Plotting logistic regressions, Part 3

If you haven’t already, check out plotting logistic regression part 1 (continuous by categorical interactions) and part 2 (continuous by continuous interactions).

All of this code is available on Rose’s github: https://github.com/rosemm/rexamples/blob/master/logistic_regression_plotting_part3.Rmd









Plotting the results of your logistic regression Part 3: 3-way interactions

If you can interpret a 3-way interaction without plotting it, go find a mirror and give yourself a big sexy wink. wink That’s impressive.

For the rest of us, looking at plots will make understanding the model and results so much easier. And even if you are one of those lucky analysts with the working memory capacity of a super computer, you may want this code so you can use plots to help communicate a 3-way interaction to your readers.

Use the model from the Part 1 code.

Here’s that model:

summary(model)
## 
## Call:
## glm(formula = DV ~ (X1 + X2 + group)^2, family = "binomial", 
##     data = data, na.action = "na.exclude")
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.44094  -0.45991   0.04136   0.52301   2.74705  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -0.5873     0.3438  -1.708 0.087631 .  
## X1            2.6508     0.5592   4.740 2.13e-06 ***
## X2           -2.2599     0.4977  -4.540 5.61e-06 ***
## groupb        2.2111     0.5949   3.717 0.000202 ***
## groupc        0.6650     0.4131   1.610 0.107456    
## X1:X2         0.1201     0.2660   0.452 0.651534    
## X1:groupb     2.7323     1.2977   2.105 0.035253 *  
## X1:groupc    -0.6816     0.7078  -0.963 0.335531    
## X2:groupb     0.8477     0.7320   1.158 0.246882    
## X2:groupc     0.4683     0.6558   0.714 0.475165    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 412.88  on 299  degrees of freedom
## Residual deviance: 205.46  on 290  degrees of freedom
## AIC: 225.46
## 
## Number of Fisher Scoring iterations: 7

Let’s add a 3-way interaction. Instead of re-running the whole model, we can use the nifty update() function. This will make the change to the model (adding the 3-way interaction), and automatically refit the whole thing. (It is also fine to just re-run the model — you’ll get the exact same results. I just wanted to show off the update() function.)

new.model <- update(model, ~ . + X1:X2:group) # the . stands in for the whole formula we had before

# if you wanted to specify the whole model from scratch instead of using update():
new.model <- glm(DV ~ X1*X2*group, 
             data=data, na.action="na.exclude",  family="binomial")  

summary(new.model)
## 
## Call:
## glm(formula = DV ~ X1 * X2 * group, family = "binomial", data = data, 
##     na.action = "na.exclude")
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.36818  -0.39475   0.02394   0.45860   2.82512  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   -0.8959     0.4037  -2.219 0.026491 *  
## X1             2.7511     0.5914   4.652 3.29e-06 ***
## X2            -2.3070     0.5076  -4.545 5.49e-06 ***
## groupb         2.5457     0.6776   3.757 0.000172 ***
## groupc         1.3403     0.5201   2.577 0.009958 ** 
## X1:X2          0.6779     0.4314   1.572 0.116057    
## X1:groupb      3.8321     1.6589   2.310 0.020882 *  
## X1:groupc     -0.6709     0.7412  -0.905 0.365349    
## X2:groupb      1.1732     0.7623   1.539 0.123806    
## X2:groupc      0.1871     0.6975   0.268 0.788483    
## X1:X2:groupb   1.1108     0.8198   1.355 0.175458    
## X1:X2:groupc  -1.4068     0.5899  -2.385 0.017082 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 412.88  on 299  degrees of freedom
## Residual deviance: 194.51  on 288  degrees of freedom
## AIC: 218.51
## 
## Number of Fisher Scoring iterations: 7

Calculate probabilities for the plot

Again, we’ll put X1 on the x-axis. That’s the only variable we’ll enter as a whole range.

X1_range <- seq(from=min(data$X1), to=max(data$X1), by=.01)

Next, compute the equations for each line in logit terms.

Pick some representative values for the other continuous variable

Just like last time, we’ll plug in some representative values for X2, so we can have separate lines for each representative level of X2.

X2_l <- mean(data$X2) - sd(data$X2) 
X2_m <- mean(data$X2)
X2_h <- mean(data$X2) + sd(data$X2)
# check that your representative values actually fall within the observed range for that variable
summary(data$X2)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -3.52900 -0.70950 -0.02922 -0.04995  0.67260  2.38000
c(X2_l, X2_m, X2_h)
## [1] -1.0621963 -0.0499475  0.9623013

Now we can go ahead and plug those values into the rest of the equation to get the expected logits across the range of X1 for each of our “groups” (hypothetical low X2 people, hypothetical average X2 people, hypothetical high X2 people). We’ll also plug in the dummy codes for each of the three groups (a, b, and c). And we’ll calculate the predicted probabilities of the DV for each combination of X2 level and group.

But instead of literally writing out all of the equations (9 of them!!), we’ll just use the fun-and-easy predict() function.

