This is a great question and answer set, and is a good match for an RCT I’m working on (currently writing the statistical analysis plan, analysis happening next year). I had written a draft post abut implementation of a non-linear specification for follow-up time in code, but Andrew’s outline corresponds closely to my scenario, so figured it was good to keep this in one location.
For additional context, our trial has four post-baseline data collections (at 3, 6, 9, and 12 months from randomisation). I feel reasonably comfortable that fluctuations in timing will be non-informative, in the sense that they should be unrelated to the outcome under study.
Modelling time effects as non-linear
My current interpretation is that specifying follow-up Time as a non-linear function can be done using standard modelling approaches described in RMS, and I can mentally visualise that this should equate to specification of different trajectories of outcomes over time for the two groups (constrained by the functional form).
I’ve adapted the syntax below from Marc’s syntax in the discussion linked below as item #1, which was my original starting point for reading up on this topic.
Here I’ve modelled Time with restricted cubic splines with 3 knots, so 2 parameters. As a reminder, we have four follow-up measurements (also discussed in the second link below).
MOD <- lme(Response ~ Group*rcs(Time, parms=3) + T1 + Group:T1,
random = ~1 | ID,
data = DF.MOD, na.action = na.omit,
correlation = corAR1(form = ~ as.numeric(Time) | ID))
Then the final step is to estimate the treatment effect between groups at the 12-month endpoint (not illustrated, but effectively the difference in means estimated at 12 months in the two groups). I will be using the emmeans package to do this (it needs a slightly fiddlier approach than the syntax above, I can provide detail in later messages if useful).
Questions:
-
Does the specification of the second model above look reasonable/appropriate for allowing for non-linear trends to estimate a difference between randomised groups at 12 months? Or is there additional complexity that I haven’t thought of (including e.g. for the correlation structure)?
-
Given that this doesn’t seem to be a widely deployed analytical strategy (based on discussions here and elsewhere), would it be preferable to state that the “primary” analysis should deploy the “time of follow-up” as occurring at the specific timed option (a factor); and then the time-as-continuous analysis as a follow-up analysis? (this is discussed at link #2 below)
Thank you in advance for any input or feedback! I’d also be interested if anyone has a link to a paper where they or others have deployed such a method.
Links to other discussions on this topic
- https://discourse.datamethods.org/t/linear-mixed-model-for-3-time-points/5753 (contains Marc’s syntax as above)
- https://discourse.datamethods.org/t/data-collected-outside-of-guidelines-window/6861 (has a great paper on the topic of (non)informative variation in measurement times.