Tag Archives: interactions

Understanding 3-way interactions between continuous and categorical variables, part ii: 2 cons, 1 cat

I posted recently (well… not that recently, now that I remember that time is linear) about how to visualise 3-way interactions between continuous and categorical variables (using 1 continuous and 2 categorical variables), which was a follow-up to my extraordinarily successful post on 3-way interactions between 3 continuous variables (by ‘extraordinarily successful’, I mean some people read it on purpose and not because they were misdirected by poorly-thought google search terms, which is what happens with the majority of my graphic insect-sex posts). I used ‘small multiples‘, and also predicting model fits when holding particular variables at distinct values.


I just had a comment on the recent post, and it got me thinking about combining these approaches:

Screen Shot 2015-06-02 at 17.32.18

There are a number of approaches we can use here, so I’ll run through a couple of examples. First, however, we need to make up some fake data! I don’t know anything about bone / muscle stuff (let’s not delve too far into what a PhD in biology really means), so I’ve taken the liberty of just making up some crap that I thought might vaguely make sense. You can see here that I’ve also pretended we have a weirdly complete and non-overlapping set of data, with one observation of bone for every combination of muscle (continuous predictor), age (continuous covariate), and group (categorical covariate). Note that the libraries you’ll need for this script include {dplyr}, {broom}, and {ggplot2}.

#### Create fake data ####
bone_dat <- data.frame(expand.grid(muscle = seq(50,99),
                                   age = seq(18, 65),
                                   groupA = c(0, 1)))
## Set up our coefficients to make the fake bone data
coef_int <- 250
coef_muscle <- 4.5
coef_age <- -1.3
coef_groupA <- -150
coef_muscle_age <- -0.07
coef_groupA_age <- -0.05
coef_groupA_muscle <- 0.3
coef_groupA_age_muscle <- 0.093
bone_dat <- bone_dat %>% 
  mutate(bone = coef_int +
  (muscle * coef_muscle) +
  (age * coef_age) +
  (groupA * coef_groupA) +
  (muscle * age * coef_muscle_age) +
  (groupA * age * coef_groupA_age) +
  (groupA * muscle * coef_groupA_muscle) +
  (groupA * muscle * age * coef_groupA_age_muscle))
       aes(x = bone)) +
  geom_histogram(color = 'black',
                 fill = 'white') +
  theme_classic() +
  facet_grid(. ~ groupA)
## Add some random noise
noise <- rnorm(nrow(bone_dat), 0, 20)
bone_dat$bone <- bone_dat$bone + noise
#### Analyse ####
mod_bone <- lm(bone ~ muscle * age * groupA,
               data = bone_dat)


While I’ve added some noise to the fake data, it should be no surprise that our analysis shows some extremely strong effects of interactions… (!)

lm(formula = bone ~ muscle * age * groupA, data = bone_dat)

Min 1Q Median 3Q Max
-71.824 -13.632 0.114 13.760 70.821

Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.382e+02 6.730e+00 35.402 < 2e-16 ***
muscle 4.636e+00 8.868e-02 52.272 < 2e-16 ***
age -9.350e-01 1.538e-01 -6.079 1.31e-09 ***
groupA -1.417e+02 9.517e+00 -14.888 < 2e-16 ***
muscle:age -7.444e-02 2.027e-03 -36.722 < 2e-16 ***
muscle:groupA 2.213e-01 1.254e-01 1.765 0.0777 .
age:groupA -3.594e-01 2.175e-01 -1.652 0.0985 .
muscle:age:groupA 9.632e-02 2.867e-03 33.599 < 2e-16 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 19.85 on 4792 degrees of freedom
Multiple R-squared: 0.9728, Adjusted R-squared: 0.9728
F-statistic: 2.451e+04 on 7 and 4792 DF, p-value: < 2.2e-16

(EDIT: Note that this post is only on how to visualise the results of your analysis; it is based on the assumption that you have done the initial data exploration and analysis steps yourself already, and are satisfied that you have the correct final model… I may write a post on this at a later date, but for now I’d recommend Zuur et al’s 2010 paper, ‘A protocol for data exploration to avoid common statistical problems‘. Or you should come on the stats course that Luc Bussière and I run).

Checking the residuals etc indicates (to nobody’s surprise) that everything is looking pretty glorious from our analysis. But how to actually interpret these interactions?

