Slopes (aka Partial derivatives, Marginal Effects, or Trends)

Description

Partial derivative of the regression equation with respect to a regressor of interest.

• slopes(): unit-level (conditional) estimates.

• avg_slopes(): average (marginal) estimates.

The newdata argument and the datagrid() function can be used to control where statistics are evaluated in the predictor space: "at observed values", "at the mean", "at representative values", etc.

See the slopes vignette and package website for worked examples and case studies:

• https://marginaleffects.com/vignettes/slopes.html

• https://marginaleffects.com/

Usage

slopes(
  model,
  newdata = NULL,
  variables = NULL,
  type = NULL,
  by = FALSE,
  vcov = TRUE,
  conf_level = 0.95,
  slope = "dydx",
  wts = FALSE,
  hypothesis = NULL,
  equivalence = NULL,
  p_adjust = NULL,
  df = Inf,
  eps = NULL,
  numderiv = "fdforward",
  ...
)

avg_slopes(
  model,
  newdata = NULL,
  variables = NULL,
  type = NULL,
  by = TRUE,
  vcov = TRUE,
  conf_level = 0.95,
  slope = "dydx",
  wts = FALSE,
  hypothesis = NULL,
  equivalence = NULL,
  p_adjust = NULL,
  df = Inf,
  eps = NULL,
  numderiv = "fdforward",
  ...
)

