This vignette covers common problems that occur while using `glmmTMB`

. The contents will expand with experience.

If your problem is not covered below, there’s a chance it has been solved in the development version; try updating to the latest version of `glmmTMB`

on GitHub.

This warning (`Model convergence problem; non-positive-definite Hessian matrix`

) states that at `glmmTMB`

’s maximum-likelihood estimate, the curvature of the negative log-likelihood surface is inconsistent with `glmmTMB`

really having found the best fit (minimum): instead, the surface is downward-curving, or flat, in some direction(s).

It will usually be accompanied by `NA`

values for the standard errors, log-likelihood, AIC, and BIC, and deviance. When you run `summary()`

on the resulting model, you’ll get the warning `In sqrt(diag(vcov)) : NaNs produced`

.

These problems are most likely:

- when a model is overparameterized (i.e. the data does not contain enough information to estimate the parameters reliably)
- when a random-effect variance is estimated to be zero, or random-effect terms are estimated to be perfectly correlated (“singular fit”: often caused by having too few levels of the random-effect grouping variable)
- when zero-inflation is estimated to be near zero (a strongly negative zero-inflation parameter)
- when dispersion is estimated to be near zero
- when
*complete separation*occurs in a binomial model: some categories in the model contain proportions that are either all 0 or all 1

How do we diagnose the problem?

Consider this example:

` glmmTMB(count~spp + (1|site), zi=~spp, Salamanders, family=nbinom2) zinbm0 =`

First, see if any of the estimated coefficients are extreme. If you’re using a non-identity link function (e.g. log, logit), then parameter values with \(|\beta|>10\) are suspect (for a logit link, this implies probabilities very close to 0 or 1; for a log link, this implies mean counts that are close to 0 or extremely large).

Inspecting the fixed-effect estimates for this model:

`fixef(zinbm0)`

**FIXME**: don’t hard-code, use inline expressions instead (search for “zi1” below)

The zero-inflation intercept parameter is tiny (\(\approx -17\)): since the parameters are estimated on the logit scale, we back-transform with `plogis(-17)`

to see the at the zero-inflation probability for the baseline level is about \(4 \times 10^{-8}\))). Many of the other ZI parameters are very large, compensating for the intercept: the estimated zero-inflation probabilities for all species are

```
fixef(zinbm0)$zi
ff <-round(plogis(c(sppGP=unname(ff[1]),ff[-1]+ff[1])),3)
```

Since the baseline probability is already effectively zero, making the intercept parameter larger or smaller will have very little effect - the likelihood is flat, which leads to the non-positive-definite warning.

Now that we suspect the problem is in the zero-inflation component, we can try to come up with ways of simplifying the model: for example, we could use a model that compared the first species (“GP”) to the rest:

```
transform(Salamanders, GP=as.numeric(spp=="GP"))
Salamanders <- update(zinbm0, ziformula=~GP) zinbm0_A =
```

This fits without a warning, although the GP zero-inflation parameter is still extreme:

`fixef(zinbm0_A)[["zi"]]`

Another possibility would be to fit the variation among species in the zero-inflation parameter as a random effect, rather than a fixed effect: this is slightly more parsimonious. This again fits without an error, although both the average level of zero-inflation and the among-species variation are estimated as very small:

```
update(zinbm0, ziformula=~(1|spp))
zinbm0_B =fixef(zinbm0_B)[["zi"]]
VarCorr(zinbm0_B)
```

The original analysis considered variation in zero-inflation by site status (mined or not mined) rather than by species - this simpler model only tries to estimate two parameters (mined + difference between mined and no-mining) rather than 7 (one per species) for the zero-inflation model.

```
glmmTMB(count~spp + (1|site), zi=~mined, Salamanders, family=nbinom2)
zinbm1 =fixef(zinbm1)[["zi"]]
```

