Note: Some of the code in this vignette requires marginaleffects
version 0.6.0 or the development version from github.
In another vignette, we introduced the “marginal effect” as a partial derivative. Since derivatives are only properly defined for continuous variables, we cannot use them to interpret the effects of changes in categorical variables. For this, we turn to contrasts between Adjusted predictions. In the context of this package, a “Contrast” is defined as:
A difference, ratio, or function of adjusted predictions, calculated for meaningfully different predictor values (e.g., College graduates vs. Others).
The marginaleffects()
function automatically calculates contrasts instead of derivatives for factor, logical, or character variables.
The comparisons()
function gives users more powerful features to compute different contrasts, such as differences, risk ratios, linear combinations, and transformations.
Consider a simple model with a logical and a factor variable:
library(marginaleffects)
library(magrittr)
mtcars
tmp <-$am <- as.logical(tmp$am)
tmp lm(mpg ~ am + factor(cyl), tmp) mod <-
The marginaleffects
function automatically computes contrasts for each level of the categorical variables, relative to the baseline category (FALSE
for logicals, and the reference level for factors), while holding all other values at their mode or mean:
marginaleffects(mod)
mfx <-summary(mfx)
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 am TRUE - FALSE 2.560 1.298 1.973 0.04851 0.01675 5.103
#> 2 cyl 6 - 4 -6.156 1.536 -4.009 6.1077e-05 -9.16608 -3.146
#> 3 cyl 8 - 4 -10.068 1.452 -6.933 4.1146e-12 -12.91359 -7.222
#>
#> Model type: lm
#> Prediction type: response
The summary printed above says that moving from the reference category 4
to the level 6
on the cyl
factor variable is associated with a change of -6.156 in the adjusted prediction. Similarly, the contrast from FALSE
to TRUE
on the am
variable is equal to 2.560.
We can obtain different contrasts by using the comparisons()
function. For example:
comparisons(mod, variables = list(cyl = "sequential")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response cyl 6 - 4 -6.156118 1.535723 -4.008612 6.107658e-05 -9.166079 -3.146156
#> 2 response cyl 8 - 6 -3.911442 1.470254 -2.660385 7.805144e-03 -6.793087 -1.029797
comparisons(mod, variables = list(cyl = "pairwise")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response cyl 6 - 4 -6.156118 1.535723 -4.008612 6.107658e-05 -9.166079 -3.146156
#> 2 response cyl 8 - 4 -10.067560 1.452082 -6.933187 4.114626e-12 -12.913589 -7.221530
#> 3 response cyl 8 - 6 -3.911442 1.470254 -2.660385 7.805144e-03 -6.793087 -1.029797
comparisons(mod, variables = list(cyl = "reference")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response cyl 6 - 4 -6.156118 1.535723 -4.008612 6.107658e-05 -9.166079 -3.146156
#> 2 response cyl 8 - 4 -10.067560 1.452082 -6.933187 4.114626e-12 -12.913589 -7.221530
For comparison, this code produces the same results using the emmeans
package:
library(emmeans)
emmeans(mod, specs = "cyl")
emm <-contrast(emm, method = "revpairwise")
#> contrast estimate SE df t.ratio p.value
#> cyl6 - cyl4 -6.16 1.54 28 -4.009 0.0012
#> cyl8 - cyl4 -10.07 1.45 28 -6.933 <.0001
#> cyl8 - cyl6 -3.91 1.47 28 -2.660 0.0331
#>
#> Results are averaged over the levels of: am
#> P value adjustment: tukey method for comparing a family of 3 estimates
emmeans(mod, specs = "am")
emm <-contrast(emm, method = "revpairwise")
#> contrast estimate SE df t.ratio p.value
#> TRUE - FALSE 2.56 1.3 28 1.973 0.0585
#>
#> Results are averaged over the levels of: cyl
Note that these commands also work on for other types of models, such as GLMs, on different scales:
glm(am ~ factor(gear), data = mtcars, family = binomial)
mod_logit <-
comparisons(mod_logit) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response gear 4 - 3 0.6666667 1.360805e-01 4.899061 9.62957e-07 0.3999538 0.9333795
#> 2 response gear 5 - 3 1.0000000 1.071403e-05 93335.529594 0.00000e+00 0.9999790 1.0000210
comparisons(mod_logit, type = "link") %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 link gear 4 - 3 21.25922 4577.962 0.004643817 0.9962948 -8951.381 8993.90
#> 2 link gear 5 - 3 41.13214 9155.924 0.004492407 0.9964156 -17904.148 17986.41
All functions of the marginaleffects
package attempt to treat character predictors as factor predictors. However, using factors instead of characters when modelling is strongly encouraged, because they are much safer and faster. This is because factors hold useful information about the full list of levels, which makes them easier to track and handle internally by marginaleffects
. Users are strongly encouraged to convert their character variables to factor before fitting their models and using marginaleffects
functions.