Arguments

model	Model object
newdata	Grid of predictor values at which we evaluate the slopes. Warning: Please avoid modifying your dataset between fitting the model and calling a marginaleffects function. This can sometimes lead to unexpected results. NULL (default): Unit-level slopes for each observed value in the dataset (empirical distribution). The dataset is retrieved using insight::get_data(), which tries to extract data from the environment. This may produce unexpected results if the original data frame has been altered since fitting the model. datagrid() call to specify a custom grid of regressors. For example: newdata = datagrid(cyl = c(4, 6)): cyl variable equal to 4 and 6 and other regressors fixed at their means or modes. See the Examples section and the datagrid() documentation. subset() call with a single argument to select a subset of the dataset used to fit the model, ex: newdata = subset(treatment == 1) dplyr::filter() call with a single argument to select a subset of the dataset used to fit the model, ex: newdata = filter(treatment == 1) string: "mean": Slopes evaluated when each predictor is held at its mean or mode. "median": Slopes evaluated when each predictor is held at its median or mode. "balanced": Slopes evaluated on a balanced grid with every combination of categories and numeric variables held at their means. "tukey": Slopes evaluated at Tukey’s 5 numbers. "grid": Slopes evaluated on a grid of representative numbers (Tukey’s 5 numbers and unique values of categorical predictors).
variables	Focal variables NULL: compute slopes or comparisons for all the variables in the model object (can be slow). Character vector: subset of variables (usually faster).
type	string indicates the type (scale) of the predictions used to compute contrasts or slopes. This can differ based on the model type, but will typically be a string such as: "response", "link", "probs", or "zero". When an unsupported string is entered, the model-specific list of acceptable values is returned in an error message. When type is NULL, the first entry in the error message is used by default.
by	Aggregate unit-level estimates (aka, marginalize, average over). Valid inputs: FALSE: return the original unit-level estimates. TRUE: aggregate estimates for each term. Character vector of column names in newdata or in the data frame produced by calling the function without the by argument. Data frame with a by column of group labels, and merging columns shared by newdata or the data frame produced by calling the same function without the by argument. See examples below. For more complex aggregations, you can use the FUN argument of the hypotheses() function. See that function’s documentation and the Hypothesis Test vignettes on the marginaleffects website.
vcov	Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values: FALSE: Do not compute standard errors. This can speed up computation considerably. TRUE: Unit-level standard errors using the default vcov(model) variance-covariance matrix. String which indicates the kind of uncertainty estimates to return. Heteroskedasticity-consistent: “HC”, “HC0”, “HC1”, “HC2”, “HC3”, “HC4”, “HC4m”, “HC5”. See ?sandwich::vcovHC Heteroskedasticity and autocorrelation consistent: “HAC” Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger" Other: “NeweyWest”, “KernHAC”, “OPG”. See the sandwich package documentation. One-sided formula which indicates the name of cluster variables (e.g., ~unit_id). This formula is passed to the cluster argument of the sandwich::vcovCL function. Square covariance matrix Function which returns a covariance matrix (e.g., stats::vcov(model))
conf_level	numeric value between 0 and 1. Confidence level to use to build a confidence interval.
slope	string indicates the type of slope or (semi-)elasticity to compute: "dydx": dY/dX "eyex": dY/dX * Y / X "eydx": dY/dX * Y "dyex": dY/dX / X Y is the predicted value of the outcome; X is the observed value of the predictor.
wts	logical, string or numeric: weights to use when computing average predictions, contrasts or slopes. These weights only affect the averaging in avg_*() or with the by argument, and not unit-level estimates. See ?weighted.mean string: column name of the weights variable in newdata. When supplying a column name to wts, it is recommended to supply the original data (including the weights variable) explicitly to newdata. numeric: vector of length equal to the number of rows in the original data or in newdata (if supplied). FALSE: Equal weights. TRUE: Extract weights from the fitted object with insight::find_weights() and use them when taking weighted averages of estimates. Warning: newdata=datagrid() returns a single average weight, which is equivalent to using wts=FALSE
hypothesis	specify a hypothesis test or custom contrast using a numeric value, vector, or matrix; a string equation; string; a formula, or a function. Numeric: Single value: the null hypothesis used in the computation of Z and p (before applying transform). Vector: Weights to compute a linear combination of (custom contrast between) estimates. Length equal to the number of rows generated by the same function call, but without the hypothesis argument. Matrix: Each column is a vector of weights, as describe above, used to compute a distinct linear combination of (contrast between) estimates. The column names of the matrix are used as labels in the output. String equation to specify linear or non-linear hypothesis tests. If the term column uniquely identifies rows, terms can be used in the formula. Otherwise, use b1, b2, etc. to identify the position of each parameter. The b* wildcard can be used to test hypotheses on all estimates. If a named vector is used, the names are used as labels in the output. Examples: hp = drat hp + drat = 12 b1 + b2 + b3 = 0 b* / b1 = 1 String: "pairwise": pairwise differences between estimates in each row. "reference": differences between the estimates in each row and the estimate in the first row. "sequential": difference between an estimate and the estimate in the next row. "meandev": difference between an estimate and the mean of all estimates. "meanotherdev": difference between an estimate and the mean of all other estimates, excluding the current one. "revpairwise", "revreference", "revsequential": inverse of the corresponding hypotheses, as described above. Formula: comparison ~ pairs | group Left-hand side determines the type of comparison to conduct: difference or ratio. If the left-hand side is empty, difference is chosen. Right-hand side determines the pairs of estimates to compare: reference, sequential, or meandev Optional: Users can supply grouping variables after a vertical bar to conduct comparisons withing subsets. Examples: ~ reference ratio ~ pairwise difference ~ pairwise | groupid Function: Accepts an argument x: object produced by a marginaleffects function or a data frame with column rowid and estimate Returns a data frame with columns term and estimate (mandatory) and rowid (optional). The function can also accept optional input arguments: newdata, by, draws. This function approach will not work for Bayesian models or with bootstrapping. In those cases, it is easy to use posterior_draws() to extract and manipulate the draws directly. See the Examples section below and the vignette: https://marginaleffects.com/vignettes/hypothesis.html
equivalence	Numeric vector of length 2: bounds used for the two-one-sided test (TOST) of equivalence, and for the non-inferiority and non-superiority tests. See Details section below.
p_adjust	Adjust p-values for multiple comparisons: "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", or "fdr". See stats::p.adjust
df	Degrees of freedom used to compute p values and confidence intervals. A single numeric value between 1 and Inf. When df is Inf, the normal distribution is used. When df is finite, the t distribution is used. See insight::get_df for a convenient function to extract degrees of freedom. Ex: slopes(model, df = insight::get_df(model))
eps	NULL or numeric value which determines the step size to use when calculating numerical derivatives: (f(x+eps)-f(x))/eps. When eps is NULL, the step size is 0.0001 multiplied by the difference between the maximum and minimum values of the variable with respect to which we are taking the derivative. Changing eps may be necessary to avoid numerical problems in certain models.
numderiv	string or list of strings indicating the method to use to for the numeric differentiation used in to compute delta method standard errors. "fdforward": finite difference method with forward differences "fdcenter": finite difference method with central differences (default) "richardson": Richardson extrapolation method Extra arguments can be specified by passing a list to the numDeriv argument, with the name of the method first and named arguments following, ex: numderiv=list(“fdcenter”, eps = 1e-5). When an unknown argument is used, marginaleffects prints the list of valid arguments for each method.
…	Additional arguments are passed to the predict() method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the marginaleffects website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the ?slopes documentation for a non-exhaustive list of available arguments.