We shall definitely have to use small multiples, because otherwise we shall quickly become overwhelmed. One method is to use a ‘heatmap’ style approach; this lets us plot in the style of a 3D surface, where our predictors / covariates are on the axes, and different colour regions within parameter space represent higher or lower values. If this sounds like gibberish, it’s really quite simple to get when you see the plot:


Here, higher values of bone are in lighter shades of blue, while lower values of bone are in darker shades. Moving horizontally, vertically or diagonally through combinations of muscle and age show you how bone changes; moreover, you can see how the relationships are different in different groups (i.e., the distinct facets).

To make this plot, I used one of my favourite new packages, ‘{broom}‘, in conjunction with the ever-glorious {ggplot2}. The code is amazingly simple, using broom’s ‘augment’ function to get predicted values from our linear regression model:

mod_bone %>% augment() %>% 
  ggplot(., aes(x = muscle,
                y = age,
                fill = .fitted)) +
  geom_tile() +
  facet_grid(. ~ groupA) +

But note that one aspect of broom is that augment just adds predicted values (and other cool stuff, like standard errors around the prediction) to your original data frame. That means that if you didn’t have such a complete data set, you would be missing predicted values because you didn’t have those original combinations of variables in your data frame. For example, if we sample 50% of the fake data, modelled it in the same way and plotted it, we would get this:


Not quite so pretty. There are ways around this (e.g. using ‘predict’ to fill all the gaps), but let’s move onto some different ways of visualising the data – not least because I still feel like it’s a little hard to get a handle on what’s really going on with these interactions.

A trick that we’ve seen before for looking at interactions between continuous variables is to look at only high/low values of one, across the whole range of another: in this case, we would show how bone changes with muscle in younger and older people separately. We could then use small multiples to view these relationships in distinct panels for each group (ethnic groups, in the example provided by the commenter above).

Here, I create a fake data set to use for predictions, where I have the full range of muscle (50:99), the full range of groups (0,1), and then age is at 1 standard deviation above or below the mean. The ‘expand.grid’ function simply creates every combination of these values for us! I use ‘predict’ to create predicted values from our linear model, and then add an additional variable to tell us whether the row is for a ‘young’ or ‘old’ person (this is really just for the sake of the legend):

#### Plot high/low values of age covariate ####
bone_pred <- data.frame(expand.grid(muscle = seq(50, 99),
                      age = c(mean(bone_dat$age) +
                              mean(bone_dat$age) -
                      groupA = c(0, 1)))
bone_pred <- cbind(bone_pred,
                     newdata = bone_pred,
                     interval = "confidence"))
bone_pred <- bone_pred %>% 
  mutate(ageGroup = ifelse(age > mean(bone_dat$age), "Old", "Young"))
       aes(x = muscle,
           y = fit)) +
  geom_line(aes(colour = ageGroup)) +
#   geom_point(data = bone_dat,
#              aes(x = muscle,
#                  y = bone)) +
  facet_grid(. ~ groupA) +

This gives us the following figure:


Here, we can quite clearly see how the relationship between muscle and bone depends on age, but that this dependency is different across groups. Cool! This is, of course, likely to be more extreme than you would find in your real data, but let’s not worry about subtlety here…

You’ll also note that I’ve commented out some lines in the specification of the plot. These show you how you would plot your raw data points onto this figure if you wanted to, but it doesn’t make a whole lot of sense here (as it would include all ages), and also our fake data set is so dense that it just obscures meaning. Good to have in your back pocket though!

Finally, what if we were more concerned with comparing the bone:muscle relationship of different groups against each other, and doing this at distinct ages? We could just switch things around, with each group a line on a single panel, with separate panels for ages. Just to make it interesting, let’s have three age groups this time: young (mean – 1SD), average (mean), old (mean + 1SD):

#### Groups on a single plot, with facets for different age values ####
avAge <- round(mean(bone_dat$age))
sdAge <- round(sd(bone_dat$age))
youngAge <- avAge - sdAge
oldAge <- avAge + sdAge
bone_pred2 <- data.frame(expand.grid(muscle = seq(50, 99),
                                      age = c(youngAge,
                                      groupA = c(0, 1)))
bone_pred2 <- cbind(bone_pred2,
                           newdata = bone_pred2,
                           interval = "confidence"))
       aes(x = muscle,
           y = fit,
           colour = factor(groupA))) +
  geom_line() +
  facet_grid(. ~ age) +

Created by Pretty R at inside-R.org

The code above gives us:


Interestingly, I think this gives us the most insightful version yet. Bone increases with muscle, and does so at a higher rate for those in group A (i.e., group A == 1). The positive relationship between bone and muscle diminishes at higher ages, but this is only really evident in non-A individuals.

