required <- c("survey", "huxtable", "sandwich", "cowplot") if (!all(sapply(required, requireNamespace, quietly = TRUE))) knitr::opts_chunk$set(eval = FALSE) knitr::opts_chunk$set(message = F, warning = F, fig.width = 6, fig.height = 5) library(jtools) library(interactions)
Understanding an interaction effect in a linear regression model is usually
difficult when using just the basic output tables and looking at the
interactions package provides several functions that can
help analysts probe more deeply.
Categorical by categorical interactions: All the tools described here
require at least one variable to be continuous. A separate vignette describes
cat_plot, which handles the plotting of interactions in which all the focal
predictors are categorical variables.
First, we use example data from
state.x77 that is built into R. Let's look
at the interaction model output with
summ as a starting point.
library(jtools) # for summ() states <- as.data.frame(state.x77) fiti <- lm(Income ~ Illiteracy * Murder + `HS Grad`, data = states) summ(fiti)
Our interaction term is significant, suggesting some more probing is warranted to see what's going on. It's worth recalling that you shouldn't focus too much on the main effects of terms included in the interaction since they are conditional on the other variable(s) in the interaction being held constant at 0.
A versatile and sometimes the most interpretable method for
understanding interaction effects is via plotting.
interact_plot as a relatively pain-free method to get good-looking plots of
ggplot2 on the backend.
interact_plot(fiti, pred = Illiteracy, modx = Murder)
Keep in mind that the default behavior of
interact_plot is to mean-center
all continuous variables not involved in the interaction so that the predicted
values are more easily interpreted. You can disable that by adding
centered = "none". You can choose specific variables by providing their names
in a vector to the
By default, with a continuous moderator you get three lines --- 1 standard
deviation above and below the mean and the mean itself. If you specify
modx.values = "plus-minus", the mean of the moderator is not plotted, just the
two +/- SD lines. You may also choose
"terciles" to split the data into
three equal-sized groups --- representing the upper, middle, and lower thirds
of the distribution of the moderator --- and get the line that represents the
median of the moderator within each of those groups.
If your moderator is a factor, each level will be plotted and you
modx.values = NULL, the default.
fitiris <- lm(Petal.Length ~ Petal.Width * Species, data = iris) interact_plot(fitiris, pred = Petal.Width, modx = Species)
If you want, you can specify a subset of a factor's levels via the
interact_plot(fitiris, pred = Petal.Width, modx = Species, modx.values = c("versicolor", "virginica"))
If you want to see the individual data points plotted to better understand how
the fitted lines related to the observed data, you can use the
plot.points = TRUE argument.
interact_plot(fiti, pred = Illiteracy, modx = Murder, plot.points = TRUE)
For continuous moderators, as you can see, the observed data points are shaded depending on the level of the moderator variable.
It can be very enlightening, too, for categorical moderators.
interact_plot(fitiris, pred = Petal.Width, modx = Species, plot.points = TRUE)
Where many points are slightly overlapping as they do here, it can be useful
to apply a random "jitter" to move them slightly to stop the overlap. Use
jitter argument to do this. If you provide a single number it will
apply a jitter of proportional magnitude both sideways and up and down. If
you provide a vector length 2, then the first is assumed to refer to the
left/right jitter and the second to the up/down jitter.
If you would like to better differentiate the points with factor moderators,
you can use
point.shape = TRUE to give a different shape to each point.
This can be especially helpful for black and white publications.
interact_plot(fitiris, pred = Petal.Width, modx = Species, plot.points = TRUE, jitter = 0.1, point.shape = TRUE)
If your original data are weighted, then the points will be sized based on the weight. For the purposes of our example, we'll weight the same model we've been using with the population of each state.
fiti <- lm(Income ~ Illiteracy * Murder, data = states, weights = Population) interact_plot(fiti, pred = Illiteracy, modx = Murder, plot.points = TRUE)
For those working with weighted data, it can be hard to use a scatterplot to explore the data unless there is some way to account for the weights. Using size is a nice middle ground.
In more complex regressions, plotting the observed data can sometimes be relatively uninformative because the points seem to be all over the place.
