Producing “flexible” marginal effect estimates with multiplicative interaction models

Reference: How Much Should We Trust Estimates from Multiplicative Interaction Models? Simple Tools to Improve Empirical Practice. Political Analysis, Vol. 27, Iss. 2, April 2019, pp. 163–192.

Authors: Jens Hainmueller, Jonathan Mummolo, Yiqing Xu, and Ziyi Liu

Maintainer: Yiqing Xu []

Date: April 23, 2020

Version: 1.1.3 (CRAN version)

Changes in Version 1.1.3:

  1. Add a function inter.test to test the difference(s) in treatment effects at two or three specific values of the moderator post estimation.

Changes in Version 1.1.0:

  1. Accommodate discrete treatments variables (>2 treatment arms) interacting with continuous moderators; users can decide whether to draw the multiple marginal effects in one plot or in several by specifying the pool option.

  2. Incorporate adaptive bandwidth search in cross validation; optimize cross-validation when fixed effects are present.

  3. Add an umbrella function interflex (S3 method) to incorporate the functionalities of both inter.binning and inter.kernel while inter.binning and inter.kernel are deprecated but still being supported (for usage, see old User’s Guide)

  4. Add a predict function, which allows users estimate and plot the expected value of Y given the values of the treatment, moderators, and controls.

  5. Allow users to test the difference in treatment effects at three different values in the moderate using the diff.values option

R source files can be found on Github. R code appearing in this demonstration can be downloaded from here.

Comments and suggestions will be greatly appreciated!

Contents

  1. Installation
  2. Simulated examples
  3. Raw plots (and GAM plots)
  4. Binning estimates
  5. Kernel estimates
  6. Fixed effects
  7. Multiple (>2) treatment arms
  8. Predicted outcomes
  9. Testing differences in treatment effects

Installation

You can install the interflex package from CRAN:

install.packages('interflex', type = "source", repos = 'http://cran.us.r-project.org')

Or you can install the up-to-date development version from Github:

install.packages('devtools', repos = 'http://cran.us.r-project.org') # if not already installed
devtools::install_github('xuyiqing/interflex')

interflex depends on the following packages, which will be installed AUTOMATICALLY when interflex is being installed; if not, please install them manually:

require(Rcpp) # for processing C++ code
require(mgcv) # for GAM
require(sandwich) # for calculating standard errors
require(pcse) # in case panel-corrected standard errors will be used
require(foreach)  # for parallel computing in kernel estimation
require(doParallel) # for parallel computing in kernel estimation
require(lmtest) # for wald test
require(lfe) # for fixed effects estimations
require(Lmoments) # for L-kurtosis measure
require(ggplot2)  # for plotting
require(plotrix) # for plotting
require(grid) # for plotting
require(gridExtra) # for plotting
require(ggplotify) # for plotting
require(RColorBrewer) # for plotting
require(grDevices) # for plotting
require(caret) # for cross-validation
require(gtable) # for plotting

Mac users who encounter “-lgfortran” or “-lquadmath” error during installation, please check out the solution here. Typing the following two lines of code in your Terminal should solve this problem.

curl -OL http://r.research.att.com/libs/gfortran-4.8.2-darwin13.tar.bz2
sudo tar fvxz gfortran-4.8.2-darwin13.tar.bz2 -C /

Mac users who encounter clang: error: unsupported option ‘-fopenmp’, please consider (1) updating your R and/or (2) installing new R macro tools from Github.

Simulated Examples

Now let’s load the package as well as four simulated toy datasets:

library(interflex)
data(interflex)
ls()
## [1] "s1" "s2" "s3" "s4" "s5"

s1 is a case of a dichotomous treatment indicator with linear marginal effects; s2 is a case of a continuous treatment indicator with linear marginal effects; s3 is a case of a dichotomous treatment indicator with nonlinear marginal effects; s4 is a case of a dichotomous treatment indicator, nonlinear marginal effects, with additive two-way fixed effects; and s5 is a case of a discrete treatment indicator, nonlinear marginal effects, with additive two-way fixed effects. They are generated using the following code:

set.seed(1234)
n<-200
d1<-sample(c(0,1),n,replace=TRUE) # dichotomous treatment
d2<-rnorm(n,3,1) # continuous treatment
x<-rnorm(n,3,1) # moderator
z<-rnorm(n,3,1) # covariate
e<-rnorm(n,0,1) # error term

