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I used the lmer function in the lme4 package in order to assess the effects of 2 categorical fixed effects (1º Animal Group: rodents and ants; 2º Microhabitat: bare soil and under cover) on seed predation (a count dependent variable). I have 2 Sites, with 10 trees per site and 4 seed stations per tree. Site and Tree are my (philosophically) random factors, but given that I have only two level for Site, it must be treated as a fixed factor. I have questions about how to interpret the results:

  1. I made a model selection criterion based on QAICc, but the best model (lower QAICc) does not result in any significant fixed effect and other models with higher QAIC (e.g. the Full Model) did find significant fixed factors. Does this make sense?
  2. Given a fixed factor that is important to the model, how do I distinguish which level of fixed factor is influencing the response variable?

Finally, correlation between the fixed factors implies an incorrect estimation of the model?

FullModel=lmer(SeedPredation ~ AnimalGroup*Microhabitat*Site + (1|Site:Tree) + 
                                   (1|obs), data=datos,  family="poisson") 

QAICc(FM)104.9896

    enterGeneralized linear mixed model fit by the Laplace approximation 
Formula: SP ~ AG * MH * Site + (1 | Site:Tree) + (1 | obs) 
   Data: datos 
   AIC   BIC logLik deviance
 101.8 125.6  -40.9     81.8
Random effects:
 Groups    Name        Variance Std.Dev.
 obs       (Intercept) 0.20536  0.45317 
 Site:Tree (Intercept) 1.19762  1.09436 
Number of obs: 80, groups: obs, 80; Site:Tree, 20

Fixed effects:
                 Estimate Std. Error z value Pr(>|z|)  
(Intercept)       0.01161    0.47608   0.024   0.9805  
AGR             -18.97679 3130.76500  -0.006   0.9952  
MHUC             -1.60704    0.63626  -2.526   0.0115 *
Site2            -0.91424    0.74506  -1.227   0.2198  
AGR:MHUC         19.92369 3130.76508   0.006   0.9949  
AGR:Site2         1.02241 4431.84919   0.000   0.9998  
MHUC:Site2        1.80029    0.86235   2.088   0.0368 *
AGR:MHUC:Site2   -3.49042 4431.84933  -0.001   0.9994  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Correlation of Fixed Effects:
            (Intr) AGR    MHUC   Site2  AGR:MHUC AGR:S2 MHUC:S
AGR          0.000                                            
MHUC        -0.281  0.000                                     
Site2       -0.639  0.000  0.180                              
AGR:MHUC     0.000 -1.000  0.000  0.000                       
AGR:Site2    0.000 -0.706  0.000  0.000  0.706                
MHUC:Site2   0.208  0.000 -0.738 -0.419  0.000    0.000       
AGR:MHUC:S2  0.000  0.706  0.000  0.000 -0.706   -1.000  0.000 code here

BestModel=lmer(SP ~ AG * MH + (1|Site:Tree) + (1|obs), data=datos,  
               family = "poisson") 

QAICc(M) 101.4419

Generalized linear mixed model fit by the Laplace approximation 
Formula: SP ~ AG + AG:MH + (1 | Site:Tree) + (1 | obs) 
   Data: datos 
   AIC   BIC logLik deviance
 100.3 114.6 -44.15     88.3
Random effects:
 Groups    Name        Variance Std.Dev.
 obs       (Intercept) 0.76027  0.87194 
 Site:Tree (Intercept) 1.14358  1.06938 
Number of obs: 80, groups: obs, 80; Site:Tree, 20

Fixed effects:
             Estimate Std. Error z value Pr(>|z|)
(Intercept)   -0.5153     0.4061  -1.269    0.205
AGR          -18.7146  2603.4397  -0.007    0.994
AGA:MHUC      -0.7301     0.5045  -1.447    0.148
AGR:MHUC      17.7221  2603.4397   0.007    0.995
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1) Yes, that makes sense. As you add random effects it is not atypical for fixed effects to become smaller. This is appropriate as estimation of the random effects in essence models some of the dependencies in the data. As you might imagine, the less you violate independence the more difficult it is to reach significance.

2) You should have a t or z statistic for each fixed effect. Evaluate those directly according to the appropriate criteria you select.

3) I think the correlation between fixed factors is normal and not a problem. R. H. Baayen's Analyzing Linguistic Data: A Practical Introduction to Statistics using R makes it clear that the correlation between fixed factors in the model results does not reflect a direct correlation between the fixed effects. It is for... something else.

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  • $\begingroup$ Is it necessary, when interpreting the GLMM results for the fixed effects, to consider the intercept value? Or, for instance, with a -1.5 for a given fixed effect reflects a negative tendence? $\endgroup$ – mtao Jul 17 '16 at 10:46
  • $\begingroup$ @Teresa: You consider the fixed effects more or less in the same way you would if you hadn't fit a random effects model. So, if the intercept would mean something if you ran the model under glm, it probably means something here as well. $\endgroup$ – russellpierce Jul 19 '16 at 5:52

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