Firstly, many thanks to the experts that take the time to post on here and answer questions, I for one really appreciate it! I apologise for the length of this post but I want to show my understanding of what I am doing so that any advice is better channelled to what I’ve missed or don’t understand.

Briefly, I am looking at a behaviour (whether it occurs or not) as a response and I am using several environmental and spatial covariates as fixed effects. The random effect that I am interested in is the individual. Ultimately, I would like to know how important the individual is in explaining the variance in the behaviour.  

 If I could just give a brief background to the initial analyses of my particular question, then I will go into the GLMM analyses with two model summaries. I first looked at GLMs with a binomial distribution and the best model was not massively overdispersed, c. 1.3. From earlier posts on numerous R forums it would seem that some people have said anything under 1.5 can be ignored others disagree with this statement and others have noted that the number of observations may play a role in (arbitrarily) deciding whether the data are overdispersed or not; ultimately it appears to be a grey area. So to be on the safe side, I used a beta-binomial distribution (package: aod) for the same model (i.e. same response and fixed-effects) and I compared the AICs of the binomial and the beta-binomial distributions, which showed that the beta-binomial distribution was better, AIC 12158.95 compared to 12558.04. Incidentally, the overdispersion coefficients for the beta-binomial were 0.05 with a se of 0.003 and it was considered significant (>0.001).

I then want to start the process all over again (i.e. starting with the ‘complex’ model) but this time I want to include individual as a random-effect (i.e. use a GLMM). Note that this is the ‘complex’ model summary below (I have excluded the correlation of the fixed effects).


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Generalized linear mixed model fit by the Laplace approximation

Formula: cbind(A_C_T, A_C_F) ~ ACT_PUP_PROP + L_DIST_PUP + L_POOL_DIST + TERRITORY + DEN_10m + PUP_AGE + RAIN_GAUGE + AIR_TEMP + HR_FAC +  YEAR_FAC + (1 | FINAL_ID)

   Data: A_DATA

  AIC  BIC logLik deviance

 7440 7588  -3698     7396

Random effects:

 Groups   Name        Variance Std.Dev.

 FINAL_ID (Intercept) 0.087889 0.29646

Number of obs: 6094, groups: FINAL_ID, 54

 

Fixed effects:

               Estimate Std. Error z value Pr(>|z|)   

(Intercept)  -2.3942840  0.1461403 -16.383  < 2e-16 ***

ACT_PUP_PROP  1.1914474  0.0569402  20.925  < 2e-16 ***

L_DIST_PUP   -0.1111116  0.0571834  -1.943 0.052007 . 

L_POOL_DIST   0.0854491  0.0298057   2.867 0.004145 **

TERRITORY    -0.0031526  0.0008232  -3.830 0.000128 ***

DEN_10m       0.0372856  0.0100775   3.700 0.000216 ***

PUP_AGE       0.0029961  0.0035653   0.840 0.400710   

RAIN_GAUGE    0.0013297  0.0033509   0.397 0.691503   

AIR_TEMP     -0.0910215  0.0197620  -4.606 4.11e-06 ***

HR_FAC2      -0.3360817  0.1121353  -2.997 0.002726 **

HR_FAC3      -0.4022991  0.1103212  -3.647 0.000266 ***

HR_FAC4      -0.5814143  0.1125690  -5.165 2.40e-07 ***

HR_FAC5      -0.8386866  0.1169818  -7.169 7.53e-13 ***

HR_FAC6      -0.7452432  0.1137925  -6.549 5.79e-11 ***

HR_FAC7      -0.6691143  0.1107415  -6.042 1.52e-09 ***

HR_FAC8      -0.7458898  0.1089098  -6.849 7.45e-12 ***

HR_FAC9      -0.6029419  0.1079628  -5.585 2.34e-08 ***

HR_FAC10     -0.6866431  0.1105382  -6.212 5.24e-10 ***

HR_FAC11     -0.6694072  0.1730285  -3.869 0.000109 ***

YEAR_FAC2008 -0.0701368  0.0735545  -0.954 0.340319   

YEAR_FAC2009  0.0856704  0.0804873   1.064 0.287150   

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

 

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>From Bolker et al.’s 2009 TREE publication it seems that there is an issue with calculating degrees of freedom and therefore getting an accurate indication of overdispersion in a GLMM, and it also seems that some regular posters are wary of the calculations that are currently used with GLMMs to test for overdispersion. So again, to be safe I looked at observation-level random effect (OLRE) (i.e. one random effect per observation), which appears to be the way to go if you have or are concerned about having overdispersion in your data. Furthermore, J. Hadfield has stated in a previous post that in most studies binomial data will be overdispersed and adding an OLRE can be a good way of modelling this. J. Maindonald and a poster (Colin) have discussed OLRE in detail in an earlier thread. The outcome appeared to be that if the OLRE accounts for a large proportion of the variance then it should be interpreted as the OLRE improving the model fit substantially. JM goes on to say that failure to model the OLRE component of variance might lead you seriously astray if you are using the information to design another study.