We can also compute contrasts for differences in numeric variables. For example, we can see what happens to the adjusted predictions when we increment the hp
variable by 1 unit (default) or by 5 units:
lm(mpg ~ hp, data = mtcars)
mod <-
comparisons(mod) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp (x + 1) - x -0.06822828 0.0101193 -6.742389 1.558037e-11 -0.08806175 -0.04839481
comparisons(
mod,variables = list(hp = 5)) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp (x + 5) - x -0.3411414 0.05059652 -6.742389 1.558038e-11 -0.4403087 -0.241974
Compare adjusted predictions for a change in the regressor between two arbitrary values:
comparisons(mod, variables = list(hp = c(90, 110))) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp 110 - 90 -1.364566 0.2023861 -6.742389 1.558038e-11 -1.761235 -0.9678961
Compare adjusted predictions when the regressor changes across the interquartile range, across one or two standard deviations about its mean, or from across its full range:
comparisons(mod, variables = list(hp = "iqr")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp Q3 - Q1 -5.697061 0.8449619 -6.742389 1.558038e-11 -7.353156 -4.040966
comparisons(mod, variables = list(hp = "sd")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp (x + sd/2) - (x - sd/2) -4.677926 0.6938085 -6.742389 1.558038e-11 -6.037766 -3.318087
comparisons(mod, variables = list(hp = "2sd")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp (x - sd) - (x + sd) -9.355853 1.387617 -6.742389 1.558038e-11 -12.07553 -6.636174
comparisons(mod, variables = list(hp = "minmax")) %>% tidy()
#> type term contrast estimate std.error statistic p.value conf.low conf.high
#> 1 response hp Max - Min -19.3086 2.863763 -6.742389 1.558038e-11 -24.92147 -13.69573
In some contexts we would like to know what happens when two (or more) predictors change at the same time. In the marginaleffects
package terminology, this is an “interaction between contrasts.”
For example, consider a model with two factor variables:
lm(mpg ~ am * factor(cyl), data = mtcars) mod <-
What happens if am
increases by 1 unit and cyl
changes from a baseline reference to another level?
comparisons(mod, variables = c("cyl", "am"))
cmp <-summary(cmp)
#> cyl am Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 4 - 4 1 - 0 5.175 2.053 2.5209 0.0117059 1.151 9.199
#> 2 6 - 4 1 - 0 -2.333 2.476 -0.9424 0.3459644 -7.186 2.519
#> 3 8 - 4 1 - 0 -7.500 2.768 -2.7095 0.0067389 -12.925 -2.075
#>
#> Model type: lm
#> Prediction type: response
When the variables
argument is used and the model formula includes interactions, the “cross-contrasts” will automatically be displayed. You can also force comparisons()
to do it by setting interactions=TRUE
and using the variables
argument to specify which variables should be manipulated simultaneously.
This section compares 4 quantities:
The ideas discussed in this section focus on contrasts, but they carry over directly to analogous types of marginal effects.
In models with interactions or non-linear components (e.g., link function), the value of a contrast or marginal effect can depend on the value of all the predictors in the model. As a result, contrasts and marginal effects are fundamentally unit-level quantities. The effect of a 1 unit increase in \(X\) can be different for Mary or John. Every row of a dataset has a different contrast and marginal effect.
