Hypothesis Tests

The marginaleffects package can conduct linear or non-linear hypothesis tests on the coefficients of any supported model class, or on the quantities generated by any of the other functions of the package: predictions(), comparisons(), or slopes().

There are two main entry points for hypothesis tests:

marginaleffects functions: Use the hypothesis argument.
Other R objects and models: Use the hypotheses() function.

Both the hypothesis argument and the hypotheses() function accept several input types, which allow a lot of flexibility in the specification of hypothesis tests:

Numeric: Value of a null hypothesis.
Formula: Equation of a (non-)linear hypothesis test.
Function: Tests on arbitrary transformations or aggregations.
Matrix: Contrast matrices and vectors.

This vignette shows how to use each of these strategies to conduct hypothesis tests on model coefficients or on the quantities estimated by the marginaleffects package. After reading it, you will be able to specify custom hypothesis tests and contrasts to assess statements like:

The coefficients \(\beta_1\) and \(\beta_2\) are equal.
The slope of \(Y\) with respect to \(X_1\) is equal to the slope with respect to \(X_2\).
The average treatment effect of \(X\) when \(W=0\) is equal to the average treatment effect of \(X\) when \(W=1\).
A non-linear function of predicted values is equal to 100.
The marginal mean in the control group is equal to the average of marginal means in the other 3 treatment arms.
Cross-level contrasts: In a multinomial model, the effect of \(X\) on the 1st outcome level is equal to the effect of \(X\) on the 2nd outcome level.

Numeric (null hypothesis)

The simplest way to modify a hypothesis test is to change the null hypothesis. By default, all functions in the marginaleffects package assume that the null is 0. This can be changed with the hypothesis argument.

Coefficients

Consider a simple logistic regression model:

library(marginaleffects)
mod <- glm(am ~ hp + drat, data = mtcars, family = binomial)

By default, the summary() function will report the results of hypothesis tests where the null is set to 0:

summary(mod)
#> 
#> Call:
#> glm(formula = am ~ hp + drat, family = binomial, data = mtcars)
#> 
#> Coefficients:
#>               Estimate Std. Error z value Pr(>|z|)  
#> (Intercept) -29.076080  12.416916  -2.342   0.0192 *
#> hp            0.010793   0.009328   1.157   0.2473  
#> drat          7.309781   3.046597   2.399   0.0164 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 43.230  on 31  degrees of freedom
#> Residual deviance: 20.144  on 29  degrees of freedom
#> AIC: 26.144
#> 
#> Number of Fisher Scoring iterations: 7

Using hypotheses(), we can easily change the null hypothesis:

hypotheses(mod, hypothesis = 6)

Term	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
(Intercept)	-29.0761	12.41692	-2.82	0.00473	7.7	-53.41279	-4.7394
hp	0.0108	0.00933	-642.04		Inf	-0.00749	0.0291
drat	7.3098	3.04660	0.43	0.66726	0.6	1.33856	13.2810

Predictions

Changing the value of the null is particularly important in the context of predictions, where the 0 baseline may not be particularly meaningful. For example, here we compute the predicted outcome for a hypothetical unit where all regressors are fixed to their sample means:

predictions(mod, newdata = "mean")

Estimate	Pr(>\|z\|)	S	2.5 %	97.5 %	hp	drat
Type: invlink(link)
Columns: rowid, estimate, p.value, s.value, conf.low, conf.high, hp, drat, am
0.231	0.135	2.9	0.0584	0.592	147	3.6

The Z statistic and p value reported above assume that the null hypothesis equals zero. We can change the null with the hypothesis argument:

predictions(mod, newdata = "mean", hypothesis = .5)

Estimate	Pr(>\|z\|)	S	2.5 %	97.5 %	hp	drat
Type: invlink(link)
Columns: rowid, estimate, p.value, s.value, conf.low, conf.high, hp, drat, am
0.231	0.0343	4.9	0.0584	0.592	147	3.6

Comparisons and slopes

When computing different quantities of interest like risk ratios, it can make sense to set the null hypothesis to 1 rather than 0:

avg_comparisons(
    mod,
    variables = "hp",
    comparison = "ratio",
    hypothesis = 1) |>
    print(digits = 5)

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
1.0027	0.0021743	1.2515	0.21074	2.2	0.99846	1.007

Equations

The hypotheses() function emulates the equation-based syntax of the well-established car::deltaMethod and car::linearHypothesis functions. However, marginaleffects supports more models, requires fewer dependencies, and offers some convenience features like the vcov argument for robust standard errors.