This again fits without a warning, but we see that the zero-inflation is effectively zero in the unmined (“minedno”) condition (`plogis(0.38-17.5)`

is approximately \(4 \times 10^{-8}\)). We can estimate the confidence interval, but it takes some extra work: the default Wald standard errors and confidence intervals are useless in this case.

```
## at present we need to specify the parameter by number; for
## extreme cases need to specify the parameter range
## (not sure why the upper bound needs to be so high ... ?)
confint(zinbm1,method="uniroot",parm=9, parm.range=c(-20,20))
cc =print(cc)
```

The lower CI is not defined; the upper CI is -2.08, i.e. we can state that the zero-inflation probability is less than `plogis(-2.08)`

= 0.11.

More broadly, general inspection of the data (e.g., plotting the response against potential covariates) should help to diagnose overly complex models.

In some cases, scaling predictor variables may help. For example, in this example from @phisanti, the results of `glm`

and `glmmTMB`

applied to a scaled version of the data set agree, while `glmmTMB`

applied to the raw data set gives a non-positive-definite Hessian warning. (**FIXME: This is no longer true now that we try harder to compute an accurate Hessian … we need another example …**)

```
## data taken from gamlss.data:plasma, originally
## http://biostat.mc.vanderbilt.edu/wiki/pub/Main/DataSets/plasma.html
load(system.file("vignette_data","plasma.rda", package="glmmTMB"))
.1 <- glm(calories ~ fat*fiber, family = Gamma(link = "log"), data = plasma)
m4.2 <- glmmTMB(calories ~ fat*fiber, family = Gamma(link = "log"), data = plasma)
m4 transform(plasma,fat=scale(fat,center=FALSE),fiber=scale(fiber,center=FALSE))
ps <-.3 <- update(m4.2, data=ps)
m4## scaling factor for back-transforming standard deviations
c(1,
ss <- 1/attr(ps$fat,"scaled:scale"),
fatsc <- 1/attr(ps$fiber,"scaled:scale"),
fibsc <-*fibsc)
fatsc## combine SEs, suppressing the warning from the unscaled model
cbind(glm=sqrt(diag(vcov(m4.1))),
s_vals <-glmmTMB_unsc=suppressWarnings(sqrt(diag(vcov(m4.2)$cond))),
glmmTMB_sc=sqrt(diag(vcov(m4.3)$cond))*ss)
print(s_vals,digits=3)
```

Here is another example (from Samantha Sherman):

` load(system.file("vignette_data","sherman.rda",package="glmmTMB")) L <-`

The first model gives the warning: “non-integer counts in an nbinom1 model” (indicating that we probably should use a different response distribution, or round the values if that seems appropriate).

`summary(mod1)`

We can immediately see that the dispersion is very small and that the zero-inflation parameter is strongly negative.

Running diagnostics on the model, these are the only problems reported.

` diagnose(mod1) d1 <-`

Let’s try dropping the zero-inflation term:

` update(mod1, ziformula=~0) mod2 <-`

We also get a “false convergence (8)” warning; see below.

**FIXME**: this anticipates/duplicates some of the discussion near the end.

The `summary()`

and `diagnose()`

functions reveal only the large, negative dispersion parameter:

`summary(mod2)`

Diagnose:

`diagnose(mod2)`

The “false convergence” warning comes from the `nlminb()`

optimizer, and is difficult to interpret and resolve. The documentation says that the cause of this warning is that:

the gradient ∇f(x) may be computed incorrectly, the other stopping tolerances may be too tight, or either f or ∇f may be discontinuous near the current iterate x.

