Data from 2 eyes and repeated measures across time

Question

I have data from two eyes that have been repeatedly measured across time (5 time points) for their intraocular pressure (IOP). Various measurements were taken including categorical and continuous data. Not all patients have both eyes eligible for the study, so there is a mixture of patients, some with 1 and some with 2 eyes. Covariates assesed at baseline include age, gender, corneal thickness and other parameters that vary between eyes of each participant.

I have formatted the data so that there are additional rows for each eye and rows for each measurement across time. I also want to investigate the interaction effect between CornealThickness and Time, as well as Gender and Time.

Is the following the correct specification of the model in R, specifically unsure how nesting should be specified:

lmer(IOP ~ Age + Gender + CornealThickness + as.factor(Time) + Gender*as.factor(Time) + CornealThickness*as.factor(Time) + (1|ID/Eye), data=data)

Alternatively, would it be the following:

lmer(IOP ~ Age + Gender + CornealThickness + as.factor(Time) + Gender*as.factor(Time) + CornealThickness*as.factor(Time) + (as.factor(Time)|ID/Eye), data=data)

UPDATE: I have tried to run the mmrm model on this dataset:

 m1 <-  mmrm::mmrm( formula = IOP ~ age + eye_visit +diabetes_duration+ us(eye_visit | id), data = data, control = mmrm_control(method = "Kenward-Roger") )

However I get the following error message:

Problem with these data entries: y_vector 5 Error in fit_single_optimizer(formula = formula, data = data, weights = weights, : Only numeric matrices, vectors, arrays, factors, lists or length-1-characters can be interfaced

What does this mean?

Thank you!

Answer 1

Another approach you can take is to look at observations from separate eyes as if they were at different times and let the data largely determine the appropriate correlations (and levels of variability) between observations from the two eyes. E.g. using the mmrm R package (assuming everything you want to be a factor is already a factor, and where eye_visit is a factor that's a concatenation of eye and visit, such as "left_week4", "right_week4", "left_week8" etc.):

mmrm::mmrm(
  formula = IOP ~ 0 + eye_visit + ...other model terms... + us(eye_visit | id),
  data = data,
  control = mmrm_control(method = "Kenward-Roger")
)

(see here for an introduction to the package).

The advantages of this approach include that

1. you don't assume the variability at each visit/for each eye is the same over time (often variability goes up over time), and
2. you flexibly estimate how correlated different timepoints (and the eyes in the same person) are from the data without imposing something like a AR(1) structure (that tends to be very wrong for any real data, but if you assumed a more structured covariance structure that is appropriate you might have a gain in efficiency).

Additionally, you can think about what variables should have an interaction with time and/or eye (e.g. a baseline starting assessment would usually be allowed to interact with the factor for time). However, that's possible in any of the models that were already mentioned.

Example:

library(tidyverse)
library(mmrm)

set.seed(42)
example <- tibble(eye=rep(c("left", "right"), 30),
                  visit=rep(rep(c("Week0", "Week4", "Week8"), each=2), 10),
                  id=rep(1L:10L, each=6),
                  random_subject_effect = rnorm(n=10, mean=0, sd=10)[id],
                  age = sample(18:85, size=10, replace=T)[id],
                  IOP = rnorm(n=60, 
                              mean=random_subject_effect+10-(eye=="left")*3+as.integer(str_extract(visit, "[0-8]+")), 
                              sd=5)) %>%
  dplyr::select(-random_subject_effect) %>%
  mutate(eye_visit = factor(paste0(eye, "_", visit)),
         id=factor(id))

mmrmfit1 <- mmrm::mmrm(
  formula = IOP ~ 0 + eye_visit + age + us(eye_visit | id),
  data = example,
  method = "Kenward-Roger")

mmrmfit1 %>%
  summary()

Answer 2

Is the following the correct specification of the model in R, specifically unsure how nesting should be specified:

lmer(IOP ~ Age + Gender + CornealThickness + as.factor(Time) + Gender*as.factor(Time) + CornealThickness*as.factor(Time) + (1|ID/Eye), data=data)

Alternatively, would it be the following:

lmer(IOP ~ Age + Gender + CornealThickness + as.factor(Time) + Gender*as.factor(Time) + CornealThickness*as.factor(Time) + (as.factor(Time)|ID/Eye), data=data)

The only difference between these models is that in the latter one, random slopes for each time point are specified.

Either model could be appropriate, but consider these issues:

• Serial correlation of residuals within subjects. The model fitted by lmer assumes that residuals are independent and identically distributed. However, as mentioned by Frank Harrell in the question comments, this type of study (longitudinal) should be expected to involve serial correlation. That is, the residuals are expected to feature a serial correlation structure in which the correlation between two measurements on the same subject decreases as the time gap widens. The lmer function cannot handle this so if would be a good idea to consider using nlme instead which does offer various correlation structures such as AR(1). Alternatively, again as mentioned by Frank Harrell, a Discrete Time Markov Model or a Generalised Least Squares model should be considered. See here for an excellent introduction to Markov Models for longitudinal data.

• The presence of random slopes in the second model is reasonable if we expect that the effect of each Time point to vary by subject, though be aware that this sometimes causes a difficulty in fitting leading to a singular model.

• Nesting structure: |ID/Eye says that Eye is nested within ID, which appears to be correct. @PBulls makes an interesting point about not needing the random intercepts for Eye, however I'm not sure that I fully agree. I would suggest fitting the nested model and a simpler model with random intercepts for just ID and compare the results.