## linear marginal effect
y1<-5 - 4 * x - 9 * d1 + 3 * x * d1 + 1 * z + 2 * e
y2<-5 - 4 * x - 9 * d2 + 3 * x * d2 + 1 * z + 2 * e
s1<-cbind.data.frame(Y = y1, D = d1, X = x, Z1 = z)
s2<-cbind.data.frame(Y = y2, D = d2, X = x, Z1 = z)

## quadratic marginal effect
x3 <- runif(n, -3,3) # uniformly distributed moderator
y3 <- d1*(x3^2-2.5) + (1-d1)*(-1*x3^2+2.5) + 1 * z + 2 * e
s3 <- cbind.data.frame(D=d1, X=x3, Y=y3, Z1 = z)

## adding two-way fixed effects
n  <- 500
d4 <-sample(c(0,1),n,replace=TRUE) # dichotomous treatment
x4 <- runif(n, -3,3) # uniformly distributed moderator
z4 <- rnorm(n, 3,1) # covariate
alpha <- 20 * rep(rnorm(n/10), each = 10)
xi <- rep(rnorm(10), n/10)
y4 <- d4*(x4^2-2.5) + (1-d4)*(-1*x4^2+2.5) + 1 * z4 + alpha + xi + 2 * rnorm(n,0,1)
s4 <- cbind.data.frame(D=d4, X=x4, Y=y4, Z1 = z4, unit = rep(1:(n/10), each = 10), year = rep(1:10, (n/10)))

## Multiple treatment arms
n <- 600
# treatment 1
d1 <- sample(c('A','B','C'),n,replace=T)
# moderator
x <- runif(n,min=-3, max = 3)
# covriates
z1 <- rnorm(n,0,3)
z2 <- rnorm(n,0,2)
# error
e <- rnorm(n,0,1)
y1 <- rep(NA,n)
y1[which(d1=='A')] <- -x[which(d1=='A')]
y1[which(d1=='B')] <- (1+x)[which(d1=='B')]
y1[which(d1=='C')] <- (4-x*x-x)[which(d1=='C')]
y1 <- y1 + e + z1 + z2
s5 <- cbind.data.frame(D=d1, X=x, Y=y1, Z1 = z1,Z2 = z2)

The implied population marginal effects for the DGPs of s1 and s2 are \[ME(X) = 3X - 9;\] the implied population marginal effects for the DGPs of s3 and s4 are \[ME(X) = 2X^{2} - 5.\] For s5, if we set treatment “A” as our base category, the implied population marginal effects for group “B” and group “C” are, respectively, \[ME(X) = 2X + 1\quad\text{and}\quad ME(X) = -X^{2} + 4.\]

The interflex package ships the following functions: interflex, inter.raw, inter.gam, plot. The functionalities of inter.binning and inter.kernel covered by interflex, but they are still supported for backward compatibility.

Raw Plots

The first step of the diagnostics is to plot raw data. We supply the function inter.raw with the variable names of the outcome Y, the treatment D, and the moderator X. You can also supply labels for these variables. If you supply a variable name to the weights option, the linear and LOESS fits will be adjusted based on the weights. Note that the correlations between covariates Z and Y are NOT partialed out. We use main to add a title to the plot and cex.main to adjust its size.

inter.raw(Y = "Y", D = "D", X = "X", data = s1, weights = NULL, Ylabel = "Outcome", Dlabel = "Treatment", Xlabel="Moderator", main = "Raw Plot", cex.main = 1.2)
## `geom_smooth()` using formula 'y ~ x'

A black-white theme is applied when we set theme.bw = TRUE. show.grid = FALSE can be used to remove grid in the plot. Both options are allowed in interflex and plot.

inter.raw(Y = "Y", D = "D", X = "X", data = s2, Ylabel = "Outcome", Dlabel = "Treatment", Xlabel="Moderator", theme.bw = TRUE, show.grid = FALSE)
## `geom_smooth()` using formula 'y ~ x'

inter.raw(Y = "Y", D = "D", X = "X", data = s3, Ylabel = "Outcome", Dlabel = "Treatment", Xlabel="Moderator")
## `geom_smooth()` using formula 'y ~ x'

For the continuous treatment case (e.g. s2), we can also draw a Generalized Additive Model (GAM) plot. You can supply a set of covariates to be controlled for by supplying Z, which takes a vector of covariate names (strings).

inter.gam(Y="Y", D="D", X="X", Z=c("Z1"), data=s2)
## Use fully moderated model.

