# lme4: Three-Level Autoregressive Model - Random Effects

I would like to fit a three level autoregressive model in lme4 to account for my longitudinal experience sampling data (beeps nested in days, nested in persons).

Several resources suggest to account for a three level data structure by including a random term with a "/":

threelevel.AR.1 <- lmer(affect ~ ∼1 + lev2pred + lev1predfor3l +
(1 + lev2pred + lev1predfor3l | PersonF/DayF),
data=df)


De Haan-Rietdijk et al. (2016), however, suggest to build the model as follows:

threelevel.AR.2 <- lmer(affect ∼1+lev2pred+ lev1predfor3l +
(1 | PersonF/DayF) + (1 + lev2pred + lev1predfor3l | PersonF),
data = ESM, REML = FALSE)


My questions are:

1 - What is the difference between the two models?

2 - Would it be correct to use the first model?

Thanks!

The full citation for the paper is: de Haan-Rietdijk, S., Kuppens, P., & Hamaker, E. L. (2016). What’s in a Day? A Guide to Decomposing the Variance in Intensive Longitudinal Data. Frontiers in Psychology, 7. https://doi.org/10.3389/fpsyg.2016.00891

First it's important to note that in lmer, (1+A+B|person/day) is a short way to write (1+A+B|person) + (1+A+B|person:day). Therefore, the first model estimates random intercepts and slopes for each person, and for each combination of person and day. This is equivalent to the assumption that each person comes with a set of random intercepts and slopes that, on average, stay the same across all days. However, each day "within" a person is also allowed to bring its own full set of random effects, so that on each day, the relationship between independent variables and response can be slightly different from the "average" response pattern of that person.