Quick start

This page explains how to interpret statistical results using the marginaleffects package for R and Python. The workflow that we propose rests on 5 conceptual pillars:

1. Quantity: What is the quantity of interest? Do we want to estimate a prediction or a function of predictions (average, difference, ratio, derivative, etc.)?
2. Predictors: What regressor values are we interested in? Do we want to produce estimates for the units in our dataset, or for hypothetical or representative individuals?
3. Aggregation: Do we report estimates for every observation in the grid or a global summary?
4. Uncertainty: How do we quantify uncertainty about our estimates?
5. Test: Which (non-)linear hypothesis or equivalence tests do we conduct?

Installation

Before we begin, let’s install the marginaleffects package, available for R and Python:

R

Install from CRAN:

install.packages("marginaleffects")

Python

Install from PyPI:

pip install marginaleffects

Quantity

The marginaleffects package allows R users to compute and plot three principal quantities of interest: (1) predictions, (2) comparisons, and (3) slopes. In addition, the package includes a convenience function to compute a fourth quantity of interest, “marginal means”, which is a special case of averaged predictions. marginaleffects can also average (or “marginalize”) unit-level (or “conditional”) estimates of all those quantities, and conduct hypothesis tests on them.

Predictions:

The outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels. a.k.a. Fitted values, adjusted predictions. predictions(), avg_predictions(), plot_predictions().

Comparisons:

Compare the predictions made by a model for different regressor values (e.g., college graduates vs. others): contrasts, differences, risk ratios, odds, etc. comparisons(), avg_comparisons(), plot_comparisons().

Slopes:

Partial derivative of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends. slopes(), avg_slopes(), plot_slopes().

Marginal Means:

Predictions of a model, averaged across a “reference grid” of categorical predictors. marginalmeans().

Hypothesis and Equivalence Tests:

Hypothesis and equivalence tests can be conducted on linear or non-linear functions of model coefficients, or on any of the quantities computed by the marginaleffects packages (predictions, slopes, comparisons, marginal means, etc.). Uncertainy estimates can be obtained via the delta method (with or without robust standard errors), bootstrap, or simulation.

Predictions, comparisons, and slopes are fundamentally unit-level (or “conditional”) quantities. Except in the simplest linear case, estimates will typically vary based on the values of all the regressors in a model. Each of the observations in a dataset is thus associated with its own prediction, comparison, and slope estimates. Below, we will see that it can be useful to marginalize (or “average over”) unit-level estimates to report an “average prediction”, “average comparison”, or “average slope”.

One ambiguous aspect of the definitions above is that the word “marginal” comes up in two different and opposite ways:

1. In “marginal effects,” we refer to the effect of a tiny (marginal) change in the regressor on the outcome. This is a slope, or derivative.
2. In “marginal means,” we refer to the process of marginalizing across rows of a prediction grid. This is an average, or integral.

On this website and in this package, we reserve the expression “marginal effect” to mean a “slope” or “partial derivative”.

The marginaleffects package includes functions to estimate, average, plot, and summarize all of the estimands described above. The objects produced by marginaleffects are “tidy”: they produce simple data frames in “long” format. They are also “standards-compliant” and work seamlessly with standard functions like summary(), head(), tidy(), and glance(), as well with external packages like modelsummary or ggplot2.

We now apply marginaleffects functions to compute each of the quantities of interest described above. First, we fit a linear regression model with multiplicative interactions:

R

library(marginaleffects)

mod <- lm(mpg ~ hp * wt * am, data = mtcars)

Python

import polars as pl
import numpy as np
import statsmodels.formula.api as smf
from marginaleffects import *

mtcars = pl.read_csv("https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv")

mod = smf.ols("mpg ~ hp * wt * am", data = mtcars).fit()

Then, we call the predictions() function. As noted above, predictions are unit-level estimates, so there is one specific prediction per observation. By default, the predictions() function makes one prediction per observation in the dataset that was used to fit the original model. Since mtcars has 32 rows, the predictions() outcome also has 32 rows:

R

pre <- predictions(mod)

nrow(mtcars)

[1] 32

nrow(pre)

[1] 32

pre

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am
22.5	0.884	25.44		471.7	20.8	24.2
20.8	1.194	17.42		223.3	18.5	23.1
25.3	0.709	35.66		922.7	23.9	26.7
20.3	0.704	28.75		601.5	18.9	21.6
17.0	0.712	23.88		416.2	15.6	18.4
29.6	1.874	15.80		184.3	25.9	33.3
15.9	1.311	12.13		110.0	13.3	18.5
19.4	1.145	16.95		211.6	17.2	21.7
14.8	2.017	7.33		42.0	10.8	18.7
21.5	1.072	20.02		293.8	19.4	23.6

