---
title: "Exploring interactions with continuous predictors in regression models"
author: "Jacob Long"
date: "`r Sys.Date()`"
output: html_vignette
vignette: >
  %\VignetteIndexEntry{Exploring interactions with continuous predictors in regression models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r echo=FALSE}
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
coefficients. 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.

```{r results = 'knitr::normal_print'}
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.

# Plotting interactions

A versatile and sometimes the most interpretable method for 
understanding interaction effects is via plotting. `interactions` provides 
`interact_plot` as a relatively pain-free method to get good-looking plots of
interactions using `ggplot2` on the backend.

```{r}
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 `centered` argument.

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 
should leave `modx.values = NULL`, the default.

```{r}
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 `modx.values`
argument:

```{r}
interact_plot(fitiris, pred = Petal.Width, modx = Species,
              modx.values = c("versicolor", "virginica"))
```


## Plotting observed data

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.

```{r}
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.

```{r}
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
the `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.

```{r}
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.

```{r}
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.

## Plotting partial residuals

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 `mpg` 
dataset from `ggplot2` and predicting the city miles per gallon (`cty`)
based on several variables, including model year, type of car, fuel type,
drive type, and an interaction between engine displacement (`displ`) and
number of cylinders in the engine (`cyl`). Let's take a look at the 
regression output:

```{r}
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.

```{r}
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 `cyl` and `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 `effects` package.

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 
values of `displ`, `cyl`, and the model error. Let's take a look.

```{r}
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.

## Confidence bands

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, 
e.g., `int.width = 0.8` corresponds to an 80% interval.

```{r}
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.

## Check linearity assumption

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.

```{r}
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.

```{r}
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.

```{r}
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. 

```{r}
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:

```{r}
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:

```{r}
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.

```{r}
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:

```{r}
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.

## Other options

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.

```{r}
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 
other `ggplot2` object. For example, using the `theme_apa` function from 
`jtools`:

```{r}
interact_plot(fitiris, pred = Petal.Width, modx = Species) + theme_apa()
```

# Simple slopes analysis and Johnson-Neyman intervals

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 `interactions` 
accepts a regression model (with an interaction term) as an input and 
automates the simple slopes procedure. The function will, by default, do the 
following:

* Mean-center all non-focal predictors (so that setting them to zero means
setting them to their mean)
* For continuous moderators, it will choose the mean as well as the mean 
plus/minus 1 standard deviation as values at which to find the slope of the
focal predictor.
* For categorical/binary moderators, it will find the slope of the focal
predictor at each level of the moderator.

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:

```{r}
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
slope of `Illiteracy` is negative and significantly different from zero. The 
value for `Illiteracy` when `Murder` is high is in the opposite direction from 
its coefficient estimate for the first version of the model fit with `lm` but 
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.

```{r}
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
`sim_slopes`.

### Visualize the coefficients

Similar to what this package's `plot_coefs`/`plot_summs` functions offer,
you can save your `sim_slopes` output to an object and call `plot` on that
object.

```{r}
ss <- sim_slopes(fiti, pred = Illiteracy, modx = Murder, 
                 modx.values = c(0, 5, 10))
plot(ss)
```

### Tabular output

You can also use the `huxtable` package to get a publication-style table
from your `sim_slopes` output.

```{r}
ss <- sim_slopes(fiti, pred = Illiteracy, modx = Murder,
                 modx.values = c(0, 5, 10))
library(huxtable)
as_huxtable(ss)
```

### Johnson-Neyman intervals

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.

```{r}
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 `johnson_neyman` function 
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 `sim_slopes`.

```{r}
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.

### False discovery rate adjustment

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 
implementation in `johnson_neyman` is based on code adapted from Esarey and
Sumner's `interactionTest` package, but any errors should be assumed to be
from `interactions`, not them.

To use this feature, simply set `control.fdr = TRUE` in the call to 
`johnson_neyman` or `sim_slopes`.

```{r}
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.

## Additional options

### Conditional intercepts

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.

```{r}
sim_slopes(fiti, pred = Illiteracy, modx = Murder, cond.int = TRUE)
```

This example shows you that while the slope associated with `Illiteracy` is 
negative when `Murder` is high, the conditional intercept is also high when 
`Murder` is high. That tells you that increases in `Illiteracy` for 
high-`Murder` observations will tend towards being equal on `Income` to 
observations with lower values of `Murder`.

### Robust standard errors

Certain models require heteroskedasticity-robust standard errors. To be 
consistent with the reporting of heteroskedasticity-robust standard errors 
offered by `summ`, `sim_slopes` will do the same with the use of the
`robust` option so you can consistently report standard errors across
models.

```{r}
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 
the `sandwich` package installed to use this feature.

### Centering variables

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.

# Do simple slopes and plotting with one function

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
both `sim_slopes()` and `interact_plot()`, taking advantage of their 
overlapping syntax.

```{r}
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:

```{r}
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
`glm`, `merMod`, and `rq` models, though you should always do your homework
to decide whether these procedures are appropriate in those cases.

# 3-way interactions

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.

```{r}
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 `mod2.labels` argument.

And don't forget that you can use `theme_apa` to format for publications or
just to make more economical use of space.

```{r}
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.

```{r fig.height = 8}
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 `plot` and `as_huxtable` methods with 3-way `sim_slopes`
objects.

```{r}
ss3 <- sim_slopes(regmodel3, pred = growth, modx = avg.ed, mod2 = enroll)
plot(ss3)
```

```{r}
as_huxtable(ss3)
```


# Generalized linear models, mixed models, et al.

`interact_plot` is designed to be as general as possible and has been tested
with `glm`, `svyglm`, `merMod`, and `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.

```{r}
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:

```{r}
summ(logit_fit)
```

Now let's plot our logit model's interaction:

```{r}
interact_plot(logit_fit, pred = x, modx = m)
```

A beautiful transverse interaction with the familiar logistic curve.

## References

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](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](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](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.