The syntax we illustrate in this section takes the form a string equation, ex: "am = 2 * vs"

Coefficients

Let’s start by estimating a simple model:

library(marginaleffects)
mod <- lm(mpg ~ hp + wt + factor(cyl), data = mtcars)

When the FUN and hypothesis arguments of hypotheses() equal NULL (the default), the function returns a data.frame of raw estimates:

hypotheses(mod)

Term	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
(Intercept)	35.8460	2.041	17.56		227.0	31.8457	39.846319
hp	-0.0231	0.012	-1.93	0.0531	4.2	-0.0465	0.000306
wt	-3.1814	0.720	-4.42		16.6	-4.5918	-1.771012
factor(cyl)6	-3.3590	1.402	-2.40	0.0166	5.9	-6.1062	-0.611803
factor(cyl)8	-3.1859	2.170	-1.47	0.1422	2.8	-7.4399	1.068169

Test of equality between coefficients:

hypotheses(mod, "hp = wt")

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
3.16	0.72	4.39		16.4	1.75	4.57

Non-linear function of coefficients

hypotheses(mod, "exp(hp + wt) = 0.1")

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
-0.0594	0.0292	-2.04	0.0418	4.6	-0.117	-0.0022

The vcov argument behaves in the same was as in all the other marginaleffects functions, allowing us to easily compute robust standard errors:

hypotheses(mod, "hp = wt", vcov = "HC3")

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
3.16	0.805	3.92		13.5	1.58	4.74

We can use shortcuts like b1, b2, ... to identify the position of each parameter in the output of FUN. For example, b2=b3 is equivalent to hp=wt because those term names appear in the 2nd and 3rd row when we call hypotheses(mod).

hypotheses(mod, "b2 = b3")

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
3.16	0.72	4.39		16.4	1.75	4.57

hypotheses(mod, hypothesis = "b* / b3 = 1")

Term	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
b1 / b3 = 1	-12.26735	2.07340	-5.9165		28.2	-16.33	-8.204
b2 / b3 = 1	-0.99273	0.00413	-240.5537		Inf	-1.00	-0.985
b3 / b3 = 1	0.00000	NA	NA	NA	NA	NA	NA
b4 / b3 = 1	0.05583	0.58287	0.0958	0.924	0.1	-1.09	1.198
b5 / b3 = 1	0.00141	0.82981	0.0017	0.999	0.0	-1.62	1.628

Term names with special characters must be enclosed in backticks:

hypotheses(mod, "`factor(cyl)6` = `factor(cyl)8`")

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
-0.173	1.65	-0.105	0.917	0.1	-3.41	3.07

Predictions

Now consider the case of adjusted predictions:

mod <- lm(mpg ~ am + vs, data = mtcars)

p <- predictions(
    mod,
    newdata = datagrid(am = 0:1, vs = 0:1))
p

am	vs	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, am, vs, mpg
0	0	14.6	0.926	15.8	183.4	12.8	16.4
0	1	21.5	1.130	19.0	266.3	19.3	23.7
1	0	20.7	1.183	17.5	224.5	18.3	23.0
1	1	27.6	1.130	24.4	435.0	25.4	29.8

Since there is no term column in the output of the predictions function, we must use parameter identifiers like b1, b2, etc. to determine which estimates we want to compare:

hypotheses(p, hypothesis = "b1 = b2")

Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-6.93	1.26	-5.49	24.6	-9.4	-4.46

Or directly:

predictions(
    mod,
    hypothesis = "b1 = b2",
    newdata = datagrid(am = 0:1, vs = 0:1))

Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-6.93	1.26	-5.49	24.6	-9.4	-4.46


p$estimate[1] - p$estimate[2]
#> [1] -6.929365

There are many more possibilities:

predictions(
    mod,
    hypothesis = "b1 + b2 = 30",
    newdata = datagrid(am = 0:1, vs = 0:1))

Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
6.12	1.64	3.74	12.4	2.91	9.32


p$estimate[1] + p$estimate[2] - 30
#> [1] 6.118254

predictions(
    mod,
    hypothesis = "(b2 - b1) / (b3 - b2) = 0",
    newdata = datagrid(am = 0:1, vs = 0:1))

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
-8.03	17	-0.473	0.636	0.7	-41.3	25.2

Comparisons and slopes

The avg_comparisons() function allows us to answer questions of this form: On average, how does the expected outcome change when I change one regressor by some amount? Consider this:

mod <- lm(mpg ~ am + vs, data = mtcars)

cmp <- avg_comparisons(mod)
cmp

Term	Estimate	Std. Error	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
am	6.07	1.27	4.76	19.0	3.57	8.57
vs	6.93	1.26	5.49	24.6	4.46	9.40

This tells us that, on average, moving from 0 to 1 on the am changes the predicted outcome by about 6.1, and changing .vs from 0 to 1 changes the predicted outcome by about 6.9.

