[R-sig-ME] correlated random effects in lmer and false convergence

[Ailene Kane]

Dear Dr. Bates and/or other expert lmer users:

We are having problems fitting a linear mixed effects model using the lmer() function in the lme4 package, and are hoping that you will be able (and willing!) to answer some of our questions.  We have read several chapters in your book on lmer, and other online resources which have helped resolve many of our questions, but our data set is pretty complex and we wanted to make very sure that we are interpreting the output (and some error messages that we get) correctly. Thank you so much for your time!

As background, we are exploring the influence of climative factors (e.g. snow, temperature, etc) on annual tree growth (“rwi”= ring width index, a detrended index for tree growth that ranges from ~-1 to +2).  We are evaluating the fit of 29 different models, including various climate variables and combinations of climate variables as fixed effects (e.g. in the model below, GST=Growing season Temperature, and GPT=growing season precipitation).  Note: the climate variables have been standardized by substracting the mean and dividing by the standard deviation. There are 2 random effects: “ind” (the individual tree; 20 trees were cored at each location and they differ in sensitivity to climate) and “yr” (the year of tree growth - this is sort of like a block effect, because you could have years with similar temperature values). A subset of our dataset is attached as a .csv, and some sample code and output is listed below (and in the attached file).

We have the following questions:

1) We originally thought that test13 and test13b were the same thing:

test13<-lmer(rwi~GST*GPT+(0+GST*GPT|ind)+(1|yrs),control=list(maxIter=5000))
test13b<-lmer(rwi~GST*GPT+(0+GST|ind)+(0+GPT|ind)+(0+GST:GPT|ind)+(1|yrs),control=list(maxIter=5000))

However, we recently noticed that they result in differences in the correlations between random effects, and different AIC values (output is below and can be generated by the code below, if you are interested).  In test13, the random effects on ind for GST,GPT, and their interaction are perfectly correlated (1.00).  In test13b the random effects are NOT PERFECTLY correlated (r= 0.6349112). Does this mean that test13 ASSUMES correlated random effects (because of the way that it is coded)? We definitely don't want this and wanted to make sure that we understood correctly how to code the model appropriately. We are comparing the 29 potential models to a null model and using the delta AIC (in part) to evaluate the best-fit model.  The null model is coded as follows:

test0<-lmer(rwi~1+(0+1|ind)+(1|yrs),control=list(maxIter=5000)

2) We have been getting warning message for some models (for example, test13): “Warning message: In mer_finalize(ans) : singular convergence (7).”  We have explored our data and get this error message only for complex models (e.g. 2 climate variables with an interaction), and mainly at locations where tree growth (of individual trees) is not correlated to any of our climate variables.  Do you think this interpretation makes sense, in which case we can conclude that the model fits poorly, indicating that other factors besides climate influence growth? Or, is there something else that you think we should do to avoid singular convergence? Again, the output generated by the code (test13b) is below and attached, in case you are interested.

Thank you so much for your help!

Best wishes,

Ailene


R OUTPUT FROM ATTACHED CODE:
> summary(test13)
Linear mixed model fit by REML 
Formula: rwi ~ GST * GPT + (0 + GST * GPT | ind) + (1 | yrs) 
    AIC  BIC logLik deviance REMLdev
 -528.4 -463  276.2     -577  -552.4
Random effects:
 Groups   Name        Variance   Std.Dev. Corr        
 yrs      (Intercept) 0.02764111 0.166256             
 ind      GST         0.00033501 0.018303             
          GPT         0.00029068 0.017049 1.000       
          GST:GPT     0.00023243 0.015246 1.000 1.000 
 Residual             0.03587474 0.189406             
Number of obs: 1727, groups: yrs, 94; ind, 20

Fixed effects:
             Estimate Std. Error t value
(Intercept)  0.999112   0.018777   53.21
GST          0.071152   0.019373    3.67
GPT          0.036372   0.019266    1.89
GST:GPT     -0.004302   0.019494   -0.22

Correlation of Fixed Effects:
        (Intr) GST    GPT   
GST      0.033              
GPT     -0.024  0.340       
GST:GPT  0.320  0.140 -0.032
> summary(test13b)
Linear mixed model fit by REML 
Formula: rwi ~ GST * GPT + (0 + GST | ind) + (0 + GPT | ind) + (0 + GST:GPT |      ind) + (1 | yrs) 
    AIC    BIC logLik deviance REMLdev
 -522.7 -473.6  270.4   -565.3  -540.7
Random effects:
 Groups   Name        Variance   Std.Dev. 
 yrs      (Intercept) 2.7626e-02 0.1662108
 ind      GST:GPT     8.7606e-05 0.0093598
 ind      GPT         3.2627e-05 0.0057120
 ind      GST         2.3199e-05 0.0048165
 Residual             3.6397e-02 0.1907813
Number of obs: 1727, groups: yrs, 94; ind, 20

Fixed effects:
             Estimate Std. Error t value
(Intercept)  0.999101   0.018778   53.21
GST          0.071329   0.018967    3.76
GPT          0.036529   0.018928    1.93
GST:GPT     -0.004107   0.019310   -0.21

Correlation of Fixed Effects:
        (Intr) GST    GPT   
GST      0.034              
GPT     -0.024  0.310       
GST:GPT  0.323  0.106 -0.069
Ailene Kane Ettinger
PhD Candidate
Biology Department
University of Washington
Box 351800
Seattle, Washington  98195-1800

ailene at u.washington.edu