Singular fit for model and want to retain random intercept

Question

I've read many posts about singular fit issues and this: https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#singular-models-random-effect-variances-estimated-as-zero-or-correlations-estimated-as---1 , however I've seen that most of the advice revolves around simplifying random effects. In my case I have just one random intercept that I believe is necessary based on the experimental design so I'm not sure what the best way to proceed is. I am looking at success on a task where the outcome is binary 1=solved, 0= not solved and how other variables contribute to success. I have 13 subjects that each have 10-12 trials so I believe it's important to have the individual as a random intercept when modeling success. The other variables are summarized here:

• Sex- categorical, 2 levels
• Zoo- categorical, 2 different zoos
• Age- continuous
• Persistence- continuous, proportion of total time engaged with task per trial
• Neophilia- continuous, latency to approach task in very first trial
• Diversity- continuous, total number of actions employed per trial.

My full model is newdmod<-glmer(Solved~Sex+Zoo+Age+Persistence+Neophilia+Diversity+(1|Individual),door, family=binomial) (output below) I get a singular fit error with this model and I can see that the variance for my random intercept is almost 0. As I understand it, this would indicate that my random intercept should be removed, but due to my experimental design I think it's important to keep it in. I've also checked for collinearity of this full model and none of the fixed effects have high VIFs.

Generalized linear mixed model fit by maximum likelihood
  (Laplace Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: 
Solved ~ Sex + Zoo + Age + Persistence + Neophilia + Diversity +  
    (1 | Individual)
   Data: door
 AIC      BIC   logLik deviance df.resid 
99.7    123.7    -41.8     83.7      141 


Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.6817  0.0171  0.0910  0.2962  3.8144
Random effects:
 Groups     Name        Variance  Std.Dev. 
 Individual (Intercept) 2.774e-18 1.665e-09
Number of obs: 149, groups:  Individual, 13
Fixed effects:
            Estimate Std. Error z value Pr(>|z|)

(Intercept)  1.94544    1.04467   1.862 0.062568 .

SexM         1.68561    0.72139   2.337 0.019459 *

ZooRGZ      -1.04102    0.59511  -1.749 0.080240 .

Age         -0.04028    0.02218  -1.816 0.069426 .

Persistence  8.19243    1.60335   5.110 3.23e-07 ***
Neophilia   -0.03757    0.01167  -3.220 0.001283 ** 
Diversity   -0.36305    0.10136  -3.582 0.000341 ***

Signif. codes:  0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
            (Intr) SexM   ZooRGZ Age    Prsstn Neophl
SexM         0.053

ZooRGZ      -0.275 -0.095

Age         -0.754 -0.219 -0.075

Persistence  0.108  0.356 -0.169 -0.291

Neophilia   -0.557 -0.564 -0.081  0.689 -0.443

Diversity   -0.508 -0.241  0.260  0.314 -0.648  0.329
optimizer (Nelder_Mead) convergence code: 0 (OK)
boundary (singular) fit: see ?isSingular

Interestingly, if I remove one of the fixed variables (Zoo, the one with the highest p value) the singularity problem disappears and the random intercept has variance again.

Generalized linear mixed model fit by maximum likelihood
  (Laplace Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: 
Solved ~ Sex + Age + Persistence + Neophilia + Diversity + (1 |  
    Individual)
   Data: door
 AIC      BIC   logLik deviance df.resid 

100.4    121.5    -43.2     86.4      142
Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.5278  0.0165  0.0816  0.2958  3.6741
Random effects:
 Groups     Name        Variance Std.Dev.
 Individual (Intercept) 0.3006   0.5483

Number of obs: 149, groups:  Individual, 13
Fixed effects:
            Estimate Std. Error z value Pr(>|z|)

(Intercept)  1.60015    1.16986   1.368 0.171372

SexM         1.64549    0.80376   2.047 0.040634 *

Age         -0.04468    0.02510  -1.780 0.075003 .

Persistence  7.89091    1.63103   4.838 1.31e-06 ***
Neophilia   -0.04018    0.01314  -3.058 0.002230 ** 
Diversity   -0.34100    0.10304  -3.309 0.000935 ***

Signif. codes:  0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
            (Intr) SexM   Age    Prsstn Neophl
SexM        -0.008

Age         -0.804 -0.199

Persistence  0.023  0.331 -0.284

Neophilia   -0.602 -0.525  0.680 -0.435

Diversity   -0.466 -0.223  0.323 -0.564  0.336

So my question is whether it is legitimate to simplify the full model by removing a fixed effect based on the p value with the justification that this variable is contributing least to variation in success? Especially since this fixes the singularity issue and given that I would like to keep the random effect. Or could my result with the full model provide enough support to argue that there is not enough individual variation so I don't have to account for repeated measures? Any and all help would be appreciated! I'm also happy to provide more information if it is helpful!

