predict_response: Adjusted predictions and estimated marginal means from...

In ggeffects: Create Tidy Data Frames of Marginal Effects for 'ggplot' from Model Outputs

Adjusted predictions and estimated marginal means from regression models

Description

After fitting a model, it is useful generate model-based estimates (expected values, or adjusted predictions) of the response variable for different combinations of predictor values. Such estimates can be used to make inferences about relationships between variables.

The ggeffects package computes marginal means and adjusted predicted values for the response, at the margin of specific values or levels from certain model terms. The package is built around three core functions: predict_response() (understanding results), test_predictions() (importance of results) and plot() (communicate results).

By default, adjusted predictions or marginal means are returned on the response scale, which is the easiest and most intuitive scale to interpret the results. There are other options for specific models as well, e.g. with zero-inflation component (see documentation of the type-argument). The result is returned as structured data frame, which is nicely printed by default. plot() can be used to easily create figures.

The main function to calculate marginal means and adjusted predictions is predict_response(), which returns adjusted predictions, marginal means or averaged counterfactual predictions depending on value of the margin-argument.

In previous versions of ggeffects, the functions ggpredict(), ggemmeans(), ggeffect() and ggaverage() were used to calculate marginal means and adjusted predictions. These functions are still available, but predict_response() as a "wrapper" around these functions is the preferred way to calculate marginal means and adjusted predictions now.

Usage

predict_response(
  model,
  terms,
  margin = "mean_reference",
  ci_level = 0.95,
  type = "fixed",
  condition = NULL,
  back_transform = TRUE,
  ppd = FALSE,
  vcov_fun = NULL,
  vcov_type = NULL,
  vcov_args = NULL,
  weights = NULL,
  interval,
  verbose = TRUE,
  ...
)