Taking a look at our table of coefficients again, this makes sense:

Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.382e+02 6.730e+00 35.402 < 2e-16 ***
muscle 4.636e+00 8.868e-02 52.272 < 2e-16 ***
age -9.350e-01 1.538e-01 -6.079 1.31e-09 ***
groupA -1.417e+02 9.517e+00 -14.888 < 2e-16 ***
muscle:age -7.444e-02 2.027e-03 -36.722 < 2e-16 ***
muscle:groupA 2.213e-01 1.254e-01 1.765 0.0777 .
age:groupA -3.594e-01 2.175e-01 -1.652 0.0985 .
muscle:age:groupA 9.632e-02 2.867e-03 33.599 < 2e-16 ***
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


There is a positive interaction between group A x muscle x bone, which – in group A individuals – overrides the negative muscle x age interaction. The main effect of muscle is to increase bone mass (positive slope), while the main effect of age is to decrease it (in this particular visualisation, you can see this because there is essentially an age-related intercept that decreases along the panels).

These are just a few of the potential solutions, but I hope they also serve to indicate how taking the time to explore options can really help you figure out what’s going on in your analysis. Of course, you shouldn’t really believe these patterns if you can’t see them in your data in the first place though!

Unfortunately, I can’t help our poor reader with her decision to use Stata, but these things happen…

Note: if you like this sort of thing, why not sign up for the ‘Advancing in statistical modelling using R‘ workshop that I teach with Luc Bussière? Not only will you learn lots of cool stuff about regression (from straightforward linear models up to GLMMs), you’ll also learn tricks for manipulating and tidying data, plotting, and visualising your model fits! Also, it’s held on the bonny banks of Loch Lomond. It is delightful.


Want to know more about understanding and visualising interactions in multiple linear regression? Check out my previous posts:

Understanding three-way interactions between continuous variables

Using small multiples to visualise three-way interactions between 1 continuous and 2 categorical variables


Understanding 3-way interactions between continuous variables

A recurrent problem I’ve found when analysing my data is that of trying to interpret 3-way interactions in multiple regression models. As always, the mantra of PLOT YOUR DATA* holds true: ggplot2 is particularly helpful for this type of visualisation, especially using facets (I will cover this in a later post). Easier said than done, though, when all three predictor variables are continuous. How do interactions between continuous variables work, and how can we plot them?

First, let’s think of a simple example: consider a response such as the time a person takes to run 100m. We might expect that height would have a significant effect, with taller people covering the ground more quickly. A continuous by continuous interaction (two-way) would mean that the effect of height on 100m time depends on another continuous variable – for example, weight. A significant height x weight interaction would mean that the slope of height on 100m time changes as weight changes: the effect of height might be mitigated at heavier weights. Already, this is getting a little tricky to visualise in your head, so what about when we add in another predictor (e.g. age, resting heart rate, volume of beer** consumed the night before, that kind of thing)?


Some quick googling brings up a nice resource from the Institute for Digital Research and Education at UCLA, showing how to visualise three-way interactions between continuous variables using the statistical software Stata. However, I don’t want to use Stata. I want to use R. Just like you probably do. And if you don’t probably do, you should definitely do, and if you don’t now definitely do, you should probably leave. I think. Anyway, I decided that I would take their example and convert it to R so that we can follow along. Hooray for me!