Edit: Adding GLM output for glm(Solved ~ Sex + Zoo + Age + Persistence + Neophilia + Diversity + Individual, family=binomial)

Call:
glm(formula = Solved ~ Sex + Zoo + Age + Persistence + Neophilia + 
    Diversity + Individual, family = binomial, data = door)
Deviance Residuals: 
     Min        1Q    Median        3Q       Max

-2.42591   0.00001   0.05371   0.34488   2.72956
Coefficients: (4 not defined because of singularities)
                   Estimate Std. Error z value Pr(>|z|)

(Intercept)         60.7768   114.3250   0.532  0.59499

SexM               -43.7212    92.9102  -0.471  0.63794

ZooRGZ             -63.5269   123.3433  -0.515  0.60652

Age                  0.1188     0.3874   0.307  0.75917

Persistence          7.8394     1.9450   4.031 5.56e-05 ***
Neophilia           -0.6940     1.2860  -0.540  0.58942

Diversity           -0.3727     0.1173  -3.178  0.00148 ** 
IndividualAjay     113.6391   229.9300   0.494  0.62114

IndividualBamboo   -65.2868   130.9487  -0.499  0.61808

IndividualBatu      97.5497   195.7502   0.498  0.61825

IndividualChandra  -61.2992   121.6868  -0.504  0.61444

IndividualDoc      105.7723   207.0135   0.511  0.60939

IndividualKandula  -15.1359    24.6599  -0.614  0.53936

IndividualKirina    20.5416  2647.9499   0.008  0.99381

IndividualMali      20.6377  2600.2857   0.008  0.99367

IndividualRex            NA         NA      NA       NA

IndividualRomani         NA         NA      NA       NA

IndividualSiri           NA         NA      NA       NA

IndividualTarga          NA         NA      NA       NA

Signif. codes:  0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 157.574  on 148  degrees of freedom

Residual deviance:  69.857  on 134  degrees of freedom
AIC: 99.857
Number of Fisher Scoring iterations: 18

Raw data here: https://github.com/sjacobson1112/Problem-solving/blob/main/doorsuccess.csv

Answer 1

You might be trying to push these data a bit too far.

First, unpenalized logistic regression models typically need about 15 cases in the least-prevalent outcome class per predictor to avoid overfitting. With only 33 trials showing Solved = 0, you should be wary of evaluating more than 2 or 3 predictors. If you count the random-intercept effect as a single predictor, you have 7 predictors in your model. Also, of the 13 Individuals, 4 account for 23 of those 33 trials, 2 have no trials with Solved = 0, and 4 more have only 1 trial each with Solved = 0. Be cautious.

Second, each Individual has a single value of each of Sex, Age, and Neophilia, in addition to Zoo, for all trials.* Removing any one of those predictors from the original full mixed model removes the singularity problem. I can't say precisely why that is the case, but it presumably has to do with the ratio of the number of such predictors (4) to the number (13) of separate individuals. Perhaps once you have identified the associations of those 4 fixed predictors with outcome, there isn't much left to further associate with the Individuals as random effects.

The implications for your questions:

whether it is legitimate to simplify the full model by removing a fixed effect based on the p value ...

You should be evaluating fewer predictors in any event. What you are proposing is similar to backward stepwise selection of predictors, probably the least objectionable of stepwise approaches. It would be better to use your knowledge of the subject matter to choose the most crucial fixed-effect predictors to include from the beginning.

Or could my result with the full model provide enough support to argue that there is not enough individual variation so I don't have to account for repeated measures?

If you perform the no-Zoo model both with the Individual random effect and by completely ignoring the Individual values (in a glm), you get very different results. The magnitudes of the fixed-effect coefficients are about 10 times larger in the no-Zoo mixed model than in a no-Zoo glm model that omits Individual. That raises red flags.

I suspect that this has to do with overfitting. For example, the fixed-effect coefficients of models restricted to Persistence, Neophilia and Diversity are quite similar whether Individual is ignored (with glm) or included as a random-effect intercept (with glmer).

Look carefully at your data and apply your knowledge of the subject matter to decide how best to proceed.

---

*That's why the last 4 coefficients for levels of Individual in the glm model without random effects are NA. Had you listed Individual as the first predictor in the model formula, then those 4 predictors would have had the NA values instead.