Details

A "slope" or "marginal effect" is the partial derivative of the regression equation with respect to a variable in the model. This function uses automatic differentiation to compute slopes for a vast array of models, including non-linear models with transformations (e.g., polynomials). Uncertainty estimates are computed using the delta method.

Numerical derivatives for the slopes function are calculated using a simple epsilon difference approach: \(\partial Y / \partial X = (f(X + \varepsilon/2) - f(X-\varepsilon/2)) / \varepsilon\), where f is the predict() method associated with the model class, and \(\varepsilon\) is determined by the eps argument.

Value

A data.frame with one row per observation (per term/group) and several columns:

• rowid: row number of the newdata data frame

• type: prediction type, as defined by the type argument

• group: (optional) value of the grouped outcome (e.g., categorical outcome models)

• term: the variable whose marginal effect is computed

• dydx: slope of the outcome with respect to the term, for a given combination of predictor values

• std.error: standard errors computed by via the delta method.

• p.value: p value associated to the estimate column. The null is determined by the hypothesis argument (0 by default), and p values are computed before applying the transform argument. For models of class feglm, Gam, glm and negbin, p values are computed on the link scale by default unless the type argument is specified explicitly.

• s.value: Shannon information transforms of p values. How many consecutive "heads" tosses would provide the same amount of evidence (or "surprise") against the null hypothesis that the coin is fair? The purpose of S is to calibrate the analyst’s intuition about the strength of evidence encoded in p against a well-known physical phenomenon. See Greenland (2019) and Cole et al. (2020).

• conf.low: lower bound of the confidence interval (or equal-tailed interval for bayesian models)

• conf.high: upper bound of the confidence interval (or equal-tailed interval for bayesian models)

See ?print.marginaleffects for printing options.

Functions

• avg_slopes(): Average slopes

Standard errors using the delta method

Standard errors for all quantities estimated by marginaleffects can be obtained via the delta method. This requires differentiating a function with respect to the coefficients in the model using a finite difference approach. In some models, the delta method standard errors can be sensitive to various aspects of the numeric differentiation strategy, including the step size. By default, the step size is set to 1e-8, or to 1e-4 times the smallest absolute model coefficient, whichever is largest.

marginaleffects can delegate numeric differentiation to the numDeriv package, which allows more flexibility. To do this, users can pass arguments to the numDeriv::jacobian function through a global option. For example:

• options(marginaleffects_numDeriv = list(method = “simple”, method.args = list(eps = 1e-6)))

• options(marginaleffects_numDeriv = list(method = “Richardson”, method.args = list(eps = 1e-5)))

• options(marginaleffects_numDeriv = NULL)