Python

pre = predictions(mod)

mtcars.shape

(32, 12)

pre.shape

(32, 20)

print(pre)

shape: (32, 7)
┌──────────┬───────────┬──────┬──────────┬─────┬──────┬───────┐
│ Estimate ┆ Std.Error ┆ z    ┆ P(>|z|)  ┆ S   ┆ 2.5% ┆ 97.5% │
│ ---      ┆ ---       ┆ ---  ┆ ---      ┆ --- ┆ ---  ┆ ---   │
│ str      ┆ str       ┆ str  ┆ str      ┆ str ┆ str  ┆ str   │
╞══════════╪═══════════╪══════╪══════════╪═════╪══════╪═══════╡
│ 22.5     ┆ 0.884     ┆ 25.4 ┆ 0        ┆ inf ┆ 20.8 ┆ 24.2  │
│ 20.8     ┆ 1.19      ┆ 17.4 ┆ 0        ┆ inf ┆ 18.5 ┆ 23.1  │
│ 25.3     ┆ 0.709     ┆ 35.7 ┆ 0        ┆ inf ┆ 23.9 ┆ 26.7  │
│ 20.3     ┆ 0.704     ┆ 28.8 ┆ 0        ┆ inf ┆ 18.9 ┆ 21.6  │
│ 17       ┆ 0.712     ┆ 23.9 ┆ 0        ┆ inf ┆ 15.6 ┆ 18.4  │
│ …        ┆ …         ┆ …    ┆ …        ┆ …   ┆ …    ┆ …     │
│ 29.6     ┆ 1.87      ┆ 15.8 ┆ 0        ┆ inf ┆ 25.9 ┆ 33.3  │
│ 15.9     ┆ 1.31      ┆ 12.1 ┆ 0        ┆ inf ┆ 13.3 ┆ 18.5  │
│ 19.4     ┆ 1.15      ┆ 16.9 ┆ 0        ┆ inf ┆ 17.2 ┆ 21.7  │
│ 14.8     ┆ 2.02      ┆ 7.33 ┆ 2.29e-13 ┆ 42  ┆ 10.8 ┆ 18.7  │
│ 21.5     ┆ 1.07      ┆ 20   ┆ 0        ┆ inf ┆ 19.4 ┆ 23.6  │
└──────────┴───────────┴──────┴──────────┴─────┴──────┴───────┘

Columns: rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

Now, we use the comparisons() function to compute the difference in predicted outcome when each of the predictors is incremented by 1 unit (one predictor at a time, holding all others constant). Once again, comparisons are unit-level quantities. And since there are 3 predictors in the model and our data has 32 rows, we obtain 96 comparisons:

R

cmp <- comparisons(mod)

nrow(cmp)

[1] 96

cmp

Term	Contrast	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am
am	1 - 0	0.325	1.68	0.193	0.8467	0.2	-2.97	3.622
am	1 - 0	-0.544	1.57	-0.347	0.7287	0.5	-3.62	2.530
am	1 - 0	1.201	2.35	0.511	0.6090	0.7	-3.40	5.802
am	1 - 0	-1.703	1.87	-0.912	0.3618	1.5	-5.36	1.957
am	1 - 0	-0.615	1.68	-0.366	0.7146	0.5	-3.91	2.680
wt	+1	-6.518	1.88	-3.462		10.9	-10.21	-2.828
wt	+1	-1.653	3.74	-0.442	0.6588	0.6	-8.99	5.683
wt	+1	-4.520	2.47	-1.830	0.0672	3.9	-9.36	0.321
wt	+1	0.635	4.89	0.130	0.8966	0.2	-8.95	10.216
wt	+1	-6.647	1.86	-3.572		11.5	-10.29	-2.999

Python

cmp = comparisons(mod)

cmp.shape

(96, 25)

print(cmp)

shape: (96, 9)
┌──────┬──────────────┬──────────┬───────────┬───┬──────────┬───────┬───────┬───────┐
│ Term ┆ Contrast     ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S     ┆ 2.5%  ┆ 97.5% │
│ ---  ┆ ---          ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---   ┆ ---   ┆ ---   │
│ str  ┆ str          ┆ str      ┆ str       ┆   ┆ str      ┆ str   ┆ str   ┆ str   │
╞══════╪══════════════╪══════════╪═══════════╪═══╪══════════╪═══════╪═══════╪═══════╡
│ am   ┆ True - False ┆ 0.325    ┆ 1.68      ┆ … ┆ 0.847    ┆ 0.24  ┆ -2.97 ┆ 3.62  │
│ am   ┆ True - False ┆ -0.544   ┆ 1.57      ┆ … ┆ 0.729    ┆ 0.457 ┆ -3.62 ┆ 2.53  │
│ am   ┆ True - False ┆ 1.2      ┆ 2.35      ┆ … ┆ 0.609    ┆ 0.715 ┆ -3.4  ┆ 5.8   │
│ am   ┆ True - False ┆ -1.7     ┆ 1.87      ┆ … ┆ 0.362    ┆ 1.47  ┆ -5.36 ┆ 1.96  │
│ am   ┆ True - False ┆ -0.615   ┆ 1.68      ┆ … ┆ 0.715    ┆ 0.485 ┆ -3.91 ┆ 2.68  │
│ …    ┆ …            ┆ …        ┆ …         ┆ … ┆ …        ┆ …     ┆ …     ┆ …     │
│ wt   ┆ +1           ┆ -6.52    ┆ 1.88      ┆ … ┆ 0.000537 ┆ 10.9  ┆ -10.2 ┆ -2.83 │
│ wt   ┆ +1           ┆ -1.65    ┆ 3.74      ┆ … ┆ 0.659    ┆ 0.602 ┆ -8.99 ┆ 5.68  │
│ wt   ┆ +1           ┆ -4.52    ┆ 2.47      ┆ … ┆ 0.0672   ┆ 3.89  ┆ -9.36 ┆ 0.321 │
│ wt   ┆ +1           ┆ 0.635    ┆ 4.89      ┆ … ┆ 0.897    ┆ 0.157 ┆ -8.95 ┆ 10.2  │
│ wt   ┆ +1           ┆ -6.65    ┆ 1.86      ┆ … ┆ 0.000355 ┆ 11.5  ┆ -10.3 ┆ -3    │
└──────┴──────────────┴──────────┴───────────┴───┴──────────┴───────┴───────┴───────┘

Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

The comparisons() function allows customized queries. For example, what happens to the predicted outcome when the hp variable increases from 100 to 120?