Is the difference between those two estimates statistically significant? In other words, Is the effect of am equal to the effect of vs? To answer this question, we use the hypothesis argument:

avg_comparisons(mod, hypothesis = "am = vs")

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
-0.863	1.94	-0.445	0.656	0.6	-4.66	2.94

The hypothesis string can include any valid R expression, so we can run some silly non-linear tests:

avg_comparisons(mod, hypothesis = "exp(am) - 2 * vs = -400")

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
817	550	1.49	0.137	2.9	-261	1896

Note that the p values and confidence intervals are calculated using the delta method and are thus based on the assumption that the hypotheses expression is approximately normally distributed. For (very) non-linear functions of the parameters, this is not realistic, and we get p values with incorrect error rates and confidence intervals with incorrect coverage probabilities. For such hypotheses, it’s better to calculate the confidence intervals using the bootstrap (see inferences for details):

While the confidence interval from the delta method is symmetric, equal to the estimate ± 1.96 times the standard error, the (perhaps) more reliable confidence interval from the bootstrap is highly skewed.

set.seed(1234)

avg_comparisons(mod, hypothesis = "exp(am) - 2 * vs = -400") |>
  inferences(method = "boot")

Estimate	Std. Error	2.5 %	97.5 %
Type: response
Columns: term, estimate, std.error, conf.low, conf.high
817	1854	414	6990

The same approach can be taken to compare slopes:

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

avg_slopes(mod, hypothesis = "10 * hp - qsec = 0")

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
0.262	0.353	0.742	0.458	1.1	-0.43	0.954

Formulas (Group-wise)

Since version 0.20.1.5 of marginaleffects, the hypothesis argument also accepts a formula to specify hypotheses to be tested on every row of the output, or within subsets. Consider this set of average predictions:

dat <- within(mtcars, {
    gear = factor(gear)
    cyl = factor(cyl)
})
dat <- sort_by(dat, ~ gear + cyl + am)
mod <- lm(mpg ~ am * wt * factor(cyl), data = dat)

avg_predictions(mod, by = "gear")

gear	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: gear, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
3	16.1	0.609	26.5	510.5	14.9	17.3
4	24.2	0.613	39.5	Inf	23.0	25.4
5	22.2	0.837	26.5	511.3	20.5	23.8

We can make “sequential”, “reference”, or “meandev” comparisons, either as differences (the default) or ratios:

avg_predictions(mod, 
    hypothesis = ~ sequential,
    by = "gear")

Hypothesis	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
(4) - (3)	8.08	0.835	9.68		71.2	6.45	9.721
(5) - (4)	-2.01	0.829	-2.43	0.0152	6.0	-3.64	-0.387


avg_predictions(mod, 
    hypothesis = ~ meandev,
    by = "gear")

Hypothesis	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
(3) - Avg	-4.72	0.563	-8.38		54.1	-5.822	-3.62
(4) - Avg	3.37	0.434	7.75		46.6	2.514	4.22
(5) - Avg	1.35	0.560	2.42	0.0156	6.0	0.257	2.45


avg_predictions(mod, 
    hypothesis = ratio ~ sequential,
    by = "gear")

Hypothesis	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
(4) / (3)	1.502	0.0662	22.7	375.6	1.372	1.632
(5) / (4)	0.917	0.0336	27.3	543.2	0.851	0.983

We can also test hypotheses by subgroup. For example, in this case, we want to compare every estimate to the reference category (first estimate) in each subset of am:

avg_predictions(mod, 
    by = c("am", "gear")
)

am	gear	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: am, gear, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
0	3	16.1	0.609	26.5	510.5	14.9	17.3
0	4	21.0	1.091	19.3	272.8	18.9	23.2
1	4	25.8	0.740	34.8	880.0	24.3	27.2
1	5	22.2	0.837	26.5	511.3	20.5	23.8


avg_predictions(mod, 
    hypothesis = ~ sequential | am,
    by = c("am", "gear")
)

Hypothesis	am	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
(4) - (3)	0	4.93	1.190	4.14	14.8	2.59	7.26
(5) - (4)	1	-3.59	0.815	-4.41	16.5	-5.19	-1.99

The grouping component of the formula is particularly useful when conducting tests on contrasts:

avg_comparisons(mod, 
    variables = "cyl",
    by = "am")