For an example, let's take a look at this model. I am using the
ggplot2 and predicting the city miles per gallon (
based on several variables, including model year, type of car, fuel type,
drive type, and an interaction between engine displacement (
number of cylinders in the engine (
cyl). Let's take a look at the
library(ggplot2) data(cars) fitc <- lm(cty ~ year + cyl * displ + class + fl + drv, data = mpg) summ(fitc)
Okay, we are explaining a lot of variance here but there is quite a bit going on. Let's plot the interaction with the observed data.
interact_plot(fitc, pred = displ, modx = cyl, plot.points = TRUE, modx.values = c(4, 5, 6, 8))
Hmm, doesn't that look...bad? You can kind of see the pattern of the
interaction, but the predicted lines don't seem to match the data very well.
But I included a lot of variables besides
displ in this model and
they may be accounting for some of this variation. This is what
partial residual plots are designed to help with. You can learn more about
the technique and theory in Fox and Weisberg (2018) and another place to
generate partial residual plots is in Fox's
Using the argument
partial.residuals = TRUE, what is plotted instead is the
observed data with the effects of all the control variables accounted for.
In other words, the value
cty for the observed data is based only on the
cyl, and the model error. Let's take a look.
interact_plot(fitc, pred = displ, modx = cyl, partial.residuals = TRUE, modx.values = c(4, 5, 6, 8))
Now we can understand how the observed data and the model relate to each other much better. One insight is how the model really underestimates the values at the low end of displacement and cylinders. You can also see how much the cylinders and displacement seem to be correlated each other, which makes it difficult to say how much we can learn from this kind of model.
Another way to get a sense of the precision of the estimates is by plotting
confidence bands. To get started, just set
interval = TRUE. To decide how
wide the confidence interval should be, express the percentile as a number,
int.width = 0.8 corresponds to an 80% interval.
interact_plot(fiti, pred = Illiteracy, modx = Murder, interval = TRUE, int.width = 0.8)
You can also use the
robust argument to plot confidence intervals based on
robust standard error calculations.
A basic assumption of linear regression is that the relationship between the predictors and response variable is linear. When you have an interaction effect, you add the assumption that relationship between your predictor and response is linear regardless of the level of the moderator.
To show a striking example of how this can work, we'll generate two simple datasets to replicate Hainmueller et al. (2017).
The first has a focal predictor $X$ that is normally distributed with mean 3 and standard deviation 1. It then has a dichotomous moderator $W$ that is Bernoulli distributed with mean probability 0.5. We also generate a normally distributed error term with standard deviation 4.
set.seed(99) x <- rnorm(n = 200, mean = 3, sd = 1) err <- rnorm(n = 200, mean = 0, sd = 4) w <- rbinom(n = 200, size = 1, prob = 0.5) y_1 <- 5 - 4*x - 9*w + 3*w*x + err
We fit a linear regression model with an interaction between x and w.
model_1 <- lm(y_1 ~ x * w) summ(model_1)
In the following plot, we use
linearity.check = TRUE argument to split the
data by the level of the moderator $W$ and plot predicted lines (black) and
a loess line (red) within each group. The predicted lines come from the full
data set. The loess line looks only at the subset of data in each pane and
will be curved if the relationship is nonlinear. What we're looking for
is whether the red loess line is straight like the predicted line.
interact_plot(model_1, pred = x, modx = w, linearity.check = TRUE, plot.points = TRUE)
In this case, it is. That means the assumption is satisfied.
Now we generate similar data, but this time the linearity assumption will be
violated. $X_2$ will now be uniformly distributed across the range of -3 to 3.
The true relationship between
y_2 and $X_2$ (
x_2) is non-linear.
x_2 <- runif(n = 200, min = -3, max = 3) y_2 <- 2.5 - x_2^2 - 5*w + 2*w*(x_2^2) + err data_2 <- as.data.frame(cbind(x_2, y_2, w)) model_2 <- lm(y_2 ~ x_2 * w, data = data_2) summ(model_2)
The regression output would lead us to believe there is no interaction.
Let's do the linearity check:
interact_plot(model_2, pred = x_2, modx = w, linearity.check = TRUE, plot.points = TRUE)
This is a striking example of the assumption being violated. At both levels of $W$, the relationship between $X_2$ and the response is non-linear. There really is an interaction, but it's a non-linear one.