The Binning Estimator

The second diagnostic tool is the binning plot. The nbins option sets the number of bins. The default number of bins is 3, and equal-sized bins are created based on the distribution of the moderator. There are four options for the choice of the variance estimator: vartype="homoscedastic", "robust", "cluster", "pcse" and "bootstrap". The default option is "robust".

Note that interflex will also automatically report a set of statistics when estimator = "binning", including: (1) the binning estimates and their standard errors and 95% confidence intervals, (2) the percentage of observations within each bin, (3) the variance of the treatment within each bin, (4) the L-kurtosis of the moderator, (5) whether the three binning estimates are in the correct order (i.e. monotonically increasing or decreasing), (6) pair-wise t-statistics for the binning estimates (when there are 2 or 3 bins), and (7) a Wald test to formally test if we can reject the linear multiplicative interaction model by comparing it with a more flexible model of multiple bins. Note that we can achieve the same goal by using inter.binning (as in v.1.0.9 or ealier versions).

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s1, estimator = 'binning', vartype = "robust", main = "Marginal Effects", ylim = c(-15, 15))
## Baseline group not specified; choose treat = 0 as the baseline group. 
## Use a fully moderated model.
out$graph

We see that the Wald test cannot reject the NULL hypothesis that the linear interaction model and the three-bin model are statistically equivalent (p = 0.509). If we only want to conduct the linear estimator, we can set estimator = "linear":

out2 <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s1, estimator = 'linear', vartype = "robust", main = "Marginal Effects", ylim = c(-15, 15))
## Baseline group not specified; choose treat = 0 as the baseline group. 
## Use a fully moderated model.
## Use a linear interaction model
out2$graph

plot allows users to adjust graphic options without re-estimating the model. The first entry must be a interflex object. Note that we use bin.labs = FALSE to hide the label on the top of each bin and Xdistr = "none" to remove the distribution of the moderator (not recommended). We use cex.axis and cex.lab to adjust the font sizes of axis numbers and labels.

plot(out, xlab = "Moderate is X", Xdistr = "none", bin.labs = FALSE, cex.axis = 0.8, cex.lab = 0.8)

Next, we use Xunif = TRUE to transform the moderator into a uniformly distributed random variable (based on the rank order in values of the orginal moderator) before estimating the marginal effects. nbins = 4 sets the number of bins to 4.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s1, estimator = 'binning', nbins = 4, theme.bw = TRUE, Xunif = TRUE)
## Baseline group not specified; choose treat = 0 as the baseline group. 
## Use a fully moderated model.
out$graph

The binning estimates for the continuous case are shown below. We now present the distribution of the moderator with a density plot using option Xdist = "density" – the default option is "hist" or "histogram". We turn off the bin labels using bin.labs = FALSE.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s2, estimator = 'binning', Xdistr = "density", bin.labs = FALSE)
## Use a fully moderated model.
out$graph

Note that you can customize the cutoff values for the bins, for example, set cutoffs = c(1, 2, 4, 5) to create five bins: [minX, 1], (1, 2], (2,4], (4, 5] and (5,maxX] (supplying N numbers will create N+1 bins). Note that the cutoffs option will override the nbins option if they are incompatible.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s2, estimator = 'binning', cutoffs = c(1,2,4,5))
## Use a fully moderated model.
out$graph

The binning estimates for the dichotomous, nonlinear case (i.e. s3) are shown below. A linear interaction model clearly gives misleading marginal effects estimates. The marginal effects plot is stored in out$graph while the estimates and standard errors are stored in out$est.lin (linear) and out$est.bin (binning). The tests results are stored in out$tests (binning).