See the "Standard Errors and Confidence Intervals" vignette on the marginaleffects website for more details on the computation of standard errors:

https://marginaleffects.com/vignettes/uncertainty.html

Note that the inferences() function can be used to compute uncertainty estimates using a bootstrap or simulation-based inference. See the vignette:

https://marginaleffects.com/vignettes/bootstrap.html

Model-Specific Arguments

Some model types allow model-specific arguments to modify the nature of marginal effects, predictions, marginal means, and contrasts. Please report other package-specific predict() arguments on Github so we can add them to the table below.

https://github.com/vincentarelbundock/marginaleffects/issues

Package	Class	Argument	Documentation
brms	brmsfit	ndraws	brms::posterior_predict
		re_formula	brms::posterior_predict
lme4	merMod	re.form	lme4::predict.merMod
		allow.new.levels	lme4::predict.merMod
glmmTMB	glmmTMB	re.form	glmmTMB::predict.glmmTMB
		allow.new.levels	glmmTMB::predict.glmmTMB
		zitype	glmmTMB::predict.glmmTMB
mgcv	bam	exclude	mgcv::predict.bam
	gam	exclude	mgcv::predict.gam
robustlmm	rlmerMod	re.form	robustlmm::predict.rlmerMod
		allow.new.levels	robustlmm::predict.rlmerMod
MCMCglmm	MCMCglmm	ndraws
sampleSelection	selection	part	sampleSelection::predict.selection

Bayesian posterior summaries

By default, credible intervals in bayesian models are built as equal-tailed intervals. This can be changed to a highest density interval by setting a global option:

options(“marginaleffects_posterior_interval” = “eti”)

options(“marginaleffects_posterior_interval” = “hdi”)

By default, the center of the posterior distribution in bayesian models is identified by the median. Users can use a different summary function by setting a global option:

options(“marginaleffects_posterior_center” = “mean”)

options(“marginaleffects_posterior_center” = “median”)

When estimates are averaged using the by argument, the tidy() function, or the summary() function, the posterior distribution is marginalized twice over. First, we take the average across units but within each iteration of the MCMC chain, according to what the user requested in by argument or tidy()/summary() functions. Then, we identify the center of the resulting posterior using the function supplied to the “marginaleffects_posterior_center” option (the median by default).

Equivalence, Inferiority, Superiority

\(\theta\) is an estimate, \(\sigma_\theta\) its estimated standard error, and \([a, b]\) are the bounds of the interval supplied to the equivalence argument.

Non-inferiority:

• \(H_0\): \(\theta \leq a\)

• \(H_1\): \(\theta > a\)

• \(t=(\theta - a)/\sigma_\theta\)

• p: Upper-tail probability

Non-superiority:

• \(H_0\): \(\theta \geq b\)

• \(H_1\): \(\theta < b\)

• \(t=(\theta - b)/\sigma_\theta\)

• p: Lower-tail probability

Equivalence: Two One-Sided Tests (TOST)

• p: Maximum of the non-inferiority and non-superiority p values.

Thanks to Russell V. Lenth for the excellent emmeans package and documentation which inspired this feature.

Prediction types

The type argument determines the scale of the predictions used to compute quantities of interest with functions from the marginaleffects package. Admissible values for type depend on the model object. When users specify an incorrect value for type, marginaleffects will raise an informative error with a list of valid type values for the specific model object. The first entry in the list in that error message is the default type.

The invlink(link) is a special type defined by marginaleffects. It is available for some (but not all) models, and only for the predictions() function. With this link type, we first compute predictions on the link scale, then we use the inverse link function to backtransform the predictions to the response scale. This is useful for models with non-linear link functions as it can ensure that confidence intervals stay within desirable bounds, ex: 0 to 1 for a logit model. Note that an average of estimates with type=“invlink(link)” will not always be equivalent to the average of estimates with type=“response”. This type is default when calling predictions(). It is available—but not default—when calling avg_predictions() or predictions() with the by argument.

