How to determine if a linear mixed model is overdetermined in lme4/nlme?

Question

In the Orthodont dataset in nlme, there are 27 subjects and each subject is measured at 4 different ages. I wish to use this data to explore at what condition the model will be overdetermined. Here are the models:

library(nlme)
library(lme4)

m1 <- lmer( distance ~ age + (age|Subject), data = Orthodont )
m2 <- lmer( distance ~ age + I(age^2) + (age|Subject), data = Orthodont )
m3 <- lmer( distance ~ age + I(age^2) + I(age^3) + (age|Subject), data = Orthodont )

m1nlme <- lme(distance ~ age, random = ~ age|Subject, data = Orthodont)
m2nlme <- lme(distance ~ age + I(age^2), random = ~ age|Subject, data = Orthodont)
m3nlme <- lme(distance ~ age + I(age^2) + I(age^3), random = ~ age|Subject, data = Orthodont)
m4nlme <- lme(distance ~ age + I(age^2) + I(age^3), random = ~ age + I(age^2) + I(age^3)|Subject, data = Orthodont)

Of all of the above models, only m3 throws a warning message:In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,:Model failed to converge with max|grad| = 0.00762984 (tol = 0.002, component 1).

Questions:

1. What does the warning message suggest and if it is sensible to ignore this message?
2. For m2, the model estimates fixed effect of intercept and fixed coefficient for age and I(age^2), together with the random effect parameter sigma^2_intercept, sigma^2_age, and sigma^2_intercept:age. So a total of 1+2+3=6 parameters are estimated for each Subject. But there are only 4 observations per subject. Why does not m2 throws an error? Isn't m2 overdetermined? Am I counting the number of paratermeters anywhere incorrectly?

Answer 1

1. The warning message means that the model fit may be a bit numerically unstable; it is done by numerically checking the scaled gradient, but as this depends in turn on the gradient and Hessian estimated by finite differences, which are themselves subject to numerical error. As I've stated in many different venues, these warnings definitely tend to be over-sensitive/likely to be false positives: see e.g. ?lme4::convergence, ?lme4::troubleshooting. The gold standard is to use allFit() to refit the model with a variety of optimizers and make sure that the results from different optimizers are close enough to the same for your purposes.

2. There are two random effects values (BLUPs or conditional modes) per subject - the subject-level deviation of the intercept and slope wrt age. For values, we will be in trouble if the number of values is greater than or equal to the number of observations per group (or, for GLMMs without a scale parameter such as the Poisson, if the number of values is strictly greater than the number of observations per group). For parameters, there are up to four fixed-effect parameters (intercept, linear, quadratic, cubic terms wrt age) and three RE parameters (variance of the intercept, variance of the slope, covariance between intercept and slope), but these 7 parameters are estimated at the population level - the appropriate comparisons are with either the total number of observations or with the number of groups, not with the number of observations per group.

In general you should probably think about the number of observations when considering the number of fixed effect parameters and the number of groups when considering the number of random effect parameters; "10 [observations/groups] per parameter" is a not-unreasonable starting rule of thumb.