Ok, so we’re going to use a simple approach to explain a three-way interaction: computing the slopes of the dependent variable (e.g. 100m time) on the independent variable (e.g. height) when moderator variables (e.g. weight and last night’s beer volume) are held constant at different combinations of high and low values. Basically, this would show the effect of height on 100m time at these combinations:

  • Heavy weight, much beer
  • Heavy weight, little beer
  • Light weight, much beer
  • Light weight, little beer

Simple, huh? Let’s take a look at the model formula, where Y is the response variable (100m time), X the predictor (height), and Z and W being the moderators (weight, last night’s beer volume):

Y = b0 + b1X + b2Z + b3W + b4XZ + b5XW + b6ZW + b7XZW

This can be reordered into two groups: the first defines the intercept (terms that do not contain X), and the second defines the simple slope (terms that do contain X):

Y = (b0 + b2Z + b3W + b6ZW) + (b1 + b4Z + b5W + b7ZW)X

We can define high values of Z and W as being one standard deviation above their respective means and will denote them as zH and wH; the low values are one standard deviation below their means (zL and wL). As in the example I gave above, this gives 4 possible combinations: zHwH, zHwL, zLwH, zLwL. As an example, here is the formula when both Z and W are at high values:

simple slope 1 = b1 + b4zH + b5wH + b7zHwH

intercept 1 = b0 + b2zH + b3wH + b6zHwH

I’m going to use the same data set given in the example that I’m converting, and you can find the Stata file here. Don’t worry, we can convert this to R data format easily enough with the R package ‘foreign’:

hsb2 <- read.dta("hsb2.dta")

As we’re not really concerned with the innards of this data set (yes, I’m afraid it’s not really the effect on running time of height, weight and beer consumption), let’s rename the variables that we’ll be using to Y, X, Z and W:

colnames(hsb2)[c(8,7,9,10)] <- c("y","x","z","w")


lm.hsb2 <- lm(y ~ x*z*w, data = hsb2)

…giving the following output:

     Min       1Q   Median       3Q      Max 
-20.0432  -4.8576   0.6129   4.0728  17.6322 
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  1.665e+02  9.313e+01   1.788   0.0753 .
x           -3.014e+00  1.803e+00  -1.672   0.0962 .
z           -3.228e+00  1.926e+00  -1.676   0.0954 .
w           -2.435e+00  1.757e+00  -1.386   0.1674  
x:z          7.375e-02  3.590e-02   2.054   0.0413 *
x:w          5.531e-02  3.252e-02   1.701   0.0906 .
z:w          6.111e-02  3.503e-02   1.745   0.0827 .
x:z:w       -1.256e-03  6.277e-04  -2.001   0.0468 *
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.896 on 192 degrees of freedom
Multiple R-squared:  0.4893,	Adjusted R-squared:  0.4707 
F-statistic: 26.28 on 7 and 192 DF,  p-value: < 2.2e-16

You can see that the three-way interaction between the continuous variables (i.e., X:Z:W) is ‘significant’, just about. Let’s save any grousing about p-values for another day, shall we? Also, I would normally centre and standardise these types of continuous predictor variables before regression (see Gelman 2007, Gelman & Hill 2007, or Schielzeth 2010 for more details on why), but in this case I’m just converting code from a given example so we’ll take things at face value…

The next thing I want to do is to create the slope and intercept values for our 4 combinations of values (high Z + high W, high Z + low W, etc). I use the original data to create high and low values for Z and W (mean plus/minus one standard deviation), then use coefficients from the linear regression to compute the intercepts and slopes:

## High / low for w and z
zH <- mean(hsb2$z) + sd(hsb2$z)
zL <- mean(hsb2$z) - sd(hsb2$z)
wH <- mean(hsb2$w) + sd(hsb2$w)
wL <- mean(hsb2$w) - sd(hsb2$w)
## Get coefficients from regression
coefs.hsb2 <- coef(lm.hsb2)
coef.int <- coefs.hsb2["(Intercept)"]
coef.x <- coefs.hsb2["x"]
coef.z <- coefs.hsb2["z"]
coef.w <- coefs.hsb2["w"]
coef.xz <- coefs.hsb2["x:z"]
coef.xw <- coefs.hsb2["x:w"]
coef.zw <- coefs.hsb2["z:w"]
coef.xzw <- coefs.hsb2["x:z:w"]
## Create slopes
zHwH <- coef.x + zH*coef.xz + wH*coef.xw + zH*wH*coef.xzw
zHwL <- coef.x + zH*coef.xz + wL*coef.xw + zH*wL*coef.xzw
zLwH <- coef.x + zL*coef.xz + wH*coef.xw + zL*wH*coef.xzw
zLwL <- coef.x + zL*coef.xz + wL*coef.xw + zL*wL*coef.xzw
## Create intercepts
i.zHwH <- coef.int + zH*coef.z + wH*coef.w + zH*wH*coef.zw
i.zHwL <- coef.int + zH*coef.z + wL*coef.w + zH*wL*coef.zw
i.zLwH <- coef.int + zL*coef.z + wH*coef.w + zL*wH*coef.zw
i.zLwL <- coef.int + zL*coef.z + wL*coef.w + zL*wL*coef.zw

