# Repeated measures mixed modeling. Why is Time (day, second etc.) a fixed effect?

Let's say I have an experiment where I repeatedly measure something over Time (1:10), say 10 days.

Here I simulate some data...

set.seed(101)

N = 10 # number of repeats
n = 10 # number of individuals

# Data
Time = rep(1:N,n) # measured over time


Note that I set the fixed effect of time to 0.

# Fixed effects
X = matrix(c(Time), ncol = 1)
B = c(0) # Set fixed effect of Time's coefficient to 0

# Random effects
id = factor(c(0L, cumsum(diff(Time) < 0))) # make id for each block of Time
Z = model.matrix(~id+id:Time) # create design matrix

u = c(rnorm(n),rnorm(n,2,1)) # random slopes and intercepts
e = rnorm(N*n) # error

y = X%*%B + Z%*%u + e # Model


Now let's say I fit two models:

library(lme4)
fit1 = lmer(y ~ 1 + Time + (1 + Time|id))
fit2 = lmer(y ~ 1 + (1+ Time|id))


Comparing the two with anova()

anova(fit1,fit2)
Models:
fit2: y ~ 1 + (1 + Time | id)
fit1: y ~ 1 + Time + (1 + Time | id)
Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
fit2  5 380.80 393.83 -185.40   370.80
fit1  6 329.41 345.05 -158.71   317.41 53.389      1  2.736e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Here we see that fit1, which includes Time as a fixed effect, performs much better, but why is this true when its coefficient is set to 0?

I have a feeling it has something to do with this line.

Z = model.matrix(~id+id:Time) # create design matrix


but my intuition is failing me.

## 1 Answer

If you look at the summary of your models, you'll see that the estimated time effect is actually greater than zero (about 2 given your seed). I think your error is in the

u = c(rnorm(n),rnorm(n,2,1)) # random slopes and intercepts


It should actually be flipped based on your design matrix, so it reads random intercepts and slopes.

u = c(rnorm(n,2,1), rnorm(n)) # random intercepts and slopes


Before you had an intercept of zero and a slope of 2, when you actually wanted it the other way around.

This yields what you're looking for I think. Now if you use summary on mod1, you'll see that the fixed effect of time is close to zero (it would be closer with more than N = 10), and it shows that mod2 provides more parsimonious model fit according to AIC.

Data:
Models:
fit2: y ~ 1 + (1 + Time | id)
fit1: y ~ 1 + Time + (1 + Time | id)
Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
fit2  5 352.85 365.88 -171.43   342.85
fit1  6 354.69 370.32 -171.35   342.69 0.1584      1     0.6907