Practical example: Logistic Mixed Effects Model with Interaction Term

Daniel Lüdecke

2019-11-08

This vignette demonstrate how to use ggeffects to compute and plot marginal effects of a logistic regression model. To cover some frequently asked questions by users, we’ll fit a mixed model, inlcuding an interaction term and a quadratic resp. spline term. A general introduction into the package usage can be found in the vignette marginal effects of regression model.

First, we load the required packages and create a sample data set with a binomial and continuous variable as predictor as well as a group factor. To avoid convergence warnings, the continuous variable is standardized.

library(ggeffects)
library(sjmisc)
library(lme4)
library(splines)

set.seed(123)

dat <- data.frame(
  outcome = rbinom(n = 100, size = 1, prob = 0.35),
  var_binom = as.factor(rbinom(n = 100, size = 1, prob = 0.2)),
  var_cont = rnorm(n = 100, mean = 10, sd = 7),
  group = sample(letters[1:4], size = 100, replace = TRUE)
)

dat$var_cont <- sjmisc::std(dat$var_cont)

Simple Logistic Mixed Effects Model

We start by fitting a simple mixed effects model.

m1 <- glmer(
  outcome ~ var_binom + var_cont + (1 | group), 
  data = dat, 
  family = binomial(link = "logit")
)

For a discrete variable, marginal effects for all levels are calculated by default. For continuous variables, a pretty range of values is generated. See more details about value ranges in the vignette marginal effects at specific values.

For logistic regression models, since ggeffects returns marginal effects on the response scale, the predicted values are predicted probabilities. Furthermore, for mixed models, the predicted values are typically at the population level, not group-specific.

ggpredict(m1, "var_binom")
#> 
#> # Predicted probabilities of outcome
#> # x = var_binom
#> 
#>  x predicted std.error conf.low conf.high
#>  0     0.369     0.290    0.249     0.508
#>  1     0.379     0.517    0.181     0.627
#> 
#> Adjusted for:
#> * var_cont = -0.00
#> *    group = 0 (population-level)

ggpredict(m1, "var_cont")
#> Data were 'prettified'. Consider using `terms="var_cont [all]"` to get smooth plots.
#> 
#> # Predicted probabilities of outcome
#> # x = var_cont
#> 
#>     x predicted std.error conf.low conf.high
#>  -2.5     0.337     0.619    0.131     0.631
#>  -2.0     0.343     0.527    0.157     0.594
#>  -1.0     0.356     0.367    0.212     0.532
#>   0.0     0.369     0.290    0.249     0.508
#>   0.5     0.376     0.306    0.248     0.523
#>   1.0     0.382     0.354    0.236     0.554
#>   2.0     0.396     0.508    0.195     0.640
#>   3.5     0.416     0.794    0.131     0.772
#> 
#> Adjusted for:
#> * var_binom = 0
#> *     group = 0 (population-level)

To plot marginal effects, simply plot the returned results or use the pipe.

# save marginal effects in an object and plot
me <- ggpredict(m1, "var_binom")
plot(me)

# plot using the pipe
ggpredict(m1, "var_cont") %>% plot()

Logistic Mixed Effects Model with Interaction Term

Next, we fit a model with an interaction between the binomial and continuous variable.

m2 <- glmer(
  outcome ~ var_binom * var_cont + (1 | group), 
  data = dat, 
  family = binomial(link = "logit")
)

To compute or plot marginal effects of interaction terms, simply specify these terms, i.e. the names of the variables, as character vector in the terms-argument. Since we have an interaction between var_binom and var_cont, the argument would be terms = c("var_binom", "var_cont"). However, the first variable in the terms-argument is used as predictor along the x-axis. Marginal effects are then plotted for specific values or at specific levels from the second variable.

If the second variable is a factor, marginal effects for each level are plotted. If the second variable is continuous, representative values are chosen (typically, mean +/- one SD, see marginal effects at specific values).

ggpredict(m2, c("var_cont", "var_binom")) %>% plot()
#> Data were 'prettified'. Consider using `terms="var_cont [all]"` to get smooth plots.

ggpredict(m2, c("var_binom", "var_cont")) %>% plot()

Logistic Mixed Effects Model with quadratic Interaction Term

Now we fit a model with interaction term, where the continuous variable is modelled as quadratic term.

m3 <- glmer(
  outcome ~ var_binom * poly(var_cont, degree = 2, raw = TRUE) + (1 | group), 
  data = dat, 
  family = binomial(link = "logit")
)

Again, ggeffect automatically plots all high-order terms when these are specified in the terms-argument. Hence, the function call is identical to the previous examples with interaction terms, which had no polynomial term included.

ggpredict(m3, c("var_cont", "var_binom")) %>% plot()
#> Model contains splines or polynomial terms. Consider using `terms="var_cont [all]"` to get smooth plots. See also package-vignette 'Marginal Effects at Specific Values'.

As you can see, ggeffects also returned a message indicated that the plot may not look very smooth due to the involvement of polynomial or spline terms:

Model contains splines or polynomial terms. Consider using terms="var_cont [all]" to get smooth plots. See also package-vignette ‘Marginal Effects at Specific Values’.

This is because for mixed models, computing marginal effects with spline or polynomial terms may lead to memory allocation problems. If you are sure that this will not happen, add the [all]-tag to the terms-argument, as described in the message:

ggpredict(m3, c("var_cont [all]", "var_binom")) %>% plot()

The above plot produces much smoother curves.

Logistic Mixed Effects Model with Three-Way Interaction

The last model does not produce very nice plots, but for the sake of demonstration, we fit a model with three interaction terms, including polynomial and spline terms.

set.seed(321)
dat <- data.frame(
  outcome = rbinom(n = 100, size = 1, prob = 0.35),
  var_binom = rbinom(n = 100, size = 1, prob = 0.5),
  var_cont = rnorm(n = 100, mean = 10, sd = 7),
  var_cont2 = rnorm(n = 100, mean = 5, sd = 2),
  group = sample(letters[1:4], size = 100, replace = TRUE)
)


m4 <- glmer(
  outcome ~ var_binom * poly(var_cont, degree = 2) * ns(var_cont2, df = 3) + (1 | group),
  data = dat, 
  family = binomial(link = "logit")
)

Since we have marginal effects for var_cont at the levels of var_cont2 and var_binom, we not only have groups, but also facets to plot all three “dimensions”. Three-way interactions are plotted simply by speficying all terms in question in the terms-argument.

ggpredict(m4, c("var_cont [all]", "var_cont2", "var_binom")) %>% plot()