Get started

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. Estimand: What is the quantity of interest? Do we want to estimate a prediction or a function of predictions (average, difference, ratio, derivative, etc.)?
  2. Grid: 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:

Install from CRAN:

install.packages("marginaleffects")

Install from PyPI:

pip install marginaleffects

Estimands

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 estimand, “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 estimands described above. First, we fit a linear regression model with multiplicative interactions:

library(marginaleffects)

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

pre <- predictions(mod)

nrow(mtcars)
[1] 32
nrow(pre)
[1] 32
pre

 Estimate Std. Error     z Pr(>|z|)     S 2.5 % 97.5 %
     22.5      0.884 25.44   <0.001 471.7  20.8   24.2
     20.8      1.194 17.42   <0.001 223.3  18.5   23.1
     25.3      0.709 35.66   <0.001 922.7  23.9   26.7
     20.3      0.704 28.75   <0.001 601.5  18.9   21.6
     17.0      0.712 23.88   <0.001 416.2  15.6   18.4
--- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- 
     29.6      1.874 15.80   <0.001 184.3  25.9   33.3
     15.9      1.311 12.13   <0.001 110.0  13.3   18.5
     19.4      1.145 16.95   <0.001 211.6  17.2   21.7
     14.8      2.017  7.33   <0.001  42.0  10.8   18.7
     21.5      1.072 20.02   <0.001 293.8  19.4   23.6
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am 
Type:  response 
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:

cmp <- comparisons(mod)

nrow(cmp)
[1] 96
cmp

 Term Contrast Estimate Std. Error      z Pr(>|z|)   S  2.5 % 97.5 %
   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
--- 86 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- 
   wt    +1      -6.518       1.88 -3.462   <0.001 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   <0.001 11.5 -10.29 -2.999
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 
Type:  response 
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?

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

 Term  Contrast Estimate Std. Error      z Pr(>|z|)   S  2.5 %  97.5 %
   hp 120 - 100   -0.738      0.370 -1.995  0.04607 4.4 -1.463 -0.0129
   hp 120 - 100   -0.574      0.313 -1.836  0.06640 3.9 -1.186  0.0388
   hp 120 - 100   -0.931      0.452 -2.062  0.03922 4.7 -1.817 -0.0460
   hp 120 - 100   -0.845      0.266 -3.182  0.00146 9.4 -1.366 -0.3248
   hp 120 - 100   -0.780      0.268 -2.909  0.00362 8.1 -1.306 -0.2547
--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- 
   hp 120 - 100   -1.451      0.705 -2.058  0.03958 4.7 -2.834 -0.0692
   hp 120 - 100   -0.384      0.270 -1.422  0.15498 2.7 -0.912  0.1451
   hp 120 - 100   -0.641      0.334 -1.918  0.05513 4.2 -1.297  0.0141
   hp 120 - 100   -0.126      0.272 -0.463  0.64360 0.6 -0.659  0.4075
   hp 120 - 100   -0.635      0.332 -1.911  0.05598 4.2 -1.286  0.0162
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 
Type:  response 
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?

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

 Term                Contrast Estimate Std. Error      z Pr(>|z|)   S 2.5 %  97.5 %
   hp (x + sd/2) - (x - sd/2)   -2.530      1.269 -1.995  0.04607 4.4 -5.02 -0.0441
   hp (x + sd/2) - (x - sd/2)   -1.967      1.072 -1.836  0.06640 3.9 -4.07  0.1332
   hp (x + sd/2) - (x - sd/2)   -3.193      1.549 -2.062  0.03922 4.7 -6.23 -0.1578
   hp (x + sd/2) - (x - sd/2)   -2.898      0.911 -3.182  0.00146 9.4 -4.68 -1.1133
   hp (x + sd/2) - (x - sd/2)   -2.675      0.919 -2.909  0.00362 8.1 -4.48 -0.8731
--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- 
   hp (x + sd/2) - (x - sd/2)   -4.976      2.418 -2.058  0.03958 4.7 -9.71 -0.2373
   hp (x + sd/2) - (x - sd/2)   -1.315      0.925 -1.422  0.15498 2.7 -3.13  0.4974
   hp (x + sd/2) - (x - sd/2)   -2.199      1.147 -1.918  0.05513 4.2 -4.45  0.0483
   hp (x + sd/2) - (x - sd/2)   -0.432      0.933 -0.463  0.64360 0.6 -2.26  1.3970
   hp (x + sd/2) - (x - sd/2)   -2.177      1.139 -1.911  0.05598 4.2 -4.41  0.0556
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 
Type:  response 
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?