Arguments

model	A model object.
terms	Names of those terms from model, for which predictions should be displayed (so called focal terms). Can be: A character vector, specifying the names of the focal terms. This is the preferred and probably most flexible way to specify focal terms, e.g. terms = "x [40:60]", to calculate predictions for the values 40 to 60. A list, where each element is a named vector, specifying the focal terms and their values. This is the "classical" R way to specify focal terms, e.g. list(x = 40:60). A formula, e.g. terms = ~ x + z, which is internally converted to a character vector. This is probably the least flexible way, as you cannot specify representative values for the focal terms. A data frame representing a "data grid" or "reference grid". Predictions are then made for all combinations of the variables in the data frame. terms at least requires one variable name. The maximum length is four terms, where the second to fourth term indicate the groups, i.e. predictions of the first term are grouped at meaningful values or levels of the remaining terms (see values_at()). It is also possible to define specific values for focal terms, at which adjusted predictions should be calculated (see details below). All remaining covariates that are not specified in terms are "marginalized", see the margin argument in ?predict_response. See also argument condition to fix non-focal terms to specific values, and argument typical for ggpredict() or ggemmeans().
margin	Character string, indicating how to marginalize over the non-focal predictors, i.e. those variables that are not specified in terms. Possible values are "mean_reference", "mean_mode", "marginalmeans" and "empirical" (or "counterfactual", aka average "counterfactual" predictions). You can set a default-option for the margin argument via options(), e.g. options(ggeffects_margin = "empirical"), so you don't have to specify your preferred marginalization method each time you call predict_response(). See details in the documentation below.
ci_level	Numeric, the level of the confidence intervals. Use ci_level = NA if confidence intervals should not be calculated (for instance, due to computation time). Typically, confidence intervals are based on the returned standard errors for the predictions, assuming a t- or normal distribution (based on the model and the available degrees of freedom, i.e. roughly ⁠+/- 1.96 * SE⁠). See introduction of this vignette for more details.
type	Character, indicating whether predictions should be conditioned on specific model components or not. Consequently, most options only apply for survival models, mixed effects models and/or models with zero-inflation (and their Bayesian counter-parts); only exception is type = "simulate", which is available for some other model classes as well (which respond to simulate()). Note 1: For brmsfit-models with zero-inflation component, there is no type = "zero_inflated" nor type = "zi_random"; predicted values for these models always condition on the zero-inflation part of the model. The same is true for MixMod-models from GLMMadaptive with zero-inflation component (see 'Details'). Note 2: If margin = "empirical", or when calling ggaverage() respectively, (i.e. counterfactual predictions), the type argument is handled differently. It is set to "response" by default, but usually accepts all possible options from the type-argument of the model's respective predict() method. E.g., passing a glm object would allow the options "response", "link", and "terms". For models with zero-inflation component, the below mentioned options "fixed", "zero_inflated" and "zi_prob" can also be used and will be "translated" into the corresponding type option of the model's respective predict()-method. "fixed" (or "fe" or "count") Predicted values are conditioned on the fixed effects or conditional model only (for mixed models: predicted values are on the population-level and confidence intervals are returned, i.e. re.form = NA when calling predict()). For instance, for models fitted with zeroinfl from pscl, this would return the predicted mean from the count component (without zero-inflation). For models with zero-inflation component, this type calls predict(..., type = "link") (however, predicted values are back-transformed to the response scale, i.e. the conditional mean of the response). "random" (or "re") This only applies to mixed models, and type = "random" does not condition on the zero-inflation component of the model. type = "random" still returns population-level predictions, however, conditioned on random effects and considering individual level predictions, i.e. re.form = NULL when calling predict(). This may affect the returned predicted values, depending on whether REML = TRUE or REML = FALSE was used for model fitting. Furthermore, unlike type = "fixed", intervals also consider the uncertainty in the variance parameters (the mean random effect variance, see Johnson et al. 2014 for details) and hence can be considered as prediction intervals. For models with zero-inflation component, this type calls predict(..., type = "link") (however, predicted values are back-transformed to the response scale). To get predicted values for each level of the random effects groups, add the name of the related random effect term to the terms-argument (for more details, see this vignette). "zero_inflated" (or "fe.zi" or "zi") Predicted values are conditioned on the fixed effects and the zero-inflation component. For instance, for models fitted with zeroinfl from pscl, this would return the predicted (or expected) response (mu*(1-p)), and for glmmTMB, this would return the expected response mu*(1-p) without conditioning on random effects (i.e. random effect variances are not taken into account for the confidence intervals). For models with zero-inflation component, this type calls predict(..., type = "response"). See 'Details'. "zi_random" (or "re.zi" or "zero_inflated_random") Predicted values are conditioned on the zero-inflation component and take the random effects uncertainty into account. For models fitted with glmmTMB(), hurdle() or