 

(All of these ‘quotes’ and points noted above can be found on this thread: https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005320.html)



For the purposes of being thorough and perhaps helping other people that are wrestling with the same/similar questions references that have been used in previous posts are:

 Section 10.5 of the 3rd edition of Analysis & Graphics Using R - it is a (rare, as far as I am aware) cite-able example of how to apply this analysis.

 Hinde J (1982) GLIM 82-109, cited in an earlier post by J. Hadfield who states that Hinde uses a log-normal mixing distribution instead of the usual gamma (i.e. negative binomial) which is the equivalent to the OLRE with log link.

 Breslow, N. 1990 Tests of hypotheses in overdispersed Poisson   regression and other quasi-likelihood models J. Am. Stat. Ass. 85   565-571. Again cited by J. Hadfield, where Breslow refers to this as the “classical” approach, and JH considers the idea to be so old that it could simply be attributed to common sense. 

 And finally (for my mini-basic-review), the best/clearest definition of OLRE I have found is from B. Bolker where he stated that the OLRE approach is equivalent to assuming that the individual observations are [x]-normal distributed, i.e. a compound of a normal distribution transformed by the inverse link function and the specified distribution family (lognormal-Poisson for Poisson with log link; logit-normal-binomial for binomial with logit link).

 (The ‘quotes’ and references noted above can be found on this thread: https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005656.html or on this thread: https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005227.html)


Given the support for this approach from several regular posters of this group I applied this to the model:

 

A_DATA$OBS_INDEX<-1:nrow(A_DATA)

 

The output is shown below:

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Generalized linear mixed model fit by the Laplace approximation

Formula: cbind(A_C_T, A_C_F) ~ ACT_PUP_PROP + L_DIST_PUP + L_POOL_DIST +TERRITORY + DEN_10m + PUP_AGE + RAIN_GAUGE + AIR_TEMP + HR_FAC + YEAR_FAC + (1 | FINAL_ID_FAC) + (1 | OBS_INDEX)

   Data: A_DATA

  AIC  BIC logLik deviance

 7156 7311  -3555     7110

Random effects:

 Groups       Name        Variance Std.Dev.

 OBS_INDEX    (Intercept) 0.580827 0.76212

 FINAL_ID_FAC (Intercept) 0.053969 0.23231

Number of obs: 6094, groups: OBS_INDEX, 6094; FINAL_ID_FAC, 54

 

Fixed effects:

              Estimate Std. Error z value Pr(>|z|)    

(Intercept)  -2.645349   0.170566 -15.509  < 2e-16 ***

ACT_PUP_PROP  1.277917   0.070394  18.154  < 2e-16 ***

L_DIST_PUP   -0.117012   0.069451  -1.685 0.092023 . 

L_POOL_DIST   0.099797   0.034874   2.862 0.004215 **

TERRITORY    -0.003562   0.001012  -3.522 0.000428 ***

DEN_10m       0.043536   0.011914   3.654 0.000258 ***

PUP_AGE       0.003645   0.004337   0.840 0.400646   

RAIN_GAUGE    0.000579   0.004080   0.142 0.887170   

AIR_TEMP     -0.092591   0.024148  -3.834 0.000126 ***

HR_FAC2      -0.344388   0.137746  -2.500 0.012414 * 

HR_FAC3      -0.405143   0.134810  -3.005 0.002653 **

HR_FAC4      -0.605626   0.136984  -4.421 9.82e-06 ***

HR_FAC5      -0.874781   0.141214  -6.195 5.84e-10 ***

HR_FAC6      -0.784525   0.137942  -5.687 1.29e-08 ***

HR_FAC7      -0.704094   0.134388  -5.239 1.61e-07 ***

HR_FAC8      -0.815604   0.132333  -6.163 7.13e-10 ***

HR_FAC9      -0.646171   0.131626  -4.909 9.15e-07 ***

HR_FAC10     -0.717495   0.134594  -5.331 9.78e-08 ***

HR_FAC11     -0.705579   0.202216  -3.489 0.000484 ***

YEAR_FAC2008 -0.031999   0.079125  -0.404 0.685911   

YEAR_FAC2009  0.124139   0.087507   1.419 0.156009   

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

 

Correlation of Fixed Effects:

             (Intr) ACT_PU L_DIST L_POOL TERRIT DEN_10 PUP_AG RAIN_G AIR_TE HR_FAC2

ACT_PUP_PRO  -0.202                                                               

L_DIST_PUP   -0.241  0.090                                                        

L_POOL_DIST  -0.148  0.009  0.335                                                 

TERRITORY    -0.427  0.047  0.193  0.061                                          

DEN_10m      -0.192 -0.003 -0.010  0.318 -0.114                                   

PUP_AGE      -0.044  0.052  0.105 -0.042 -0.237 -0.082                            

RAIN_GAUGE   -0.223 -0.007  0.020 -0.050  0.094  0.103 -0.029                     

AIR_TEMP     -0.014 -0.039  0.009  0.071 -0.057 -0.104  0.187 -0.071              

HR_FAC2      -0.584  0.025 -0.005  0.007  0.000 -0.042 -0.066  0.014  0.028       

HR_FAC3      -0.621  0.057 -0.003  0.002  0.007 -0.028 -0.083  0.000  0.016  0.752

HR_FAC4      -0.608  0.060 -0.008  0.005 -0.001 -0.018 -0.100  0.008 -0.012  0.741

HR_FAC5      -0.583  0.068 -0.009 -0.005 -0.008 -0.023 -0.103 -0.009 -0.019  0.719

HR_FAC6      -0.603  0.046  0.000 -0.002 -0.002 -0.004 -0.106  0.000 -0.043  0.734

HR_FAC7      -0.616  0.055  0.005 -0.004  0.002 -0.013 -0.096 -0.004 -0.032  0.753

HR_FAC8      -0.616  0.042  0.002 -0.016 -0.009 -0.021 -0.104 -0.010 -0.030  0.765

HR_FAC9      -0.618  0.043 -0.004 -0.018 -0.020 -0.023 -0.093  0.002 -0.005  0.770

HR_FAC10     -0.592  0.023 -0.012 -0.032 -0.016 -0.032 -0.052  0.009 -0.020  0.747

HR_FAC11     -0.406  0.013  0.008 -0.015  0.009 -0.011  0.042 -0.003 -0.020  0.483

YEAR_FAC2008 -0.331 -0.102 -0.093 -0.121 -0.029  0.001 -0.012  0.093  0.169  0.049

YEAR_FAC2009 -0.326 -0.111 -0.160 -0.154  0.066  0.203 -0.205  0.222 -0.090 -0.002

             HR_FAC3 HR_FAC4 HR_FAC5 HR_FAC6 HR_FAC7 HR_FAC8 HR_FAC9 HR_FAC10

ACT_PUP_PRO                                                                 

L_DIST_PUP                                                                  

L_POOL_DIST                                                                  

TERRITORY                                                                   

DEN_10m                                                                     

PUP_AGE                                                                     

RAIN_GAUGE                                                                   

AIR_TEMP                                                                    

HR_FAC2                                                                     

HR_FAC3                                                                      

HR_FAC4       0.765                                                         

HR_FAC5       0.744   0.735                                                 

HR_FAC6       0.759   0.750   0.730                                          

HR_FAC7       0.778   0.768   0.747   0.764                                 

HR_FAC8       0.790   0.780   0.759   0.776   0.794                         

HR_FAC9       0.795   0.784   0.763   0.779   0.798   0.811                 

HR_FAC10      0.768   0.758   0.737   0.753   0.772   0.784   0.789         

HR_FAC11      0.495   0.486   0.472   0.483   0.496   0.503   0.506   0.498 

YEAR_FAC2008  0.082   0.079   0.072   0.074   0.065   0.071   0.081   0.050 

YEAR_FAC2009  0.030   0.037   0.026   0.040   0.027   0.033   0.028  -0.001 

             HR_FAC11 YEAR_FAC2008

ACT_PUP_PRO                      

L_DIST_PUP                       

L_POOL_DIST                      

TERRITORY                        

DEN_10m                           

PUP_AGE                          

RAIN_GAUGE                       

AIR_TEMP                         

HR_FAC2                          

HR_FAC3                          

HR_FAC4                          

HR_FAC5                          

HR_FAC6                          

HR_FAC7                          

HR_FAC8                          

HR_FAC9                          

HR_FAC10                         

HR_FAC11                         

YEAR_FAC2008  0.013              

YEAR_FAC2009 -0.013    0.704     

 

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The purpose of presenting the initial GLM analyses was to gain an opinion on whether it is possible to gain any inference on overdispersion of a GLMM based on a GLM (that is 'the same' but without the random effect)?

 Irrespective, it is clear that the OLRE explains a significant amount of the variation in the random effects (0.580827). Also, comparing the AIC it is the better model (7156 compared to 7440). So based on these two facts it seems to me that irrespective of overdispersion I should include the OLRE in the model?

 If that is the case, would I be correct in interpreting the OLRE as a within-individual variance measure, and that in this case there is a lot of variation with individuals’ behaviour? And if I look at the other random effect (FINAL_ID), which explains 0.053969 of the variance in the random effects, which I take to be the between-individual variation, then proportionally we can say that there is little evidence for variation between individuals?

 

Any help or advice would be greatly appreciated,

 

Best wishes,

 

Ross

 

Ross Culloch

PhD Candidate

School of Biological & Biomedical Sciences

University of Durham

South Road

Durham

UK

DH1 3LE
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