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
mean(6) - mean(4)	0	-11.33	6.04	-1.875	0.0608	4.0	-23.2	0.512
mean(6) - mean(4)	1	-4.39	4.94	-0.889	0.3739	1.4	-14.1	5.291
mean(8) - mean(4)	0	-9.59	3.84	-2.495	0.0126	6.3	-17.1	-2.057
mean(8) - mean(4)	1	-7.85	8.46	-0.928	0.3535	1.5	-24.4	8.730


avg_comparisons(mod, 
    variables = "cyl",
    by = "am", 
    hypothesis = ~ sequential | contrast)

Contrast	Hypothesis	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
mean(6) - mean(4)	(cyl, 1) - (cyl, 0)	6.93	7.80	0.889	0.374	1.4	-8.36	22.2
mean(8) - mean(4)	(cyl, 1) - (cyl, 0)	1.74	9.29	0.188	0.851	0.2	-16.46	20.0


avg_comparisons(mod, 
    variables = "cyl",
    by = "am", 
    hypothesis = ~ sequential | am)

Hypothesis	am	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
(cyl, mean(8) - mean(4)) - (cyl, mean(6) - mean(4))	0	1.74	4.79	0.363	0.717	0.5	-7.64	11.1
(cyl, mean(8) - mean(4)) - (cyl, mean(6) - mean(4))	1	-3.45	9.65	-0.358	0.720	0.5	-22.37	15.5

It is also a powerful tool in categorical outcome models, to compare estimates for different outcome levels:

library("MASS")

mod <- polr(gear ~ cyl + hp, dat)

avg_comparisons(mod, variables = "cyl")

Group	Contrast	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: term, group, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
3	mean(6) - mean(4)	0.2679	0.0787	3.403		10.6	0.1136	0.4221
3	mean(8) - mean(4)	0.8582	0.0432	19.886		289.9	0.7737	0.9428
4	mean(6) - mean(4)	0.0108	0.0474	0.228	0.81962	0.3	-0.0822	0.1038
4	mean(8) - mean(4)	-0.1971	0.0651	-3.028	0.00246	8.7	-0.3246	-0.0695
5	mean(6) - mean(4)	-0.2787	0.0717	-3.887		13.3	-0.4192	-0.1382
5	mean(8) - mean(4)	-0.6612	0.0694	-9.533		69.2	-0.7971	-0.5253

Is the “8 vs. 4” contrast equal to the “6 vs. 4” contrast, within each outcome level?

avg_comparisons(mod, 
    variables = "cyl",
    hypothesis = ~ sequential | group)

Group	Hypothesis	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: group, hypothesis, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
3	(cyl, mean(8) - mean(4)) - (cyl, mean(6) - mean(4))	0.590	0.0815	7.24		41.0	0.431	0.7502
4	(cyl, mean(8) - mean(4)) - (cyl, mean(6) - mean(4))	-0.208	0.0717	-2.90	0.00376	8.1	-0.348	-0.0673
5	(cyl, mean(8) - mean(4)) - (cyl, mean(6) - mean(4))	-0.383	0.0575	-6.65		35.0	-0.495	-0.2698

Is the “8 vs. 4” contrast for outcome level 3 equal to the “8 vs. 4” contrast for outcome level 4?

avg_comparisons(mod, 
    variables = "cyl",
    hypothesis = ~ sequential | contrast)

Contrast	Hypothesis	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: hypothesis, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
mean(6) - mean(4)	(4, cyl) - (3, cyl)	-0.257	0.1084	-2.37	0.01775	5.8	-0.470	-0.0445
mean(6) - mean(4)	(5, cyl) - (4, cyl)	-0.289	0.0927	-3.12	0.00178	9.1	-0.471	-0.1079
mean(8) - mean(4)	(4, cyl) - (3, cyl)	-1.055	0.0859	-12.28		112.7	-1.224	-0.8869
mean(8) - mean(4)	(5, cyl) - (4, cyl)	-0.464	0.1274	-3.64		11.9	-0.714	-0.2145

Functions

Hypothesis tests can also be conducted using arbitrary R functions. This allows users to test hypotheses on complex aggregations or transformations. To achieve this, we define a custom function which accepts a fitted model or marginaleffects objects, and returns a data frame with at least two columns: term (or hypothesis) and estimate.

Predictions

When supplying a function to the hypothesis argument, that function must accept an argument x which is a data frame with columns rowid and estimate (and optional columns for other elements of newdata). That function must return a data frame with columns term (or hypothesis) and estimate.