R

comparisons(mod, variables = list(hp = c(120, 100)))

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am
-0.738	0.370	-1.995	0.04607	4.4	-1.463	-0.0129
-0.574	0.313	-1.836	0.06640	3.9	-1.186	0.0388
-0.931	0.452	-2.062	0.03922	4.7	-1.817	-0.0460
-0.845	0.266	-3.182	0.00146	9.4	-1.366	-0.3248
-0.780	0.268	-2.909	0.00362	8.1	-1.306	-0.2547
-1.451	0.705	-2.058	0.03958	4.7	-2.834	-0.0692
-0.384	0.270	-1.422	0.15498	2.7	-0.912	0.1451
-0.641	0.334	-1.918	0.05513	4.2	-1.297	0.0141
-0.126	0.272	-0.463	0.64360	0.6	-0.659	0.4075
-0.635	0.332	-1.911	0.05598	4.2	-1.286	0.0162

Python

cmp = comparisons(mod, variables = {"hp": [120, 100]})
print(cmp)

shape: (32, 9)
┌──────┬───────────┬──────────┬───────────┬───┬─────────┬───────┬─────────┬───────┐
│ Term ┆ Contrast  ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|) ┆ S     ┆ 2.5%    ┆ 97.5% │
│ ---  ┆ ---       ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---   │
│ str  ┆ str       ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str     ┆ str   │
╞══════╪═══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═════════╪═══════╡
│ hp   ┆ 100 - 120 ┆ 0.738    ┆ 0.37      ┆ … ┆ 0.0461  ┆ 4.44  ┆ 0.0129  ┆ 1.46  │
│ hp   ┆ 100 - 120 ┆ 0.574    ┆ 0.313     ┆ … ┆ 0.0664  ┆ 3.91  ┆ -0.0388 ┆ 1.19  │
│ hp   ┆ 100 - 120 ┆ 0.931    ┆ 0.452     ┆ … ┆ 0.0392  ┆ 4.67  ┆ 0.046   ┆ 1.82  │
│ hp   ┆ 100 - 120 ┆ 0.845    ┆ 0.266     ┆ … ┆ 0.00146 ┆ 9.42  ┆ 0.325   ┆ 1.37  │
│ hp   ┆ 100 - 120 ┆ 0.78     ┆ 0.268     ┆ … ┆ 0.00362 ┆ 8.11  ┆ 0.255   ┆ 1.31  │
│ …    ┆ …         ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …       ┆ …     │
│ hp   ┆ 100 - 120 ┆ 1.45     ┆ 0.705     ┆ … ┆ 0.0396  ┆ 4.66  ┆ 0.0692  ┆ 2.83  │
│ hp   ┆ 100 - 120 ┆ 0.384    ┆ 0.27      ┆ … ┆ 0.155   ┆ 2.69  ┆ -0.145  ┆ 0.912 │
│ hp   ┆ 100 - 120 ┆ 0.641    ┆ 0.334     ┆ … ┆ 0.0551  ┆ 4.18  ┆ -0.0141 ┆ 1.3   │
│ hp   ┆ 100 - 120 ┆ 0.126    ┆ 0.272     ┆ … ┆ 0.644   ┆ 0.636 ┆ -0.408  ┆ 0.659 │
│ hp   ┆ 100 - 120 ┆ 0.635    ┆ 0.332     ┆ … ┆ 0.056   ┆ 4.16  ┆ -0.0162 ┆ 1.29  │
└──────┴───────────┴──────────┴───────────┴───┴─────────┴───────┴─────────┴───────┘

Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

What happens to the predicted outcome when the hp variable increases by 1 standard deviation about its mean?

R

comparisons(mod, variables = list(hp = "sd"))

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am
-2.530	1.269	-1.995	0.04607	4.4	-5.02	-0.0441
-1.967	1.072	-1.836	0.06640	3.9	-4.07	0.1332
-3.193	1.549	-2.062	0.03922	4.7	-6.23	-0.1578
-2.898	0.911	-3.182	0.00146	9.4	-4.68	-1.1133
-2.675	0.919	-2.909	0.00362	8.1	-4.48	-0.8731
-4.976	2.418	-2.058	0.03958	4.7	-9.71	-0.2373
-1.315	0.925	-1.422	0.15498	2.7	-3.13	0.4974
-2.199	1.147	-1.918	0.05513	4.2	-4.45	0.0483
-0.432	0.933	-0.463	0.64360	0.6	-2.26	1.3970
-2.177	1.139	-1.911	0.05598	4.2	-4.41	0.0556

