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

distributionsfittingheavy-tailedr

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.

Best Answer

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)
Related Question