how to interpret a generalised linear mixed model with binomial data

Question

I know there has been a similar question before but I'm struggling to use the answers there to help interpret my data. I'm new to statistics so am very keen for help! I have data where each row represents a habitat patch, and some of my columns include 'urban' (whether the patch is bordered by urban habitat or not), 'Pre_postdev' (whether the habitat came from a pre-development map or a post-development map), and 'project_ID' (I randomly sampled projects and then within those systematically sampled every habitat patch, hence why I'm using a mixed model). I have used the GLMM:

glmm_urban <- glmer(urban ~ Pre_postdev + (1 | project_ID), 
                family = binomial(link = "logit"),
                data = size3)

In order to investigate the difference in proportion of habitat patches being bordered by urban habitat, between pre and post-development habitats. I've accounted for my sampling method with the random effect of project ID. I have two levels to my Pre_postdev variable, "pre" and "post" (which I reordered to be in that order), and "urban" is coded in binary where 1 is Yes and 0 is no.

However, I'm really struggling to interpret the output of my GLMM. Here it is:

Generalized linear mixed model fit by maximum likelihood (Laplace 
    Approximation) ['glmerMod']
 Family: binomial  ( logit )
 Formula: urban ~ Pre_postdev + (1 | project_ID)
   Data: size3

     AIC      BIC   logLik deviance df.resid 
   263.1    273.0   -128.6    257.1      198 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.6526 -0.6816 -0.3952  0.7494  1.8038 

Random effects:
 Groups     Name        Variance Std.Dev.
 project_ID (Intercept) 1.415    1.19    
Number of obs: 201, groups:  project_ID, 12

Fixed effects:
              Estimate Std. Error z value Pr(>|z|)
(Intercept)    0.01086    0.41891   0.026    0.979
Pre_postdev.L  0.29207    0.24846   1.176    0.240

Correlation of Fixed Effects:
            (Intr)
Pre_pstdv.L -0.140

1. Could you help explain how I should interpret this output? I don
2. How can I interpret 'log odds' into answering my research question (Are a higher proportion of patches bordered by urban habitats in post-development maps compared to pre-development maps?).

Answer 1

Answering just based on the output presented, without any real understanding of the data or model diagnostics:

Also assuming the null hypothesis was that the Pre_postdev effect was not a significant contributor to the probability of a habitat patch being bordered by an urban area, and that the researchers intended to test this at a 0.05 type I error cutoff:

1. After accounting for the variance due to the Project_ID, there is insufficient evidence to conclude that the effect of Pre_postdev "post" is different from Pre_postdev "pre" in the probability of a habitat patch being bordered by an urban area. (p-value on Pre_postdev.L of 0.240 > 0.05)

2. The best way to understand how to interpret the coefficients is to write down the regression equation:

$$\textbf{ln}\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 \textbf{Pre_postdev} + \alpha_{\textbf{Project_ID}} + \epsilon$$

$$E(\alpha_{\textbf{Project_ID}}) = 0$$

$$E(\epsilon) = 0$$

$$\hat{p}|\textbf{Pre} = \frac{e^{\beta_0}}{1 + e^{\beta_0}}$$

$$\hat{p}|\textbf{Post} = \frac{e^{\beta_0 + \beta_1}}{1 + e^{\beta_0 + \beta_1}}$$

Difference in log-odds

$$\textbf{ln}\left(\frac{p}{1-p}\right)_{\textbf{Post}} - \textbf{ln}\left(\frac{p}{1-p}\right)_{\textbf{Pre}} = \beta_1 = 0.29207$$

Ratio of odds

$$\frac{\left(\frac{p}{1-p}\right)_{\textbf{Post}}}{\left(\frac{p}{1-p}\right)_{\textbf{Pre}}} = e^{\beta_1} = 1.339$$