In this example, we test if the mean predicted value is different from 2:

dat <- transform(mtcars, gear = factor(gear), cyl = factor(cyl))

mod <- lm(wt ~ mpg * hp * cyl, data = dat)

hyp <- function(x) {
    data.frame(
        hypothesis = "Avg(Ŷ) = 2",
        estimate = mean(x$estimate) - 2
    )
}
predictions(mod, hypothesis = hyp)

Hypothesis	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: hypothesis, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
Avg(Ŷ) = 2	1.22	0.0914	13.3	132.0	1.04	1.4

In this ordinal logit model, the predictions() function returns one row per observation and per level of the outcome variable:

library(MASS)
library(dplyr)

mod <- polr(gear ~ cyl + hp, dat)

avg_predictions(mod)

Group	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: probs
Columns: group, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
3	0.471	0.0584	8.05	50.2	0.3561	0.585
4	0.366	0.0715	5.12	21.7	0.2263	0.507
5	0.163	0.0478	3.41	10.6	0.0692	0.257

We can use a function in the hypothesis argument to collapse the rows, displaying the average predicted values in groups 3-4 vs. 5:

fun <- function(x) {
    out <- x |> 
        mutate(term = ifelse(group %in% 3:4, "3 & 4", "5")) |>
        summarize(estimate = mean(estimate), .by = term)
    return(out)
}
avg_predictions(mod, hypothesis = fun)

Term	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: probs
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
3 & 4	0.419	0.0239	17.51	225.5	0.3717	0.465
5	0.163	0.0478	3.41	10.6	0.0692	0.257

And we can compare the two categories by doing:

fun <- function(x) {
    out <- x |> 
        mutate(term = ifelse(group %in% 3:4, "3 & 4", "5")) |>
        summarize(estimate = mean(estimate), .by = term) |>
        summarize(estimate = diff(estimate), term = "5 - (3 & 4)")
    return(out)
}
avg_predictions(mod, hypothesis = fun)

Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: probs
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
-0.256	0.0717	-3.56	11.4	-0.396	-0.115

Comparisons and slopes

In the same ordinal logit model, we can estimate the average effect of an increase of 1 unit in hp on the expected probability of every level of the outcome:

avg_comparisons(mod, variables = "hp")

Group	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: term, group, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
3	-0.00774	0.002294	-3.38		10.4	-0.01224	-0.00325
4	0.00258	0.002201	1.17	0.24	2.1	-0.00173	0.00690
5	0.00516	0.000882	5.85		27.6	0.00343	0.00689

Compare estimate for different outcome levels:

fun <- function(x) {
    x |> 
    mutate(estimate = (estimate - lag(estimate)),
           group = sprintf("%s - %s", group, lag(group))) |>
    filter(!is.na(estimate))
}
avg_comparisons(mod, variables = "hp", hypothesis = fun)

Group	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: term, group, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
4 - 3	0.01033	0.00441	2.34	0.0191	5.7	0.00169	0.01897
5 - 4	0.00258	0.00245	1.05	0.2922	1.8	-0.00222	0.00737

Now suppose we want to compare the effect of hp for different levels of the outcome, but this time we do the computation within levels of cyl:

avg_comparisons(mod, variables = "hp", by = "cyl")

Group	cyl	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: term, group, contrast, cyl, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
3	4	-0.007862	0.003445	-2.282	0.02248	5.5	-1.46e-02	-0.00111
3	6	-0.013282	0.005038	-2.637	0.00838	6.9	-2.32e-02	-0.00341
3	8	-0.004882	0.001457	-3.350		10.3	-7.74e-03	-0.00203
4	4	-0.003024	0.003714	-0.814	0.41560	1.3	-1.03e-02	0.00426
4	6	0.008573	0.006431	1.333	0.18251	2.5	-4.03e-03	0.02118
4	8	0.003995	0.001627	2.455	0.01409	6.1	8.06e-04	0.00719
5	4	0.010886	0.002652	4.104		14.6	5.69e-03	0.01608
5	6	0.004709	0.001929	2.442	0.01462	6.1	9.29e-04	0.00849
5	8	0.000887	0.000431	2.059	0.03947	4.7	4.27e-05	0.00173

fun <- function(x) {
    x |> 
    mutate(estimate = (estimate - lag(estimate)),
           group = sprintf("%s - %s", group, lag(group)), 
           .by = "cyl") |>
    filter(!is.na(estimate))
}
avg_comparisons(mod, variables = "hp", by = "cyl", hypothesis = fun)

Group	cyl	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: term, group, contrast, cyl, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
4 - 3	4	0.00484	0.00666	0.727	0.46719	1.1	-0.008205	0.017882
4 - 3	6	0.02185	0.01139	1.919	0.05503	4.2	-0.000471	0.044180
4 - 3	8	0.00888	0.00306	2.902	0.00371	8.1	0.002882	0.014874
5 - 4	4	0.01391	0.00546	2.548	0.01082	6.5	0.003211	0.024607
5 - 4	6	-0.00386	0.00805	-0.480	0.63117	0.7	-0.019638	0.011911
5 - 4	8	-0.00311	0.00188	-1.651	0.09869	3.3	-0.006798	0.000581

Fitted models

The hypothesis argument can be used to compute standard errors for arbitrary functions of model parameters. This user-supplied function must accept a single model object, and return a data.frame with two columns named term and estimate.