The only practical options we have for satisfying ourselves that a false convergence warning is really a false positive are the standard brute-force solutions of (1) making sure the gradients are small and the Hessian is positive definite (these are already checked internally); (2) trying different starting conditions, including re-starting at the current optimum; and (3) trying a different optimizer. We’ll try option 3 and refit with the BFGS option from `optim()`

:

```
update(mod2,
mod2_optim <-control=glmmTMBControl(optimizer=optim,
optArgs=list(method="BFGS")))
```

```
## Warning in glmmTMB(formula = taeCPUE ~ Avg.Relief + (1 | site.name/reef.name), :
## non-integer counts in a nbinom1 model
```

BFGS doesn’t give us any warnings. Comparing the parameter estimates:

```
cbind(nlminb=unlist(fixef(mod2)),
(parcomp <-optim=unlist(fixef(mod2_optim))))
```

```
## nlminb optim
## cond.(Intercept) -4.6150738 -4.6152081
## cond.Avg.Relief 0.1674507 0.1674755
## disp.(Intercept) -17.3339441 -11.4800820
```

The conditional-model parameters are practically identical. The dispersion parameters *look* quite different (-17.3 vs. -11.5), but if we back-transform from the log scale (via `exp()`

) we can see that they are both extremely small (\(2.96\times 10^{-8}\) vs. \(1.03\times 10^{-5}\)).

Simplify the model by switching from NB1 to Poisson:

` update(mod2, family=poisson) mod3 <-`

`summary(mod3)`

`diagnose(mod3)`

You can also check directly whether the Hessian (curvature) of the model is OK by examining the `pdHess`

(“positive-definite Hessian”) component of the `sdr`

(“standard deviation report”) component of the model:

`$sdr$pdHess mod3`

In general models with non-positive definite Hessian matrices should be excluded from further consideration.

` glmmTMB(count~spp + mined + (1|site), zi=~spp + mined, Salamanders, family=genpois) m1 <-`

`diagnose(m1)`

In this example, the fixed-effect covariance matrix is `NaN`

. It may have to do with the generalized Poisson (`genpois`

) distribution, which is known to have convergence problems; luckily, the negative binomial (`nbinom1`

and `nbinom2`

) and/or Conway-Maxwell Poisson (`compois`

) are good alternatives.

Models with convergence problems should be excluded from further consideration, in general.

In some cases, extreme eigenvalues may be caused by having predictor variables that are on very different scales: try rescaling, and centering, continuous predictors in the model.

Warning in nlminb(start = par, objective = fn, gradient = gr) : NA/NaN function evaluation

This warning occurs when the optimizer visits a region of parameter space that is invalid. It is not a problem as long as the optimizer has left that region of parameter space upon convergence, which is indicated by an absence of the model convergence warnings described above.

The following warnings indicate possibly-transient numerical problems with the fit, and can be treated in the same way (i.e. ignored if there are no errors or convergence warnings about the final fitted model).

Cholmod warning ‘matrix not positive definite’

In older versions of R (< 3.6.0):

Warning in f(par, order = order, …) : value out of range in ‘lgamma’

This warning:

false convergence: the gradient ∇f(x) may be computed incorrectly, the other stopping tolerances may be too tight, or either f or ∇f may be discontinuous near the current iterate x

comes from the `nlminb`

optimizer used by default in `glmmTMB`

. It’s usually hard to diagnose the source of this warning (this Stack Overflow answer explains a bit more about what it means). Reasonable methods for making sure your model is OK are:

- restart the model at the estimated fitted values
- try using a different optimizer, e.g.
`control=glmmTMBControl(optimizer=optim, optArgs=list(method="BFGS"))`

and see if the results are sufficiently similar to the original fit.

```
expand.grid(y=-1:1, rep=1:10)
dat1 = glmmTMB(y~1, dat1, family=nbinom2) m1 =
```

**FIXME**: this is no longer a “gradient evaluation” error …

The error occurs here because the negative binomial distribution is inappropriate for data with negative values.

If you see this error, check that the response variable meets the assumptions of the specified distribution.

Error in

`nlminb(start = par, objective = fn, gradient = gr)`

: gradient function must return a numeric vector of length x

Error in

`optimHess(par.fixed, obj$fn, obj$gr)`

: gradient in optim evaluated to length x

Try rescaling predictor variables. Try a simpler model and build up. (If you have a simple reproducible example of these errors, please post them to the issues list.)