problems with transforming proportion data in GLMM model

Question

I'm comparing the effects of a single treatment on multiple response variables from a BACI study.

Most of my data are normally distributed, but I also have some proportion data (binomial) and count data (poisson). For example the proportion of fish displaying behavior X, before and after treatment.

Since all my variables are in different units... and since I want to show the relative effects of treatment on each variable.... I used the scale() function to center and scale my variables -- such that the coefficients in the models represent change in units of standard deviations for each variable.

This is fine with the normally distributed data. BUT... for the proportion data -- if I scale()... I end up with some negative values and also ranges great than +/- 1... since scale() subtracts the mean divides by the standard deviation.

For my normally distributed data I built linear mixed effects models (lme4) with in the form: lmer(Variable ~ Time* Location + (1|Site), data = data)

My understanding is that the interaction of “Time” (before or after impact) and “Location” (Control or Impact) would be significant “when change occurs at the impact sites but not the control sites” (e.g. Popescu et al., 2012 ; Smokorowski and Randal 2017).

For my proportion data I used glmer with binomial distributions... BUT after scaling I can't use a glmer with a binomial distribution, since the transformed data is now no longer a proportion between 0 and 1. But it is also still not normally distributed, so I can't use an lmer model.

I know there is something basic I am missing here... Is there anyway I can show the effect of a treatment on a response variable that is a proportion...in units of SD's?

Answer 1

Welcome to the site, Gabe. I'll try to provide some suggestions and comments about each of your questions:

My understanding is that the interaction of “Time” (before or after impact) and “Location” (Control or Impact) would be significant “when change occurs at the impact sites but not the control sites”

Let's assume you have coded the variables as such, and have them as factor variables:

time==0 if before
time==1 if after
location==0 if control
location==1 if impact

If these are all the variables in your mixed model, you will get four fixed effects coefficients, which have the following interpretation:

(Intercept) - outcome mean at time==0 (before) and location==0 (control)
time        - outcome mean for time==1 (after) and location==0 (control)
location    - outcome mean for location==1 (treatment) and time==0 (before)
time*location - outcome mean for time==1 (after) and location==1 (treatment) 

Set it up as a grid and it should be more obvious what you are getting, a set of means for different groups of fish:

You can also view the interaction as a difference in means for two groups of fish - 1) fish at time 1 in treatment vs time 1 control fish (thus compared to the time coefficient) and 2) fish in treatment at time 1 vs time 0 treatment fish (thus compared to the location coefficient). I always find it helpful to plot the marginal means that come from the model. You can use the ggeffects() package for this:

m1 <- lmer(Variable ~ Time* Location + (1|Site), data = data)
require(ggeffects)
ggpredict(m1, c("Time", "Location") %>% plot()

This will give you a plot of the marginal means with standard error bars for each group, making it much easier to visually inspect the differences.

I know there is something basic I am missing here... Is there anyway I can show the effect of a treatment on a response variable that is a proportion...in units of SD's?

This is not a straightforward topic. Effect size in linear multilevel models is still somewhat debated, but there are approaches. Effect size in generalized multilevel models is not even close to solved. For count data, I suggest viewing this presentation, however note that it is not necessarily for a multilevel Poisson model. For the binomial data, many people interpret the odds ratio ($exp(\hat{\beta})$) as an effect size. See this thread on CV. So do not use any scaling of outcomes or predictors, just run the binomial mixed model as normal and report exponentiated coefficients.