# Binomial glmer() singular despite "lots" of data

I have been reading different questions about how easy it is to bump into singularities when fitting mixed effects models with glmer(). In general, the idea is that singularities might arise from very complex random structures. If the random structure is simple, it might also happen when the data is not enough to calculate the variance-covariance matrix... see for example this page by Ben Bolker, Robert Long's answer to this post or the help page of isSingular().

However, the model I'm trying to fit is very simple:

mod.detection_rand <- glmer(reaction ~ Pedra + (1|Channel), family="binomial", data = garotes)
boundary (singular) fit: see ?isSingular


... and apparently I have enough data for the different (fixed and random) predictor variable combinations:

library(tidyverse)
garotes %>%
group_by(Channel, Pedra) %>%
summarise(n = n())
# A tibble: 16 x 3
# Groups:   Channel [8]
Channel Pedra     n
<int> <fct> <int>
1       1 No       13
2       1 Yes      13
3       2 No       14
4       2 Yes      12
5       3 No       12
6       3 Yes      14
7       4 No       13
8       4 Yes      13
9       5 No       13
10       5 Yes      13
11       6 No       14
12       6 Yes      12
13       7 No       13
14       7 Yes      13
15       8 No       14
16       8 Yes      12


What do you think?

EDIT: Here's the summary of the model, summary(mod.detection_rand)

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial  ( logit )
Formula: reaction ~ Pedra + (1 | Channel)
Data: garotes

AIC      BIC   logLik deviance df.resid
261.5    271.5   -127.7    255.5      205

Scaled residuals:
Min      1Q  Median      3Q     Max
-1.8533 -0.9449  0.5396  0.5396  1.0583

Random effects:
Groups  Name        Variance Std.Dev.
Channel (Intercept) 0        0
Number of obs: 208, groups:  Channel, 8

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  -0.1133     0.1946  -0.582     0.56
PedraYes      1.3473     0.3066   4.394 1.11e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr)
PedraYes -0.635
convergence code: 0
boundary (singular) fit: see ?isSingular


EDIT2: Following Billy's comment:

bobyqa : boundary (singular) fit: see ?isSingular
[OK]
Nelder_Mead : boundary (singular) fit: see ?isSingular
[OK]
nlminbwrap : boundary (singular) fit: see ?isSingular
[OK]
nmkbw : boundary (singular) fit: see ?isSingular
[OK]
optimx.L-BFGS-B : boundary (singular) fit: see ?isSingular
[OK]
nloptwrap.NLOPT_LN_NELDERMEAD : boundary (singular) fit: see ?isSingular
[OK]
nloptwrap.NLOPT_LN_BOBYQA : boundary (singular) fit: see ?isSingular
[OK]


EDIT3: Following Isabella's answer:

I checked the structure of the outcome variable (reaction). Here's the table of outcomes:

library(tidyverse)
garotes %>%
group_by(Channel, Pedra, reaction) %>%
summarise(n = n()) %>%
print(n = Inf)
# A tibble: 32 x 4
# Groups:   Channel, Pedra [16]
Channel Pedra   reaction  n
<int> <fct>    <int>  <int>
1       1 No           0     6
2       1 No           1     7
3       1 Yes          0     3
4       1 Yes          1    10
5       2 No           0     7
6       2 No           1     7
7       2 Yes          0     2
8       2 Yes          1    10
9       3 No           0     8
10       3 No           1     4
11       3 Yes          0     6
12       3 Yes          1     8
13       4 No           0     7
14       4 No           1     6
15       4 Yes          0     3
16       4 Yes          1    10
17       5 No           0     8
18       5 No           1     5
19       5 Yes          0     1
20       5 Yes          1    12
21       6 No           0     6
22       6 No           1     8
23       6 Yes          0     2
24       6 Yes          1    10
25       7 No           0     6
26       7 No           1     7
27       7 Yes          0     2
28       7 Yes          1    11
29       8 No           0     8
30       8 No           1     6
31       8 Yes          0     4
32       8 Yes          1     8


Apparently, there are both types of outcomes for all Channels and all Pedratreatments... so it is not like the example Isabella presented... furthermore, I tried to model this GLMM with the library(GLMMadaptive) and it did not converge either.

EDIT4: The data set I'm using, in case someone's curious.