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s3, estimator = 'binning')
## Baseline group not specified; choose treat = 0 as the baseline group. 
## Use a fully moderated model.
out$tests
## $treat.type
## [1] "discrete"
## 
## $bin.size
## [1] "0.335" "0.330" "0.335"
## 
## $X.Lkurtosis
## [1] "-0.002"
## 
## $correctOrder
## $correctOrder$`1`
## [1] FALSE
## 
## 
## $p.twosided
## $p.twosided$`1`
##  p.1v2  p.2v3  p.1v3 
## 0.0000 0.0000 0.1693 
## 
## 
## $fmt
## [1] "%#.3f"
## 
## $p.wald
## [1] "0.000"

This time the NULL hypothesis that the linear interaction model and the three-bin model are statistically equivalent is safely rejected (p.wald = 0.00).

Kernel Estimates

With the kernel method, a bandwidth is first selected via 10-fold least-squares cross-validation. There are three options for the choice of the cross-validation: CV.method="simple", "cluster", and "stratify". The default setting is "simple".

The standard errors are produced by a non-parametric bootstrap (you can adjust the number of bootstrap iterations using the nboots option). The grid option can either be an integer, which specifies the number of bandwidths to be cross-validated, or a vector of candidate bandwidths – the default is grid = 20. You can also specify a clustered group structure by using the cl option, in which case a block bootstrap procedure will be performed.

Starting from v.1.1.0, we introduce adaptive bandwidth selection to interflex. When bw.adaptive = TRUE (default), a locally adaptive bandwidth will be applied. This means that the bandwidth will be smaller in regions where there are more data and larger in regions where there are few data points (based on the moderator). When adaptive bandwidth is being used, bw refers to the bandwidth applied in the region with the highest density of observations.

Even with an optimized algorithm, the bootstrap procedure for large datasets can be slow. We incorporate parallel computing (parallel=TRUE) to speed it up. You can choose the number of cores to be used for parallel computing; otherwise the program will detect the number of logical cores in your computer and use as many cores as possible (warning: this will significantly slow down your computer!). Note that we can also use inter.kernel as in the earlier versions.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s1, estimator = 'kernel', bw.adaptive = TRUE, , nboots = 200, parallel = TRUE, cores = 4)
## Baseline group not specified; choose treat = 0 as the baseline group. 
## Use a fully moderated model.
## Use adaptive bandwidth.
## Parallel computing with 4 cores...
## Cross-validating bandwidth ... 
## #folds = 10
## Bandwidth = 6.592 
## Bootstrapping ...
## 
out$graph

If you specify a bandwidth manually (for example, by setting bw = 1), the cross-validation step will be skipped. The main option controls the title of the plot while the xlim and ylim options control the ranges of the x-axis and y-axis, respectively.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s1, estimator = 'kernel', nboots = 200, bw = 1, main = "The Linear Case", xlim = c(-0.5,6.5), ylim = c(-15, 15))
out$graph

For dataset s3, the kernel method produces non-linear marginal effects estimates that are much closer to the effects implied by the true DGP.

out <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s3, estimator = 'kernel', theme.bw = TRUE)
out$graph

Again, if we only want to change the look of a marginal effect plot, we can use plot to save time. CI = FALSE removes the confidence interval ribbon.

plot(out, main = "Nonlinear Marginal Effects", ylab = "Coefficients", Xdistr = "density", xlim = c(-3,3), ylim = c(-10,12), CI = FALSE, cex.main = 0.8, cex.lab = 0.7, cex.axis = 0.7)

The semi-parametric marginal effect estimates are stored in out$est.

Note that we can use the file option to save a plot to a file in interflex (e.g. by setting file = "myplot.pdf" or file = "myplot.png").

Fixed Effects

We move on to linear fixed effects models. Remember in s4, a large chunk of the variation in the outcome variable is driven by group fixed effects. Below is a scatterplot of the raw data (group index vs. outcome). Red and green dots represent treatment and control units, respectively. We can see that outcomes are highly correlated within a group.

library(ggplot2)
ggplot(s4, aes(x=group, y = Y, colour = as.factor(D))) + geom_point() + guides(colour=FALSE)

When fixed effects are present, it is possible that we cannot observe a clear pattern of marginal effects in the raw plot as before, while binning estimates have wide confidence intervals:

inter.raw(Y = "Y", D = "D", X = "X", data = s4, weights = NULL)

We can use the option Z and FE in inter.raw to residualize the original y values.

