Formal definition of the qqline used in a Q-Q plot

Question

I'm doing some distribution fitting work and I'm looking at Q-Q plots and how they can be used visually to interpret goodness of fit.

My data is heavy-tailed so I am looking at Weibull, log-normal, Pareto and log-logistic distributions initially.

For a Weibull distribution, I understand how the points on the Q-Q plot are constructed (using the quantiles of observed data vs. the quantiles of an estimated Weibull distribution). The piece I am not clear on is how the line used in Q-Q plots is calculated/constructed.

The R documentation for the qqplot() function provides the following description:

qqnorm is a generic function the default method of which produces a normal QQ plot of the values in y. qqline adds a line to a “theoretical”, by default normal, quantile-quantile plot which passes through the probs quantiles, by default the first and third quartiles.

Another post on Cross Validated seems to indicate that the line is essentially a line constructed from the parameters of the theoretical (estimated) distribution. Is this a true statement and correct interpretation?

If a link to a formal definition could be provided I'd very much appreciate it.

Answer 1

Sort of "both" - the line depends both on the observed quantiles (which define the y-axis of the QQ plot) and the expected/theoretical/reference quantiles (which the define the x-axis). The documentation (which you quote) should always be taken as the canonical reference:

‘qqline’ adds a line to a “theoretical”, by default normal, quantile-quantile plot which passes through the ‘probs’ quantiles, by default the first and third quartiles.

If in doubt, USTL ("Use the Source, Luke") , which can be found here: here's a slightly abridged and commented version

 ## quantiles (.25 and 0.75 by default) of data
 y <- quantile(y, probs, names=FALSE, type=qtype, na.rm = TRUE)
 ## quantiles of reference/theoretical distribution
 x <- distribution(probs)
 ## ...
 slope <- diff(y)/diff(x)  ## observed slope between quantiles
 int <- y[1L]-slope*x[1L]  ## intercept
 abline(int, slope, ...)   ## draw the line

For what it's worth, I believe that this approach (line connecting central quantiles) is used because it fulfills the following criteria for exploratory/diagnostic approaches:

• quick (e.g. no need to run a linear regression, just find the quantiles and draw a straight line)
• robust (it only depends on the behavior of the central part of the distribution, won't be thrown off by weird tails)

Answer 2

I think it simply adds a line segment between the points (x1, y1) & (x2, y2) for given probabilities (p1, p2)

(x1, x2) are the quantiles of the theoretical distribution; (y1, y2) for the data comparison. Function qline has simple code under the hood. This is a simple e.g. in R

# sample data
set.seed(2)
y <- rt(100, df = 5)

# get the values
probs <- c(0.25, 0.75)
x1 <- qnorm(probs[1])
x2 <- qnorm(probs[2])
y1 <- quantile(y, probs[1])
y2 <- quantile(y, probs[2])

# plot
qqnorm(y)
segments(x1, y1, x2, y2, col = "red", lwd = 2)
qqline(y, lty = 2)
# theoretical match is straight line. If you add more samples, qqline should 
# converge to this
abline(0,1)