Modelling longitudinal data with crossed random effects

Question

Let's say I have 40 participants. They are each measured three times (session). On each visit they see ten stimuli (item), which can be one of two types (type). They then get some score for each stimuli. How could I model the effect of stimulus type, session number, and their interaction, with lme4?

I've included some example data, and my best guess at the model with the full random effects structure below.

library("lme4")

participant <- rep(1:40, each = 30)
session <- rep(rep(1:3, each = 10), times = 40)
item <- rep(1:10, times = 120)
type <- rep(1:2, times = 600)
score <- rnorm(1200)

data <- cbind(participant, session, item, type, score)
data <- as.data.frame(data)

data$participant <- factor(data$participant)
data$session <- factor(data$session)
data$item <- factor(data$item)
data$type <- factor(data$type)

m <- lmer(score ~ type * session + (1 + type | participant / session) + 
                  (1 | item / session), data = data)

Answer 1

First, note that the simulated data above results in a singular model fit because there is no variation in the response among any of the random factors. This can be overcome with a simple modification:

library(lme4)

set.seed(15)
participant <- rep(1:40, each = 30)
session <- rep(rep(1:3, each = 10), times = 40)
item <- rep(1:10, times = 120)
type <- rep(1:2, times = 600)
# score <- rnorm(1200)                        ##### This line removed
score <- participant + item + rnorm(1200)     ##### This line added

data <- cbind(participant, session, item, type, score)
data <- as.data.frame(data)

data$participant <- factor(data$participant)
data$session <- factor(data$session)
data$item <- factor(data$item)
data$type <- factor(data$type)

m <- lmer(score ~ type * session + (1 + type | participant / session) + 
              (1 | item / session), data = data)

Second, now note that the model m above does not converge. This is because the fixed effect session is also included as a random grouping factor, which does not make any sense. A more sensible alternative is:

m0 <- lmer(score ~ type * session + (1 | participant) + (1 | item), data = data)

where I have also removed the random slope/coefficient for type by participant which also does not make sense given that the data are simple random draws - that is, there is no systematic association of score and type for each participant.

> summary(m0)
Linear mixed model fit by REML ['lmerMod']
Formula: score ~ type * session + (1 | participant) + (1 | item)
   Data: data

REML criterion at convergence: 3828.3

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.2095 -0.6430  0.0437  0.6908  3.1569 

Random effects:
 Groups      Name        Variance Std.Dev.
 participant (Intercept) 136.623  11.689  
 item        (Intercept)   9.856   3.139  
 Residual                  1.024   1.012  
Number of obs: 1200, groups:  participant, 40; item, 10

Fixed effects:
                Estimate Std. Error t value
(Intercept)    25.557744   2.322048  11.007
type2           0.988240   1.988126   0.497
session2       -0.025759   0.101185  -0.255
session3       -0.042207   0.101185  -0.417
type2:session2 -0.028411   0.143098  -0.199
type2:session3 -0.002558   0.143098  -0.018

Correlation of Fixed Effects:
            (Intr) type2  sessn2 sessn3 typ2:2
type2       -0.428                            
session2    -0.022  0.025                     
session3    -0.022  0.025  0.500              
type2:sssn2  0.015 -0.036 -0.707 -0.354       
type2:sssn3  0.015 -0.036 -0.354 -0.707  0.500

This model estimates the fixed effect of stimulus type, session number, and their interaction, while controlling for crossed random effects for participant and item, as requested.

Random slopes for type, session and their interaction could potentially be included, provided that the data supports such a structure (the simulated data do not) and provided these are indicated by the underlying theory of the data generation process. It is not generally a good idea to begin with a maximal random effects structure.