zeroinfl(), this would return the expected value mu*(1-p). For glmmTMB, prediction intervals also consider the uncertainty in the random effects variances. This type calls predict(..., type = "response"). See 'Details'. "zi_prob" (or "zi.prob") Predicted zero-inflation probability. For glmmTMB models with zero-inflation component, this type calls predict(..., type = "zlink"); models from pscl call predict(..., type = "zero") and for GLMMadaptive, predict(..., type = "zero_part") is called. "simulate" (or "sim") Predicted values and confidence resp. prediction intervals are based on simulations, i.e. calls to simulate(). This type of prediction takes all model uncertainty into account, including random effects variances. Currently supported models are objects of class lm, glm, glmmTMB, wbm, MixMod and merMod. See ... for details on number of simulations. "survival" and "cumulative_hazard" (or "surv" and "cumhaz") Applies only to coxph-objects from the survial-package and calculates the survival probability or the cumulative hazard of an event. When margin = "empirical" (or when calling ggaverage()), the type argument accepts all values from the type-argument of the model's respective predict()-method.
condition	Named character vector, which indicates covariates that should be held constant at specific values. Unlike typical, which applies a function to the covariates to determine the value that is used to hold these covariates constant, condition can be used to define exact values, for instance condition = c(covariate1 = 20, covariate2 = 5). See 'Examples'.
back_transform	Logical, if TRUE (the default), predicted values for log-, log-log, exp, sqrt and similar transformed responses will be back-transformed to original response-scale. See insight::find_transformation() for more details.
ppd	Logical, if TRUE, predictions for Stan-models are based on the posterior predictive distribution rstantools::posterior_predict(). If FALSE (the default), predictions are based on posterior draws of the linear predictor rstantools::posterior_epred(). This is roughly comparable to the distinction between confidence and prediction intervals. ppd = TRUE incorporates the residual variance and hence returned intervals are similar to prediction intervals. Consequently, if interval = "prediction", ppd is automatically set to TRUE. The ppd argument will be deprecated in a future version. Please use interval = "prediction" instead.
vcov_fun	Variance-covariance matrix used to compute uncertainty estimates (e.g., for confidence intervals based on robust standard errors). This argument accepts a covariance matrix, a function which returns a covariance matrix, or a string which identifies the function to be used to compute the covariance matrix. A (variance-covariance) matrix A function which returns a covariance matrix (e.g., stats::vcov()) A string which indicates the estimation type for the heteroscedasticity-consistent variance-covariance matrix, e.g. vcov_fun = "HC0". Possible values are "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", and "HC5", which will then call the vcovHC()-function from the sandwich package, using the specified type. Further possible values are "CR0", "CR1", "CR1p", "CR1S", "CR2", and "CR3", which will call the vcovCR()-function from the clubSandwich package. A string which indicates the name of the ⁠vcov*()⁠-function from the sandwich or clubSandwich packages, e.g. vcov_fun = "vcovCL", which is used to compute (cluster) robust standard errors for predictions. If NULL, standard errors (and confidence intervals) for predictions are based on the standard errors as returned by the predict()-function. Note that probably not all model objects that work with predict_response() are also supported by the sandwich or clubSandwich packages. See details in this vignette.
vcov_type	Character vector, specifying the estimation type for the robust covariance matrix estimation (see ?sandwich::vcovHC or ?clubSandwich::vcovCR for details). Only used when vcov_fun is a character string indicating one of the functions from those packages. When vcov_fun is a function, a possible type argument must be provided via the vcov_args argument.
vcov_args	List of named vectors, used as additional arguments that are passed down to vcov_fun.
weights	This argument is used in two different ways, depending on the margin argument. When margin = "empirical", weights can either be a character vector, naming the weigthing variable in the data, or a vector of weights (of same length as the number of observations in the data). This variable will be used to weight adjusted predictions. When margin = "marginalmeans", weights must be a character vector and is passed to emmeans::emmeans(), specifying weights to use in averaging non-focal categorical predictors. See https://rvlenth.github.io/emmeans/reference/emmeans.html for details.
interval	Type of interval calculation, can either be "confidence" (default) or "prediction". May be abbreviated. Unlike confidence intervals, prediction intervals include the residual variance (sigma^2) to account for the uncertainty of predicted values. For mixed models, interval = "prediction" is the default for type = "random". When type = "fixed", the default is interval = "confidence". Note that prediction intervals are not available for all models, but only for models that work with insight::get_sigma(). For Bayesian models, when interval = "confidence", predictions are based on posterior draws of the linear predictor rstantools::posterior_epred(). If interval = "prediction", rstantools::posterior_predict() is called.
verbose	Toggle messages or warnings.
...	If margin is set to "mean_reference" or "mean_mode", arguments are passed down to ggpredict() (further down to predict()); for margin = "marginalmeans", further arguments passed down to ggemmeans() and thereby to emmeans::emmeans(); if margin = "empirical", further arguments are passed down to marginaleffects::avg_predictions(). If type = "simulate", ... may also be used to set the number of simulation, e.g. nsim = 500. When calling ggeffect(), further arguments passed down to effects::Effect().