inter.raw(Y = "Y", D = "D", X = "X", Z="Z1", FE = c("group", "year"), data = s4, weights = NULL)

Here, we can see if we don’t include FE or Z in interflex, the results are not quite informative.

s4.binning <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s4, estimator = 'binning', FE = NULL, cl = "group")
s4.binning$graph

The binning estimates are much more informative when fixed effects are included, by using the FE option. Note that the number of group indicators can exceed 2. Our algorithm is optimized for a large number of fixed effects or many group indicators. The cl option controls the level at which standard errors are clustered.

s4$wgt <- 1
s4.binning <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s4, estimator = 'binning', FE = c("group","year"), cl = "group", weights = "wgt")
s4.binning$graph

When fixed effects are not taken into account, the kernel estimates are also less precisely estimated. Because the model is incorrectly specified, cross-validated bandwidths also tend to be bigger than optimal.

s4.kernel <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s4, estimator = 'kernel', FE = NULL, cl = "group", weights = "wgt")
s4.kernel$graph

Controlling for fixed effects by using the FE option solves this problem. The estimates are now much closer to the population truth. Note that all uncertainty estimates are produced via bootstrapping. When the cl option is specified, a block bootstrap procedure will be performed.

s4.kernel <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s4, estimator = 'kernel', FE = c("group","year"), cl = "group")
s4.kernel$graph

With large datasets, cross-validation or bootstrapping can take a while. One way to check the result quickly is to shut down the bootstrap procedure (using CI = FALSE). interflex will then present the point estimates only. Another way is to supply a reasonable bandwidth manually by using the bw option such that cross-validation will be skipped.

s4.kernel <- interflex(Y = "Y", D = "D", X = "X", Z = "Z1", data = s4, estimator = 'kernel', bw = 0.62, FE = c("group","year"), cl = "group", CI = FALSE, ylim = c(-9, 15), theme.bw = TRUE)
s4.kernel$graph

Multiple (>2) Treatment Arms

Next, we will show how interflex can be applied when there are more than two condition. First, we plot the raw data:

inter.raw(Y = "Y", D = "D", X = "X", data = s5)

By default, interflex will produce \((n-1)\) marginal effects plots, taking one category as the baseline (if not specified, the base group is selected based on the numeric/character order of the treatment values). Here we also specify vartype = "bootstrap" in order to estimate the confidence intervals.

s5.binning <- interflex(Y = "Y", D = "D", X = "X", Z = c("Z1","Z2"), data = s5, estimator = 'binning',vartype = 'bootstrap')
s5.binning$graph

By setting the option order and subtitles (in either interflex or plot), users can also specify the order of plots they prefer and the subtitles they want:

plot(s5.binning, order=c("C", "B"), subtitles = c("Group C","Group B"))

By setting pool = TRUE (in either interflex or plot), we can combine these plots. Users can also specify the colors they like, :

plot(s5.binning, order=c("C", "B"), subtitles = c("Control Group","Group C","Group B"), pool = TRUE, color = c("Salmon","#7FC97F", "#BEAED4"))

We specify the base group using the option base:

s5.binning2 <- interflex(Y = "Y", D = "D", X = "X",Z = c("Z1", "Z2"), data = s5, estimator = 'binning', base='B', vartype = 'bootstrap')
s5.binning2$graph

For dataset s5, because of the nonlinear interaction effects, the kernel estimator will produce marginal effects estimates that are much closer to the effects implied by the true DGP.

s5.kernel <- interflex(Y = "Y", D = "D", X = "X",Z = c("Z1", "Z2"), data = s5, estimator = 'kernel')
s5.kernel$graph

plot(s5.kernel, pool = TRUE)

One can also plot a sub-plot or sub-plots only by using the to.show option:

plot(s5.kernel, to.show = c("C"))

Predicted Outcomes

Starting from v.1.1.0, interflex allows users to estimate and visualize predicted outcomes given fixed values of the treatment and moderator based on flexible interaction models. All predictions are made by setting covariates equal to their sample means. In order to do so, users need to specify predict = TRUE when applying interflex (the default is FALSE for speedy estimation).

s5.kernel <-interflex(Y = "Y", D = "D", X = "X", Z = c("Z1", "Z2"),data = s5, estimator = 'kernel', predict = TRUE)
predict(s5.kernel)