You could try to fit this true model with the polynomial term like this:
model_3 <- lm(y_2 ~ poly(x_2, 2) * w, data = data_2) summ(model_3)
interact_plot can plot these kinds of effects, too. Just provide the
untransformed predictor's name (in this case,
x_2) and also include the
data in the
data argument. If you don't include the data, the function will
try to find the data you used but it will warn you about it and it could cause
problems under some circumstances.
interact_plot(model_3, pred = x_2, modx = w, data = data_2)
Look familiar? Let's do the linearity.check, which in this case is more like a non-linearity check:
interact_plot(model_3, pred = x_2, modx = w, data = data_2, linearity.check = TRUE, plot.points = TRUE)
The red loess line almost perfectly overlaps the black predicted line, indicating the assumption is satisfied. As a note of warning, in practice the lines will rarely overlap so neatly and you will have to make more difficult judgment calls.
There are a number of other options not mentioned, many relating to the appearance.
For instance, you can manually specify the axis labels, add a main title, choose a color scheme, and so on.
interact_plot(fiti, pred = Illiteracy, modx = Murder, x.label = "Custom X Label", y.label = "Custom Y Label", main.title = "Sample Plot", legend.main = "Custom Legend Title", colors = "seagreen")
Because the function uses
ggplot2, it can be modified and extended like any
ggplot2 object. For example, using the
theme_apa function from
interact_plot(fitiris, pred = Petal.Width, modx = Species) + theme_apa()
Simple slopes analysis gives researchers a way to express the interaction effect in terms that are easy to understand to those who know how to interpret direct effects in regression models. This method is designed for continuous variable by continuous variable interactions, but can work when the moderator is binary.
In simple slopes analysis, researchers are interested in the conditional slope of the focal predictor; that is, what is the slope of the predictor when the moderator is held at some particular value? The regression output we get when including the interaction term tells us what the slope is when the moderator is held at zero, which is often not a practically/theoretically meaningful value. To better understand the nature of the interaction, simple slopes analysis allows the researcher to specify meaningful values at which to hold the moderator value.
While the computation behind doing so isn't exactly rocket science, it is
inconvenient and prone to error. The
sim_slopes function from
accepts a regression model (with an interaction term) as an input and
automates the simple slopes procedure. The function will, by default, do the
In its most basic use case,
sim_slopes needs three arguments: a regression
model, the name of the focal predictor as the argument for
pred =, and the
name of the moderator as the argument for
modx =. Let's go through an example.
Now let's do the most basic simple slopes analysis:
sim_slopes(fiti, pred = Illiteracy, modx = Murder, johnson_neyman = FALSE)
So what we see in this example is that when the value of
Murder is high, the
Illiteracy is negative and significantly different from zero. The
Murder is high is in the opposite direction from
its coefficient estimate for the first version of the model fit with
this result makes sense considering the interaction coefficient was negative;
it means that as one of the variables goes up, the other goes down. Now we
know the effect of
Illiteracy only exists when
Murder is high.
You may also choose the values of the moderator yourself with the
modx.values = argument.
sim_slopes(fiti, pred = Illiteracy, modx = Murder, modx.values = c(0, 5, 10), johnson_neyman = FALSE)
Bear in mind that these estimates are managed by refitting the models. If the
model took a long time to fit the first time, expect a long run time for
Similar to what this package's
plot_summs functions offer,
you can save your
sim_slopes output to an object and call
plot on that
ss <- sim_slopes(fiti, pred = Illiteracy, modx = Murder, modx.values = c(0, 5, 10)) plot(ss)
You can also use the
huxtable package to get a publication-style table
ss <- sim_slopes(fiti, pred = Illiteracy, modx = Murder, modx.values = c(0, 5, 10)) library(huxtable) as_huxtable(ss)
Did you notice how I was adding the argument
johnson_neyman = FALSE above?
That's because by default,
sim_slopes will also calculate what is called the
Johnson-Neyman interval. This tells you all the values of the moderator for
which the slope of the predictor will be statistically significant. Depending
on the specific analysis, it may be that all values of the moderator
outside of the interval will have a significant slope for the predictor.
Other times, it will only be values inside the interval---you will have to
look at the output to see.
It can take a moment to interpret this correctly if you aren't familiar with the Johnson-Neyman technique. But if you read the output carefully and take it literally, you'll get the hang of it.
sim_slopes(fiti, pred = Illiteracy, modx = Murder, johnson_neyman = TRUE)
In the example above, we can see that the Johnson-Neyman interval and the simple slopes analysis agree---they always will. The benefit of the J-N interval is it will tell you exactly where the predictor's slope becomes significant/insignificant at a specified alpha level.