The mtcars
dataset has 32 rows, so the comparisons()
function produces 32 contrast estimates:
library(marginaleffects)
glm(vs ~ factor(gear) + mpg, family = binomial, data = mtcars)
mod <- comparisons(mod, variables = "mpg")
cmp <-nrow(cmp)
#> [1] 32
By default, the marginaleffects()
and comparisons()
functions compute marginal effects and contrasts for every row of the original dataset. These unit-level estimates can be unwieldy and hard to interpret. To help interpretation, the summary()
function computes the “Average Marginal Effect” or “Average Contrast,” by taking the mean of all the unit-level estimates.
summary(cmp)
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg (x + 1) - x 0.06081 0.01284 4.737 2.1714e-06 0.03565 0.08597
#>
#> Model type: glm
#> Prediction type: response
which is equivalent to:
mean(cmp$comparison)
#> [1] 0.06080995
We could also show the full distribution of contrasts across our dataset with a histogram:
library(ggplot2)
comparisons(mod, variables = "gear")
cmp <-
ggplot(cmp, aes(comparison)) +
geom_histogram(bins = 30) +
facet_wrap(~contrast, scale = "free_x") +
labs(x = "Distribution of unit-level contrasts")
This graph display the effect of a change of 1 unit in the mpg
variable, for each individual in the observed data.
An alternative which used to be very common but has now fallen into a bit of disfavor is to compute “Contrasts at the mean.” The idea is to create a “synthetic” or “hypothetical” individual (row of the dataset) whose characteristics are completely average. Then, we compute and report the contrast for this specific hypothetical individual.
This can be achieved by setting newdata="mean"
or to newdata=datagrid()
, both of which fix variables to their means or modes:
comparisons(mod, variables = "mpg", newdata = "mean")
#> rowid type term contrast comparison std.error statistic p.value conf.low conf.high gear
#> 1 1 response mpg (x + 1) - x 0.1664787 0.06245542 2.66556 0.007686022 0.04406829 0.288889 3
#> mpg eps
#> 1 20.09062 0.00235
Contrasts at the mean can differ substantially from average contrasts.
The advantage of this approach is that it is very cheap and fast computationally. The disadvantage is that the interpretation is somewhat ambiguous. Often times, there simply does not exist an individual who is perfectly average across all dimensions of the dataset. It is also not clear why the analyst should be particularly interested in the contrast for this one, synthetic, perfectly average individual.
Yet another type of contrast is the “Contrast between marginal means.” This type of contrast is closely related to the “Contrast at the mean”, with a few wrinkles. It is the default approach used by the emmeans
package for R
.
Roughly speaking, the procedure is as follows:
btype
(focal variable) and resp
(group by
variable).btype
.The contrast obtained through this approach has two critical characteristics:
With respect to (a), the analyst should ask themselves: Is my quantity of interest the contrast for a perfectly average hypothetical individual? With respect to (b), the analyst should ask themselves: Is my quantity of interest the contrast in a model estimated using (potentially) unbalanced data, but interpreted as if the data were perfectly balanced?
For example, imagine that one of the control variables in your model is a variable measuring educational attainment in 4 categories: No high school, High school, Some college, Completed college. The contrast between marginal is a weighted average of contrasts estimated in the 4 cells, and each of those contrasts will be weighted equally in the overall estimate. If the population of interest is highly unbalanced in the educational categories, then the estimate computed in this way will not be most useful.
