to make sure that my math was correct (my calculated confidence bands from the models covariance matrix agree with the parametric bootstrapped confidence bands).
Trying to simulate new datasets, I realised it's not clear to me exactly what the continuous-time AR(1) process is.
help(corCAR1) doesn't give any mathematical details, but only references to three textbooks.
- Does anyone know how continuous-time AR(1) process are implemented in the
nlmepackage? I'm aware that an Ornstein-Uhlenbeck process is considered an continuous time analogue of the AR(1) process... but I'm not sure if there are other implementations I'm not aware of which
nlmemay be using.
In case more information is needed, below I've put the model summary.
Linear mixed-effects model fit by REML Data: data AIC BIC logLik -6472.216 -6415.398 3244.108 Random effects: Formula: ~1 | fish (Intercept) Residual StdDev: 0.09100886 0.2251129 Correlation Structure: Continuous AR(1) Formula: ~time.2 | fish Parameter estimate(s): Phi 0.6738598 Fixed effects: distance_log ~ period + current_speed + tide_m Value Std.Error DF t-value p-value (Intercept) 1.2148390 0.02487684 8957 48.83414 0.0000 periodDuring 0.0228981 0.01208525 8957 1.89471 0.0582 periodAfter 0.0519890 0.01285010 8957 4.04581 0.0001 current_speed 0.1018782 0.01850602 8957 5.50514 0.0000 tide_m -0.0088376 0.00490187 8957 -1.80290 0.0714 Correlation: (Intr) prdDrn prdAft crrnt_ periodDuring -0.292 periodAfter -0.249 0.451 current_speed -0.432 0.178 0.092 tide_m 0.083 -0.054 0.015 -0.197 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -6.30361758 -0.56513258 -0.01809098 0.46685852 4.81926579 Number of Observations: 8980 Number of Groups: 19 ```