Here, we test if the sum of the hp and mpg coefficients is equal to 2:

mod <- glm(am ~ hp + mpg, data = mtcars, family = binomial)

fun <- function(x) {
    b <- coef(x)
    out <- data.frame(
        term = "hp + mpg = 2",
        estimate = b["hp"] + b["mpg"] - 2,
        row.names = NULL
    )
    return(out)
}

hypotheses(mod, hypothesis = fun)

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
-0.685	0.593	-1.16	0.248	2.0	-1.85	0.476

Test of equality between two two predictions:

fun <- function(x) {
    p <- predict(x, newdata = mtcars)
    out <- data.frame(term = "pred[2] = pred[3]", estimate = p[2] - p[3])
    return(out)
}
hypotheses(mod, hypothesis = fun)

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
-1.33	0.616	-2.16	0.0305	5.0	-2.54	-0.125

We can also use more complex aggregation patterns. In this ordinal logistic regression model, we model the number of gears for each ar. If we compute fitted values with the predictions() function, we obtain one predicted probability for each individual car and for each level of the response variable:

library(MASS)
library(dplyr)

dat <- transform(mtcars, 
    gear = factor(gear),
    cyl = factor(cyl))
mod <- polr(gear ~ cyl + hp, dat)

predictions(mod)

Group	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Type: probs
Columns: rowid, group, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, gear, cyl, hp
3	0.3931	0.19125	2.06	0.03982	4.7	0.0183	0.768
3	0.3931	0.19125	2.06	0.03982	4.7	0.0183	0.768
3	0.0440	0.04256	1.03	0.30081	1.7	-0.0394	0.127
3	0.3931	0.19125	2.06	0.03982	4.7	0.0183	0.768
3	0.9963	0.00721	138.17		Inf	0.9822	1.010
5	0.6969	0.18931	3.68		12.1	0.3258	1.068
5	0.0555	0.06851	0.81	0.41775	1.3	-0.0788	0.190
5	0.8115	0.20626	3.93		13.5	0.4073	1.216
5	0.9111	0.16818	5.42		24.0	0.5815	1.241
5	0.6322	0.19648	3.22	0.00129	9.6	0.2471	1.017

There are three levels to the outcome: 3, 4, and 5. Imagine that, for each car in the dataset, we want to collapse categories of the output variable into two categories (“3 & 4” and “5”) by taking sums of predicted probabilities. Then, we want to take the average of those predicted probabilities for each level of cyl. To do so, we define a custom function, and pass it to the hypothesis argument of the hypotheses() function:

fun <- function(x) {
    predictions(x, vcov = FALSE) |>
        # label the new categories of outcome levels
        mutate(group = ifelse(group %in% c("3", "4"), "3 & 4", "5")) |>
        # sum of probabilities at the individual level
        summarize(estimate = sum(estimate), .by = c("rowid", "cyl", "group")) |>
        # average probabilities for each value of `cyl`
        summarize(estimate = mean(estimate), .by = c("cyl", "group")) |>
        # the `FUN` argument requires a `term` column
        rename(term = cyl)
}

hypotheses(mod, hypothesis = fun)

Group	Term	Estimate	Std. Error	z	Pr(>\|z\|)	S	2.5 %	97.5 %
Columns: term, group, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
3 & 4	6	0.8390	0.0651	12.89		123.9	0.7115	0.967
3 & 4	4	0.7197	0.1099	6.55		34.0	0.5044	0.935
3 & 4	8	0.9283	0.0174	53.45		Inf	0.8943	0.962
5	6	0.1610	0.0651	2.47	0.0134	6.2	0.0334	0.289
5	4	0.2803	0.1099	2.55	0.0108	6.5	0.0649	0.496
5	8	0.0717	0.0174	4.13		14.7	0.0377	0.106

Note that this workflow will not work for bayesian models or with bootstrap. However, with those models it is trivial to do the same kind of aggregation by calling posterior_draws() and operating directly on draws from the posterior distribution. See the vignette on bayesian analysis for examples with the posterior_draws() function.

Vectors and Matrices

The predictions() function can estimate marginal means. The hypothesis argument of that function offers a powerful mechanism to estimate custom contrasts between marginal means, by way of linear combination.

