glmer and nAGQ
Thanks again for your feedback. Much appreciated. With the canonical link in the Gamma model, once a neg infinity result is returned for the logLik (n=4 for my simulation data), increasing the value of nAGQ doesn't change that result and the warning message about the estimated variance-covariance matrix not being positive definite is constant also. If I use the log link, there is a large increase (-678.1 to -5.6) in the LogLik in going from the default value of n=1 to n=2, but it then doesn't change for larger n values and the model results are stable. So I assume then that the Gamma model with the canonical link cannot be fit to this data while with the log link one does get a robust result? There was no detectable delay in the time to run the model with nAGQ = 25 versus the default value. Is that suspicious? Peter Sent with ProtonMail Secure Email. ??????? Original Message ???????
On Friday, June 18th, 2021 at 8:16 PM, Ben Bolker <bbolker at gmail.com> wrote:
On 6/18/21 6:39 PM, Peter R Law wrote:
Thanks for your response. I'm sorry the formatting of the R output got messed up and hard to read when I forwarded the original message. I do understand that changing the value of nAGQ is changing the approximation of the likelihood, not changing the model itself. I wondered whether the change in likelihood from -643.8 (n=1) to -5.9 (n=2) to -inf (n=3) was a reason to suspect the model was badly behaved in that none of these approximations of the logLik should be trusted. The parameter estimates for the fixed effects don't change that much, basically just the logLik and consequently the AIC. I wanted to compare the glmer with a Gamma distribution to the lmer to see whether AIC favoured one over the other. My intent was to start with the glmer with default n=1, then increase n until model parameters seemed stable, then use that n value for the glmer model to obtain a logLik and then AIC to compare to the lmer version. But the dramatic differences in logLik in going from n=1 to n= 2 and the negative infinite value already for n=3 I thought might suggest that none of these logLiks should be trusted and the the Gamma model simply abandoned. Does that sound plausible in light of the warning messages? In particular, that the negative infinity answer is a failuer in the estimation procedure rather than an indication that that the Gamma model has zero likelihood?
As I suggested in my answer, you shouldn't worry about the big jump in log-likelihood from nAGQ=1 to nAGQ=2. I would try n=5, 10, 15, 20 and see how it goes (unless n=20 is too slow to be practical). (Yes, I think -Inf is a problem with the fitting procedure)
I will repeat the analyses with the log link. As far as the data is concerned, it's simulated data I created without using any distributional assumptions but rather based on ecological assumptions for how the response might depend on the fixed predictors. My purpose here is to explore the glmer function in preparation for a study with real data (still being collected). Because the responses are positive values, I thought it would be worth investigating how a Gamma glmer compares with an lmer (and compare both with a log-normal model, after making the adjustment in the latter so that its AIC is comparable to the others, as was pointed out in a previous posting).