Value

A data frame (with ggeffects class attribute) with consistent data columns:

• "x": the values of the first term in terms, used as x-position in plots.

• "predicted": the predicted values of the response, used as y-position in plots.

• "std.error": the standard error of the predictions. Note that the standard errors are always on the link-scale, and not back-transformed for non-Gaussian models!

• "conf.low": the lower bound of the confidence interval for the predicted values.

• "conf.high": the upper bound of the confidence interval for the predicted values.

• "group": the grouping level from the second term in terms, used as grouping-aesthetics in plots.

• "facet": the grouping level from the third term in terms, used to indicate facets in plots.

  The estimated marginal means (or predicted values) are always on the response scale!

  For proportional odds logistic regression (see ?MASS::polr) resp. cumulative link models (e.g., see ?ordinal::clm), an additional column "response.level" is returned, which indicates the grouping of predictions based on the level of the model's response.

  Note that for convenience reasons, the columns for the intervals are always named "conf.low" and "conf.high", even though for Bayesian models credible or highest posterior density intervals are returned.

  There is an as.data.frame() method for objects of class ggeffects, which has an terms_to_colnames argument, to use the term names as column names instead of the standardized names "x" etc.

Supported Models

A list of supported models can be found at the package website. Support for models varies by marginalization method (the margin argument), i.e. although predict_response() supports most models, some models are only supported exclusively by one of the four downstream functions (ggpredict(), ggemmeans(), ggeffect() or ggaverage()). This means that not all models work for every margin option of predict_response().

Holding covariates at constant values, or how to marginalize over the non-focal predictors

predict_response() is a wrapper around ggpredict(), ggemmeans() and ggaverage(). Depending on the value of the margin argument, predict_response() calls one of those functions. The margin argument indicates how to marginalize over the non-focal predictors, i.e. those variables that are not specified in terms. Possible values are:

• "mean_reference" and "mean_mode": For "mean_reference", non-focal predictors are set to their mean (numeric variables), reference level (factors), or "most common" value (mode) in case of character vectors. For "mean_mode", non-focal predictors are set to their mean (numeric variables) or mode (factors, or "most common" value in case of character vectors).

  These predictons represent a rather "theoretical" view on your data, which does not necessarily exactly reflect the characteristics of your sample. It helps answer the question, "What is the predicted value of the response at meaningful values or levels of my focal terms for a 'typical' observation in my data?", where 'typical' refers to certain characteristics of the remaining predictors.

• "marginalmeans": non-focal predictors are set to their mean (numeric variables) or averaged over the levels or "values" for factors and character vectors. Averaging over the factor levels of non-focal terms computes a kind of "weighted average" for the values at which these terms are hold constant. Thus, non-focal categorical terms are conditioned on "weighted averages" of their levels. There are different weighting options that can be altered using the weights argument.

  These predictions come closer to the sample, because all possible values and levels of the non-focal predictors are taken into account. It would answer the question, "What is the predicted value of the response at meaningful values or levels of my focal terms for an 'average' observation in my data?". It refers to randomly picking a subject of your sample and the result you get on average.

• "empirical" (or "counterfactual"): non-focal predictors are averaged over the observations in the sample. The response is predicted for each subject in the data and predicted values are then averaged across all subjects, aggregated/grouped by the focal terms. In particular, averaging is applied to counterfactual predictions (Dickerman and Hernan 2020). There is a more detailed description in this vignette.

  Counterfactual predictions are useful, insofar as the results can also be transferred to other contexts. It answers the question, "What is the predicted value of the response at meaningful values or levels of my focal terms for the 'average' observation in the population?". It does not only refer to the actual data in your sample, but also "what would be if" we had more data, or if we had data from a different population. This is where "counterfactual" refers to.

You can set a default-option for the margin argument via options(), e.g. options(ggeffects_margin = "empirical"), so you don't have to specify your "default" marginalization method each time you call predict_response(). Use options(ggeffects_margin = NULL) to remove that setting.

The condition argument can be used to fix non-focal terms to specific values.

Marginal Means and Adjusted Predictions at Specific Values

Meaningful values of focal terms can be specified via the terms argument. Specifying meaningful or representative values as string pattern is the preferred way in the ggeffects package. However, it is also possible to use a list() for the focal terms if prefer the "classical" R way. terms can also be a data (or reference) grid provided as data frame. All options are described in this vignette.

Indicating levels in square brackets allows for selecting only certain groups or values resp. value ranges. The term name and the start of the levels in brackets must be separated by a whitespace character, e.g. terms = c("age", "education [1,3]"). Numeric ranges, separated with colon, are also allowed: terms = c("education", "age [30:60]"). The stepsize for ranges can be adjusted using by, e.g. terms = "age [30:60 by=5]".

The terms argument also supports the same shortcuts as the values argument in values_at(). So terms = "age [meansd]" would return predictions for the values one standard deviation below the mean age, the mean age and one SD above the mean age. terms = "age [quart2]" would calculate predictions at the value of the lower, median and upper quartile of age.

Furthermore, it is possible to specify a function name. Values for predictions will then be transformed, e.g. terms = "income [exp]". This is useful when model predictors were transformed for fitting the model and should be back-transformed to the original scale for predictions. It is also possible to define own functions (see this vignette).

Instead of a function, it is also possible to define the name of a variable with specific values, e.g. to define a vector v = c(1000, 2000, 3000) and then use terms = "income [v]".