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

 Term  Contrast Estimate Std. Error    z Pr(>|z|)     S 2.5 % 97.5 %
   hp mean(+50)    0.905     0.0319 28.4   <0.001 586.8 0.843  0.968

Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted 
Type:  response 
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:

mfx <- slopes(mod)

nrow(mfx)
[1] 96
mfx

 Term Contrast Estimate Std. Error      z Pr(>|z|)   S  2.5 % 97.5 %
   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
--- 86 rows omitted. See ?avg_slopes and ?print.marginaleffects --- 
   wt    dY/dX   -6.518       1.88 -3.462   <0.001 10.9 -10.21 -2.828
   wt    dY/dX   -1.653       3.74 -0.442   0.6588  0.6  -8.99  5.682
   wt    dY/dX   -4.520       2.47 -1.829   0.0673  3.9  -9.36  0.322
   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   <0.001 11.5 -10.29 -3.000
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 
Type:  response 
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      ┆ … ┆ 0.000535 ┆ 10.9  ┆ -10.2 ┆ -2.83 │
│ wt   ┆ dY/dX        ┆ -1.65    ┆ 3.74      ┆ … ┆ 0.658    ┆ 0.603 ┆ -8.98 ┆ 5.67  │
│ wt   ┆ dY/dX        ┆ -4.52    ┆ 2.47      ┆ … ┆ 0.0671   ┆ 3.9   ┆ -9.36 ┆ 0.318 │
│ wt   ┆ dY/dX        ┆ 0.635    ┆ 4.89      ┆ … ┆ 0.897    ┆ 0.157 ┆ -8.95 ┆ 10.2  │
│ wt   ┆ dY/dX        ┆ -6.65    ┆ 1.86      ┆ … ┆ 0.000358 ┆ 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

Grid

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:

predictions(mod, newdata = "mean")

 Estimate Std. Error    z Pr(>|z|)     S 2.5 % 97.5 %  hp   wt am
     18.7      0.649 28.8   <0.001 603.8  17.4     20 147 3.22  0

Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am 
Type:  response 
predictions(mod, newdata = "median")

 Estimate Std. Error  z Pr(>|z|)     S 2.5 % 97.5 %  hp   wt am
     19.4      0.646 30   <0.001 653.2  18.1   20.6 123 3.33  0

Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am 
Type:  response 
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   │
╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡
│ 19.9     ┆ 0.723     ┆ 27.5 ┆ 0       ┆ inf ┆ 18.5 ┆ 21.3  │
└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘

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:

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

 am   wt Estimate Std. Error    z Pr(>|z|)    S 2.5 % 97.5 %  hp
  0 1.51     23.3       2.71 8.60   <0.001 56.7 17.96   28.6 147
  0 5.42     12.8       2.98 4.30   <0.001 15.8  6.96   18.6 147
  1 1.51     27.1       2.85 9.52   <0.001 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

Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, am, wt 
Type:  response 
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 ┆ 21.4     ┆ 2.41      ┆ … ┆ 0        ┆ inf  ┆ 16.7  ┆ 26.1  │
│ 0   ┆ 5.42 ┆ 12.5     ┆ 1.96      ┆ … ┆ 1.75e-10 ┆ 32.4 ┆ 8.66  ┆ 16.3  │
│ 1   ┆ 1.51 ┆ 25.1     ┆ 3.77      ┆ … ┆ 2.76e-11 ┆ 35.1 ┆ 17.7  ┆ 32.5  │
│ 1   ┆ 5.42 ┆ 7.41     ┆ 6.12      ┆ … ┆ 0.225    ┆ 2.15 ┆ -4.57 ┆ 19.4  │
└─────┴──────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

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:

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

 Term am Estimate Std. Error     z Pr(>|z|)   S 2.5 % 97.5 %
   wt  0    -2.68       1.42 -1.89   0.0593 4.1 -5.46  0.105
   wt  1    -5.43       2.15 -2.52   0.0116 6.4 -9.65 -1.214

Columns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, am, predicted_lo, predicted_hi, predicted, mpg, hp, wt 
Type:  response 
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    ┆ -2.19    ┆ … ┆ 0.0307  ┆ 5.03 ┆ -4.18 ┆ -0.204 │
│ 1   ┆ wt   ┆ dY/dX    ┆ -4.36    ┆ … ┆ 0.0851  ┆ 3.55 ┆ -9.32 ┆ 0.603  │
└─────┴──────┴──────────┴──────────┴───┴─────────┴──────┴───────┴────────┘

