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

Douglas Bates bates at stat.wisc.edu
Tue Jun 22 00:04:08 CEST 2010

```A quick response as I must catch a bus soon.

On Mon, Jun 21, 2010 at 4:18 PM, Ailene Kane <ailene at u.washington.edu> wrote:
> 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))

No, they are not the same as you have seen in your later
investigation.  The first model generates a vector-valued random
effect for each level of ind with possible correlation in the
variance-covariance matrix.  In the second model the components of the
random effects are independent.

It seems to me that there is very little variation associated with the
levels of ind.  I enclose a transcript of fitting a series of models
ending up with one that doesn't have any random effects for ind and it
seems to compare well to the others.

I must go to the bus.  I'll explain about the model-building strategy later.
> 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
>
>
>
>
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> library(lme4a)

Attaching package: 'Matrix'

The following object(s) are masked from 'package:base':

det

Attaching package: 'lme4a'

The following object(s) are masked from 'package:stats':

AIC

+            {
+                ind <- factor(inds)
+                yrs <- factor(yrs)
+            })
> xyplot(rwi ~ SWE | inds, hdat, pch=16, type =c("p","r"),
+        main=paste("Abam-snow"), lwd=2, ylim=c(0.5,1.5))
>
> (test13b <- lmer(rwi ~ GST*GPT+(1|inds)+(0+GST|inds)+(0+GPT|inds)+(0+GST:GPT|inds)+(1|yrs),
+                  hdat, REML=FALSE))
Linear mixed model fit by maximum likelihood ['merMod']
Formula: rwi ~ GST * GPT + (1 | inds) + (0 + GST | inds) + (0 + GPT |      inds) + (0 + GST:GPT | inds) + (1 | yrs)
Data: hdat
AIC       BIC    logLik  deviance
-545.3849 -490.8435  282.6925 -565.3849

Random effects:
Groups   Name        Variance  Std.Dev.
yrs      (Intercept) 2.637e-02 0.162397
inds     GST:GPT     8.551e-05 0.009247
inds     GPT         3.040e-05 0.005514
inds     GST         2.080e-05 0.004560
inds     (Intercept) 0.000e+00 0.000000
Residual             3.640e-02 0.190796
Number of obs: 1727, groups: yrs, 94; inds, 20

Fixed effects:
Estimate Std. Error t value
(Intercept)  0.999094   0.018377   54.37
GST          0.071329   0.018559    3.84
GPT          0.036532   0.018522    1.97
GST:GPT     -0.004108   0.018898   -0.22

Correlation of Fixed Effects:
(Intr) GST    GPT
GST      0.034
GPT     -0.025  0.310
GST:GPT  0.323  0.106 -0.069
> (test13c <- lmer(rwi ~ GST*GPT+(0+GST|inds)+(0+GPT|inds)+(0+GST:GPT|inds)+(1|yrs),
+                  hdat, REML=FALSE))
Linear mixed model fit by maximum likelihood ['merMod']
Formula: rwi ~ GST * GPT + (0 + GST | inds) + (0 + GPT | inds) + (0 +      GST:GPT | inds) + (1 | yrs)
Data: hdat
AIC       BIC    logLik  deviance
-547.3849 -498.2977  282.6925 -565.3849

Random effects:
Groups   Name        Variance  Std.Dev.
yrs      (Intercept) 2.637e-02 0.162397
inds     GST:GPT     8.551e-05 0.009247
inds     GPT         3.040e-05 0.005514
inds     GST         2.080e-05 0.004560
Residual             3.640e-02 0.190796
Number of obs: 1727, groups: yrs, 94; inds, 20

Fixed effects:
Estimate Std. Error t value
(Intercept)  0.999094   0.018377   54.37
GST          0.071329   0.018559    3.84
GPT          0.036532   0.018522    1.97
GST:GPT     -0.004108   0.018898   -0.22

Correlation of Fixed Effects:
(Intr) GST    GPT
GST      0.034
GPT     -0.025  0.310
GST:GPT  0.323  0.106 -0.069
> (test13d <- lmer(rwi ~ GST*GPT+(0+GST|inds)+(0+GPT|inds)+(1|yrs),
+                  hdat, REML=FALSE))
Linear mixed model fit by maximum likelihood ['merMod']
Formula: rwi ~ GST * GPT + (0 + GST | inds) + (0 + GPT | inds) + (1 |      yrs)
Data: hdat
AIC       BIC    logLik  deviance
-549.0160 -505.3829  282.5080 -565.0160

Random effects:
Groups   Name        Variance  Std.Dev.