You can take a random sample of any size with sample=n, e.g terms = "income [sample=8]", which will sample eight values from all possible values of the variable income. This option is especially useful for plotting predictions at certain levels of random effects group levels, where the group factor has too many levels to be completely plotted. For more details, see this vignette.

Finally, numeric vectors for which no specific values are given, a "pretty range" is calculated (see pretty_range()), to avoid memory allocation problems for vectors with many unique values. If a numeric vector is specified as second or third term (i.e. if this focal term is used for "stratification"), representative values (see values_at()) are chosen (unless other values are specified), which are typically the mean value, as well as one standard deviation below and above the mean. If all values for a numeric vector should be used to compute predictions, you may use e.g. terms = "age [all]". See also package vignettes.

To create a pretty range that should be smaller or larger than the default range (i.e. if no specific values would be given), use the n tag, e.g. terms="age [n=5]" or terms="age [n=12]". Larger values for n return a larger range of predicted values.

Bayesian Regression Models

predict_response() also works with Stan-models from the rstanarm or brms-packages. The predicted values are the median value of all drawn posterior samples. Standard errors are the median absolute deviation of the posterior samples. The confidence intervals for Stan-models are Bayesian predictive intervals. By default, the predictions are based on rstantools::posterior_epred() and hence have the limitations that the uncertainty of the error term (residual variance) is not taken into account. The recommendation is to use the posterior predictive distribution (rstantools::posterior_predict()), i.e. setting interval = "prediction".

Zero-Inflated and Zero-Inflated Mixed Models with brms

Models of class brmsfit always condition on the zero-inflation component, if the model has such a component. Hence, there is no type = "zero_inflated" nor type = "zi_random" for brmsfit-models, because predictions are based on draws of the posterior distribution, which already account for the zero-inflation part of the model.

Zero-Inflated and Zero-Inflated Mixed Models with glmmTMB

If model is of class glmmTMB, hurdle, zeroinfl or zerotrunc, and margin is not set to ⁠"empirical⁠, simulations from a multivariate normal distribution (see ?MASS::mvrnorm) are drawn to calculate mu*(1-p). Confidence intervals are then based on quantiles of these results. For type = "zi_random", prediction intervals also take the uncertainty in the random-effect paramters into account (see also Brooks et al. 2017, pp.391-392 for details).

An alternative for models fitted with glmmTMB that take all model uncertainties into account are simulations based on simulate(), which is used when type = "simulate" (see Brooks et al. 2017, pp.392-393 for details).

Finally, if margin = "empirical", the returned predictions are already conditioned on the zero-inflation part (and possible random effects) of the model, thus these are most comparable to the type = "simulate" option. In other words, if all model components should be taken into account for predictions, you should consider using margin = "empirical".

MixMod-models from GLMMadaptive

Predicted values for the fixed effects component (type = "fixed" or type = "zero_inflated") are based on predict(..., type = "mean_subject"), while predicted values for random effects components (type = "random" or type = "zi_random") are calculated with predict(..., type = "subject_specific") (see ?GLMMadaptive::predict.MixMod for details). The latter option requires the response variable to be defined in the newdata-argument of predict(), which will be set to its typical value (see values_at()).

Multinomial Models

polr, clm models, or more generally speaking, models with ordinal or multinominal outcomes, have an additional column response.level, which indicates with which level of the response variable the predicted values are associated.

Averaged model predictions (package MuMIn)

For averaged model predictions, i.e. when the input model is an object of class "averaging" (MuMIn::model.avg()), predictions are made with the full averaged coefficients.

Note

Printing Results

The print() method gives a clean output (especially for predictions by groups), and indicates at which values covariates were held constant. Furthermore, the print() method has several arguments to customize the output. See this vignette for details.

Limitations

The support for some models, for example from package MCMCglmm, is not fully tested and may fail for certain models. If you encounter any errors, please file an issue at Github.

References

• Brooks ME, Kristensen K, Benthem KJ van, Magnusson A, Berg CW, Nielsen A, et al. glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling. The R Journal. 2017;9: 378-400.

• Johnson PC. 2014. Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models. Methods Ecol Evol, 5: 944-946.

• Dickerman BA, Hernan, MA. Counterfactual prediction is not only for causal inference. Eur J Epidemiol 35, 615–617 (2020).

