GLMER warning: variance-covariance matrix [...] is not positive definite or contains NA values

Question

I sometimes find that my GLMMs from glmer, package lme4, show the following warning messages, when their summary is called:

Warning messages:
1: In vcov.merMod(object, use.hessian = use.hessian) :
  variance-covariance matrix computed from finite-difference Hessian is
not positive definite or contains NA values: falling back to var-cov estimated from RX
2: In vcov.merMod(object, correlation = correlation, sigm = sig) :
  variance-covariance matrix computed from finite-difference Hessian is
not positive definite or contains NA values: falling back to var-cov estimated from RX

Similar questions I found here on Stackoverflow refer to other functions, not glmer, and the LME4 Wiki does not elaborate on that either. In this question, the problem was solved before that kind of error messages were tackled, and here the discussion focuses on a particular model rather than on the meaning of the warning message.

So the question is: should I worry about that message, or is it OK because it is simply a warning and not an error, and as it says, it is "falling back to var-cov estimated from RX" (whatever RX is) anyway.

Interestingly, although the summary states that the model failed to converge, I do not get the usual convergence warnings in red.

Here comes a (minimal) dataset:

testdata=structure(list(Site = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L), .Label = c("EO1", "EO2", 
"EO3", "EO4", "EO5", "EO6"), class = "factor"), Treatment = structure(c(1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L), .Label = c("control", 
"no ants", "no birds", "no birds no ants"), class = "factor"), 
    Tree = structure(c(2L, 3L, 4L, 16L, 12L, 13L, 14L, 15L, 5L, 
    6L, 7L, 8L, 1L, 9L, 10L, 11L, 28L, 29L, 30L, 31L, 17L, 25L, 
    26L, 27L, 18L, 19L, 20L, 32L, 21L, 22L, 23L, 24L, 33L, 41L, 
    42L, 43L, 37L, 38L, 39L, 40L, 44L, 45L, 46L, 47L, 34L, 35L, 
    36L, 48L, 49L, 57L, 58L, 59L, 50L, 51L, 52L, 64L, 53L, 54L, 
    55L, 56L, 60L, 61L, 62L, 63L, 66L, 67L, 68L, 80L, 69L, 70L, 
    71L, 72L, 76L, 77L, 78L, 79L, 65L, 73L, 74L, 75L, 82L, 83L, 
    84L, 96L, 92L, 93L, 94L, 95L, 85L, 86L, 87L, 88L, 81L, 89L, 
    90L, 91L), .Label = c("EO1 1", "EO1 10", "EO1 11", "EO1 12", 
    "EO1 13", "EO1 14", "EO1 15", "EO1 16", "EO1 2", "EO1 3", 
    "EO1 4", "EO1 5", "EO1 6", "EO1 7", "EO1 8", "EO1 9", "EO2 1", 
    "EO2 10", "EO2 11", "EO2 12", "EO2 13", "EO2 14", "EO2 15", 
    "EO2 16", "EO2 2", "EO2 3", "EO2 4", "EO2 5", "EO2 6", "EO2 7", 
    "EO2 8", "EO2 9", "EO3 1", "EO3 10", "EO3 11", "EO3 12", 
    "EO3 13", "EO3 14", "EO3 15", "EO3 16", "EO3 2", "EO3 3", 
    "EO3 4", "EO3 5", "EO3 6", "EO3 7", "EO3 8", "EO3 9", "EO4 1", 
    "EO4 10", "EO4 11", "EO4 12", "EO4 13", "EO4 14", "EO4 15", 
    "EO4 16", "EO4 2", "EO4 3", "EO4 4", "EO4 5", "EO4 6", "EO4 7", 
    "EO4 8", "EO4 9", "EO5 1", "EO5 10", "EO5 11", "EO5 12", 
    "EO5 13", "EO5 14", "EO5 15", "EO5 16", "EO5 2", "EO5 3", 
    "EO5 4", "EO5 5", "EO5 6", "EO5 7", "EO5 8", "EO5 9", "EO6 1", 