If the contrasts between marginal means is really the quantity of interest, it is easy to use the comparisons()
to estimate contrasts between marginal means. The newdata
determines the values of the predictors at which we want to compute contrasts. We can set newdata="marginalmeans"
to emulate the emmeans
behavior. For example, here we compute contrasts in a model with an interaction:
read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/palmerpenguins/penguins.csv")
dat <- lm(bill_length_mm ~ species * sex + island + body_mass_g, data = dat)
mod <-
comparisons(
cmp <-
mod,newdata = "marginalmeans",
variables = c("species", "island"))
summary(cmp)
#> species island Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 Adelie - Adelie Dream - Biscoe -0.45571 0.4533 -1.005 0.31472 -1.3441 0.4327
#> 2 Adelie - Adelie Torgersen - Biscoe 0.08507 0.4701 0.181 0.85639 -0.8363 1.0064
#> 3 Chinstrap - Adelie Biscoe - Biscoe 10.26934 0.4067 25.252 < 2.22e-16 9.4723 11.0664
#> 4 Chinstrap - Adelie Dream - Biscoe 9.81362 0.4336 22.630 < 2.22e-16 8.9637 10.6636
#> 5 Chinstrap - Adelie Torgersen - Biscoe 10.35441 0.6217 16.656 < 2.22e-16 9.1360 11.5728
#> 6 Gentoo - Adelie Biscoe - Biscoe 5.89568 0.6773 8.705 < 2.22e-16 4.5683 7.2231
#> 7 Gentoo - Adelie Dream - Biscoe 5.43996 0.9413 5.779 7.504e-09 3.5951 7.2849
#> 8 Gentoo - Adelie Torgersen - Biscoe 5.98075 0.9542 6.268 3.667e-10 4.1105 7.8510
#>
#> Model type: lm
#> Prediction type: response
Which is equivalent to this in emmeans
:
emmeans(
emm <-
mod,specs = c("species", "island"))
contrast(emm, method = "trt.vs.ctrl1")
#> contrast estimate SE df t.ratio p.value
#> Chinstrap Biscoe - Adelie Biscoe 10.2693 0.407 324 25.252 <.0001
#> Gentoo Biscoe - Adelie Biscoe 5.8957 0.677 324 8.705 <.0001
#> Adelie Dream - Adelie Biscoe -0.4557 0.453 324 -1.005 0.8274
#> Chinstrap Dream - Adelie Biscoe 9.8136 0.434 324 22.630 <.0001
#> Gentoo Dream - Adelie Biscoe 5.4400 0.941 324 5.779 <.0001
#> Adelie Torgersen - Adelie Biscoe 0.0851 0.470 324 0.181 0.9994
#> Chinstrap Torgersen - Adelie Biscoe 10.3544 0.622 324 16.656 <.0001
#> Gentoo Torgersen - Adelie Biscoe 5.9808 0.954 324 6.268 <.0001
#>
#> Results are averaged over the levels of: sex
#> P value adjustment: dunnettx method for 8 tests
The emmeans
section of the Alternative Software vignette shows further examples.
The excellent vignette of the emmeans
package discuss the same issues in a slightly different (and more positive) way:
The point is that the marginal means of cell.means give equal weight to each cell. In many situations (especially with experimental data), that is a much fairer way to compute marginal means, in that they are not biased by imbalances in the data. We are, in a sense, estimating what the marginal means would be, had the experiment been balanced. Estimated marginal means (EMMs) serve that need.
All this said, there are certainly situations where equal weighting is not appropriate. Suppose, for example, we have data on sales of a product given different packaging and features. The data could be unbalanced because customers are more attracted to some combinations than others. If our goal is to understand scientifically what packaging and features are inherently more profitable, then equally weighted EMMs may be appropriate; but if our goal is to predict or maximize profit, the ordinary marginal means provide better estimates of what we can expect in the marketplace.
Consider a model with an interaction term. What happens to the dependent variable when the hp
variable increases by 10 units?
library(marginaleffects)
lm(mpg ~ hp * wt, data = mtcars)
mod <-
plot_cco(
mod,effect = list(hp = 10),
condition = "wt")
So far we have focused on simple differences between adjusted predictions. Now, we show how to use ratios, back transformations, and arbitrary functions to estimate a slew of quantities of interest. Powerful transformations and custom contrasts are made possible by using three arguments which act at different stages of the computation process:
transform_pre
transform_post
transform_avg
Consider the case of a model with a single predictor \(x\). To compute average contrasts, we proceed as follows:
transform_pre
: Compute unit-level contrasts by taking the difference between (or some other function of) adjusted predictions: \(\hat{y}_{x+1} - \hat{y}_x\)transform_post
: Transform the unit-level contrasts or return them as-is.transform_avg
: Transform the average contrast or return them as-is.The transform_pre
argument of the comparisons()
function determines how adjusted predictions are combined to create a contrast. By default, we take a simple difference between predictions with hi
value of \(x\), and predictions with a lo
value of \(x\): function(hi, lo) hi-lo
.
The transform_post
argument of the comparisons()
function applies a custom transformation to the unit-level contrasts.
The transform_avg
argument is available in the tidy()
and summary()
functions. It applies a custom transformation to the average contrast.