Examples

library(sjlabelled)
data(efc)
fit <- lm(barthtot ~ c12hour + neg_c_7 + c161sex + c172code, data = efc)

predict_response(fit, terms = "c12hour")
predict_response(fit, terms = c("c12hour", "c172code"))
# more compact table layout for printing
out <- predict_response(fit, terms = c("c12hour", "c172code", "c161sex"))
print(out, collapse_table = TRUE)

# specified as formula
predict_response(fit, terms = ~ c12hour + c172code + c161sex)

# only range of 40 to 60 for variable 'c12hour'
predict_response(fit, terms = "c12hour [40:60]")

# terms as named list
predict_response(fit, terms = list(c12hour = 40:60))

# covariate "neg_c_7" is held constant at a value of 11.84 (its mean value).
# To use a different value, use "condition"
predict_response(fit, terms = "c12hour [40:60]", condition = c(neg_c_7 = 20))

# to plot ggeffects-objects, you can use the 'plot()'-function.
# the following examples show how to build your ggplot by hand.


# plot predicted values, remaining covariates held constant
library(ggplot2)
mydf <- predict_response(fit, terms = "c12hour")
ggplot(mydf, aes(x, predicted)) +
  geom_line() +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.1)

# three variables, so we can use facets and groups
mydf <- predict_response(fit, terms = c("c12hour", "c161sex", "c172code"))
ggplot(mydf, aes(x = x, y = predicted, colour = group)) +
  stat_smooth(method = "lm", se = FALSE) +
  facet_wrap(~facet, ncol = 2)

# select specific levels for grouping terms
mydf <- predict_response(fit, terms = c("c12hour", "c172code [1,3]", "c161sex"))
ggplot(mydf, aes(x = x, y = predicted, colour = group)) +
  stat_smooth(method = "lm", se = FALSE) +
  facet_wrap(~facet) +
  labs(
    y = get_y_title(mydf),
    x = get_x_title(mydf),
    colour = get_legend_title(mydf)
  )

# level indication also works for factors with non-numeric levels
# and in combination with numeric levels for other variables
data(efc)
efc$c172code <- sjlabelled::as_label(efc$c172code)
fit <- lm(barthtot ~ c12hour + neg_c_7 + c161sex + c172code, data = efc)
predict_response(fit, terms = c("c12hour",
  "c172code [low level of education, high level of education]",
  "c161sex [1]"))

# when "terms" is a named list
predict_response(fit, terms = list(
  c12hour = seq(0, 170, 30),
  c172code = c("low level of education", "high level of education"),
  c161sex = 1)
)

# use categorical value on x-axis, use axis-labels, add error bars
dat <- predict_response(fit, terms = c("c172code", "c161sex"))
ggplot(dat, aes(x, predicted, colour = group)) +
  geom_point(position = position_dodge(0.1)) +
  geom_errorbar(
    aes(ymin = conf.low, ymax = conf.high),
    position = position_dodge(0.1)
  ) +
  scale_x_discrete(breaks = 1:3, labels = get_x_labels(dat))

# 3-way-interaction with 2 continuous variables
data(efc)
# make categorical
efc$c161sex <- as_factor(efc$c161sex)
fit <- lm(neg_c_7 ~ c12hour * barthtot * c161sex, data = efc)
# select only levels 30, 50 and 70 from continuous variable Barthel-Index
dat <- predict_response(fit, terms = c("c12hour", "barthtot [30,50,70]", "c161sex"))
ggplot(dat, aes(x = x, y = predicted, colour = group)) +
  stat_smooth(method = "lm", se = FALSE, fullrange = TRUE) +
  facet_wrap(~facet) +
  labs(
    colour = get_legend_title(dat),
    x = get_x_title(dat),
    y = get_y_title(dat),
    title = get_title(dat)
  )

# or with ggeffects' plot-method
plot(dat, show_ci = FALSE)


# predictions for polynomial terms
data(efc)
fit <- glm(
  tot_sc_e ~ c12hour + e42dep + e17age + I(e17age^2) + I(e17age^3),
  data = efc,
  family = poisson()
)
predict_response(fit, terms = "e17age")

ggeffects documentation built on Sept. 12, 2024, 7:41 a.m.