What is the difference between probability plots, PP-plots and QQ-plots when trying to analyse a fitted distribution to data?
As @vector07 notes, probability plot is the more abstract category of which pp-plots and qq-plots are members. Thus, I will discuss the distinction between the latter two. The best way to understand the differences is to think about how they are constructed, and to understand that you need to recognize the difference between the quantiles of a distribution and the proportion of the distribution that you have passed through when you reach a given quantile. You can see the relationship between these by plotting the cumulative distribution function (CDF) of a distribution. For example, consider the standard normal distribution:
We see that approximately 68% of the y-axis (region between red lines) corresponds to 1/3 of the x-axis (region between blue lines). That means that when we use the proportion of the distribution we have passed through to evaluate the match between two distributions (i.e., we use a pp-plot), we will get a lot of resolution in the center of the distributions, but less at the tails. On the other hand, when we use the quantiles to evaluate the match between two distributions (i.e., we use a qq-plot), we will get very good resolution at the tails, but less in the center. (Because data analysts are typically more concerned about the tails of a distribution, which will have more effect on inference for example, qq-plots are much more common than pp-plots.)
To see these facts in action, I will walk through the construction of a pp-plot and a qq-plot. (I also walk through the construction of a qq-plot verbally / more slowly here: QQ-plot does not match histogram.) I don't know if you use R, but hopefully it will be self-explanatory:
set.seed(1) # this makes the example exactly reproducible N = 10 # I will generate 10 data points x = sort(rnorm(n=N, mean=0, sd=1)) # from a normal distribution w/ mean 0 & SD 1 n.props = pnorm(x, mean(x), sd(x)) # here I calculate the probabilities associated # w/ these data if they came from a normal # distribution w/ the same mean & SD # I calculate the proportion of x we've gone through at each point props = 1:N / (N+1) n.quantiles = qnorm(props, mean=mean(x), sd=sd(x)) # this calculates the quantiles (ie # z-scores) associated w/ the props my.data = data.frame(x=x, props=props, # here I bundle them together normal.proportions=n.props, normal.quantiles=n.quantiles) round(my.data, digits=3) # & display them w/ 3 decimal places # x props normal.proportions normal.quantiles # 1 -0.836 0.091 0.108 -0.910 # 2 -0.820 0.182 0.111 -0.577 # 3 -0.626 0.273 0.166 -0.340 # 4 -0.305 0.364 0.288 -0.140 # 5 0.184 0.455 0.526 0.043 # 6 0.330 0.545 0.600 0.221 # 7 0.487 0.636 0.675 0.404 # 8 0.576 0.727 0.715 0.604 # 9 0.738 0.818 0.781 0.841 # 10 1.595 0.909 0.970 1.174
Unfortunately, these plots aren't very distinctive, because there are few data and we are comparing a true normal to the correct theoretical distribution, so there isn't anything special to see in either the center or the tails of the distribution. To better demonstrate these differences, I plot a (fat-tailed) t-distribution with 4 degrees of freedom, and a bi-modal distribution below. The fat tails are much more distinctive in the qq-plot, whereas the bi-modality is more distinctive in the pp-plot.
Here is a definition from v8doc.sas.com:
A P-P plot compares the empirical cumulative distribution function of a data set with a specified theoretical cumulative distribution function F(·). A Q-Q plot compares the quantiles of a data distribution with the quantiles of a standardized theoretical distribution from a specified family of distributions.
In the text, they also mention:
- differences regarding the way P-P plots and Q-Q plots are constructed and interpreted.
- advantages of using one or another, regarding comparing empirical and theoretical distributions.
SAS Institute Inc., SAS OnlineDoc®, Version 8, Cary, NC: SAS Institute Inc., 1999