If you ran your model in SPSS, so you only have the coefficients and not the whole model as an R object, you can still make the plots — you just need to spend some quality time writing out those equations. For examples of how to do this (for just 3 equations, but you get the idea) see Part 1 and Part 2 in this series.

To use predict(), you make a new data frame with the predictor values you want to use (i.e. the whole range for X1, group a, and the representative values we picked for X2), and then when you run predict() on it, for each row in the data frame it will generate the predicted value for your DV from the model you saved. The expand.grid() function is a quick and easy way to make a data frame out of all possible combinations of the variables provided. Perfect for this situation!

#make a new data frame with the X values you want to predict 
generated_data <- as.data.frame(expand.grid(X1=X1_range, X2=c(X2_l, X2_m, X2_h), group=c("a", "b", "c") ))
head(generated_data)
##          X1        X2 group
## 1 -2.770265 -1.062196     a
## 2 -2.760265 -1.062196     a
## 3 -2.750265 -1.062196     a
## 4 -2.740265 -1.062196     a
## 5 -2.730265 -1.062196     a
## 6 -2.720265 -1.062196     a
#use `predict` to get the probability using type='response' rather than 'link' 
generated_data$prob <- predict(new.model, newdata=generated_data, type = 'response')
head(generated_data) 
##          X1        X2 group       prob
## 1 -2.770265 -1.062196     a 0.01675787
## 2 -2.760265 -1.062196     a 0.01709584
## 3 -2.750265 -1.062196     a 0.01744050
## 4 -2.740265 -1.062196     a 0.01779198
## 5 -2.730265 -1.062196     a 0.01815042
## 6 -2.720265 -1.062196     a 0.01851595
# let's make a factor version of X2, so we can do gorgeous plotting stuff with it later :)
generated_data$X2_level <- factor(generated_data$X2, labels=c("low (-1SD)", "mean", "high (+1SD)"), ordered=T)
summary(generated_data)
##        X1                 X2           group         prob        
##  Min.   :-2.77027   Min.   :-1.06220   a:1683   Min.   :0.00000  
##  1st Qu.:-1.37027   1st Qu.:-1.06220   b:1683   1st Qu.:0.02416  
##  Median : 0.02974   Median :-0.04995   c:1683   Median :0.56304  
##  Mean   : 0.02974   Mean   :-0.04995            Mean   :0.51727  
##  3rd Qu.: 1.42973   3rd Qu.: 0.96230            3rd Qu.:0.99162  
##  Max.   : 2.82973   Max.   : 0.96230            Max.   :1.00000  
##         X2_level   
##  low (-1SD) :1683  
##  mean       :1683  
##  high (+1SD):1683  
##                    
##                    
## 

Plot time!

This kind of situation is exactly when ggplot2 really shines. We want multiple plots, with multiple lines on each plot. Of course, this is totally possible in base R (see Part 1 and Part 2 for examples), but it is so much easier in ggplot2. To do this in base R, you would need to generate a plot with one line (e.g. group a, low X2), then add the additional lines one at a time (group a, mean X2; group a, high X2), then generate a new plot (group b, low X2), then add two more lines, then generate a new plot, then add two more lines. Sigh.

Not to go down too much of a rabbit hole, but this illustrates what is (in my opinion) the main difference between base R graphics and ggplot2: base graphics are built for drawing, whereas ggplot is built for visualizing data. It’s the difference between specifying each line and drawing them on your plot vs. giving a whole data frame to the plotting function and telling it which variables to use and how. Depending on your needs and preferences, base graphics or ggplot may be a better choice for you. For plotting complex model output, like a 3-way interaction, I think you’ll generally find that ggplot2 saves the day.

library(ggplot2)

plot.data <- generated_data

# check out your plotting data
head(plot.data)
##          X1        X2 group       prob   X2_level
## 1 -2.770265 -1.062196     a 0.01675787 low (-1SD)
## 2 -2.760265 -1.062196     a 0.01709584 low (-1SD)
## 3 -2.750265 -1.062196     a 0.01744050 low (-1SD)
## 4 -2.740265 -1.062196     a 0.01779198 low (-1SD)
## 5 -2.730265 -1.062196     a 0.01815042 low (-1SD)
## 6 -2.720265 -1.062196     a 0.01851595 low (-1SD)
ggplot(plot.data, aes(x=X1, y=prob, color=X2_level)) + 
  geom_line(lwd=2) + 
  labs(x="X1", y="P(outcome)", title="Probability of super important outcome") +
  facet_wrap(~group) # i love facet_wrap()! it's so great. you should fall in love, too, and use it all the time.

# let's try flipping it, so the facets are by X2 level and the lines are by group
ggplot(plot.data, aes(x=X1, y=prob, color=group)) + 
  geom_line(lwd=2) + 
  labs(x="X1", y="P(outcome)", title="Probability of super important outcome") +
  facet_wrap(~X2_level)