    "EO6 10", "EO6 11", "EO6 12", "EO6 13", "EO6 14", "EO6 15", 
    "EO6 16", "EO6 2", "EO6 3", "EO6 4", "EO6 5", "EO6 6", "EO6 7", 
    "EO6 8", "EO6 9"), class = "factor"), predators_trunk = c(7L, 
    10L, 9L, 15L, 18L, 11L, 5L, 7L, 15L, 12L, 6L, 12L, 7L, 13L, 
    24L, 17L, 3L, 0L, 0L, 2L, 4L, 3L, 0L, 6L, 2L, 3L, 5L, 1L, 
    5L, 12L, 18L, 15L, 7L, 0L, 5L, 1L, 17L, 7L, 13L, 19L, 7L, 
    3L, 5L, 10L, 11L, 7L, 13L, 7L, 7L, 0L, 4L, 2L, 5L, 7L, 4L, 
    7L, 8L, 7L, 9L, 20L, 13L, 2L, 12L, 7L, 0L, 7L, 2L, 2L, 2L, 
    4L, 17L, 2L, 3L, 1L, 1L, 1L, 11L, 1L, 1L, 8L, 8L, 18L, 5L, 
    6L, 6L, 5L, 6L, 5L, 9L, 2L, 8L, 13L, 13L, 5L, 3L, 5L), pH_H2O = c(4.145, 
    4.145, 4.145, 4.145, 4.1825, 4.1825, 4.1825, 4.1825, 4.1325, 
    4.1325, 4.1325, 4.1325, 4.14125, 4.14125, 4.14125, 4.14125, 
    4.265, 4.265, 4.265, 4.265, 4.21, 4.21, 4.21, 4.21, 4.18375, 
    4.18375, 4.18375, 4.18375, 4.09625, 4.09625, 4.09625, 4.09625, 
    4.1575, 4.1575, 4.1575, 4.1575, 4.1125, 4.1125, 4.1125, 4.1125, 
    4.20875, 4.20875, 4.20875, 4.20875, 3.97125, 3.97125, 3.97125, 
    3.97125, 4.025, 4.025, 4.025, 4.025, 4.005, 4.005, 4.005, 
    4.005, 4.04, 4.04, 4.04, 4.04, 4.03125, 4.03125, 4.03125, 
    4.03125, 4.4575, 4.4575, 4.4575, 4.4575, 4.52, 4.52, 4.52, 
    4.52, 4.505, 4.505, 4.505, 4.505, 4.34875, 4.34875, 4.34875, 
    4.34875, 4.305, 4.305, 4.305, 4.305, 4.32, 4.32, 4.32, 4.32, 
    4.35, 4.35, 4.35, 4.35, 4.445, 4.445, 4.445, 4.445), ant_mean_abundance = c(53.85714, 
    53.85714, 53.85714, 53.85714, 24.28571, 24.28571, 24.28571, 
    24.28571, 45.5, 45.5, 45.5, 45.5, 51.14286, 51.14286, 51.14286, 
    51.14286, 66.28571, 66.28571, 66.28571, 66.28571, 76.5, 76.5, 
    76.5, 76.5, 65.71429, 65.71429, 65.71429, 65.71429, 8.642857, 
    8.642857, 8.642857, 8.642857, 109.3571, 109.3571, 109.3571, 
    109.3571, 25.14286, 25.14286, 25.14286, 25.14286, 101.3571, 
    101.3571, 101.3571, 101.3571, 31.78571, 31.78571, 31.78571, 
    31.78571, 78.64286, 78.64286, 78.64286, 78.64286, 93.28571, 
    93.28571, 93.28571, 93.28571, 63.14286, 63.14286, 63.14286, 
    63.14286, 67.14286, 67.14286, 67.14286, 67.14286, 44.0625, 
    44.0625, 44.0625, 44.0625, 23.875, 23.875, 23.875, 23.875, 
    95.8125, 95.8125, 95.8125, 95.8125, 49.125, 49.125, 49.125, 
    49.125, 57, 57, 57, 57, 38.125, 38.125, 38.125, 38.125, 40.6875, 
    40.6875, 40.6875, 40.6875, 22, 22, 22, 22), bird_activity = c(153.24, 
    153.24, 153.24, 153.24, 153.24, 153.24, 153.24, 153.24, 0, 
    0, 0, 0, 0, 0, 0, 0, 240.96, 240.96, 240.96, 240.96, 240.96, 
    240.96, 240.96, 240.96, 0, 0, 0, 0, 0, 0, 0, 0, 154.54, 154.54, 
    154.54, 154.54, 154.54, 154.54, 154.54, 154.54, 0, 0, 0, 
    0, 0, 0, 0, 0, 107.68, 107.68, 107.68, 107.68, 107.68, 107.68, 
    107.68, 107.68, 0, 0, 0, 0, 0, 0, 0, 0, 172.42, 172.42, 172.42, 
    172.42, 172.42, 172.42, 172.42, 172.42, 0, 0, 0, 0, 0, 0, 
    0, 0, 113.8, 113.8, 113.8, 113.8, 113.8, 113.8, 113.8, 113.8, 
    0, 0, 0, 0, 0, 0, 0, 0)), .Names = c("Site", "Treatment", 
"Tree", "predators_trunk", "pH_H2O", "ant_mean_abundance", "bird_activity"
), class = "data.frame", row.names = c(NA, -96L))

