Question

Is there a difference between the functions fitted() and predict()? I’ve noticed that mixed models from lme4 work with fitted() but not predict().

Answer 1

Yes, there is. If there is a link function relating the linear predictor to the expected value of the response (such as log for Poisson regression or logit for logistic regression), predict returns the fitted values before the inverse of the link function is applied (to return the data to the same scale as the response variable), and fitted shows it after it is applied.

For example:

x = rnorm(10)
y = rpois(10, exp(x))
m = glm(y ~ x, family="poisson")

print(fitted(m))
#         1         2         3         4         5         6         7         8 
# 0.3668989 0.6083009 0.4677463 0.8685777 0.8047078 0.6116263 0.5688551 0.4909217 
#         9        10 
# 0.5583372 0.6540281 
print(predict(m))
#          1          2          3          4          5          6          7 
# -1.0026690 -0.4970857 -0.7598292 -0.1408982 -0.2172761 -0.4916338 -0.5641295 
#          8          9         10 
# -0.7114706 -0.5827923 -0.4246050 
print(all.equal(log(fitted(m)), predict(m)))
# [1] TRUE

This does mean that for models created by linear regression (lm), there is no difference between fitted and predict.

In practical terms, this means that if you want to compare the fit to the original data, you should use fitted.