Python

cmp = comparisons(mod, variables = {"hp": "sd"})
print(cmp)

shape: (32, 9)
┌──────┬─────────────────────┬──────────┬───────────┬───┬─────────┬───────┬───────┬─────────┐
│ Term ┆ Contrast            ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|) ┆ S     ┆ 2.5%  ┆ 97.5%   │
│ ---  ┆ ---                 ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---   ┆ ---     │
│ str  ┆ str                 ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str   ┆ str     │
╞══════╪═════════════════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════╪═════════╡
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -2.53    ┆ 1.27      ┆ … ┆ 0.0461  ┆ 4.44  ┆ -5.02 ┆ -0.0441 │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -1.97    ┆ 1.07      ┆ … ┆ 0.0664  ┆ 3.91  ┆ -4.07 ┆ 0.133   │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -3.19    ┆ 1.55      ┆ … ┆ 0.0392  ┆ 4.67  ┆ -6.23 ┆ -0.158  │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -2.9     ┆ 0.911     ┆ … ┆ 0.00146 ┆ 9.42  ┆ -4.68 ┆ -1.11   │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -2.68    ┆ 0.919     ┆ … ┆ 0.00362 ┆ 8.11  ┆ -4.48 ┆ -0.873  │
│ …    ┆ …                   ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …     ┆ …       │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -4.98    ┆ 2.42      ┆ … ┆ 0.0396  ┆ 4.66  ┆ -9.71 ┆ -0.237  │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -1.32    ┆ 0.925     ┆ … ┆ 0.155   ┆ 2.69  ┆ -3.13 ┆ 0.497   │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -2.2     ┆ 1.15      ┆ … ┆ 0.0551  ┆ 4.18  ┆ -4.45 ┆ 0.0483  │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -0.432   ┆ 0.933     ┆ … ┆ 0.644   ┆ 0.636 ┆ -2.26 ┆ 1.4     │
│ hp   ┆ (x+sd/2) - (x-sd/2) ┆ -2.18    ┆ 1.14      ┆ … ┆ 0.056   ┆ 4.16  ┆ -4.41 ┆ 0.0556  │
└──────┴─────────────────────┴──────────┴───────────┴───┴─────────┴───────┴───────┴─────────┘

Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

The comparisons() function also allows users to specify arbitrary functions of predictions, with the comparison argument. For example, what is the average ratio between predicted Miles per Gallon after an increase of 50 units in Horsepower?

R

comparisons(
  mod,
  variables = list(hp = 50),
  comparison = "ratioavg")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
0.905	0.0319	28.4		586.8	0.843	0.968

Python

cmp = comparisons(
  mod,
  variables = {"hp": 50},
  comparison = "ratioavg")
print(cmp)

shape: (1, 9)
┌──────┬──────────┬──────────┬───────────┬───┬─────────┬─────┬───────┬───────┐
│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|) ┆ S   ┆ 2.5%  ┆ 97.5% │
│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ --- ┆ ---   ┆ ---   │
│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str ┆ str   ┆ str   │
╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═════╪═══════╪═══════╡
│ hp   ┆ +50      ┆ 0.91     ┆ 0.0291    ┆ … ┆ 0       ┆ inf ┆ 0.853 ┆ 0.966 │
└──────┴──────────┴──────────┴───────────┴───┴─────────┴─────┴───────┴───────┘

Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

See the Comparisons vignette for detailed explanations and more options.

The slopes() function allows us to compute the partial derivative of the outcome equation with respect to each of the predictors. Once again, we obtain a data frame with 96 rows:

R

mfx <- slopes(mod)

nrow(mfx)

[1] 96

mfx

Term	Contrast	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am
am	1 - 0	0.325	1.68	0.193	0.8467	0.2	-2.97	3.622
am	1 - 0	-0.544	1.57	-0.347	0.7287	0.5	-3.62	2.530
am	1 - 0	1.201	2.35	0.511	0.6090	0.7	-3.40	5.802
am	1 - 0	-1.703	1.87	-0.912	0.3618	1.5	-5.36	1.957
am	1 - 0	-0.615	1.68	-0.366	0.7146	0.5	-3.91	2.680
wt	dY/dX	-6.518	1.88	-3.462		10.9	-10.21	-2.828
wt	dY/dX	-1.653	3.74	-0.442	0.6588	0.6	-8.99	5.683
wt	dY/dX	-4.520	2.47	-1.830	0.0673	3.9	-9.36	0.321
wt	dY/dX	0.635	4.89	0.130	0.8966	0.2	-8.95	10.216
wt	dY/dX	-6.647	1.86	-3.572		11.5	-10.29	-2.999

Python

mfx = slopes(mod)

mfx.shape

(96, 25)

print(mfx)

shape: (96, 9)
┌──────┬──────────────┬──────────┬───────────┬───┬──────────┬───────┬───────┬───────┐
│ Term ┆ Contrast     ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S     ┆ 2.5%  ┆ 97.5% │
│ ---  ┆ ---          ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---   ┆ ---   ┆ ---   │
│ str  ┆ str          ┆ str      ┆ str       ┆   ┆ str      ┆ str   ┆ str   ┆ str   │
╞══════╪══════════════╪══════════╪═══════════╪═══╪══════════╪═══════╪═══════╪═══════╡
│ am   ┆ True - False ┆ 0.325    ┆ 1.68      ┆ … ┆ 0.847    ┆ 0.24  ┆ -2.97 ┆ 3.62  │
│ am   ┆ True - False ┆ -0.544   ┆ 1.57      ┆ … ┆ 0.729    ┆ 0.457 ┆ -3.62 ┆ 2.53  │
│ am   ┆ True - False ┆ 1.2      ┆ 2.35      ┆ … ┆ 0.609    ┆ 0.715 ┆ -3.4  ┆ 5.8   │
│ am   ┆ True - False ┆ -1.7     ┆ 1.87      ┆ … ┆ 0.362    ┆ 1.47  ┆ -5.36 ┆ 1.96  │
│ am   ┆ True - False ┆ -0.615   ┆ 1.68      ┆ … ┆ 0.715    ┆ 0.485 ┆ -3.91 ┆ 2.68  │
│ …    ┆ …            ┆ …        ┆ …         ┆ … ┆ …        ┆ …     ┆ …     ┆ …     │
│ wt   ┆ dY/dX        ┆ -6.52    ┆ 1.88      ┆ … ┆ 0.000537 ┆ 10.9  ┆ -10.2 ┆ -2.83 │
│ wt   ┆ dY/dX        ┆ -1.65    ┆ 3.74      ┆ … ┆ 0.659    ┆ 0.602 ┆ -8.99 ┆ 5.68  │
│ wt   ┆ dY/dX        ┆ -4.52    ┆ 2.47      ┆ … ┆ 0.0676   ┆ 3.89  ┆ -9.37 ┆ 0.328 │
│ wt   ┆ dY/dX        ┆ 0.635    ┆ 4.89      ┆ … ┆ 0.897    ┆ 0.157 ┆ -8.95 ┆ 10.2  │
│ wt   ┆ dY/dX        ┆ -6.65    ┆ 1.86      ┆ … ┆ 0.00036  ┆ 11.4  ┆ -10.3 ┆ -3    │
└──────┴──────────────┴──────────┴───────────┴───┴──────────┴───────┴───────┴───────┘

Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

Predictors

Predictions, comparisons, and slopes are typically “conditional” quantities which depend on the values of all the predictors in the model. By default, marginaleffects functions estimate quantities of interest for the empirical distribution of the data (i.e., for each row of the original dataset). However, users can specify the exact values of the predictors they want to investigate by using the newdata argument.

newdata accepts data frames, shortcut strings, or a call to the datagrid() function. For example, to compute the predicted outcome for a hypothetical car with all predictors equal to the sample mean or median, we can do:

R

predictions(mod, newdata = "mean")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %	hp	wt	am
Type: response
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, hp, wt, am, mpg
18.7	0.649	28.8		603.8	17.4	20	147	3.22	0

predictions(mod, newdata = "median")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %	hp	wt	am
Type: response
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, hp, wt, am, mpg
19.4	0.646	30		653.2	18.1	20.6	123	3.33	0

Python

p = predictions(mod, newdata = "mean")
print(p)

shape: (1, 7)
┌──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐
│ Estimate ┆ Std.Error ┆ z    ┆ P(>|z|) ┆ S   ┆ 2.5% ┆ 97.5% │
│ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │
│ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │
╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡
│ 17.5     ┆ 0.83      ┆ 21.1 ┆ 0       ┆ inf ┆ 15.9 ┆ 19.1  │
└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘

Columns: rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

p = predictions(mod, newdata = "median")
print(p)

shape: (1, 7)
┌──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐
│ Estimate ┆ Std.Error ┆ z    ┆ P(>|z|) ┆ S   ┆ 2.5% ┆ 97.5% │
│ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │
│ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │
╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡
│ 17.1     ┆ 0.808     ┆ 21.1 ┆ 0       ┆ inf ┆ 15.5 ┆ 18.6  │
└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘

Columns: rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb

The datagrid function gives us a powerful way to define a grid of predictors. All the variables not mentioned explicitly in datagrid() are fixed to their mean or mode:

R

predictions(
  mod,
  newdata = datagrid(
    am = c(0, 1),
    wt = range))

am	wt	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %	hp
Type: response
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, hp, am, wt, mpg
0	1.51	23.3	2.71	8.60		56.7	17.96	28.6	147
0	5.42	12.8	2.98	4.30		15.8	6.96	18.6	147
1	1.51	27.1	2.85	9.52		69.0	21.56	32.7	147
1	5.42	5.9	5.81	1.01	0.31	1.7	-5.50	17.3	147

Python

p = predictions(
  mod,
  newdata = datagrid(
    am = [0, 1],
    wt = [mtcars["wt"].min(), mtcars["wt"].max()]))
print(p)

shape: (4, 9)
┌─────┬──────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐
│ am  ┆ wt   ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │
│ --- ┆ ---  ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │
│ str ┆ str  ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │
╞═════╪══════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡
│ 0   ┆ 1.51 ┆ 25.7     ┆ 3.54      ┆ … ┆ 3.82e-13 ┆ 41.3 ┆ 18.8  ┆ 32.7  │
│ 0   ┆ 5.42 ┆ 13.2     ┆ 4.44      ┆ … ┆ 0.00302  ┆ 8.37 ┆ 4.47  ┆ 21.9  │
│ 1   ┆ 1.51 ┆ 29.8     ┆ 1.8       ┆ … ┆ 0        ┆ inf  ┆ 26.3  ┆ 33.3  │
│ 1   ┆ 5.42 ┆ 3.94     ┆ 5.72      ┆ … ┆ 0.491    ┆ 1.03 ┆ -7.28 ┆ 15.2  │
└─────┴──────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

Columns: am, wt, rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, qsec, vs, gear, carb

The same mechanism is available in comparisons() and slopes(). To estimate the partial derivative of mpg with respect to wt, when am is equal to 0 and 1, while other predictors are held at their means:

R

slopes(
  mod,
  variables = "wt",
  newdata = datagrid(am = 0:1))

am	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, am, predicted_lo, predicted_hi, predicted, hp, wt, mpg
0	-2.68	1.42	-1.89	0.0594	4.1	-5.46	0.106
1	-5.43	2.15	-2.52	0.0116	6.4	-9.65	-1.213