Fair enough; "without any distributional assumptions" is a bit odd though. Unless your ecological model is stochastic (in which case the conditional distribution will emerge naturally from the model), you have to make some assumptions about how the observations will vary around the expected mean ... cheers Ben Bolker
Peter Sent with ProtonMail Secure Email. ??????? Original Message ??????? On Friday, June 18th, 2021 at 5:44 PM, Ben Bolker bbolker at gmail.com wrote:
This is kind of hard to read, but I'll try. First of all, log-likelihoods with nAGQ>1 (adaptive Gauss-Hermite quadrature) **are not currently commensurate with log-likelihoods with nAGQ==1 (Laplace approximation). This is hinted at in the "Deviance and log-likelihood of GLMMs" section under "Deviance and log-likelihood of GLMMs". In any case, the log-likelihoods computed under different numbers of quadrature points aren't comparable as though they were different statistical models (AIC, likelihood ratio tests, etc.). Rather, they are based on different approximations to the same model, and we know that larger numbers of quadrature points (increasing nAGQ) are more accurate (although slower) approximations. The main thing is to see if results (parameter estimates etc.) change as nAGQ increases by a magnitude that is important for the current analysis ... if they do, then you should use an nAGQ that is large enough that it approximates "infinity", i.e. increasing nAGQ further doesn't change the answers by an important amount. In general if it is not important to you to use the inverse link (for interpretation/because you think this will be the natural scale on which to measure effects of continuous parameters and/or interactions), Gamma(link="log") is generally more robust. The "rescale variables?" suggestion is just that; if the parameters are already rescaled, then it's not going to help. A large eigenvalue may not actually be a problem, it just indicates the possibility of numerical instability. In this case my guess is that the very small random effects variance might be responsible. It's also a little surprising that the normRain coefficient is very small in magnitude and yet but has a very large Z-statistic, given this sample size. The results with increasing AGQ do indeed look kind of wonky. If this were my problem I would want to look more carefully at the data/think about where they came from. How do the model diagnostics look? On 6/18/21 3:52 PM, Peter R Law via R-sig-mixed-models wrote:
I am re-sending this query that I originally emailed June 13'th. I did not receive a copy of the sent email as I have with previous postings and my query doesn't appear in the archive of postings so does not seem to have been received. Peter R Law Sent with ProtonMail Secure Email. ??????? Original Message ??????? On Monday, June 14th, 2021 at 10:48 PM, Peter R Law prldb at protonmail.com wrote:
Any help with the following query is much appreciated. I used some simulated data (not generated under any specific distributional assumption but all responses are positive quantities) to investigate the nAGQ argument in glmer, running a Gamma-distribution model. With nAAGQ=2 the logLik is dramatically different to the default value of nAGQ=1, while nAGQ=5 returned minus infinity for the logLik, but the estimates of the fixed effect parameters are somewhat consistent across each computation. Are the differences in the estimated logLik surprising or do they reflect the warnings glmer returns for this attempted modelling? I got similar results for a real dataset too. data.frame':500 obs. of8 variables: $ IBI: num25.5 25.4 25.2 25.6 25.8 ... $ MatID: Factor w/ 99 levels "M1","M10","M11",..: 1 1 1 1 1 12 12 12 12 12 ... $ Pop: Factor w/ 5 levels "P1","P2","P3",..: 1 1 1 1 1 1 1 1 1 1 ... $ density : int11 13 15 19 28 11 13 15 19 28 ... $ rain: num41.1 36.6 31.6 40 40.6 ... $ normIBI : num-1.28 -1.29 -1.31 -1.26 -1.24 ... $ normDens: num-1.72 -1.66 -1.61 -1.49 -1.23 ... $ normRain: num-0.249 -0.64 -1.073 -0.345 -0.287 ...
M61 <- glmer(IBI~normDens+normRain + (1|MatID), family=Gamma, data=Sim)
Warning message: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,: Model is nearly unidentifiable: very large eigenvalue - Rescale variables?
summary(M61)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod'] Family: Gamma( inverse ) Formula: IBI ~ normDens + normRain + (1 | MatID) Data: Sim AICBIClogLik deviance df.resid 1297.61318.7-643.81287.6495 Scaled residuals: Min1QMedian3QMax -1.58054 -0.306190.041690.367441.31012 Random effects: GroupsNameVarianceStd.Dev. MatID(Intercept) 5.802e-06 0.002409 Residual1.368e-03 0.036989 Number of obs: 500, groups:MatID, 99 Fixed effects: Estimate Std. Error t value Pr(>|z|) (Intercept)3.083e-026.090e-0450.63<2e-16 *** normDens -1.720e-037.946e-05-21.65<2e-16 *** normRain4.889e-042.463e-0519.85<2e-16 *** Signif. codes:0 ?? 0.001 ?? 0.01 ?? 0.05 ?.? 0.1 ? ? 1 Correlation of Fixed Effects: (Intr) nrmDns normDens0.028 normRain0.000 -0.044 convergence code: 0 Model is nearly unidentifiable: very large eigenvalue - Rescale variables? What kind of scaling is being suggested in the case of the default value of nAGQ? The predictors are already normalized.
