R modeling relationship between binary outcome and binary predictor among different categories

Question

I could use some advice. I am trying to model the relationship between a binary outcome (impact present/impact absent) and a binary predictor (threat present/threat absent), and see if that relationship varies among threat categories. In other words, does the correlation between threats and impacts vary significantly among threat categories. Since I am modeling a binary outcome, it seems like logistic regression would be a reasonable approach, and I will discuss what I have so far below. If anyone has suggestions for other approaches they would be welcome. My end goal is to be able to say, for each threat category, the likelihood of an impact being present if a threat present is x.

My dataframe looks like this: species = a categorical variable with 128 levels threat_category = a categorical variable with 17 levels impact.pres = a binomial variable with present = 1 and not_present = 0 threat.pres = a binomial variable with present = 1 and not_present = 0

Example data (smaller than actual dataset):

dat <- cbind(Species = rep(letters[1:10], each = 5),
             threat_cat = rep(c("recreation", "climate", "pollution", "fire", "invasive_spp"), 10),
             impact.pres = sample(0:1, size = 50, replace = T),
             threat.pres = sample(0:1, size = 50, replace = T))

I am running a no-intercept model because I am interested in the true coefficients for each threat, not the difference between each threat and a reference threat.

My model and output looks something like this:

mod<- glm(impact.pres ~ 0 + threat.pres*threat_cat, data = dat, family = "binomial")
summary(mod)

Call:
glm(formula = impact.pres ~ 0 + threat.pres * threat_cat, family = "binomial", 
    data = dat)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.66511  -0.90052  -0.00022   0.90052   1.89302  

Coefficients:
                                     Estimate Std. Error z value Pr(>|z|)
threat.pres                        -6.931e-01  1.323e+00  -0.524    0.600
threat_catclimate                   6.931e-01  8.660e-01   0.800    0.423
threat_catfire                      6.513e-16  1.000e+00   0.000    1.000
threat_catinvasive_spp              1.099e+00  1.155e+00   0.951    0.341
threat_catpollution                -9.163e-01  8.367e-01  -1.095    0.273
threat_catrecreation                6.931e-01  8.660e-01   0.800    0.423
threat.pres:threat_catfire         -9.163e-01  1.987e+00  -0.461    0.645
threat.pres:threat_catinvasive_spp -1.099e+00  1.958e+00  -0.561    0.575
threat.pres:threat_catpollution    -1.596e+01  2.284e+03  -0.007    0.994
threat.pres:threat_catrecreation    1.099e+00  1.958e+00   0.561    0.575

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 69.315  on 50  degrees of freedom
Residual deviance: 56.785  on 40  degrees of freedom
AIC: 76.785

Number of Fisher Scoring iterations: 16

In my real data, threat.pres and most of the individual impacts are significant, but none of the interaction terms are. I am wondering if I have specified the model correctly to be able to answer my question, and if so, how best to interpret these coefficients, and the significance levels associated with them.

Thank you for your time.

Answer 1

I think you actually need different syntax for your model. Here is the same model with a slightly different parameterization:

set.seed(100)
dat <- data.frame(Species = rep(letters[1:10], each = 5),
                  threat_cat = rep(c("recreation", "climate", "pollution", "fire", "invasive_spp"), 10),
                  impact.pres = sample(0:1, size = 50, replace = T),
                  threat.pres = sample(0:1, size = 50, replace = T))

mod <- glm(impact.pres ~ 0 + threat_cat/threat.pres, 
           data = dat, family = "binomial")
summary(mod)
#> 
#> Call:
#> glm(formula = impact.pres ~ 0 + threat_cat/threat.pres, family = "binomial", 
#>     data = dat)
#> 
#> Coefficients:
#>                                      Estimate Std. Error z value Pr(>|z|)
#> threat_catclimate                   5.108e-01  7.303e-01   0.699    0.484
#> threat_catfire                      1.609e+00  1.095e+00   1.469    0.142
#> threat_catinvasive_spp             -1.386e+00  1.118e+00  -1.240    0.215
#> threat_catpollution                 1.386e+00  1.118e+00   1.240    0.215
#> threat_catrecreation               -1.386e+00  1.118e+00  -1.240    0.215
#> threat_catclimate:threat.pres      -5.108e-01  1.592e+00  -0.321    0.748
#> threat_catfire:threat.pres         -2.018e+01  3.261e+03  -0.006    0.995
#> threat_catinvasive_spp:threat.pres  1.792e+00  1.443e+00   1.241    0.214
#> threat_catpollution:threat.pres     8.770e-16  1.581e+00   0.000    1.000
#> threat_catrecreation:threat.pres    1.995e+01  2.917e+03   0.007    0.995
#> 
#>     Null deviance: 69.315  on 50  degrees of freedom
#> Residual deviance: 45.511  on 40  degrees of freedom
#> AIC: 65.511
#> 

^{Created on 2022-03-02 by the reprex package (v2.0.1)}

(Note that because you did not set a seed the results will differ slightly.)

With this parameterization, you can interpret the "main effect" of each category of the categorical variable as the log odds of impact.pres for that category when threat.pres is 0. For example, the coefficient on threat_catclimate (.5108) means that the log odds of impact.pres for the climate group of threat_cat is .5108 when threat.pres is 0, or, the odds of impact.pres for the climate group of threat_cat is $\exp(.5108)=1.67$ when threat.pres is 0.

You can interpret the interaction term between threat.pres and each level of threat_cat as the difference in the log odds of impact.pres when threat.pres is 1 vs. when threat.pres is 0 for that group of threat_cat. For example, the coefficient on threat_catclimate:threat.pres (-.5108) means the log odds of impact.pres for the climate group of threat_cat is -.5108 lower when threat.pres is 1 than when threat.pres is 0.

From this model (but not the standard parameterization or your attempt), it is easy to see the "effect" of the binary variable at each level of the categorical variable without requiring a reference category.