Simple contrast

Consider a simple example:

library(marginaleffects)
library(emmeans)
library(nnet)

dat <- mtcars
dat$carb <- factor(dat$carb)
dat$cyl <- factor(dat$cyl)
dat$am <- as.logical(dat$am)
dat <- sort_by(dat, ~carb)

mod <- lm(mpg ~ carb + cyl, dat)

mm <- predictions(mod,
    by = "carb",
    newdata = "balanced")
mm

carb	Estimate	Std. Error	z	S	2.5 %	97.5 %
Type: response
Columns: carb, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
1	21.7	1.44	15.06	167.8	18.8	24.5
2	21.3	1.23	17.29	220.0	18.9	23.8
3	21.4	2.19	9.77	72.5	17.1	25.7
4	18.9	1.21	15.59	179.7	16.5	21.3
6	19.8	3.55	5.56	25.2	12.8	26.7
8	20.1	3.51	5.73	26.6	13.2	27.0

The contrast between marginal means for carb==1 and carb==2 is:

21.66232 - 21.34058 
#> [1] 0.32174

21.66232 + -(21.34058)
#> [1] 0.32174

sum(c(21.66232, 21.34058) * c(1, -1))
#> [1] 0.32174

c(21.66232, 21.34058) %*% c(1, -1)
#>         [,1]
#> [1,] 0.32174

The last two commands express the contrast of interest as a linear combination of marginal means.

In the predictions() function, we can supply a hypothesis argument to compute linear combinations of marginal means. This argument must be a numeric vector of the same length as the number of rows in the output. For example, in the previous there were six rows, and the two marginal means we want to compare are at in the first two positions:

lc <- c(1, -1, 0, 0, 0, 0)
predictions(mod,
    by = "carb",
    newdata = "balanced",
    hypothesis = lc)

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
0.322	1.77	0.181	0.856	0.2	-3.15	3.8

Complex contrast

Of course, we can also estimate more complex contrasts:

lc <- c(0, -2, 1, 1, -1, 1)
predictions(mod,
    by = "carb",
    newdata = "balanced",
    hypothesis = lc)

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
-2.02	6.32	-0.32	0.749	0.4	-14.4	10.4

emmeans produces similar results:

library(emmeans)
em <- emmeans(mod, "carb")
lc <- data.frame(custom_contrast = c(0, -2, 1, 1, -1, 1))
contrast(em, method = lc)
#>  contrast        estimate   SE df t.ratio p.value
#>  custom_contrast    -2.02 6.32 24  -0.320  0.7516
#> 
#> Results are averaged over the levels of: cyl

Multiple contrasts

Users can also compute multiple linear combinations simultaneously by supplying a numeric matrix to hypotheses. This matrix must have the same number of rows as the output of slopes(), and each column represents a distinct set of weights for different linear combinations. The column names of the matrix become labels in the output. For example:

lc <- matrix(c(
    -2, 1, 1, 0, -1, 1,
    1, -1, 0, 0, 0, 0
    ), ncol = 2)
colnames(lc) <- c("Contrast A", "Contrast B")
lc
#>      Contrast A Contrast B
#> [1,]         -2          1
#> [2,]          1         -1
#> [3,]          1          0
#> [4,]          0          0
#> [5,]         -1          0
#> [6,]          1          0

predictions(mod,
    by = "carb",
    newdata = "balanced",
    hypothesis = lc)

Term	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
Contrast A	-0.211	6.93	-0.0304	0.976	0.0	-13.79	13.4
Contrast B	0.322	1.77	0.1814	0.856	0.2	-3.15	3.8

Arbitrary quantities

marginaleffects can also compute uncertainty estimates for arbitrary quantities hosted in a data frame, as long as the user can supply a variance-covariance matrix. (Thanks to Kyle F Butts for this cool feature and example!)

Say you run a monte-carlo simulation and you want to perform hypothesis of various quantities returned from each simulation. The quantities are correlated within each draw:

# simulated means and medians
draw <- function(i) { 
  x <- rnorm(n = 10000, mean = 0, sd = 1)
  out <- data.frame(median = median(x), mean =  mean(x))
  return(out)
}
sims <- do.call("rbind", lapply(1:25, draw))

# average mean and average median 
coeftable <- data.frame(
  term = c("median", "mean"),
  estimate = c(mean(sims$median), mean(sims$mean))
)

# variance-covariance
vcov <- cov(sims)

# is the median equal to the mean?
hypotheses(
  coeftable,
  vcov = vcov,
  hypothesis = "median = mean"
)

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
-0.000968	0.00804	-0.12	0.904	0.1	-0.0167	0.0148