You can also call the
directly if you want to do something like tweak the alpha level. The
johnson_neyman function will also create a plot by default --- you can get
them by setting
jnplot = TRUE with
johnson_neyman(fiti, pred = Illiteracy, modx = Murder, alpha = .05)
A note on Johnson-Neyman plots: Once again, it is easy to misinterpret the meaning. Notice that the y-axis is the conditional slope of the predictor. The plot shows you where the conditional slope differs significantly from zero. In the plot above, we see that from the point Murder (the moderator) = 9.12 and greater, the slope of Illiteracy (the predictor) is significantly different from zero and in this case negative. The lower bound for this interval (about -32) is so far outside the observed data that it is not plotted. If you could have -32 as a value for Murder rate, though, that would be the other threshold before which the slope of Illiteracy would become positive.
The purpose of reminding you both within the plot and the printed output of the range of observed data is to help you put the results in context; in this case, the only justifiable interpretation is that Illiteracy has no effect on the outcome variable except when Murder is higher than 8.16. You wouldn't interpret the lower boundary because your dataset doesn't contain any values near it.
A recent publication (Esarey & Sumner, 2017) explored ways to calculate the Johnson-Neyman interval that properly manages the Type I and II error rates. Others have noted that the alpha level implied by the Johnson-Neyman interval won't be quite right (e.g., Bauer & Curran, 2005), but there hasn't been any general solution that has gotten wide acceptance in the research literature just yet.
The basic problem is that the Johnson-Neyman interval is essentially
doing a bunch of comparisons across all the values of the moderator,
each one inflating the Type I error rate. The issue isn't so much that
you can't possibly address it, but many solutions are far too conservative and
others aren't broadly applicable. Esarey and Sumner (2017), among other
contributions, suggested an adjustment that seems to do a good job of
balancing the desire to be a conservative test without missing a lot of
true effects in the process. I won't go into the details here. The
johnson_neyman is based on code adapted from Esarey and
interactionTest package, but any errors should be assumed to be
interactions, not them.
To use this feature, simply set
control.fdr = TRUE in the call to
sim_slopes(fiti, pred = Illiteracy, modx = Murder, johnson_neyman = TRUE, control.fdr = TRUE)
In this case, you can see that the interval is just a little bit wider. The output reports the adjusted test statistic, which is 2.33, not much different than the (approximately) 2 that would be used otherwise. In other cases it may be quite a bit larger.
Sometimes it is informative to know the conditional intercepts in addition to the slopes. It might be interesting to you that individuals low on the moderator have a positive slope and individuals high on it don't, but that doesn't mean that individuals low on the moderator will have higher values of the dependent variable. You would only know that if you know the conditional intercept. Plotting is an easy way to notice this, but you can do it with the console output as well.
You can print the conditional intercepts with the
cond.int = TRUE argument.
sim_slopes(fiti, pred = Illiteracy, modx = Murder, cond.int = TRUE)
This example shows you that while the slope associated with
Murder is high, the conditional intercept is also high when
Murder is high. That tells you that increases in
Murder observations will tend towards being equal on
observations with lower values of
Certain models require heteroskedasticity-robust standard errors. To be
consistent with the reporting of heteroskedasticity-robust standard errors
sim_slopes will do the same with the use of the
robust option so you can consistently report standard errors across
sim_slopes(fiti, pred = Illiteracy, modx = Murder, robust = "HC3")
These data are a relatively rare case in which the robust standard errors are
even more efficient than typical OLS standard errors. Note that you must have
sandwich package installed to use this feature.
By default, all non-focal variables are mean-centered. You
may also request that no variables be centered with
centered = "none". You
may also request specific variables to center by providing a vector of quoted
variable names --- no others will be centered in that case.
Note that the moderator is centered around the specified values. Factor variables are ignored in the centering process and just held at their observed values.
You won't have to use these functions long before you may find yourself using
both of them for each model you want to explore. To streamline the process,
this package offers
probe_interaction() as a convenience function that calls
interact_plot(), taking advantage of their
library(survey) data(api) dstrat <- svydesign(id = ~1, strata = ~stype, weights = ~pw, data = apistrat, fpc = ~fpc) regmodel <- svyglm(api00 ~ avg.ed * growth, design = dstrat) probe_interaction(regmodel, pred = growth, modx = avg.ed, cond.int = TRUE, interval = TRUE, jnplot = TRUE)
Note in the above example that you can provide arguments that only apply to one
function and they will be used appropriately. On the other hand, you cannot
apply their overlapping functions selectively. That is, you can't have one
centered = "all" and the other
centered = "none". If you want that
level of control, just call each function separately.