Now, I want to create a data frame that uses these intercepts and slopes to ‘predict’ lines: because I know that the lines are going to be straight, I simply compute the value of Y for the minimum and maximum values of X for each Z:W combination. I’m pretty sure it can be done with far less code than I’ve used here, but there you go:

## a set of values of x
x0 <- seq(min(hsb2$x), max(hsb2$x), length.out = 2)
df.HH <- data.frame(x0 = x0)
df.HH$y0 <- i.zHwH + df.HH$x0*zHwH
df.HH$type <- rep("zHwH", nrow(df.HH))
df.HL <- data.frame(x0 = x0)
df.HL$y0 <- i.zHwL + df.HL$x0*zHwL
df.HL$type <- rep("zHwL", nrow(df.HL))
df.LH <- data.frame(x0 = x0)
df.LH$y0 <- i.zLwH + df.LH$x0*zLwH
df.LH$type <- rep("zLwH", nrow(df.LH))
df.LL <- data.frame(x0 = x0)
df.LL$y0 <- i.zLwL + df.LL$x0*zLwL
df.LL$type <- rep("zLwL", nrow(df.LL))
## Create final data frame
df.pred <- rbind(df.HH, df.HL, df.LH, df.LL)
## Remove unnecessary data frames
rm(df.HH, df.HL, df.LH, df.LL)

Now, it’s time to plot the data! I’ve used ggplot2, and specified different colours and line types for each of our 4 combinations: red denotes high Z-value, and solid line denotes high W-value. I’ve also plotted the original data over the top:

## Call library
## Convert 'type' to factor
df.pred$type <- factor(df.pred$type)
## Draw plot
ggplot(df.pred, aes(x = x0,
                    y = y0)) +
  geom_line(aes(colour = type,
                linetype = type)) +
  geom_jitter(data = hsb2, 
            aes(x = x,
                y = y),
             size = 3,
             alpha = 0.7) + 
  theme_bw() +
  theme(panel.background = element_rect(fill="ivory")) +
  theme(legend.key = element_blank()) +
  theme(text = element_text(size = 15)) +
  scale_colour_manual(name = "Partial effect",
                      labels = c("High z High w", 
                                 "High z Low w", 
                                 "Low z High w", 
                                 "Low z Low w"),
                      values = c("#E41A1C", 
                                 "#377EB8")) +   
  scale_linetype_manual(name = "Partial effect",
                     labels = c("High z High w", 
                                "High z Low w", 
                                "Low z High w", 
                                "Low z Low w"),
                     values = c("solid", 

The finished article:


Clearly, three of the slopes are very similar (differing mostly in terms of intercept), while one is very different to the rest. The relationship between Y and X is changed dramatically when there are high values of Z in combination with low values of W. It can still be hard to grasp these in abstract terms, so let’s refer back to my initial example (just for ‘fun’): this would mean that when weight is high [Z] and beer consumption is low [W], there is a steep relationship between height [X] and running time [Y]: short, heavy teetotallers are slow, but tall, heavy teetotallers are really fast.

Remember, these lines are theoretical, and drawn from the regression coefficients rather than the data itself; it’s up to you to make sure that short, heavy teetotallers are represented within your data set. I, for one, drink quite a lot of beer.


Want to know more about understanding and visualising interactions in multiple linear regression? Check out my follow-up posts:

Using ‘small multiples’ to visualise 3-way interactions between 1 continuous and 2 categorical variables

Three-way interactions between 2 continuous and 1 categorical variable


* My supervisor calls this the ‘ADF method’, as in ‘ANY DAMN FOOL (can see what’s happening if you just actually plot your data)’

** I originally had beer count, but my esteemed colleague Lilly Herridge (who knows what she is talking about much more than I do) pointed out that counts are discrete, not continuous, hence the change to volume. SMARTS.

Code in this post was adapted from the UCLA Statistical Consulting Group’s page on three-way continuous interactions, and highlighted using inside-R’s ‘Pretty-R‘ tool.