Error bars overlap but effect is significant in glmm. What's going on?

Question

I have run the following glmm:

mod<-glmer(data=newdata, total_flr_vis ~ treatment + flr_num + (1|individual), family=poisson)

and get this output from summary()

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: poisson  ( log )
Formula: total_flr_vis ~ treatment + flr_num + (1 | individual)
Data: newdata

 AIC      BIC   logLik deviance df.resid 
706.8    718.9   -349.4    698.8      148 

Scaled residuals: 
Min      1Q  Median      3Q     Max 
-3.1461 -0.8155 -0.3522 -0.2022 14.1669 

Random effects:
Groups     Name        Variance Std.Dev.
individual (Intercept) 4.066    2.016   
Number of obs: 152, groups:  individual, 42

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.610176   0.432666  -3.722 0.000198 ***
treatmentR  -0.457492   0.121054  -3.779 0.000157 ***
flr_num      0.037063   0.005064   7.319  2.5e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
             (Intr) trtmnR
  treatmentR -0.117       
  flr_num    -0.416  0.040

But I get the following plot for treatment using plot(allEffects(mod))

I don't understand why the effects plot shows overlapping error bars while the summary() output tells me that the effect of treatment is significant. Is there a problem with the model, or is it the plot? How can I troubleshoot this?

Here is the residual plot (which I got using plot(mod))

I'm not totally sure how to interpret this plot, but it does not look random to me, thus I suspect that there is something wrong with the model.

I am happy to post data if someone can tell me where to do that.

Note: I posted this on Stack Overflow and someone suggested that I am misinterpreting the first figure. This is very likely, but, unfortunately, the user did not suggest an alternative way to interpret the figure. If anyone here can do so, I would be very grateful.

Any help would be very welcome.

Answer 1

Your model has a random effect for individual which effects the marginal distribution of your observations when split into treatment groups, which is what you see in the first plot (I think, because you don't describe the data.) In the reproducible example below, there is a very obvious effect of the factor 'x', but it there is a lot of overlap in the 'allEffects' plot because this plot seems to average over the grouping factor.

The short answer is that effective blocking reduces error variance.

effect <- 3
ngroups <- 4
nper <- 5
total <- ngroups*nper
groups <- rnorm(ngroups, 0, 3)
offset <- 2
library(dplyr)
df <- data.frame(x = as.factor(rep(c(0, 1), each=ngroups/2*nper)),
             groupid = as.factor(rep(1:ngroups, times=nper))) %>%
  mutate(y = rpois(total, exp(offset + effect*(x==1) + groups[groupid] + rnorm(total))))
library(lme4)
library(effects)
fit <- glmer(y ~ x + (1|groupid), family=poisson("log"), data=df)
summary(fit)
plot(allEffects(fit))

library(ggplot2)
ggplot(df, aes(x=groupid, y=y, fill=x)) + geom_boxplot(position="dodge") + 
scale_y_continuous(trans="log")