A common way of illustrating the idea behind statistical power in null hypothesis significance testing, is by plotting the sampling distributions of the null hypothesis () and the alternative hypothesis (). Typically, these illustrations highlight the regions that correspond to making a type II error (), type I error () and correctly rejecting the null hypothesis (i.e. the test’s power; ).

In this post I will show how to create such “power plots” using R. Typically, I prefer to use ggplot for plotting, but tasks such as this is one of the few times were I think R’s base graphics have some merit—especially for creating black and white plots, since ggplot does not support using patterns. Thus, I will present code both for ggplot and base graphics.

Creating these plots is pretty straight forwards. You only need to be vaguely familiar with the mechanics of plotting polygons. For instance, say we want to plot a triangle with the following coordinates.

       (2,3)
        /\
       /  \
      /    \
     /      \
(1,2) ------ (3,2)

Then we just specify x and y as vectors, like this:

# ggplot polygon example
ggplot(data.frame(x=c(1,2,3),y=c(2,3,2)), aes(x,y)) + geom_polygon()

So, let us begin by creating the data for the two distributions and three polygons that we will need.

library(ggplot2)
library(grid) # need for arrow()
m1 <- 0  # mu H0
sd1 <- 1.5 # sigma H0
m2 <- 3.5 # mu HA
sd2 <- 1.5 # sigma HA

z_crit <- qnorm(1-(0.05/2), m1, sd1)

# set length of tails
min1 <- m1-sd1*4
max1 <- m1+sd1*4
min2 <- m2-sd2*4
max2 <- m2+sd2*4          
# create x sequence
x <- seq(min(min1,min2), max(max1, max2), .01)
# generate normal dist #1
y1 <- dnorm(x, m1, sd1)
# put in data frame
df1 <- data.frame("x" = x, "y" = y1)
# generate normal dist #2
y2 <- dnorm(x, m2, sd2)
# put in data frame
df2 <- data.frame("x" = x, "y" = y2)

# Alpha polygon
y.poly <- pmin(y1,y2)
poly1 <- data.frame(x=x, y=y.poly)
poly1 <- poly1[poly1$x >= z_crit, ] 
poly1<-rbind(poly1, c(z_crit, 0))  # add lower-left corner

# Beta polygon
poly2 <- df2
poly2 <- poly2[poly2$x <= z_crit,] 
poly2<-rbind(poly2, c(z_crit, 0))  # add lower-left corner

# power polygon; 1-beta
poly3 <- df2
poly3 <- poly3[poly3$x >= z_crit,] 
poly3 <-rbind(poly3, c(z_crit, 0))  # add lower-left corner

# combine polygons. 
poly1$id <- 3 # alpha, give it the highest number to make it the top layer
poly2$id <- 2 # beta
poly3$id <- 1 # power; 1 - beta
poly <- rbind(poly1, poly2, poly3)
poly$id <- factor(poly$id,  labels=c("power","beta","alpha"))

Now that we have all the data that we need, let us create the first plot using ggplot. The annotation is set manually, so it will be a bit tedious to change these plots.

 
# plot with ggplot2
ggplot(poly, aes(x,y, fill=id, group=id)) +
  geom_polygon(show_guide=F, alpha=I(8/10)) +
  # add line for treatment group
  geom_line(data=df1, aes(x,y, color="H0", group=NULL, fill=NULL), size=1.5, show_guide=F) + 
  # add line for treatment group. These lines could be combined into one dataframe.
  geom_line(data=df2, aes(color="HA", group=NULL, fill=NULL),size=1.5, show_guide=F) +
  # add vlines for z_crit
  geom_vline(xintercept = z_crit, size=1, linetype="dashed") +
  # change colors 
  scale_color_manual("Group", 
                     values= c("HA" = "#981e0b","H0" = "black")) +
  scale_fill_manual("test", values= c("alpha" = "#0d6374","beta" = "#be805e","power"="#7cecee")) +
  # beta arrow
  annotate("segment", x=0.1, y=0.045, xend=1.3, yend=0.01, arrow = arrow(length = unit(0.3, "cm")), size=1) +
  annotate("text", label="beta", x=0, y=0.05, parse=T, size=8) +
  # alpha arrow
  annotate("segment", x=4, y=0.043, xend=3.4, yend=0.01, arrow = arrow(length = unit(0.3, "cm")), size=1) +
  annotate("text", label="frac(alpha,2)", x=4.2, y=0.05, parse=T, size=8) +
  # power arrow
  annotate("segment", x=6, y=0.2, xend=4.5, yend=0.15, arrow = arrow(length = unit(0.3, "cm")), size=1) +
  annotate("text", label="1-beta", x=6.1, y=0.21, parse=T, size=8) +
  # H_0 title
  annotate("text", label="H[0]", x=m1, y=0.28, parse=T, size=8) +
  # H_a title
  annotate("text", label="H[a]", x=m2, y=0.28, parse=T, size=8) +
  ggtitle("Statistical Power Plots, Textbook-style") +
  # remove some elements
  theme(panel.grid.minor = element_blank(),
             panel.grid.major = element_blank(),
             panel.background = element_blank(),
             plot.background = element_rect(fill="#f9f0ea"),
             panel.border = element_blank(),
             axis.line = element_blank(),
             axis.text.x = element_blank(),
             axis.text.y = element_blank(),
             axis.ticks = element_blank(),
             axis.title.x = element_blank(),
             axis.title.y = element_blank(),
             plot.title = element_text(size=22))

ggsave("stat_power_ggplot.png", height=8, width=13, dpi=72)

Now, if we want a more “classical looking” black and white-plot, we need to use base graphics.

# example with base graphics
png("stat_power_base.png", width=900, height=600, units="px") # save as png
# reset
plot.new()
# set window size
plot.window(xlim=range(x), ylim=c(-0.01,0.3))
# add polygons
polygon(poly3,  density=10) # 1-beta
polygon(poly2, density=3, angle=-45, lty="dashed") # beta
polygon(poly1, density=10, angle=0) # alpha
# add h_a dist
lines(df2,lwd=3)
# add h_0 dist
lines(df1,lwd=3)
### annotations
# h_0 title
text(m1, 0.3, expression(H[0]), cex=1.5)
# h_a title
text(m2, 0.3, expression(H[a]), cex=1.5)
# beta annotation
arrows(x0=-1, y0=0.045, x1=1, y1=0.01,lwd=2,length=0.15)
text(-1.2, 0.045, expression(beta), cex=1.5)
# beta annotation
arrows(x0=4, y0=-0.01, x1=3.5, y1=0.01, lwd=2, length=0.15)
text(x=4.1, y=-0.015, expression(alpha/2), cex=1.5)
# 1-beta 
arrows(x0=6, y0=0.15, x1=5, y1=0.1, lwd=2,length=0.15)
text(x=7, y=0.155, expression(paste(1-beta, "  (\"power\")")), cex=1.5)
# show z_crit; start of rejection region
abline(v=z_crit)
# add bottom line
abline(h=0)
title("Statistical Power")
dev.off()