The difference between transform_post
and transform_avg
is that the former is applied before we take the average, and the latter is applied to the average. This seems like a subtle distinction, but it can be important practical implications, since a function of the average is rarely the same as the average of a function:
set.seed(1024)
rnorm(100)
x <-exp(mean(x))
#> [1] 0.9806912
mean(exp(x))
#> [1] 1.587238
The default contrast calculate by the comparisons()
function is a (untransformed) difference between two adjusted predictions. For instance, to estimate the effect of a change of 1 unit, we do:
library(marginaleffects)
library(magrittr)
glm(vs ~ mpg, data = mtcars, family = binomial)
mod <-
# construct data
mtcars_plus <- mtcars
mtcars_minus <-$mpg <- mtcars_minus$mpg - 0.5
mtcars_minus$mpg <- mtcars_plus$mpg + 0.5
mtcars_plus
# adjusted predictions
predict(mod, newdata = mtcars_minus, type = "response")
yhat_minus <- predict(mod, newdata = mtcars_plus, type = "response")
yhat_plus <-
# unit-level contrasts
yhat_plus - yhat_minus
con <-
# average contrasts
mean(con)
#> [1] 0.05540227
We can use the comparisons()
and summary()
functions to obtain the same results:
comparisons(mod)
con <-summary(con)
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg (x + 1) - x 0.0554 0.008327 6.653 2.8699e-11 0.03908 0.07172
#>
#> Model type: glm
#> Prediction type: response
Instead of taking simple differences between adjusted predictions, it can sometimes be useful to compute ratios or other functions of predictions. For example, the adjrr
function the Stata
software package can compute “adjusted risk ratios”, which are ratios of adjusted predictions. To do this in R
, we use the transform_pre
argument:
comparisons(mod, transform_pre = "ratio") %>% summary()
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg (x + 1) / x 1.287 0.1328 9.697 < 2.22e-16 1.027 1.548
#>
#> Model type: glm
#> Prediction type: response
This result is the average adjusted risk ratio, that is, the adjusted predictions when the mpg
are incremented by 1, divided by the adjusted predictions when mpg
is at its original value.
The transform_pre
accepts different values for common types of contrasts: ‘difference’, ‘ratio’, ‘lnratio’, ‘ratioavg’, ‘lnratioavg’, ‘lnoravg’, ‘differenceavg’. These strings are shortcuts for functions that accept two vectors of adjusted predictions and returns a single vector of contrasts. For example, these two commands yield identical results:
comparisons(mod, transform_pre = "ratio") %>% summary()
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg (x + 1) / x 1.287 0.1328 9.697 < 2.22e-16 1.027 1.548
#>
#> Model type: glm
#> Prediction type: response
comparisons(mod, transform_pre = function(hi, lo) hi / lo) %>% summary()
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg custom 1.287 0.1328 9.697 < 2.22e-16 1.027 1.548
#>
#> Model type: glm
#> Prediction type: response
#> Pre-transformation: function(hi, lo) hi/lo
This mechanism is powerful, because it lets users create fully customized contrasts. Here is a non-sensical example:
comparisons(mod, transform_pre = function(hi, lo) sqrt(hi) / log(lo + 10)) %>% summary()
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg custom 0.2641 0.02614 10.1 < 2.22e-16 0.2128 0.3153
#>
#> Model type: glm
#> Prediction type: response
#> Pre-transformation: function(hi, lo) sqrt(hi)/log(lo + 10)
The same arguments work in the plotting function plot_cco()
as well, which allows us to plot various custom contrasts. Here is a comparison of Adjusted Risk Ratio and Adjusted Risk Difference in a model of the probability of survival aboard the Titanic:
library(ggplot2)
library(patchwork)
"https://vincentarelbundock.github.io/Rdatasets/csv/Stat2Data/Titanic.csv"
titanic <- read.csv(titanic)
titanic <- glm(
mod_titanic <-~ Sex * PClass + Age + I(Age^2),
Survived family = binomial,
data = titanic)
comparisons(mod_titanic)
cmp <-summary(cmp)