Channel Pedra   reaction
1   No  1
2   No  0
3   No  0
4   No  0
5   No  0
6   No  1
7   No  0
8   No  0
1   No  1
2   No  1
3   No  1
4   No  1
5   No  0
6   No  0
7   No  0
8   No  0
1   No  0
2   No  1
3   No  0
4   No  0
5   No  0
6   No  0
7   No  0
8   No  1
1   No  0
2   No  1
3   Yes 0
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 0
1   Yes 1
2   Yes 1
3   Yes 0
4   Yes 0
5   No  0
6   No  1
7   Yes 1
8   Yes 1
1   Yes 0
2   Yes 1
3   Yes 1
4   Yes 1
5   Yes 1
6   Yes 0
7   No  1
8   No  1
1   Yes 1
2   Yes 1
3   Yes 1
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 1
1   Yes 1
2   Yes 1
3   Yes 1
4   Yes 1
5   Yes 0
6   Yes 1
7   Yes 1
8   Yes 1
1   Yes 1
2   Yes 1
3   Yes 0
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 0
8   Yes 0
1   Yes 1
2   Yes 1
3   Yes 0
4   Yes 0
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 0
1   Yes 1
2   Yes 1
3   Yes 0
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 0
8   Yes 0
1   Yes 1
2   Yes 0
3   Yes 1
4   Yes 0
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 1
1   Yes 1
2   Yes 1
3   Yes 0
4   Yes 1
5   Yes 1
6   Yes 0
7   Yes 1
8   Yes 1
1   Yes 1
2   Yes 1
3   Yes 1
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 1
1   Yes 0
2   Yes 0
3   Yes 1
4   Yes 1
5   Yes 1
6   Yes 1
7   Yes 1
8   Yes 1
1   Yes 1
2   No  0
3   Yes 1
4   No  1
5   Yes 1
6   No  1
7   Yes 1
8   No  1
1   No  0
2   Yes 1
3   No  0
4   Yes 1
5   No  1
6   Yes 1
7   No  1
8   Yes 1
1   Yes 0
2   No  1
3   Yes 1
4   No  0
5   Yes 1
6   No  1
7   Yes 1
8   No  0
1   No  0
2   No  1
3   No  1
4   No  0
5   No  1
6   No  0
7   No  0
8   No  0
1   No  1
5   No  0
3   No  1
4   No  1
2   No  1
6   No  0
7   No  1
8   No  0
1   No  0
5   No  0
3   No  0
4   No  0
2   No  1
6   No  0
7   No  0
8   No  0
1   No  1
5   No  1
3   No  1
4   No  0
2   No  0
6   No  1
7   No  1
8   No  0
1   No  1
5   No  0
3   No  0
4   No  1
2   No  0
6   No  1
7   No  1
8   No  1
1   No  1
5   No  1
3   No  0
4   No  1
2   No  0
6   No  1
7   No  1
8   No  1
1   No  1
5   No  1
3   No  0
4   No  0
2   No  0
6   No  1
7   No  0
8   No  0
1   No  0
5   No  0
3   No  0
4   No  1
2   No  0
6   No  0
7   No  1
8   No  1


Thank you very much for all of your responses, in any case! Learning a lot from them!

• If you include the output of summary(mod.detection_rand), we can usually see why it singular.
– Eoin
Sep 18, 2020 at 14:24
• Thanks Eoin, just edited the main question to include the summary(mod.detection_rand). Sep 18, 2020 at 14:45
• The following seems to work: mixed_model(reaction ~ Pedra , random = ~ 1 | Channel, data = DF, family = binomial(), iter_EM = 200) Sep 21, 2020 at 14:11
• @DimitrisRizopoulos it works! Ha! But... why was it so difficult to make it converge? Any hypothesis? Thanks a lot! Sep 21, 2020 at 15:37

A further comment: I took a look at your data, and it's clear, again, that there is no evidence of systematic variance between the different channels. This is why the mixed model estimates the between-channel variance to be $$0$$, making the model singular.

You can see this in the figure below, where the standard errors for almost every channel overlap...

...and can confirm it by ANOVA decomposition of a fixed-effects GLM, showing that there is no significant main effect of Channel (p = .986).

m_fixed_effects = glm(cbind(n, total) ~ Pedra + factor(Channel),
data=positive, family=binomial)
car::Anova(m_fixed_effects)
# Analysis of Deviance Table (Type II tests)
#
# Response: cbind(n, total)
#                 LR Chisq Df Pr(>Chisq)
# Pedra             4.9148  1    0.02663 *
# factor(Channel)   1.3859  7    0.98600
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


## Code

library(tidyverse)
#   Channel Pedra reaction  n
# 1       1    No        0  6
# 2       1    No        1  7
# 3       1   Yes        0  3
# 4       1   Yes        1 10
# 5       2    No        0  7
# 6       2    No        1  7

df = df %>%
group_by(Channel, Pedra) %>%
mutate(total = sum(n),
prop = n / total,
se = sqrt((prop * (1-prop)) / n)) %>%
ungroup()
positive = filter(df, reaction==1)

ggplot(positive, aes(Pedra, prop, group=Channel, color=factor(Channel))) +
geom_path(position = position_dodge(width=.1)) +
geom_point(position = position_dodge(width=.1)) +
stat_summary(fun.data=mean_se, group=1, color='black',
position = position_nudge(x=c(-.2, .2))) +
geom_linerange(mapping=aes(ymin=prop-se, ymax=prop+se),
position = position_dodge(width=.1)) +
geom_hline(linetype='dashed', yintercept=.5) +
coord_cartesian(ylim=c(0, 1)) +
labs(color='Channel',  y='Proportion positive reactions',
caption='Error bars show SEM')

m_fixed_effects = glm(cbind(n, total) ~ Pedra + factor(Channel),
data=positive, family=binomial)
car::Anova(m_fixed_effects)
# Analysis of Deviance Table (Type II tests)
#
# Response: cbind(n, total)
#                 LR Chisq Df Pr(>Chisq)
# Pedra             4.9148  1    0.02663 *
# factor(Channel)   1.3859  7    0.98600
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