M62 <- glmer(IBI~normDens+normRain + (1|MatID), family=Gamma, nAGQ=2,data=Sim)
boundary (singular) fit: see ?isSingular
summary(M62)
Generalized linear mixed model fit by maximum likelihood (Adaptive Gauss-Hermite Quadrature, nAGQ = 2) ['glmerMod'] Family: Gamma( inverse ) Formula: IBI ~ normDens + normRain + (1 | MatID) Data: Sim AICBIClogLik deviance df.resid 21.742.8-5.911.7495 Scaled residuals: Min1QMedian3QMax -2.3532 -0.6617 -0.13050.51013.4180 Random effects: GroupsNameVariance Std.Dev. MatID(Intercept) 0.000000.0000 Residual0.024160.1554 Number of obs: 500, groups:MatID, 99 Fixed effects: Estimate Std. Error t value Pr(>|z|) (Intercept)0.02882820.001300822.161< 2e-16 *** normDens-0.00392740.0012153-3.2320.00123 ** normRain0.00027830.00129940.2140.83042 Signif. codes:0 ?? 0.001 ?? 0.01 ?? 0.05 ?.? 0.1 ? ? 1 Correlation of Fixed Effects: (Intr) nrmDns normDens -0.275 normRain0.019 -0.019 convergence code: 0 boundary (singular) fit: see ?isSingular
M65 <- glmer(IBI~normDens+normRain + (1|MatID), family=Gamma, nAGQ=5,data=Sim)
summary(M65)
Generalized linear mixed model fit by maximum likelihood (Adaptive Gauss-Hermite Quadrature, nAGQ = 5) ['glmerMod'] Family: Gamma( inverse ) Formula: IBI ~ normDens + normRain + (1 | MatID) Data: Sim AICBIClogLik deviance df.resid InfInf-InfInf495 Scaled residuals: Min1QMedian3QMax -2.2488 -0.39020.03160.34851.7127 Random effects: GroupsNameVarianceStd.Dev. MatID(Intercept) 6.007e-06 0.002451 Residual8.831e-04 0.029716 Number of obs: 500, groups:MatID, 99 Fixed effects: Estimate Std. Error t value Pr(>|z|) (Intercept)2.936e-022.495e-04117.68<2e-16 *** normDens-2.182e-031.163e-04-18.76<2e-16 *** normRain4.883e-043.973e-0512.29<2e-16 *** Signif. codes:0 ?? 0.001 ?? 0.01 ?? 0.05 ?.? 0.1 ? ? 1 Correlation of Fixed Effects: (Intr) nrmDns normDens -0.013 normRain0.004 -0.041 convergence code: 0 Gradient contains NAs 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 For comparison, the linear model seems to be well behaved: M1 <- lmer(IBI~normDens+normRain +(1|MatID), REML=FALSE, data=Sim) summary(M1) Linear mixed model fit by maximum likelihood ['lmerMod'] Formula: IBI ~ normDens + normRain + (1 | MatID) Data: Sim AIC BIC logLik deviance df.resid 1804.2 1825.2 -897.1 1794.2 495 Scaled residuals: Min 1Q Median 3Q Max -3.2970 -0.5304 0.0153 0.5484 3.4798 Random effects: Groups Name Variance Std.Dev. MatID (Intercept) 43.5694 6.6007 Residual 0.6737 0.8208 Number of obs: 500, groups: MatID, 99 Fixed effects: Estimate Std. Error t value (Intercept) 35.47121 0.66443 53.39 normDens 2.00732 0.11816 16.99 normRain -0.71747 0.04237 -16.93 Correlation of Fixed Effects: (Intr) nrmDns normDens -0.002 normRain 0.000 -0.044 Should I just conclude that the data is not well modelled by a Gamma GLMM? Peter R Law Research Associate Center for African Conservation Ecology Nelson Mandela University South Africa Sent with ProtonMail Secure Email. [[alternative HTML version deleted]]
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