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 Sex male - female -0.484676 0.030607 -15.835 < 2.22e-16 -0.544665 -0.424687
#> 2 PClass 2nd - 1st -0.205782 0.039374 -5.226 1.7296e-07 -0.282954 -0.128609
#> 3 PClass 3rd - 1st -0.404283 0.039839 -10.148 < 2.22e-16 -0.482367 -0.326199
#> 4 Age (x + 1) - x -0.006504 0.001072 -6.069 1.2904e-09 -0.008605 -0.004403
#>
#> Model type: glm
#> Prediction type: response
plot_cco(
p1 <-
mod_titanic,effect = "Age",
condition = "Age",
transform_pre = "ratio") +
ylab("Adjusted Risk Ratio\nP(Survival | Age + 1) / P(Survival | Age)")
plot_cco(
p2 <-
mod_titanic,effect = "Age",
condition = "Age") +
ylab("Adjusted Risk Difference\nP(Survival | Age + 1) - P(Survival | Age)")
+ p2 p1
By default, the standard errors around contrasts are computed using the delta method on the scale determined by the type
argument (e.g., “link” or “response”). Some analysts may prefer to proceed differently. For example, in Stata
, the adjrr
computes adjusted risk ratios (ARR) in two steps:
Step 1 is easy to achieve with the transform_pre
argument described above. Step 2 can be achieved with the transform_post
argument:
comparisons(
mod,transform_pre = function(hi, lo) log(hi / lo),
transform_post = exp) |>
summary()
#> Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg custom 1.287 0.09313 13.82 < 2.22e-16 1.105 1.47
#>
#> Model type: glm
#> Prediction type: response
#> Pre-transformation: function(hi, lo) log(hi/lo)
#> Post-transformation: exp
Note that we can use the lnratioavg
shortcut instead of defining the function ourselves.
The order of operations in previous command was:
summary()
functionThere is a very subtle difference between the procedure above and this code:
comparisons(
mod,transform_pre = function(hi, lo) log(hi / lo)) %>%
summary(transform_avg = exp)
#> Term Contrast Effect Pr(>|z|) 2.5 % 97.5 %
#> 1 mpg custom 1.274 0.0093462 1.061 1.529
#>
#> Model type: glm
#> Prediction type: response
#> Pre-transformation: function(hi, lo) log(hi/lo)
#> Average-transformation:
Since the exp
function is now passed to the transform_avg
argument of summary()
function, the exponentiation is now done only after unit-level contrasts have been averaged. This is what Stata
appears to does under the hood, and the results are slightly different.
comparisons(
mod,transform_pre = function(hi, lo) log(mean(hi) / mean(lo)),
transform_post = exp)
#> type term contrast comparison p.value conf.low conf.high
#> 1 response mpg custom 1.135065 2.380805e-10 1.091432 1.180442
Note that equivalent results can be obtained using shortcut strings in the transform_pre
argument: “ratio”, “lnratio”, “lnratioavg”.
comparisons(
mod,transform_pre = "lnratioavg",
transform_post = exp)
#> type term contrast comparison p.value conf.low conf.high
#> 1 response mpg ln(mean(x + 1) / mean(x)) 1.135065 2.380805e-10 1.091432 1.180442
All the same arguments apply to the plotting functions of the marginaleffects
package as well. For example we can plot the Adjusted Risk Ratio in a model with a quadratic term:
library(ggplot2)
glm(vs ~ mpg + mpg^2, data = mtcars, family = binomial)
mod2 <-
plot_cco(
mod2,effect = list("mpg" = 10),
condition = "mpg",
transformation_pre = "ratio") +
ylab("Adjusted Risk Ratio\nP(vs = 1 | mpg + 10) / P(vs = 1 | mpg)")
With hurdle models, we can fit two separate models simultaneously:
We can calculate predictions and marginal effects for each of these hurdle model processes, but doing so requires some variable transformation since the stages of these models use different link functions.
The hurdle_lognormal()
family in brms
uses logistic regression (with a logit link) for the hurdle part of the model and lognormal regression (where the outcome is logged before getting used in the model) for the non-hurdled part. Let’s look at an example of predicting GDP per capita (which is distributed exponentially) using life expectancy. We’ll add some artificial zeros so that we can work with a hurdle stage of the model.