We can also set pool = TRUE to combine them.

predict(s5.kernel,order = c('A','B','C'),subtitle = c("Group A", "Group B", "Group C"), pool = T, legend.title = "Three Different Groups")

We can also use the linear model or the binning model to estimate and visualize predicted outcomes. For the latter, it is expected to see some “zigzags” at the bin boundaries in a plot:

s5.binning <-interflex(Y = "Y", D = "D", X = "X", Z = c("Z1","Z2"),data = s5, estimator = 'binning',vartype = 'bootstrap' , predict = TRUE, nbins = 4)
predict(s5.binning)

s5.linear <-interflex(Y = "Y", D = "D", X = "X", Z = c("Z1","Z2"), data = s5, estimator = 'linear', vartype = 'robust' , predict = TRUE)
predict(s5.linear)

Obviously, the kernel estimator characterizes the data more better than a linear or binning estimator.

Testing Difference(s) in Treatment Effects

Starting from v.1.1.0, interflex allows users to compare treatment effects at two or three specific values of the moderator using linear or kernel model (but not the binning estimator). In order to do so, users can pass two or three real-numbers into diff.values when applying linear or kernel estimator. If not specified, this programme will by default choose values at the 25,50,75 percentile of the moderator as the diff.values.

s5.kernel <-interflex(Y = "Y", D = "D", X = "X", Z = c("Z1", "Z2"), data = s5, estimator = 'kernel', diff.values = c(-2, 0, 2))
s5.kernel$graph

s5.kernel$t.test.diffs
## $B
##          Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 0 vs -2 4.467    0.278  16.062   0.000         3.832         4.916
## 2 vs 0  3.864    0.249  15.490   0.000         3.445         4.336
## 2 vs -2 8.332    0.256  32.531   0.000         7.830         8.777
## 
## $C
##           Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 0 vs -2  4.100    0.267  15.335   0.000         3.605         4.638
## 2 vs 0  -4.061    0.257 -15.775   0.000        -4.513        -3.601
## 2 vs -2  0.039    0.277   0.139   0.889        -0.444         0.598

interflex now also allows users to compare treatment effects at two or three values of the moderator after a linear or kernel estimator is applied using a new function called inter.test. Users can pass the output of inter.linear, inter.binning or inter.kernel and specify the values of interest of the moderator.

Note that we use a GAM model to approximate the variance-covariance matrix, thus saving time. Because it is an approximation, the results may be slightly different from directly specifying diff.values when applying inter.linear or inter.kernel. We also limit the elements passed in diff.values within the range of the moderator to avoid extrapolation bias.

inter.test(s5.kernel,diff.values=c(-2, 0, 1))
## $B
##          Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 0 vs -2 4.471    0.279  16.034   0.000         3.925         5.018
## 1 vs 0  1.850    0.239   7.741   0.000         1.382         2.318
## 1 vs -2 6.321    0.262  24.118   0.000         5.807         6.835
## 
## $C
##           Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 0 vs -2  4.091    0.265  15.461   0.000         3.573         4.610
## 1 vs 0  -1.056    0.234  -4.518   0.000        -1.514        -0.598
## 1 vs -2  3.035    0.260  11.694   0.000         2.527         3.544

We can also compare treatment effects at different percentiles of the moderator by setting percentile = TRUE:

inter.test(s5.kernel,diff.values=c(0.1, 0.9), percentile=TRUE)
## $B
##             Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 90% vs 10% 9.729    0.286  33.969   0.000         9.168        10.291
## 
## $C
##             Diff Std.Err. z-score p-value CI_lower(95%) CI_upper(95%)
## 90% vs 10% 0.240    0.288   0.833   0.405        -0.324         0.804

Note:

Starting from v.1.1.0, we incorporate the interaction terms between the moderator X and all covariates Z in our models to reduce biases induced by their correlations (Blackwell and Olson 2019). Though not recommended, users can turn off this option by setting full.moderate = FALSE. In 22 published papers that we look into, we find the differences between the original model and the fully moderated model to be small.

Reference: Matthew Blackwell and Michael Olson (2019). “Reducing Model Misspecification and Bias in the Estimation of Interactions.” Mimeo, Harvard University.