• Good work, nice answer (+1) Sep 22, 2020 at 10:07
• Wow, thanks a lot @Eoin, that's a very clear way to show it. Thanks again for your answer and for your time! Sep 23, 2020 at 7:34

Isabella made some excellent points. This can also happen when there is very little variation at the channel level. Perhaps channels are just very similar to each other so their variance really is close to zero and therefore not needed in the model. You can assess this by fitting a glm and see if the inferences are similar.

• In fact, this is what the output of summary(mod.detection_rand) is telling us. The model estimates that the variance in intercepts across the different channels (in the random effects block) is $0$. In other words, there are apparently no systematic differences between channels once the effect of Pedra has been taken into account.
– Eoin
Sep 20, 2020 at 17:52
• @Eoin yes, that's exactly what I meant, but Isabella also pointed out that this can happen due to data sparsity and in my experience it can happen for unknown reasons when there is actual variation. Sep 20, 2020 at 18:12
• Agreed, but we can see from the fixed effects and the balanced design that sparsity isn't the problem here (unless some very exotic contrasts were used for Pedra).
– Eoin
Sep 21, 2020 at 8:47
• @Eoin and @Robert, I don't think variance at the Channel level is 0. This must be some kind of artifact. In fact, there are big differences across channels, that's why we decided to include them as a random effect... weird... anyway, thanks a lot for your help! Sep 21, 2020 at 10:29
• @JordiPagès yes that's what I meant in my last comment. Have you tried increasing nAGQ > 1 ? Also have you tried other packages apart from GLMMAdaptive, or setting different start points in the optimizer ? Can you share the data itself ? Sep 21, 2020 at 10:59

Because this is a mixed effects binary logistic regression model, it assumes that your outcome variable is binary with values coded as either 0 or 1.

What you need to investigate is whether you have enough 1's present in your response variable for a sufficient number of 'subjects'. (In your case, subject stands for channel.)

Here is a made-up example which produces the same warning as wnat you got:

SubjectID <- rep(1:5, each = 3)
SubjectID

Outcome <- rep(0, 15)
Outcome[1] <- 1

Data <- data.frame(Outcome, SubjectID)
str(Data)

Data

library(lme4)

glmer(Outcome ~ 1 + (1|SubjectID), family="binomial", data = Data)


In this example, there are 5 subjects such that 4 of them have only 0 outcome values and one of them has outcome values which include a single value of 1. (Each subject has 3 outcome values in total.)

Even if you give each of the subject in this made-up example a value of 1 for their first outcome value, you will still get the same error message when fitting the model:

Outcome <- rep(0, 15)

Outcome[c(1, 4, 7, 10, 13)] <- 1


However, if all 4 subjects who initially had only 0 values are allowed to keep these values and the first subject receives two values of 1, the error message disappears:

Outcome <- rep(0, 15)

Outcome[c(1,2)] <- 1


Once you understand better the pattern of 0 and 1 values for the outcome variable among your study subjects, the other thing you can try is fitting your model with the mixed_model() function from the GLMMadaptive package in R.

For the small example provided here, this function would be used like this:

library(GLMMadaptive)

m <- mixed_model(fixed = Outcome ~ 1,
random = ~ 1 | SubjectID,
data = Data,
family = binomial())
summary(m)

• Fantastic! I will definitively try that! Cheers! Sep 18, 2020 at 15:48
• the only big difference between lme4::glmer and GLMMadaptive::mixed_model is the default number of quadrature points (1 for glmer, 11 for mixed_model in this case); you can adjust this with nAGQ (in either package). Sep 18, 2020 at 23:33
• Thank you very much for this complete response Isabella. I have checked the outcome variable (reaction in this case) and the GLMMadaptive::mixed_model function. See EDIT3 in my original question. Cheers! Sep 21, 2020 at 10:30

Out of curiosity, does the error arise when you use an alternative estimator? It could be that the estimator is for some reason getting stuck at a singularity. You may just try the following: mod.alt_est <- allFit(mod.detection_rand). Alternatively, you may need a Bayesian solution to help regularize the estimation and force it away from a singularity (try blme package if the allFit function doesn't produce an estimator that works).

• Thanks Billy. It does give the same warning for all optimizers. See edit in the main question. Sep 18, 2020 at 15:43