library(dplyr)
library(ggplot2)
library(patchwork)
library(brms)
library(marginaleffects)
library(gapminder)
# Build some 0s into the GDP column
set.seed(1234)
gapminder::gapminder %>%
gapminder <- filter(continent != "Oceania") %>%
# Make a bunch of GDP values 0
mutate(prob_zero = ifelse(lifeExp < 50, 0.3, 0.02),
will_be_zero = rbinom(n(), 1, prob = prob_zero),
gdpPercap0 = ifelse(will_be_zero, 0, gdpPercap)) %>%
select(-prob_zero, -will_be_zero)
brm(
mod <-bf(gdpPercap0 ~ lifeExp,
~ lifeExp),
hu data = gapminder,
family = hurdle_lognormal(),
chains = 4, cores = 4, seed = 1234)
We have two different sets of coefficients here for the two different processes. The hurdle part (hu
) uses a logit link, and the non-hurdle part (mu
) uses an identity link. However, that’s a slight misnomer—a true identity link would show the coefficients on a non-logged dollar value scale. Because we’re using a lognormal
family, GDP per capita is pre-logged, so the “original” identity scale is actually logged dollars.
summary(mod)
#> Family: hurdle_lognormal
#> Links: mu = identity; sigma = identity; hu = logit
#> Formula: gdpPercap0 ~ lifeExp
#> hu ~ lifeExp
#> Data: gapminder (Number of observations: 1680)
#> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup draws = 4000
#>
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 3.47 0.09 3.29 3.65 1.00 4757 3378
#> hu_Intercept 3.16 0.40 2.37 3.96 1.00 2773 2679
#> lifeExp 0.08 0.00 0.08 0.08 1.00 5112 3202
#> hu_lifeExp -0.10 0.01 -0.12 -0.08 1.00 2385 2652
#> ...
We can get predictions for the hu
part of the model on the link (logit) scale:
predictions(mod, dpar = "hu", type = "link",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type predicted conf.low conf.high lifeExp
#> 1 1 link -0.817487 -1.033982 -0.6043308 40
#> 2 2 link -2.805488 -3.062906 -2.5550801 60
#> 3 3 link -4.790200 -5.337808 -4.2745563 80
…or on the response (percentage point) scale:
predictions(mod, dpar = "hu", type = "response",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type predicted conf.low conf.high lifeExp
#> 1 1 response 0.306297360 0.262312829 0.35335351 40
#> 2 2 response 0.057028334 0.044663565 0.07208594 60
#> 3 3 response 0.008242295 0.004783404 0.01372716 80
We can also get slopes for the hu
part of the model on the link (logit) or response (percentage point) scales:
marginaleffects(mod, dpar = "hu", type = "link",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type term dydx conf.low conf.high predicted predicted_hi predicted_lo lifeExp
#> 1 1 link lifeExp -0.09930925 -0.1157859 -0.08366088 -0.817487 -0.8180725 -0.817487 40
#> 2 2 link lifeExp -0.09930925 -0.1157859 -0.08366088 -2.805488 -2.8060666 -2.805488 60
#> 3 3 link lifeExp -0.09930925 -0.1157859 -0.08366088 -4.790200 -4.7908031 -4.790200 80
#> eps
#> 1 0.0059004
#> 2 0.0059004
#> 3 0.0059004
marginaleffects(mod, dpar = "hu", type = "response",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type term dydx conf.low conf.high predicted predicted_hi predicted_lo
#> 1 1 response lifeExp -0.0210776902 -0.025913450 -0.0165879119 0.306297360 0.306172973 0.306297360
#> 2 2 response lifeExp -0.0053208087 -0.006148655 -0.0045608559 0.057028334 0.056997229 0.057028334
#> 3 3 response lifeExp -0.0008118892 -0.001154388 -0.0005429417 0.008242295 0.008237367 0.008242295
#> lifeExp eps
#> 1 40 0.0059004
#> 2 60 0.0059004
#> 3 80 0.0059004
Working with the mu
part of the model is trickier. Switching between type = "link"
and type = "response"
doesn’t change anything, since the outcome is pre-logged:
predictions(mod, dpar = "mu", type = "link",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type predicted conf.low conf.high lifeExp
#> 1 1 link 6.612435 6.542113 6.685787 40
