26 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:
-
marginaleffectsfunctions: Use thehypothesisargument. - Other
Robjects and models: Use thehypotheses()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.
27 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.
27.1 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: 7Using 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 |
27.2 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 |
27.3 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 |
28 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"
28.1 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 |
28.2 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 | Pr(>|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 | Pr(>|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 | Pr(>|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] -6.929365There are many more possibilities:
predictions(
mod,
hypothesis = "b1 + b2 = 30",
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 | ||||||
| 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 |
28.3 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 | 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 | |||||||
| 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 |
29 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 | Pr(>|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 | 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) | 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 | Pr(>|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 | Pr(>|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 | |
30 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.
30.1 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 | Pr(>|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:
| Group | Estimate | Std. Error | z | Pr(>|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 | Pr(>|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 | Pr(>|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 | |
30.2 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 |
30.3 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.
31 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.
31.1 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 | Pr(>|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] 0.32174or
21.66232 + -(21.34058)
#> [1] 0.32174or
or
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 |
31.2 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:
31.3 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 |
32 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 |
33 Joint hypotheses tests
The hypotheses() function can also test multiple hypotheses jointly. For example, consider this model:
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))34 More
34.1 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 -1.27089689Then, 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 |