Python

s = slopes(
  mod,
  variables = "wt",
  newdata = datagrid(mod, am = [0, 1]))
print(s)

shape: (2, 10)
┌─────┬──────┬──────────┬──────────┬───┬──────────┬──────┬───────┬───────┐
│ am  ┆ Term ┆ Contrast ┆ Estimate ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │
│ --- ┆ ---  ┆ ---      ┆ ---      ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │
│ str ┆ str  ┆ str      ┆ str      ┆   ┆ str      ┆ str  ┆ str   ┆ str   │
╞═════╪══════╪══════════╪══════════╪═══╪══════════╪══════╪═══════╪═══════╡
│ 0   ┆ wt   ┆ dY/dX    ┆ -3.21    ┆ … ┆ 0.11     ┆ 3.18 ┆ -7.14 ┆ 0.729 │
│ 1   ┆ wt   ┆ dY/dX    ┆ -6.61    ┆ … ┆ 0.000393 ┆ 11.3 ┆ -10.3 ┆ -2.96 │
└─────┴──────┴──────────┴──────────┴───┴──────────┴──────┴───────┴───────┘

Columns: am, rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, gear, carb

We can also plot how predictions, comparisons, or slopes change across different values of the predictors using three powerful plotting functions:

• plot_predictions: Conditional Adjusted Predictions
• plot_comparisons: Conditional Comparisons
• plot_slopes: Conditional Marginal Effects

For example, this plot shows the outcomes predicted by our model for different values of the wt and am variables:

R

plot_predictions(mod, condition = list("hp", "wt" = "threenum", "am"))

Python

cond = {
  "hp": None,
  "wt": [mtcars["wt"].mean() - mtcars["wt"].std(),
         mtcars["wt"].mean(),
         mtcars["wt"].mean() + mtcars["wt"].std()],
  "am": None
}
plot_predictions(mod, condition = cond)

<string>:1: FutureWarning: Using repr(plot) to draw and show the plot figure is deprecated and will be removed in a future version. Use plot.show().
<Figure Size: (640 x 480)>

This plot shows how the derivative of mpg with respect to am varies as a function of wt and hp:

R

plot_slopes(mod, variables = "am", condition = list("hp", "wt" = "minmax"))

Python

plot_slopes(mod,
  variables = "am",
  condition = {"hp": None, "wt": [mtcars["wt"].min(), mtcars["wt"].max()]}
)

<string>:1: FutureWarning: Using repr(plot) to draw and show the plot figure is deprecated and will be removed in a future version. Use plot.show().
<Figure Size: (640 x 480)>

See this vignette for more information: Plots, interactions, predictions, contrasts, and slopes

Aggregation

Since predictions, comparisons, and slopes are conditional quantities, they can be a bit unwieldy. Often, it can be useful to report a one-number summary instead of one estimate per observation. Instead of presenting “conditional” estimates, some methodologists recommend reporting “marginal” estimates, that is, an average of unit-level estimates.

(This use of the word “marginal” as “averaging” should not be confused with the term “marginal effect” which, in the econometrics tradition, corresponds to a partial derivative, or the effect of a “small/marginal” change.)

To marginalize (average over) our unit-level estimates, we can use the by argument or the one of the convenience functions: avg_predictions(), avg_comparisons(), or avg_slopes(). For example, both of these commands give us the same result: the average predicted outcome in the mtcars dataset:

R

avg_predictions(mod)

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
20.1	0.39	51.5		Inf	19.3	20.9

Python

p = avg_predictions(mod)
print(p)

shape: (1, 7)
┌──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐
│ Estimate ┆ Std.Error ┆ z    ┆ P(>|z|) ┆ S   ┆ 2.5% ┆ 97.5% │
│ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │
│ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │
╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡
│ 20.1     ┆ 0.39      ┆ 51.5 ┆ 0       ┆ inf ┆ 19.3 ┆ 20.9  │
└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘

Columns: estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

This is equivalent to manual computation by:

R

mean(predict(mod))

[1] 20.09062

Python

np.mean(mod.predict())

np.float64(20.09062499999996)

The main marginaleffects functions all include a by argument, which allows us to marginalize within sub-groups of the data. For example,

R

avg_comparisons(mod, by = "am")

Term	Contrast	am	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: term, contrast, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
am	mean(1) - mean(0)	0	-1.3830	2.5250	-0.548	0.58388	0.8	-6.3319	3.56589
am	mean(1) - mean(0)	1	1.9029	2.3086	0.824	0.40980	1.3	-2.6219	6.42773
hp	mean(+1)	0	-0.0343	0.0159	-2.160	0.03079	5.0	-0.0654	-0.00317
hp	mean(+1)	1	-0.0436	0.0213	-2.050	0.04039	4.6	-0.0854	-0.00191
wt	mean(+1)	0	-2.4799	1.2316	-2.014	0.04406	4.5	-4.8939	-0.06595
wt	mean(+1)	1	-6.0718	1.9762	-3.072	0.00212	8.9	-9.9451	-2.19846