#> 2 2 link 8.183520 8.145944 8.220893 60
#> 3 3 link 9.753512 9.687209 9.820665 80
predictions(mod, dpar = "mu", type = "response",
newdata = datagrid(lifeExp = seq(40, 80, 20)))
#> rowid type predicted conf.low conf.high lifeExp
#> 1 1 response 6.612435 6.542113 6.685787 40
#> 2 2 response 8.183520 8.145944 8.220893 60
#> 3 3 response 9.753512 9.687209 9.820665 80
For predictions, we need to exponentiate the results to scale them back up to dollar amounts. We can do this by post-processing the results (e.g. with dplyr::mutate(predicted = exp(predicted))
), or we can use the transform_post
argument in predictions()
to pass the results to exp()
after getting calculated:
predictions(mod, dpar = "mu",
newdata = datagrid(lifeExp = seq(40, 80, 20)),
transform_post = exp)
#> rowid type predicted conf.low conf.high lifeExp
#> 1 1 response 744.2932 693.7513 800.9406 40
#> 2 2 response 3581.4392 3449.3601 3717.8204 60
#> 3 3 response 17214.5804 16110.2130 18410.2831 80
We can pass transform_post = exp
to plot_cap()
too:
plot_cap(
mod,dpar = "hu",
type = "link",
condition = "lifeExp") +
labs(y = "hu",
title = "Hurdle part (hu)",
subtitle = "Logit-scale predictions") +
plot_cap(
mod,dpar = "hu",
type = "response",
condition = "lifeExp") +
labs(y = "hu",
subtitle = "Percentage point-scale predictions") +
plot_cap(
mod,dpar = "mu",
condition = "lifeExp") +
labs(y = "mu",
title = "Non-hurdle part (mu)",
subtitle = "Log-scale predictions") +
plot_cap(
mod,dpar = "mu",
transform_post = exp,
condition = "lifeExp") +
labs(y = "mu",
subtitle = "Dollar-scale predictions")
For marginal effects, we need to transform the predictions before calculating the instantaneous slopes. We also can’t use the marginaleffects()
function directly—we need to use comparisons()
and compute the numerical derivative ourselves (i.e. predict gdpPercap
at lifeExp
of 40 and 40.001 and calculate the slope between those predictions). We can use the transform_pre
argument to pass the pair of predicted values to exp()
before calculating the slopes:
# step size of the numerical derivative
0.001
eps <-
comparisons(
mod,dpar = "mu",
variables = list(lifeExp = eps),
newdata = datagrid(lifeExp = seq(40, 80, 20)),
# rescale the elements of the slope
# (exp(40.001) - exp(40)) / exp(0.001)
transform_pre = function(hi, lo) ((exp(hi) - exp(lo)) / exp(eps)) / eps
)#> rowid type term contrast comparison conf.low conf.high predicted predicted_hi predicted_lo
#> 1 1 response lifeExp custom 58.39448 55.84743 61.02206 6.612435 6.612474 6.612396
#> 2 2 response lifeExp custom 280.89410 266.57621 295.50894 8.183520 8.183559 8.183481
#> 3 3 response lifeExp custom 1349.40503 1222.58608 1490.38119 9.753512 9.753551 9.753473
#> lifeExp eps
#> 1 40 0.0059004
#> 2 60 0.0059004
#> 3 80 0.0059004
We can visually confirm that these are the instantaneous slopes at each of these levels of life expectancy:
predictions(
predictions_data <-
mod,newdata = datagrid(lifeExp = seq(30, 80, 1)),
dpar = "mu",
transform_post = exp) |>
select(lifeExp, predicted)
comparisons(
slopes_data <-
mod,dpar = "mu",
variables = list(lifeExp = eps),
newdata = datagrid(lifeExp = seq(40, 80, 20)),
transform_pre = function(hi, lo) ((exp(hi) - exp(lo)) / exp(eps)) / eps) %>%
select(lifeExp, comparison) %>%
left_join(predictions_data, by = "lifeExp") %>%
# Point-slope formula: (y - y1) = m(x - x1)
mutate(intercept = comparison * (-lifeExp) + predicted)
ggplot(predictions_data, aes(x = lifeExp, y = predicted)) +
geom_line(size = 1) +
geom_abline(data = slopes_data, aes(slope = comparison, intercept = intercept),
size = 0.5, color = "red") +
geom_point(data = slopes_data) +
geom_label(data = slopes_data, aes(label = paste0("Slope: ", round(comparison, 1))),
nudge_x = -1, hjust = 1) +
theme_minimal()