QQ-Plots

Effect of right-skew on qq-plot

Let’s create a data set that is clearly skewed right. We’ll name our variable right, and put 500 observations in it using random draws from the exponential distribution (rate = 2).

right = rexp(500, rate = 2)

The histogram shows this data set appears to have been drawn from an approximately normal distribution with massive skew to the right.

histogram(right, width = .25)

The qq-plot is clearly curved with the middle portion below the 45-degree line and the ends above it.

qqmath(right)

Effect of left-skew on qq-plot

Let’s do the same thing to create a left-skewed data set. However, to keep the data in the positive range, let’s take 5 minus the randomized values from the exponential distribution.

left = 5 - rexp(500, rate = 2)

Let’s display a histogram to see that it could be thought of as having been drawn from an approximately normal distribution with massive skew to the right.

histogram(left, width = .25)

The qq-plot is clearly curved with the middle portion above the 45-degree line and the ends below it.

qqmath(left)

Effect of uniform distribution on qq-plot

Now we need to create a uniform distribution, so we can simply use the randomization function which generates real numbers between 0 and 1. To convert to a similar range as above, we’ll multiply each random value by 5.

uni = runif(500) * 5

Let’s display a histogram to see that it could be thought of as having been drawn from an approximately normal distribution with massive skew to the right.

histogram(uni, width = .25)

The qq-plot has a clear S-shape.

qqmath(uni)

Effect of heteroskedasticity on qq-plot

It takes a bit of work to create a distribution where the variance increase as we move to the right along the \(x\)-axis. Notice that it’s actually easier to detect the heteroskedasticity in the fitted values plot.

x = runif(50, min = 0 , max = 5)
y = 5 + 10 * x
yHat = 5 + 10 * x + rnorm(50) * 5 * x
mod = lm (yHat ~ x)
plot(mod, which = 1)

The qq-plot has a very odd look at the extremes, but the diagnostic plot that will be more likely to show a problem is the fitted values plot.

res = y - yHat
qqmath(res)