Python

cmp = avg_comparisons(mod, by = "am")
print(cmp)

shape: (6, 10)
┌─────┬──────┬──────────────────────────┬──────────┬───┬─────────┬───────┬─────────┬──────────┐
│ am  ┆ Term ┆ Contrast                 ┆ Estimate ┆ … ┆ P(>|z|) ┆ S     ┆ 2.5%    ┆ 97.5%    │
│ --- ┆ ---  ┆ ---                      ┆ ---      ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │
│ str ┆ str  ┆ str                      ┆ str      ┆   ┆ str     ┆ str   ┆ str     ┆ str      │
╞═════╪══════╪══════════════════════════╪══════════╪═══╪═════════╪═══════╪═════════╪══════════╡
│ 0   ┆ am   ┆ mean(True) - mean(False) ┆ -1.38    ┆ … ┆ 0.584   ┆ 0.776 ┆ -6.33   ┆ 3.57     │
│ 1   ┆ am   ┆ mean(True) - mean(False) ┆ 1.9      ┆ … ┆ 0.41    ┆ 1.29  ┆ -2.62   ┆ 6.43     │
│ 0   ┆ hp   ┆ +1                       ┆ -0.0343  ┆ … ┆ 0.0308  ┆ 5.02  ┆ -0.0654 ┆ -0.00317 │
│ 1   ┆ hp   ┆ +1                       ┆ -0.0436  ┆ … ┆ 0.0404  ┆ 4.63  ┆ -0.0854 ┆ -0.00191 │
│ 0   ┆ wt   ┆ +1                       ┆ -2.48    ┆ … ┆ 0.0441  ┆ 4.5   ┆ -4.89   ┆ -0.066   │
│ 1   ┆ wt   ┆ +1                       ┆ -6.07    ┆ … ┆ 0.00212 ┆ 8.88  ┆ -9.95   ┆ -2.2     │
└─────┴──────┴──────────────────────────┴──────────┴───┴─────────┴───────┴─────────┴──────────┘

Columns: am, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

Marginal Means are a special case of predictions, which are marginalized (or averaged) across a balanced grid of categorical predictors. To illustrate, we estimate a new model with categorical predictors:

R

dat <- mtcars
dat$am <- as.logical(dat$am)
dat$cyl <- as.factor(dat$cyl)
mod_cat <- lm(mpg ~ am + cyl + hp, data = dat)

Python

dat = pl.read_csv("https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv") \
  .with_columns(pl.col("am").cast(pl.Boolean),
                pl.col("cyl").cast(pl.Utf8))
mod_cat = smf.ols('mpg ~ am + cyl + hp', data=dat.to_pandas()).fit()

We can compute marginal means manually using the functions already described:

R

avg_predictions(
  mod_cat,
  newdata = "balanced",
  by = "am")

am	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
TRUE	22.5	0.834	26.9		528.6	20.8	24.1
FALSE	18.3	0.785	23.3		397.4	16.8	19.9

Python

predictions(
  mod_cat,
  newdata = datagrid(grid_type = "balanced"),
  by = "am")
print(p)

The Marginal Means vignette offers more detail.

Uncertainty

The marginaleffects package reports uncertainty estimates for all the quantities it computes: predictions, comparisons, slopes, etc. By default, standard errors are computed using the delta method and classical standard errors. These standard errors are fast to compute, and have appealing properties in some, but not all cases. marginaleffects supports several alternatives, including: Huber-White Heteroskedasticity Robust, Cluster-Robust, Bootstrap, and Simulation-based uncertainty estimates.

The Standard Errors vignette offers more detail. For now, it suffices to show two examples. First, we use the vcov argument to report “HC3” (heteroskedasticity-consistent) standard errors.

R

avg_predictions(mod, by = "am", vcov = "HC3")

am	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
1	24.4	0.782	31.2		707.3	22.9	25.9
0	17.1	0.614	27.9		568.5	15.9	18.3

Python

Not supported yet.

Second, we use the inferences() function to compute bootstrap intervals using 500 resamples:

R

avg_predictions(mod, by = "am") |>
  inferences(method = "boot", R = 500)

am	Estimate	Std. Error	2.5 %	97.5 %
Type: response
Columns: am, estimate, std.error, conf.low, conf.high
1	24.4	3.038	15.4	26.9
0	17.1	0.586	16.2	18.5

Python

Not supported yet.

Tests

The hypotheses() function and the hypothesis argument can be used to conduct linear and non-linear hypothesis tests on model coefficients, or on any of the quantities computed by the functions introduced above.

Consider this model:

R

mod <- lm(mpg ~ qsec * drat, data = mtcars)
coef(mod)

(Intercept)        qsec        drat   qsec:drat 
 12.3371987  -1.0241183  -3.4371461   0.5973153 

Python

mod = smf.ols('mpg ~ qsec * drat', data=mtcars).fit()
print(mod.params)

Intercept    12.337199
qsec         -1.024118
drat         -3.437146
qsec:drat     0.597315
dtype: float64

Can we reject the null hypothesis that the drat coefficient is 2 times the size of the qsec coefficient?

hypotheses(mod, "drat = 2 * qsec")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-1.39	10.8	-0.129	0.897	0.2	-22.5	19.7

We can ask the same question but refer to parameters by position, with indices b1, b2, b3, etc.:

R

hypotheses(mod, "b3 = 2 * b2")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-1.39	10.8	-0.129	0.897	0.2	-22.5	19.7

Python

h = hypotheses(mod, "b3 = 2 * b2")
print(h)

shape: (1, 8)
┌─────────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐
│ Term    ┆ Estimate ┆ Std.Error ┆ z      ┆ P(>|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │
│ ---     ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │
│ str     ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │
╞═════════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡
│ b3=2*b2 ┆ -1.39    ┆ 10.8      ┆ -0.129 ┆ 0.897   ┆ 0.156 ┆ -22.5 ┆ 19.7  │
└─────────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘

Columns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

The main functions in marginaleffects all have a hypothesis argument, which means that we can do complex model testing. For example, consider two slope estimates:

R

slopes(
  mod,
  variables = "drat",
  newdata = datagrid(qsec = range))

qsec	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, qsec, predicted_lo, predicted_hi, predicted, drat, mpg
14.5	5.22	3.79	1.38	0.1682	2.6	-2.206	12.7
22.9	10.24	5.15	1.99	0.0469	4.4	0.139	20.3

Python

s = slopes(
  mod,
  variables = "drat",
  newdata = datagrid(qsec = [mtcars["qsec"].min(), mtcars["qsec"].max()]))
print(s)

shape: (2, 10)
┌──────┬──────┬──────────┬──────────┬───┬─────────┬──────┬────────┬───────┐
│ qsec ┆ Term ┆ Contrast ┆ Estimate ┆ … ┆ P(>|z|) ┆ S    ┆ 2.5%   ┆ 97.5% │
│ ---  ┆ ---  ┆ ---      ┆ ---      ┆   ┆ ---     ┆ ---  ┆ ---    ┆ ---   │
│ str  ┆ str  ┆ str      ┆ str      ┆   ┆ str     ┆ str  ┆ str    ┆ str   │
╞══════╪══════╪══════════╪══════════╪═══╪═════════╪══════╪════════╪═══════╡
│ 14.5 ┆ drat ┆ dY/dX    ┆ 5.22     ┆ … ┆ 0.167   ┆ 2.58 ┆ -2.18  ┆ 12.6  │
│ 22.9 ┆ drat ┆ dY/dX    ┆ 10.2     ┆ … ┆ 0.0481  ┆ 4.38 ┆ 0.0871 ┆ 20.4  │
└──────┴──────┴──────────┴──────────┴───┴─────────┴──────┴────────┴───────┘

Columns: qsec, rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, vs, am, gear, carb

Are these two slopes significantly different from one another? To test this, we can use the hypothesis argument:

R

slopes(
  mod,
  hypothesis = "b1 = b2",
  variables = "drat",
  newdata = datagrid(qsec = range))

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-5.02	8.51	-0.589	0.556	0.8	-21.7	11.7

Python

s = slopes(
  mod,
  hypothesis = "b1 = b2",
  variables = "drat",
  newdata = datagrid(qsec = [mtcars["qsec"].min(), mtcars["qsec"].max()]))
print(s)

shape: (1, 8)
┌───────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐
│ Term  ┆ Estimate ┆ Std.Error ┆ z      ┆ P(>|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │
│ ---   ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │
│ str   ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │
╞═══════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡
│ b1=b2 ┆ -5.02    ┆ 8.53      ┆ -0.588 ┆ 0.556   ┆ 0.846 ┆ -21.7 ┆ 11.7  │
└───────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘

Columns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

Alternatively, we can also refer to values with term names (when they are unique):

R

avg_slopes(mod)

Term	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
drat	7.22	1.365	5.29		23.0	4.548	9.90
qsec	1.12	0.433	2.60	0.00945	6.7	0.275	1.97

avg_slopes(mod, hypothesis = "drat = qsec")

Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
6.1	1.45	4.2		15.2	3.25	8.95

Python

s = avg_slopes(mod)
print(s)

shape: (2, 9)
┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐
│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │
│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │
│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │
╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡
│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.21e-07 ┆ 23   ┆ 4.55  ┆ 9.9   │
│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.434     ┆ … ┆ 0.00957  ┆ 6.71 ┆ 0.274 ┆ 1.97  │
└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

s = avg_slopes(mod, hypothesis = "drat = qsec")
print(s)

shape: (1, 8)
┌───────────┬──────────┬───────────┬─────┬──────────┬──────┬──────┬───────┐
│ Term      ┆ Estimate ┆ Std.Error ┆ z   ┆ P(>|z|)  ┆ S    ┆ 2.5% ┆ 97.5% │
│ ---       ┆ ---      ┆ ---       ┆ --- ┆ ---      ┆ ---  ┆ ---  ┆ ---   │
│ str       ┆ str      ┆ str       ┆ str ┆ str      ┆ str  ┆ str  ┆ str   │
╞═══════════╪══════════╪═══════════╪═════╪══════════╪══════╪══════╪═══════╡
│ drat=qsec ┆ 6.1      ┆ 1.45      ┆ 4.2 ┆ 2.66e-05 ┆ 15.2 ┆ 3.25 ┆ 8.95  │
└───────────┴──────────┴───────────┴─────┴──────────┴──────┴──────┴───────┘

Columns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

Now, imagine that for theoretical (or substantive or clinical) reasons, we only care about slopes larger than 2. We can use the equivalence argument to conduct an equivalence test:

R

avg_slopes(mod, equivalence = c(-2, 2))

Term	Estimate	Std. Error	z	Pr(>|z|)	S	2.5 %	97.5 %	p (NonSup)	p (NonInf)	p (Equiv)
Type: response
Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv
drat	7.22	1.365	5.29		23.0	4.548	9.90	0.9999		0.9999
qsec	1.12	0.433	2.60	0.00945	6.7	0.275	1.97	0.0216		0.0216

Python

s = avg_slopes(mod, equivalence = [-2., 2.])
print(s)

shape: (2, 9)
┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐
│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │
│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │
│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │
╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡
│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.21e-07 ┆ 23   ┆ 4.55  ┆ 9.9   │
│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.434     ┆ … ┆ 0.00957  ┆ 6.71 ┆ 0.274 ┆ 1.97  │
└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, statistic_noninf, statistic_nonsup, p_value_noninf, p_value_nonsup, p_value_equiv

See the Hypothesis Tests and Custom Contrasts vignette for background, details, and for instructions on how to conduct hypothesis tests in more complex situations.