yrs      (Intercept) 2.637e-02 0.162376
inds     GPT         3.646e-05 0.006038
inds     GST         1.146e-05 0.003385
Residual             3.649e-02 0.191035
Number of obs: 1727, groups: yrs, 94; inds, 20

Fixed effects:
Estimate Std. Error t value
(Intercept)  0.999095   0.018377   54.37
GST          0.071351   0.018545    3.85
GPT          0.036539   0.018530    1.97
GST:GPT     -0.004121   0.018782   -0.22

Correlation of Fixed Effects:
(Intr) GST    GPT
GST      0.034
GPT     -0.025  0.310
GST:GPT  0.325  0.107 -0.069
> (test13e <- lmer(rwi ~ GST + GPT+(0+GST|inds)+(0+GPT|inds)+(1|yrs),
+                  hdat, REML=FALSE))
Linear mixed model fit by maximum likelihood ['merMod']
Formula: rwi ~ GST + GPT + (0 + GST | inds) + (0 + GPT | inds) + (1 |      yrs)
Data: hdat
AIC       BIC    logLik  deviance
-550.9679 -512.7889  282.4840 -564.9679

Random effects:
Groups   Name        Variance  Std.Dev.
yrs      (Intercept) 2.638e-02 0.162421
inds     GPT         3.656e-05 0.006046
inds     GST         1.150e-05 0.003391
Residual             3.649e-02 0.191035
Number of obs: 1727, groups: yrs, 94; inds, 20

Fixed effects:
Estimate Std. Error t value
(Intercept)  1.00041    0.01738   57.56
GST          0.07179    0.01844    3.89
GPT          0.03626    0.01849    1.96

Correlation of Fixed Effects:
(Intr) GST
GST -0.001
GPT -0.002  0.320
> (test13f <- lmer(rwi ~ GST + GPT +(1|yrs), hdat, REML=FALSE))
Linear mixed model fit by maximum likelihood ['merMod']
Formula: rwi ~ GST + GPT + (1 | yrs)
Data: hdat
AIC       BIC    logLik  deviance
-554.8997 -527.6289  282.4498 -564.8997

Random effects:
Groups   Name        Variance Std.Dev.
yrs      (Intercept) 0.02637  0.1624
Residual             0.03654  0.1912
Number of obs: 1727, groups: yrs, 94

Fixed effects:
Estimate Std. Error t value
(Intercept)  1.00040    0.01738   57.56
GST          0.07180    0.01843    3.90
GPT          0.03631    0.01844    1.97

Correlation of Fixed Effects:
(Intr) GST
GST -0.001
GPT -0.002  0.321
>
> anova(test13f, test13e, test13d, test13c, test13b)
Data: hdat
Models:
test13f: rwi ~ GST + GPT + (1 | yrs)
test13e: rwi ~ GST + GPT + (0 + GST | inds) + (0 + GPT | inds) + (1 |
test13e:     yrs)
test13d: rwi ~ GST * GPT + (0 + GST | inds) + (0 + GPT | inds) + (1 |
test13d:     yrs)
test13c: rwi ~ GST * GPT + (0 + GST | inds) + (0 + GPT | inds) + (0 +
test13c:     GST:GPT | inds) + (1 | yrs)
test13b: rwi ~ GST * GPT + (1 | inds) + (0 + GST | inds) + (0 + GPT |
test13b:     inds) + (0 + GST:GPT | inds) + (1 | yrs)
Df     AIC     BIC logLik  Chisq Chi Df Pr(>Chisq)
test13f  5 -554.90 -527.63 282.45
test13e  7 -550.97 -512.79 282.48 0.0683      2     0.9664
test13d  8 -549.02 -505.38 282.51 0.0481      1     0.8263
test13c  9 -547.38 -498.30 282.69 0.3689      1     0.5436
test13b 10 -545.38 -490.84 282.69 0.0000      1     1.0000
> qqmath(ranef(test13f, postVar = TRUE))
\$yrs

>
> proc.time()
user  system elapsed
10.980   0.210  11.414
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