It returns an object with each of the two functions' return objects:
out <- probe_interaction(regmodel, pred = growth, modx = avg.ed, cond.int = TRUE, interval = TRUE, jnplot = TRUE) names(out)
Also, the above example comes from the survey package as a means to show that,
yes, these tools can be used with
svyglm objects. It is also tested with
rq models, though you should always do your homework
to decide whether these procedures are appropriate in those cases.
If 2-way interactions can be hard to grasp by looking at regular regression output, then 3-way interactions are outright inscrutable. The aforementioned functions also support 3-way interactions, however. Plotting these effects is particularly helpful.
Note that Johnson-Neyman intervals are still provided, but only insofar as you get the intervals for chosen levels of the second moderator. This does go against some of the distinctiveness of the J-N technique, which for 2-way interactions is a way to avoid having to choose points of the moderator to check whether the predictor has a significant slope.
fita3 <- lm(rating ~ privileges * critical * learning, data = attitude) probe_interaction(fita3, pred = critical, modx = learning, mod2 = privileges, alpha = .1)
You can change the labels for each plot via the
And don't forget that you can use
theme_apa to format for publications or
just to make more economical use of space.
mtcars$cyl <- factor(mtcars$cyl, labels = c("4 cylinder", "6 cylinder", "8 cylinder")) fitc3 <- lm(mpg ~ hp * wt * cyl, data = mtcars) interact_plot(fitc3, pred = hp, modx = wt, mod2 = cyl) + theme_apa(legend.pos = "bottomright")
You can get Johnson-Neyman plots for 3-way interactions as well, but keep in
mind what I mentioned earlier in this section about the J-N technique for 3-way
interactions. You will also need the
cowplot package, which is used on the
backend to mush together the separate J-N plots.
regmodel3 <- svyglm(api00 ~ avg.ed * growth * enroll, design = dstrat) sim_slopes(regmodel3, pred = growth, modx = avg.ed, mod2 = enroll, jnplot = TRUE)
Notice that at one of the three values of the second moderator, there were no Johnson-Neyman interval values so it wasn't plotted. The more levels of the second moderator you want to plot, the more likely that the resulting plot will be unwieldy and hard to read. You can resize your window to help, though.
You can also use the
as_huxtable methods with 3-way
ss3 <- sim_slopes(regmodel3, pred = growth, modx = avg.ed, mod2 = enroll) plot(ss3)
interact_plot is designed to be as general as possible and has been tested
rq models. When dealing with generalized
linear models, it can be immensely useful to get a look at the predicted
values on their response scale (e.g., the probability scale for logit models).
To give an example of how such a plot might look, I'll generate some example data.
set.seed(5) x <- rnorm(100) m <- rnorm(100) prob <- binomial(link = "logit")$linkinv(.25 + .3*x + .3*m + -.5*(x*m) + rnorm(100)) y <- rep(0, 100) y[prob >= .5] <- 1 logit_fit <- glm(y ~ x * m, family = binomial)
Here's some summary output, for reference:
Now let's plot our logit model's interaction:
interact_plot(logit_fit, pred = x, modx = m)
A beautiful transverse interaction with the familiar logistic curve.
Bauer, D. J., & Curran, P. J. (2005). Probing interactions in fixed and multilevel regression: Inferential and graphical techniques. Multivariate Behavioral Research, 40, 373–400.
Esarey, J., & Sumner, J. L. (2017). Marginal effects in interaction models: Determining and controlling the false positive rate. Comparative Political Studies, 1–33. https://doi.org/10.1177/0010414017730080
Fox, J., & Weisberg, S. (2018). Visualizing fit and lack of fit in complex regression models with predictor effect plots and partial residuals. Journal of Statistical Software, 87, 1–27. https://doi.org/10.18637/jss.v087.i09
Hainmueller, J., Mummolo, J., & Xu, Y. (2017). How much should we trust estimates from multiplicative interaction models? Simple tools to improve empirical practice. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2739221
Johnson, P. O., & Fay, L. C. (1950). The Johnson-Neyman technique, its theory and application. Psychometrika, 15, 349–367.
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