Some of the most common type values are:

response, link, E, Ep, average, class, conditional, count, cum.prob, cumhaz, cumprob, density, detection, disp, ev, expected, expvalue, fitted, hazard, invlink(link), latent, latent_N, linear, linear.predictor, linpred, location, lp, mean, numeric, p, ppd, pr, precision, prediction, prob, probability, probs, quantile, risk, rmst, scale, survival, unconditional, utility, variance, xb, zero, zlink, zprob

Parallel computation

The slopes() and comparisons() functions can use parallelism to speed up computation. Operations are parallelized for the computation of standard errors, at the model coefficient level. There is always considerable overhead when using parallel computation, mainly involved in passing the whole dataset to the different processes. Thus, parallel computation is most likely to be useful when the model includes many parameters and the dataset is relatively small.

Warning: In many cases, parallel processing will not be useful at all.

To activate parallel computation, users must load the future.apply package, call plan() function, and set a global option. For example:

library(future.apply)
plan("multicore", workers = 4)
options(marginaleffects_parallel = TRUE)

slopes(model)

To disable parallelism in marginaleffects altogether, you can set a global option:

options(marginaleffects_parallel = FALSE)

Order of operations

Behind the scenes, the arguments of marginaleffects functions are evaluated in this order:

1. newdata

2. variables

3. comparison and slopes

4. by

5. vcov

6. hypothesis

7. transform

Global options

The behavior of marginaleffects functions can be modified by setting global options.

Disable some safety checks:

options(marginaleffects_safe = FALSE)

Omit some columns from the printed output:

options(marginaleffects_print_omit = c("p.value", "s.value"))`

References

• Greenland S. 2019. "Valid P-Values Behave Exactly as They Should: Some Misleading Criticisms of P-Values and Their Resolution With S-Values." The American Statistician. 73(S1): 106–114.

• Cole, Stephen R, Jessie K Edwards, and Sander Greenland. 2020. "Surprise!" American Journal of Epidemiology 190 (2): 191–93. https://doi.org/10.1093/aje/kwaa136

Examples

library("marginaleffects")



# Unit-level (conditional) Marginal Effects
mod <- glm(am ~ hp * wt, data = mtcars, family = binomial)
mfx <- slopes(mod)
head(mfx)

# Average Marginal Effect (AME)
avg_slopes(mod, by = TRUE)


# Marginal Effect at the Mean (MEM)
slopes(mod, newdata = datagrid())

# Marginal Effect at User-Specified Values
# Variables not explicitly included in `datagrid()` are held at their means
slopes(mod, newdata = datagrid(hp = c(100, 110)))

# Group-Average Marginal Effects (G-AME)
# Calculate marginal effects for each observation, and then take the average
# marginal effect within each subset of observations with different observed
# values for the `cyl` variable:
mod2 <- lm(mpg ~ hp * cyl, data = mtcars)
avg_slopes(mod2, variables = "hp", by = "cyl")

# Marginal Effects at User-Specified Values (counterfactual)
# Variables not explicitly included in `datagrid()` are held at their
# original values, and the whole dataset is duplicated once for each
# combination of the values in `datagrid()`
mfx <- slopes(mod,
    newdata = datagrid(
        hp = c(100, 110),
        grid_type = "counterfactual"))
head(mfx)

# Heteroskedasticity robust standard errors
mfx <- slopes(mod, vcov = sandwich::vcovHC(mod))
head(mfx)

# hypothesis test: is the `hp` marginal effect at the mean equal to the `drat` marginal effect
mod <- lm(mpg ~ wt + drat, data = mtcars)

slopes(
    mod,
    newdata = "mean",
    hypothesis = "wt = drat")

# same hypothesis test using row indices
slopes(
    mod,
    newdata = "mean",
    hypothesis = "b1 - b2 = 0")

# same hypothesis test using numeric vector of weights
slopes(
    mod,
    newdata = "mean",
    hypothesis = c(1, -1))

# two custom contrasts using a matrix of weights
lc <- matrix(
    c(
        1, -1,
        2, 3),
    ncol = 2)
colnames(lc) <- c("Contrast A", "Contrast B")
slopes(
    mod,
    newdata = "mean",
    hypothesis = lc)