And here is the code leading to the warnings:

library(lme4)
summary(glmer.nb(predators_trunk ~ scale(ant_mean_abundance) + scale(bird_activity) + scale(pH_H2O) + (1 | Site/Treatment), testdata, na.action = na.fail))
summary(glmer(predators_trunk ~ scale(ant_mean_abundance) + scale(bird_activity) + scale(pH_H2O) + (1 | Site/Treatment), testdata, family = negative.binomial(theta = 4.06643400243645), na.action = na.fail))

Interestingly to me, the summary of the glmer.nb does not yield any warnings, but the call to glmer, using the theta that was estimated by glmer.nb, does give me the warnings. The latter is the model call that is generated by using dredge (MuMIn) on the corresponding glmer.nb full model.

Answer 1

This warning suggests that your standard error estimates might be less accurate. But as with all warnings, it's hard to know for sure and the best thing is to try to cross-check if you can.

In this case I saved your two fits, from glmer.nb and glmer, as g1 and g2. You can see that the estimates (point estimates, SEs, Z values ...) have changed a little bit, but not very much, so at the very least that should reassure you.

printCoefmat(coef(summary(g1)),digits=2)
                          Estimate Std. Error z value Pr(>|z|)    
(Intercept)                  1.844      0.111    16.7   <2e-16 ***
scale(ant_mean_abundance)   -0.347      0.077    -4.5    7e-06 ***
scale(bird_activity)        -0.122      0.076    -1.6    0.107    
scale(pH_H2O)               -0.275      0.104    -2.6    0.008 ** 

> printCoefmat(coef(summary(g2)),digits=2)
                          Estimate Std. Error z value Pr(>|z|)    
(Intercept)                  1.846      0.108    17.1   <2e-16 ***
scale(ant_mean_abundance)   -0.347      0.077    -4.5    6e-06 ***
scale(bird_activity)        -0.122      0.075    -1.6    0.102    
scale(pH_H2O)               -0.275      0.102    -2.7    0.007 ** 

I have a development version of lme4 on Github (the test_mods branch, hopefully integrated into the master branch soon: if you want to install it, you can use devtools::install_github("lme4/lme4",ref="test_mods")) which allows you to pick a more accurate (but slower) calculation for the standard errors: this gets us back to (nearly) the same standard errors as glmer.nb.

g3 <- update(g2, control=glmerControl(deriv.method="Richardson"))
printCoefmat(coef(summary(g3)),digits=2)
                          Estimate Std. Error z value Pr(>|z|)    
(Intercept)                  1.846      0.111    16.7   <2e-16 ***
scale(ant_mean_abundance)   -0.347      0.077    -4.5    6e-06 ***
scale(bird_activity)        -0.122      0.076    -1.6    0.106    
scale(pH_H2O)               -0.275      0.104    -2.6    0.008 ** 

all.equal(coef(summary(g1))[,"Std. Error"],
          coef(summary(g3))[,"Std. Error"])
[1] "Mean relative difference: 0.001597978"

The glmmTMB package (on Github) also gives almost the same results:

library(glmmTMB)
g5 <- glmmTMB(predators_trunk ~ scale(ant_mean_abundance) +
                     scale(bird_activity) + scale(pH_H2O) +
                     (1 | Site/Treatment), testdata,
              family=nbinom2)
printCoefmat(coef(summary(g5))[["cond"]],digits=2)
                          Estimate Std. Error z value Pr(>|z|)    
(Intercept)                  1.852      0.110    16.8   <2e-16 ***
scale(ant_mean_abundance)   -0.348      0.077    -4.5    7e-06 ***
scale(bird_activity)        -0.123      0.076    -1.6    0.106    
scale(pH_H2O)               -0.276      0.105    -2.6    0.008 **