Hi Vincent Nadon

I am assuming time is entered as a continuous variable and you intend to have random slopes over time for individual participants (ID). I also assume that the same participants were assessed over the days. In this case, you can use the following syntax to fit separate slopes for individual participants over different days by nesting the days under the participants.

random = ~ time|ID/Day

This will also estimate a random intercept for each participant, and each participant, day combination. I would also suggest adding day as a fixed effect. Looking at raw data and residual plots to make sure that the linear trends can explain the variance in outcome across time may be needed. Here are a few examples for nested designs with explanations: https://rpsychologist.com/r-guide-longitudinal-lme-lmer

I hope it was helpful and my assumptions are correct.

Do you expect two different time trends?

1 the trend over the 4 days : the secular trend as people say get steadily better.

2 around this trend, you could then have a diurnal pattern, which may be cyclical, say low in morning, high by lunchtime, then lowering towards the evening, and this may asymmetric with a faster rise in the morning and a slower fall in the evening.

around these two trends could be stochastic fluctuations that could pertain to the individual, the day and the occasion.

i would start by doing some theorising about likely processes and then averaging and graphing the response to give you and idea before embarking on the modelling following Usman Rashid ‘s suggestions.

Usman Rashid indeed the same participants were assessed over the 4 days and time is continuous (expressed in minutes since beginning of the day).

Adding Day as a fixed effect is also a good idea.

Thanks for the info.

Kelvyn Jones I would expect that my time-varying predictors (such as cumulative noise level) would explain my Outcome more than time on its own. I would expect similar trends based on the time-varying predictor values. So if the participant is tested in similar conditions as previous day, the outcome should have the same trend considering I'm looking at short-term effects here.

Updated: sorry that I was not clear. For me this a two stage procedure. First I would find out what needs to be explained and therefore thinking, averaging, graphing and then an unconditional model with just some function of the two types of time. And to consider the nature of the unexplained variation . Stage 2, having grasped the patterns I would then proceed to a conditional model with explanatory variables to account for this variation.

This way you see what needs to be accounted for, and you can see if your predictions are an improvement on simply elapsed time .

Have a look at

Longitudinal Analysis: Modeling Within-Person Fluctuation and Change (by Lesa Hoffman)

http://www.pilesofvariance.com/

By the two types of time , I mean 0 1 2 3 etc for days and

0 1 2 3 4 ... 15; 0 ,1 2 3 4 ....15 etc. This could be continuous time as long as it starts back each day at 0.

The intercept is then the mean response on day 1 measurement 1.

I am presuming that there is a diurnal change as you would not have otherwise gone to the effort of so many repeated measures within the day.