# Interpretation of binomial GLM (glmer) with interaction and results description

I would like to confirm if I am analysing the results of my model correctly and get some advise if I am missing something!

I conducted the following model to analyse factors that describe the feeding behaviour of whales. My dataset includes a set of dives performed by whales where foraging takes the value 1 if the whale fed during the dive and 0 if it didn't. The explanatory variables are basically dive metrics (maximum depth of dive, duration, etc.) and time of the day (Day and Night).

g_model <- glmer(foraging~max_depths*diel_2+duration+pd_times+d_rate+a_rate+bottom_prop+(1|whale),
data=data, control=glmerControl(optimizer="bobyqa",optCtrl=list(maxfun=2e5)),
family="binomial", na.action = na.fail)

summary(g_model)

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial  ( logit )
Formula: foraging ~ max_depths * diel_2 + duration + pd_times + d_rate +
a_rate + bottom_prop + (1 | whale)
Data: data
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))

AIC      BIC   logLik deviance df.resid
1244.6   1301.4   -612.3   1224.6     2164

Scaled residuals:
Min      1Q  Median      3Q     Max
-99.364  -0.195   0.083   0.285   4.311

Random effects:
Groups Name        Variance Std.Dev.
whale  (Intercept) 0.2201   0.4692
Number of obs: 2174, groups:  whale, 18

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)              0.5621     0.1600   3.514 0.000441 ***
max_depths               1.9123     0.1863  10.266  < 2e-16 ***
diel_2Night              1.5491     0.2577   6.012 1.83e-09 ***
duration                -0.3535     0.1516  -2.331 0.019739 *
pd_times                -0.6118     0.1531  -3.996 6.45e-05 ***
d_rate                   0.6536     0.1099   5.949 2.70e-09 ***
a_rate                  -0.3688     0.1248  -2.955 0.003122 **
bottom_prop              2.8876     0.1343  21.502  < 2e-16 ***
max_depths:diel_2Night   1.9049     0.3184   5.984 2.18e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) mx_dpt dl_2Ng duratn pd_tms d_rate a_rate bttm_p
max_depths  -0.083
diel_2Night -0.313  0.206
duration    -0.115 -0.678  0.066
pd_times     0.034 -0.037 -0.020 -0.098
d_rate       0.079 -0.131 -0.042  0.409 -0.015
a_rate       0.095 -0.659 -0.139  0.423  0.093 -0.134
bottom_prop  0.086  0.628  0.158 -0.449 -0.083  0.078 -0.248
mx_dpth:_2N  0.065  0.034  0.600 -0.165  0.066 -0.138 -0.081  0.168
1. I am wondering if the interpretation is as simply as, for example, there is a significant difference in the maximum depth (max_depths) reached between feeding and non-feeding dives, with maximum depth taking higher (deeper) values in feeding dives.

And for diel2_Night, does it means there are more feeding dives at night?

1. How do I interpret the interaction? Maximum depth is higher in feeding dives at night? What about during the day?

2. What is the typical form of presentation of these results. I guess I should refer the p-value but do I have to state the estimate as well for example?

Please try not to be too concerned with p-values. They don't tell you anything about practical significance.

I am wondering if the interpretation is as simply as, for example, there is a significant difference in the maximum depth (max_depths) reached between feeding and non-feeding dives, with maximum depth taking higher (deeper) values in feeding dives.

Unfortunately not. max_depths is involved in an interaction with diel_2 so it's main effect, 1.91, is conditional on diel_2 being zero (or since it's a categorical variable, at it's reference level, which is Day).

That is, for every 1 unit increase in max_depths, there is an estimated expected 1.91 increase in the log-odds of feeding being 1, for whales feeding in the day time.

And for diel2_Night, does it means there are more feeding dives at night?

No. Again, because diel2 is ivolved in the same interaction, with max_depths, this is conditional on max_depths being zero.

That is, there is an estimated expected 1.55 increase in the log-odds of feeding being 1, for whales feeding at night compared to whales feeding at daytime, when max_depths is zero. For this reason alone you may want to centre the max_depths variable so that a value of zero makes more sense.

How do I interpret the interaction? Maximum depth is higher in feeding dives at night? What about during the day?

Not quite. The slope for max_depths is 1.90 higher for whales feeding at night, compared to whales feeding in the daytime. In other words, every 1 unit increase in max_depths is associated with a 1.90 higher increase in the log-odds of feeding being 1 for whales feeding at night, compared to whales feeding in the daytime.

All of the raw estimates are based on the log-odds scale. You can exponentiate them to obtain odds ratios if you wish.

What is the typical form of presentation of these results. I guess I should refer the p-value but do I have to state the estimate as well for example?

Unfortunately there is nothing typical. If you are preparing a paper for publication then my practical advice is to read the submission guidelines for the journal you are submitting to, as well as looking at previously published articles in your field in that journal. Focus on the effect sizes. Rather than quoting p values, remark that the effect size is significant (at whatever significance level you choose). If you have some "non signifcant" findings, these might also be interesting.

Also, compute confidence intervals and quote them alongside the effect sizes (rather than quoting p values)

• hello @Robert Long, could you explain what is the meaning in the estimate value of duration? If I average the dive duration of feeding and non-feeding dives, average duration of feeding dives its much higher. Boxplots show me the same thing. Therefore, I dont understand why the estimate value is negative, is it due to the random effect? Jul 24, 2020 at 11:08
• Sure. The estimate for duration is the association of a 1 unit change with the outcome - so every 1 unit increase in duration is associated with an decrease of 0.35 in the log odds of feeding being 1. The phonomenon you describe could be an example of Simpson's paradox where subject-level associations can be reversed in the population. You might explore this with plots, and to model it better you might want to consider mean-centering duration for each whale and including the whale mean duration as a level 2 variable. You will then get seperate estimates for between and within. Jul 25, 2020 at 8:25
• Hi again @Robert Long! I'm working in r and I don't know how to mean-center duration, do you have any tip? Thanks again! Jul 26, 2020 at 9:41
• also, I scaled all variables before the model using scale(), shouldn't that do the trick? Jul 26, 2020 at 9:48
• If you applied scale to the whole variable it would scale it globally. You have to do for each whale and also make a variable for the mean duration of each whale. Jul 26, 2020 at 10:31