# lme() and lmer() give different estimates for crossed random effects [closed]

I have a question on statistical mixed linear model analysis with nlme in R. There are a couple of posts that are related to my question, I think (see here and here), but I'm not sure they give the solution to my problem.

I would like to fit a linear mixed model to data from an experiment with crossed random effects for subjects and stimuli. Here are simplified data simulated based on my data and with great help from this paper and the accompanying R code.

library(tidyverse)
library(lme4)
library(nlme)

set.seed(1234)

# parameters for group
beta_0  <- 55.145 # intercept
omega_0 <-  3.644 # by-item random intercept sd
tau_0  <- 7.671 # by-subject random intercept sd
sigma  <- 26.683 # residual (error) sd

# number of subjects and items
n_subj <- 150
n_items  <-  36

items <- data.frame(
item_id = seq_len(n_items),
O_0i = rnorm(n = n_items, mean = 0, sd = omega_0)
)

subjects <- data.frame(
subj_id = seq_len(n_subj),
T_0i = rnorm(n = n_subj, mean = 0, sd = tau_0)
)

# add subject IDs
subjects1$subj_id <- seq_len(n_subj) trials <- crossing(subjects, items) %>% mutate(e_si = rnorm(nrow(.), mean = 0, sd = sigma)) dat_sim <- trials %>% mutate(y = beta_0 + T_0i + O_0i + e_si) %>% select(subj_id, item_id, y)  I would typically use lme4::lmer() to fit a model to this data but I have a between subjects factor in my experiment and the data turned out to be heteroscedastic. As far as I know, I need to use nlme::lme() to fit different residual variances for each between-subjects group, so lme4::lmer() is not an option. Now, I would like to fit a model in nlme::lme() that matches my original model in lme4::lmer(). For demonstration, I fit intercept-only models. (In reality I would have fixed effects, too. But this is not relevant for the question I have.) My lmer() model would look like this: m.lmer <- lmer(y ~ 1 + (1|subj_id) + (1|item_id), data = dat_sim, REML = T, control = lmerControl(optimizer="nloptwrap"))  With the help of this post, I pieced this model together for lme(): dat_sim$dummy <- factor(1)

m.lme <- lme(y ~ 1,
random = list(dummy =
pdBlocked(list(pdIdent(~subj_id-1),
pdIdent(~item_id-1)))),
data = dat_sim,
method = "REML")


And here is my problem: The two models give different estimates for fixed and especially random effects:

summary(m.lmer)\$coefficients
#             Estimate  Std. Error     df  t value     Pr(>|t|)
# (Intercept) 54.08182  0.8576796 102.7496 63.05597 5.819007e-84

VarCorr(m.lmer)
# Groups   Name        Std.Dev.
# subj_id  (Intercept)  7.6215
# item_id  (Intercept)  2.8036
# Residual             26.4984

summary(m.lme)
# Fixed effects:  y ~ 1
#             Value    Std.Error    DF   t-value     p-value
# (Intercept) 53.66397 0.9505988 5399 56.45281       0

VarCorr(m.lme)
# dummy = pdIdent(subj_id - 1), pdIdent(item_id - 1)
#           Variance     StdDev
# subj_id  6.408743e-04  0.02531550
# item_id  6.743363e-03  0.08211798
# Residual 7.655888e+02 27.66927555



But if I remove one of the random effects, my results are identical across lmer() and lme():

m.lmer.s <- update(m.lmer, ~ 1 + (1|subj_id))
VarCorr(m.lmer.s)
# Groups   Name        Std.Dev.
# subj_id  (Intercept)  7.6072
# Residual             26.6463

m.lme.s <- update(m.lme, random = ~1|subj_id)
VarCorr(m.lme.s)
# subj_id = pdLogChol(1)
# Variance  StdDev
# (Intercept)  57.86926  7.607185
# Residual    710.02435 26.646282

m.lmer.i <- update(m.lmer, ~ 1 + (1|item_id))
VarCorr(m.lmer.i)
# Groups   Name        Std.Dev.
# item_id  (Intercept)  2.7336
# Residual             27.5727

m.lme.i <- update(m.lme, random = ~1|item_id)
VarCorr(m.lme.i)
# item_id = pdLogChol(1)
# Variance   StdDev
# (Intercept)   7.472733  2.73363
# Residual    760.252039 27.57267



Why do the models with both random effects give so such different results and why does this seem to depend on whether only one or both random intercepts are fitted? Is my model specification for lme() wrong? Or is there an issue with a local maximum when fitting two random effects with lmer() (see this post)? Could anyone explain to me what is happening here?

• The nlme results are nowhere near the simulated variances. This seems to suggest that the model you specified with lme does not do what you want it to do. I am more familiar with lme4, which to me looks correctly specified. Perhaps you should remove the -1 part in the lme one since you don't have a random intercept like in the linked post? Dec 3, 2021 at 10:30
• Too late to edit, but I meant to say you don't have a random slope Dec 3, 2021 at 17:06
• @FransRodenburg Thank you for your comment. Yes, that makes sense, the results from lme() are not at all close to the intended variances. I tried removing the -1 but this returns an error Error in pdConstruct.pdBlocked(object, form = form, nam = nam, data = data, : cannot have duplicated column names in a "pdMat" object
– Jass
Dec 6, 2021 at 9:41
• Your codes for both lme and lmer are fine. The disparity is because subj_id and item_id are generated as integers. They must be factors for lme to recognize as grouping levels instead of a continuous variable, unless they appear after | which is forbidden inside pdBlocked(). See more at stackoverflow.com/questions/36643713. 2 days ago