Dear all,

 

I am currently trying to model the behavioural response of individual
seabirds (in terms of foraging efficiency) to the variation in sea ice cover
(SICdr) of their foraging environment. I have 13 years of data, birds are
individually marked and followed, I have several records (= foraging
efficiency data = CPUEr in my code) per individual (IDr) for each year
(YEARr) and individuals are followed across years.

 

I am trying to find the right random effect structure (biologically
meaningful and dealing with problems of independence) and to deal with
heterogeneity of the residual variance at the same time (for all my models,
the variance of the residuals increases with increasing fitted values).
Regarding the random effect structure, would you say that crossed random
effects of the form (1|IDr) + (1|YEARr) would correctly reflect the study
design? Is there any way to model the variance heterogeneity in lmer that
would be analogous to the varIdent or varFixed functions in nlme? So far, I
can model the variance heterogeneity with nlme only and the (hopefully)
appropriate random effect structure with lmer only. Would you have other
suggestions for dealing with this heteroscedasticity?

 

Here are a couple of examples regarding the random effect structure with
some associated questions: 

 

> M1 <- lmer(CPUEr~SEXr+SICdr+(1|IDr))

> summary(M1)

 

Linear mixed model fit by REML

Formula: CPUEr ~ SEXr + SICdr + (1 | IDr) 

   AIC   BIC logLik deviance REMLdev

 270.2 297.6 -130.1    234.5   260.2

Random effects:

 Groups   Name        Variance Std.Dev.

 IDr      (Intercept) 0.010906 0.10443 

 Residual             0.060610 0.24619 

Number of obs: 1759, groups: IDr, 229

 

Fixed effects:

             Estimate Std. Error t value

(Intercept) 0.3070164  0.0155734  19.714

SEXrM       0.0961795  0.0195420   4.922

SICdr       0.0026240  0.0008478   3.095

 

Correlation of Fixed Effects:

      (Intr) SEXrM 

SEXrM -0.612       

SICdr -0.478 -0.006

 

Here, the correlation between 2 observations from the same individual
(irrespective of year) is: 0.010906/(0.010906+0.060610)=0.15

 

> M2 <- lmer(CPUEr~SEXr+SICdr+(1|YEARr))

> summary(M2)

Linear mixed model fit by REML

Formula: CPUEr ~ SEXr + SICdr + (1 | YEARr) 

   AIC   BIC logLik deviance REMLdev

 117.1 144.5 -53.55     84.8   107.1

Random effects:

 Groups   Name        Variance Std.Dev.

 YEARr    (Intercept) 0.020395 0.14281 

 Residual             0.059892 0.24473 

Number of obs: 1759, groups: YEARr, 13

 

Fixed effects:

            Estimate Std. Error t value

(Intercept)  0.36443    0.04367   8.345

SEXrM        0.10819    0.01175   9.207

SICdr       -0.00920    0.00192  -4.793

 

Correlation of Fixed Effects:

      (Intr) SEXrM 

SEXrM -0.134       

SICdr -0.367  0.009

 

Here, the correlation between 2 observations from the same year
(irrespective of the bird) is: 0.020395/(0.020395+0.059892)=0.25 How do I
get the correlation of 2 observations from the same individual within a
year? By modeling CPUEr~SEXr+SICdr+(1|YEARr/IDr)?

 

> M3 <- lmer(CPUEr~SEXr+SICdr+(1|YEARr/IDr))

> summary(M3)

Linear mixed model fit by REML

Formula: CPUEr ~ SEXr + SICdr + (1 | YEARr/IDr) 

   AIC   BIC logLik deviance REMLdev

 51.29 84.12 -19.64    17.21   39.29

Random effects:

 Groups    Name        Variance  Std.Dev.

 IDr:YEARr (Intercept) 0.0097178 0.09858 

 YEARr     (Intercept) 0.0188065 0.13714 

 Residual              0.0500727 0.22377 

Number of obs: 1759, groups: IDr:YEARr, 543; YEARr, 13

 

Fixed effects:

             Estimate Std. Error t value

(Intercept)  0.357318   0.042408   8.426

SEXrM        0.104650   0.014207   7.366

SICdr       -0.008960   0.001855  -4.831

 

Correlation of Fixed Effects:

      (Intr) SEXrM 

SEXrM -0.166       

SICdr -0.365  0.004

 

Then, would the correlation of 2 observations from the same individual
within a year be 0.0097178/(0.0097178+0.0500727)=0.16?

 

My best model (in terms of AIC) so far is the following:

 

> M4 <- lmer(CPUEr~SEXr+SICdr+(SICdr|IDr)+(1|YEARr))

> summary(M4)

Linear mixed model fit by REML

Formula: CPUEr ~ SEXr + SICdr + (SICdr | IDr) + (1 | YEARr) 

   AIC   BIC logLik deviance REMLdev

 12.88 56.66  1.559   -24.55  -3.119

Random effects:

 Groups   Name        Variance   Std.Dev.  Corr   

 IDr      (Intercept) 8.9314e-03 0.0945058        

          SICdr       2.3781e-05 0.0048766 -0.464 

 YEARr    (Intercept) 2.1401e-02 0.1462922        

 Residual             5.0765e-02 0.2253112        

Number of obs: 1759, groups: IDr, 229; YEARr, 13

 

Fixed effects:

             Estimate Std. Error t value

(Intercept)  0.363366   0.045471   7.991

SEXrM        0.100215   0.017188   5.830

SICdr       -0.009910   0.001974  -5.021

 

Correlation of Fixed Effects:

      (Intr) SEXrM 

SEXrM -0.189       

SICdr -0.357  0.010

 

How should I interpret the random effects?

 

I am using the R package version 0.999375-31 of lme4 and R version 2.9.2.

 

Thanks in advance for your help!

 

Cheers,

 

Amelie

 

 

 

 


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