Joint hypotheses tests

The hypotheses() function can also test multiple hypotheses jointly. For example, consider this model:

model <- lm(mpg ~ as.factor(cyl) * hp, data = mtcars)
coef(model)
#>        (Intercept)    as.factor(cyl)6    as.factor(cyl)8                 hp as.factor(cyl)6:hp as.factor(cyl)8:hp 
#>        35.98302564       -15.30917451       -17.90295193        -0.11277589         0.10516262         0.09853177

We may want to test the null hypothesis that two of the coefficients are jointly (both) equal to zero.

hypotheses(model, joint = c("as.factor(cyl)6:hp", "as.factor(cyl)8:hp"))

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
2.11	0.142	2	26

The joint argument allows users to flexibly specify the parameters to be tested, using character vectors, integer indices, or Perl-compatible regular expressions. We can also specify the null hypothesis for each parameter individually using the hypothesis argument.

Naturally, the hypotheses function also works with marginaleffects objects.

# ## joint hypotheses: regular expression
hypotheses(model, joint = "cyl")

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
5.7	0.00197	4	26


# joint hypotheses: integer indices
hypotheses(model, joint = 2:3)

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
6.12	0.00665	2	26


# joint hypotheses: different null hypotheses
hypotheses(model, joint = 2:3, hypothesis = 1)

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
6.84	0.00411	2	26

hypotheses(model, joint = 2:3, hypothesis = 1:2)

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
7.47	0.00273	2	26


# joint hypotheses: marginaleffects object
cmp <- avg_comparisons(model)
hypotheses(cmp, joint = "cyl")

F	Pr(>\|F\|)	Df 1	Df 2
Columns: statistic, p.value, df1, df2
1.6	0.221	2	26

We can also combine multiple calls to hypotheses to execute a joint test on linear combinations of coefficients:

# fit model
mod <- lm(mpg ~ factor(carb), mtcars)

# hypothesis matrix for linear combinations
H <- matrix(0, nrow = length(coef(mod)), ncol = 2)
H[2:3, 1] <- H[4:6, 2] <- 1

# test individual linear combinations
hyp <- hypotheses(mod, hypothesis = H)
hyp

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
-12.0	4.92	-2.44	0.01477	6.1	-21.6	-2.35
-25.5	9.03	-2.83	0.00466	7.7	-43.2	-7.85


# test joint hypotheses
#hypotheses(hyp, joint = TRUE, hypothesis = c(-10, -20))

Difference-in-Differences

Now we illustrate how to use the machinery described above to do pairwise comparisons between contrasts, a type of analysis often associated with a “Difference-in-Differences” research design.

First, we simulate data with two treatment groups and pre/post periods:

library(data.table)

N <- 1000
did <- data.table(
    id = 1:N,
    pre = rnorm(N),
    trt = sample(0:1, N, replace = TRUE))
did$post <- did$pre + did$trt * 0.3 + rnorm(N)
did <- melt(
    did,
    value.name = "y",
    variable.name = "time",
    id.vars = c("id", "trt"))
head(did)
#>       id   trt   time           y
#>    <int> <int> <fctr>       <num>
#> 1:     1     1    pre  0.92924817
#> 2:     2     1    pre -1.66043941
#> 3:     3     0    pre -0.74964444
#> 4:     4     1    pre  0.08894837
#> 5:     5     0    pre -0.66554261
#> 6:     6     1    pre -1.27089689

Then, we estimate a linear model with a multiple interaction between the time and the treatment indicators. We also compute contrasts at the mean for each treatment level:

did_model <- lm(y ~ time * trt, data = did)

comparisons(
    did_model,
    newdata = datagrid(trt = 0:1),
    variables = "time")

trt	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, trt, predicted_lo, predicted_hi, predicted, time, y
0	0.00075	0.0792	0.00948	0.992	0.0	-0.154	0.156
1	0.26946	0.0788	3.41776		10.6	0.115	0.424

Finally, we compute pairwise differences between contrasts. This is the Diff-in-Diff estimate:

comparisons(
    did_model,
    variables = "time",
    newdata = datagrid(trt = 0:1),
    hypothesis = "pairwise")

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
-0.269	0.112	-2.41	0.0162	6.0	-0.488	-0.0497

Numeric (null hypothesis)

Coefficients

Predictions

Comparisons and slopes

Equations

Coefficients

Predictions

Comparisons and slopes

Formulas (Group-wise)

Functions

Predictions

Comparisons and slopes

Fitted models

Vectors and Matrices

Simple contrast

Complex contrast

Multiple contrasts

Arbitrary quantities

Joint hypotheses tests

More

Difference-in-Differences