Marginal Effects of Regression Models
Daniel Lüdecke
2019-03-17
Aim of the ggeffects-package
Results of regression models are typically presented as tables that are easy to understand. For more complex models that include interaction or quadratic / spline terms, tables with numbers are less helpful and difficult to interpret. In such cases, the visualization of marginal effects is far easier to understand and allows to intuitively get the idea of how predictors and outcome are associated, even for complex models.
The aim of this package is similar to the broom-package: transforming “untidy” input into a tidy data frame, especially for further use with ggplot. However, ggeffects does not return model-summaries; rather, this package computes marginal effects at the mean or average marginal effects from statistical models and returns the result as tidy data frame.
Since the focus lies on plotting the data (the marginal effects), at least one model term needs to be specified for which the effects are computed. It is also possible to compute marginal effects for model terms, grouped by the levels of another model’s predictor. The package also allows plotting marginal effects for two- or three-way-interactions, or for specific values of a model term only. Examples are shown below.
Short technical note
ggpredict(), ggeffect(), ggemmeans() and ggaverage() always return predicted values for the response of a model (or response distribution for Bayesian models).
Typically, ggpredict() returns confidence intervals based on the standard errors (+/- 1.96 * SE) as returned by the predict()-function. If predict() for a certain class does not return standard errors, these are calculated manually, by following steps: matrix-multiply X by the parameter vector B to get the predictions, then extract the variance-covariance matrix V of the parameters and compute XVX' to get the variance-covariance matrix of the predictions. The square-root of the diagonal of this matrix represent the standard errors of the predictions.
For mixed models, if type = "re" or type = "re.zi", the uncertainty in the random effects is accounted for when calculating the standard errors. Hence, in such cases, the confidence intervals may be considered as “prediction intervals”.
Consistent and tidy structure
The returned data frames always have the same, consistent structure and column names, so it’s easy to create ggplot-plots without the need to re-write the arguments to be mapped in each ggplot-call. x and predicted are the values for the x- and y-axis. conf.low and conf.high could be used as ymin and ymax aesthetics for ribbons to add confidence bands to the plot. group can be used as grouping-aesthetics, or for faceting.
The examples shown here mostly use ggplot2-code for the plots, however, there is also a plot()-method, which is described in the vignette Plotting Marginal Effects.
Marginal effects at the mean
ggpredict() computes predicted values for all possible levels and values from a model’s predictors. In the simplest case, a fitted model is passed as first argument, and the term in question as second argument. Use the raw name of the variable for the terms-argument only - you don’t need to write things like poly(term, 3) or I(term^2) for the terms-argument.
As you can see, ggpredict() (and ggeffect() or ggemmeans()) has a nice print()-method, which takes care of printing not too many rows (but always an equally spaced range of values, including minimum and maximum value of the term in question) and giving some extra information. This is especially useful when predicted values are shown depending on the levels of other terms (see below).
The output shows the predicted values for the response at each value from the term c12hour. The data is already in shape for ggplot:
Marginal effects at the mean for different groups
The terms-argument accepts up to three model terms, where the second and third term indicate grouping levels. This allows predictions for the term in question at different levels for other model terms:
Creating a ggplot is pretty straightforward: the colour-aesthetics is mapped with the group-column:
Finally, a second grouping structure can be defined, which will create another column named facet, which - as the name implies - might be used to create a facted plot:
Marginal effects for each model term
If the term argument is either missing or NULL, marginal effects for each model term are calculated. The result is returned as a list, which can be plotted manually (or using the plot() function).
Average marginal effects
ggaverage() compute average marginal effects. While ggpredict() creates a data-grid (using expand.grid()) for all possible combinations of values (even if some combinations are not present in the data), ggaverage() computes predicted values based on the given data. This means that different predicted values for the outcome may occure at the same value or level for the term in question. The predicted values are then averaged for each value of the term in question and the linear trend is smoothed accross the averaged predicted values. This means that the line representing the marginal effects may cross or diverge, and are not necessarily in paralell to each other.
Since ggaverage() produces different predicted values for the same value or level of terms, linear-smoothing for gaussian models and loess-smoothing for non-gaussian models is used to produce “smooth” lines. This may lead to misleading plots, especially if polynomial or spline terms are included in the model. It’s preferred to use ggpredict() or ggeffect() resp. ggemmeans().
