lme4, failure to converge with a range of optimisers, trust the fitted model anyway?

Dear list,

I know, the failure to converge problem is boring, but still I would
like your input on my situation.

I have tried four optimizers/methods, and they all fail; glmer used.

1. Nelder_Mead: Model failed to converge with max|grad| = 0.00116526
   (tol = 0.001, component 6)

2. bobyqa: Model failed to converge with max|grad| = 0.00117064 (tol =
   0.001, component 7)

3. optimx, nlminb: Model failed to converge: degenerate Hessian with 4
   negative eigenvalues

4. optimx, L-BFGS-B: Model failed to converge with max|grad| =
   0.012963 (tol = 0.001, component 7)" "Model failed to converge:
   degenerate Hessian with 3 negative eigenvalues

The sample size is large: 1.833.793

The estimates resulting from fitting the model to data with the
different optimizers are similar:

                                     NM     bobyqa      nlmin       BFGS
(Intercept)                   4.9857379  3.7283744  4.9477121  3.2138480
QoG                           0.7866227  0.5962816  0.7534208  0.5991817
GDPLog                       -1.5161825 -1.3643422 -1.5111097 -1.2940261
Ruralyes                      4.3436228  4.3422641  4.3419199  4.3415551
KilledPerMillion5Log          0.6632158  0.6005677  0.6276264  0.5216984
Ruralyes:KilledPerMillion5Log 0.7313136  0.7316543  0.7314746  0.7329137

My theoretical focus is on the last two rows.

1. Is this likely to be a false positive? I'm willing to share data if
   that can help the development of lme4.

2. If the fits are bad, then what are my alternatives? continue with
   glmer and increase nAGQ? Other ideas? Or do I need to use other
   packages? I really love lme4, so I hope this will not be necessary.


Kind regards,

Hans Ekbrand







Postscript.

Here is the output of summary for the Nelder_Mead fit:

Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) ['glmerMod']
 Family: binomial  ( logit )
Formula: SanitationDeprivation ~ (1 | Country) + (1 | ClusterID) + QoG +
    GDPLog + Rural * KilledPerMillion5Log
   Data: my.df.aid

      AIC       BIC    logLik  deviance  df.resid
1013298.5 1013397.9 -506641.2 1013282.5   1833785

Scaled residuals:
    Min      1Q  Median      3Q     Max
-9.6390 -0.2883 -0.0508  0.0666 14.6730

Random effects:
 Groups    Name        Variance Std.Dev.
 ClusterID (Intercept)  9.675   3.110
 Country   (Intercept) 11.115   3.334
Number of obs: 1833793, groups:  ClusterID, 38177; Country, 65

Fixed effects:
                              Estimate Std. Error z value Pr(>|z|)
(Intercept)                    4.98574    0.08764   56.89  < 2e-16 ***
QoG                            0.78662    0.13190    5.96 2.46e-09 ***
GDPLog                        -1.51618    0.04841  -31.32  < 2e-16 ***
Ruralyes                       4.34362    0.04655   93.31  < 2e-16 ***
KilledPerMillion5Log           0.66322    0.15070    4.40 1.08e-05 ***
Ruralyes:KilledPerMillion5Log  0.73131    0.06059   12.07  < 2e-16 ***
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

Correlation of Fixed Effects:
            (Intr) QoG    GDPLog Rurlys KlPM5L
QoG         -0.042
GDPLog      -0.153  0.129
Ruralyes    -0.020  0.067 -0.037
KlldPrMll5L -0.081 -0.113 -0.091 -0.131
Rrlys:KPM5L -0.007 -0.088 -0.044 -0.426  0.205