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:

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

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)
<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:

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

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

avg_predictions(mod)

 Estimate Std. Error    z Pr(>|z|)   S 2.5 % 97.5 %
     20.1       0.39 51.5   <0.001 Inf  19.3   20.9

Columns: estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 
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:

mean(predict(mod))
[1] 20.09062
np.mean(mod.predict())
20.090625000000014

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

avg_comparisons(mod, by = "am")

 Term          Contrast am Estimate Std. Error      z Pr(>|z|)   S   2.5 %   97.5 %
   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

Columns: term, contrast, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted 
Type:  response 
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:

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

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

    am Estimate Std. Error    z Pr(>|z|)     S 2.5 % 97.5 %
 FALSE     18.3      0.785 23.3   <0.001 397.4  16.8   19.9
  TRUE     22.5      0.834 26.9   <0.001 528.6  20.8   24.1

Columns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 
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 some 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 to examples. First, we use the vcov argument to report “HC3” (heteroskedasticity-consistent) standard errors.

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

 am Estimate Std. Error    z Pr(>|z|)     S 2.5 % 97.5 %
  0     17.1      0.614 27.9   <0.001 568.5  15.9   18.3
  1     24.4      0.782 31.2   <0.001 707.3  22.9   25.9

Columns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 

Not supported yet.

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

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

 am Estimate Std. Error 2.5 % 97.5 %
  0     17.1      0.594  16.2   18.4
  1     24.4      2.978  12.1   26.2

Columns: am, estimate, std.error, conf.low, conf.high 
Type:  response 

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:

mod <- lm(mpg ~ qsec * drat, data = mtcars)
coef(mod)
(Intercept)        qsec        drat   qsec:drat 
 12.3371987  -1.0241183  -3.4371461   0.5973153 
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")

            Term Estimate Std. Error      z Pr(>|z|)   S 2.5 % 97.5 %
 drat = 2 * qsec    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7

Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 

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

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

        Term Estimate Std. Error      z Pr(>|z|)   S 2.5 % 97.5 %
 b3 = 2 * b2    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7

Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
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:

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

 Term qsec Estimate Std. Error    z Pr(>|z|)   S  2.5 % 97.5 %
 drat 14.5     5.22       3.79 1.38   0.1678 2.6 -2.199   12.6
 drat 22.9    10.24       5.17 1.98   0.0477 4.4  0.106   20.4

Columns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, qsec, predicted_lo, predicted_hi, predicted, mpg, drat 
Type:  response 
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     ┆ … ┆ 0.168   ┆ 2.57 ┆ -2.21 ┆ 12.7  │
│ 22.9 ┆ drat ┆ dY/dX    ┆ 10.2     ┆ … ┆ 0.0477  ┆ 4.39 ┆ 0.102 ┆ 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:

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

  Term Estimate Std. Error      z Pr(>|z|)   S 2.5 % 97.5 %
 b1=b2    -5.02       8.53 -0.588    0.556 0.8 -21.7   11.7

Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 
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):

avg_slopes(mod)

 Term Estimate Std. Error    z Pr(>|z|)    S 2.5 % 97.5 %
 drat     7.22      1.365 5.29  < 0.001 23.0 4.549   9.90
 qsec     1.12      0.433 2.59  0.00946  6.7 0.275   1.97

Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 
avg_slopes(mod, hypothesis = "drat = qsec")

      Term Estimate Std. Error   z Pr(>|z|)    S 2.5 % 97.5 %
 drat=qsec      6.1       1.45 4.2   <0.001 15.2  3.25   8.95

Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
Type:  response 
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      ┆ … ┆ 1.22e-07 ┆ 23   ┆ 4.55  ┆ 9.9   │
│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.435     ┆ … ┆ 0.00971  ┆ 6.69 ┆ 0.272 ┆ 1.98  │
└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

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:

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)
 drat     7.22      1.365 5.29  < 0.001 23.0 4.549   9.90     0.9999     <0.001    0.9999
 qsec     1.12      0.433 2.59  0.00946  6.7 0.275   1.97     0.0216     <0.001    0.0216

Columns: term, 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 
Type:  response 
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      ┆ … ┆ 1.22e-07 ┆ 23   ┆ 4.55  ┆ 9.9   │
│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.435     ┆ … ┆ 0.00971  ┆ 6.69 ┆ 0.272 ┆ 1.98  │
└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘

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.