Two- and Three-Way-Interactions
To plot the marginal effects of interaction terms, simply specify these terms in the terms-argument.
Since the terms-argument accepts up to three model terms, you can also compute marginal effects for a 3-way-interaction. To plot the marginal effects of three interaction terms, just like before, specify all three terms in the terms-argument.
# fit model with 3-way-interaction
fit <- lm(neg_c_7 ~ c12hour * barthtot * c161sex, data = efc)
# select only levels 30, 50 and 70 from continuous variable Barthel-Index
mydf <- ggpredict(fit, terms = c("c12hour", "barthtot [30,50,70]", "c161sex"))
ggplot(mydf, aes(x, predicted, colour = group)) +
geom_line() +
facet_wrap(~facet)
Polynomial terms and splines
ggpredict() also works for models with polynomial terms or splines. Following code reproduces the plot from ?splines::bs:
Survival models
ggpredict() also supports coxph-models from the survival-package and is able to either plot risk-scores (the default), probabilities of survival (type = "surv") or cumulative hazards (type = "cumhaz").
Since probabilities of survival and cumulative hazards are changing accross time, the time-variable is automatically used as x-axis in such cases, so the terms-argument only needs up to two variables for type = "surv" or type = "cumhaz".
data("lung", package = "survival")
# remove category 3 (outlier)
lung <- subset(lung, subset = ph.ecog %in% 0:2)
lung$sex <- factor(lung$sex, labels = c("male", "female"))
lung$ph.ecog <- factor(lung$ph.ecog, labels = c("good", "ok", "limited"))
m <- survival::coxph(survival::Surv(time, status) ~ sex + age + ph.ecog, data = lung)
# predicted risk-scores
ggpredict(m, c("sex", "ph.ecog"))
#>
#> # Predicted risk scores
#> # x = sex
#>
#> # ph.ecog = good
#> x predicted std.error conf.low conf.high
#> 1 0.829 0.148 0.621 1.107
#> 2 0.481 0.176 0.340 0.679
#>
#> # ph.ecog = limited
#> x predicted std.error conf.low conf.high
#> 1 2.044 0.157 1.503 2.781
#> 2 1.185 0.172 0.846 1.658
#>
#> # ph.ecog = ok
#> x predicted std.error conf.low conf.high
#> 1 1.249 0.107 1.013 1.540
#> 2 0.724 0.125 0.566 0.925
#>
#> Adjusted for:
#> * age = 62.42
Printing factor levels instead of numeric values
By default, to quickly create plot where data points can be connected with lines (i.e. to use a continuous x-axis-scale), the x-value is numeric, even if the first variable in terms (which is represented by the x-column in the returned data frame) is categorical. Since ggeffects supports labelled data (see also next section), plots are always correctly annotated, and a discrete scale is automatically used (see vignette Plotting Marginal Effects).
If you want to preserve the factor-class for x, use the argument x.as.factor = TRUE. This will also print the factor levels instead of numeric values in the console. An alternative would be to use the x.lab = TRUE-argument in the print()-method.
Here are some examples, first we fit a model where two variables are factors with character factor levels (we use sjmisc::to_label() to convert the labelled data into factors):
By default, x is numeric.
As value labels and factor levels are saved as attributes in the returned data frame, we still can print these value labels:
Again, printing the results will cause factors as x to become numeric…
… unless you change this behaviour with x.as.factor = TRUE.
Labelling the data
ggeffects makes use of the sjlabelled-package and supports labelled data. If the data from the fitted models is labelled, the value and variable label attributes are usually copied to the model frame stored in the model object. ggeffects provides various getter-functions to access these labels, which are returned as character vector and can be used in ggplot’s lab()- or scale_*()-functions.
get_title() - a generic title for the plot, based on the model family, like “predicted values” or “predicted probabilities”get_x_title() - the variable label of the first model term in terms.get_y_title() - the variable label of the response.get_legend_title() - the variable label of the second model term in terms.get_x_labels() - value labels of the first model term in terms.get_legend_labels() - value labels of the second model term in terms.
The data frame returned by ggpredict() or ggaverage() must be used as argument to one of the above function calls.
