Percentages as the response variable in GLMM (glmer), proportional binomial or not?

Question

After looking through several questions on stackexchange, and even applying many of the suggested methods I have come to a point where I would like to have an expert's advice if the methods applied on my dataset are correct. This is for my master-dissertation and up untill 1,5 months ago I never worked with R.

I studied the micro-habitat characteristics for the ovipostion of the Pyrgus Malvae butterfly. I measured several environmental variables as the percentage of the hostplant site area, more specifically with an example: the percentage of dwarf-shrub cover per host-plant site, or the percentage of wild-boar digging per hostplant site. Each site equals a circle with a radius of 25 cm with the hostplant in the center. For each occupied hostplant I selected a paired unoccupied hostplant.

The test I chose to use is a GLMM (glmer from the package lme4) since I want to account for several random effects such as the hostplant species (HP_spp), the date on which I measured these variables (VS_Date) and the pair number (Pair_nr) of each occupied and unoccupied hostplant pair. In the GLMM I want to use my environmental variables as the response variable and use Occupancy (0 = unoccupied = no egg found; 1 = occupied) as the independent variable to check for differences within each environmental variable between the two levels of occupancy.

I have performed 3 different kind of glmer: I will use one of my variables as an example here: VS_G = vegetation structure grass = the percentage of grass cover within a hostplant site.

1) standard glmer with family = binomial
Note: VS_G is expressed as a decimal (i.e. 0.7 for 70% grass cover)

GLMMS106_VS_G_Occ <- glmer(VS_G~Occupancy + 
                             (1|VS_Date) + (1|Pair_nr) + (1|HP_spp), family = binomial, data = PM_data106)

with this test I obviously get the error message:

Warning message: In eval(expr, envir, enclos) : non-integer #successes in a binomial glm!

And a resulting p-value of:

#R              Estimate Std. Error z   value Pr(>|z|)    
#R (Intercept)    -0.665        0.290  -2.293 0.021869 *  
#R Occupancy      -2.574        0.777  -3.312 0.000925 ***

2) After looking up this error I found out I had to change my test since I am working with proportional data, so I applied 2 tests with the cbind command and the weights command:

2a) glmer with cbind
Note: VS_G is expressed as a integer (i.e. 70 for 70% grass cover), I created VS_G_inv = 100-VS_G; so in case of VS_G == 70, then VS_G_inv == 30.

GLMMS106_VS_G_Occ <- glmer(cbind(VS_G, VS_G_inv)~Occupancy +   
                              (1|VS_Date) + (1|Pair_nr) + (1|HP_spp), family = binomial, data = PM_data106)

Hooray! no error anymore however the p-value seems of:

#R             Estimate Std. Error z value Pr(>|z|)    
#R (Intercept) -0.58302    0.22478  -2.594  0.00949 **   
#R Occupancy   -0.96390    0.04594 -20.982  < 2e-16 ***

2b) glmer with weights
Note: VS_G is expressed as an integer (i.e. 70 for 70% grass cover) and VS_Weights was created (= 100 for each site)

GLMMS106_VS_G_Occ <- glmer(VS_G/VS_Weights~Occupancy +   
                              (1|VS_Date) + (1|Pair_nr) + (1|HP_spp), family = binomial, data = PM_data106, weights = VS_Weights)

Same result as the cbind test:

#R             Estimate Std. Error z value Pr(>|z|)    
#R (Intercept) -0.58302    0.22478  -2.594  0.00949 **  
#R Occupancy   -0.96390    0.04594 -20.982  < 2e-16 ***

So after discussing this with my supervisors I found out that via using the cbind or weights command my sample size is increased by a factor 100 in this case. This in it's turn results in these very low p-values. So now I am wondering if the first test is ok to use after all?

Answer 1

I think that You might use simply a discrete binomial GLMM rather than continuous, which is just slightly different from the model You described. By the way, the warning message You've mentioned is no error: it simply notifies You, that the binomial response variable was continuous (having non-integer, i.e. non-zero and non-one values). If the non-integer values are between zero and one, there should be no problem.

If I understand correctly, You are interested in whether the structural vegetation coverage influences the presence of P. malvae eggs, correct? In that case, Occupancy should be the response variable, because You expect change in the presence of eggs in response to other environmental variables. Having the cause and causation in the right order helps to make sense of the results of such ecological models. In my opinion, the model You might want to use would look something like this:

glmer(Occupancy~VS_G+HP_spp+(1|VS_Date)+(1|Pair_nr), family="binomial", data=PM_data106)

In this model, You can add all 3 variables representing vegetation-coverages, although You should keep an eye out for the possible interactions between the three coverage variables. Also, i think it would make more ecological sense, to specify the hostplant species as fixed effect, because it has relevance to see, whether one plant species or another is preferred by the animal for laying eggs.

Cheers,

ZR