In ZellW/LearnPlotting:

See http://www.r-bloggers.com/exploratory-data-analysis-quantile-quantile-plots-for-new-yorks-ozone-pollution-data/ See http://www.statmethods.net/stats/rdiagnostics.html See http://www.statmethods.net/advgraphs/probability.html

A quantile-quantile plot, or Q-Q plot, is a plot of the sorted quantiles of one data set against the sorted quantiles of another data set. It is used to visually inspect the similarity between the underlying distributions of 2 data sets. Each point (x, y) is a plot of a quantile of one distribution along the vertical axis (y-axis) against the corresponding quantile of the other distribution along the horizontal axis (x-axis). If the 2 distributions are similar, then the points would lie close to the identity line, y = x.

Quantile-Quantile Plots of Ozone Pollution Data

By Eric Cai - The Chemical Statistician

clear all variables

rm(list = ls(all.names = TRUE))

view first 6 entries of the "Ozone" data frame

head(airquality)

extract "Ozone" data vector

ozone = airquality$Ozone

sample size of "ozone"

length(ozone)

summary of "ozone"

summary(ozone)

remove missing values from "ozone"

ozone = ozone[!is.na(ozone)]

having removed missing values, find the number of non-missing values in "ozone"

n = length(ozone)

calculate mean, variance and standard deviation of "ozone"

mean.ozone = mean(ozone) var.ozone = var(ozone) sd.ozone = sd(ozone)

Now, let’s set the n points in the interval (0,1) for the n equi-probable point-wise probabilities, each of which is assigned to the correspondingly ranked quantile. (The smallest probability is assigned to the smallest quantile, and the largest probability is assigned to the largest quantile.) These probabilities will be used to calculate the quantiles for each hypothesized theoretical distribution.

set n points in the interval (0,1)

use the formula k/(n+1), for k = 1,..,n

this is a vector of the n probabilities

probabilities = (1:n)/(n+1)

Since “ozone” is a continuous variable, let’s try fitting it to the normal distribution – the most commonly used distribution for continuous variables. Notice my use of the qnorm() function to calculate the quantiles from the normal distribution.
Specifically, I used the sample mean and the sample standard deviation of “ozone” to specify the parameters of the normal distribution.

calculate normal quantiles using mean and standard deviation from "ozone"

normal.quantiles = qnorm(probabilities, mean(ozone, na.rm = T), sd(ozone, na.rm = T))

Finally, let’s plot the theoretical quantiles on the horizontal axis and the sample quantiles on the vertical axis.
Notice my use of the abline() function to add the identify line. The two parameters call for a line with an intercept of 0 and a slope of 1.

normal quantile-quantile plot for "ozone"

png('INSERT YOUR DIRECTORY PATH HERE') plot(sort(normal.quantiles), sort(ozone) , xlab = 'Theoretical Quantiles from Normal Distribution', ylab = 'Sample Quqnatiles of Ozone', main = 'Normal Quantile-Quantile Plot of Ozone') abline(0,1) dev.off()

The plotted points do not fall closely onto the identity line, so the data do not seem to come from the normal distribution.
Is there another distribution that would work better?

Recall from my earlier post on kernel density estimation that the “ozone” data are right-skewed. A gamma distribution with a small shape parameter tends to be right-skewed, so let’s try this instead. Notice my use of the sample mean and sample variance to estimate the shape and the scale parameters.

calculate gamma quantiles using mean and standard deviation from "ozone" to calculate shape and scale parameters

gamma.quantiles = qgamma(probabilities, shape = mean.ozone^2/var.ozone, scale = var.ozone/mean.ozone)

gamma quantile-quantile plot for "ozone"

png('INSERT YOUR DIRECTORY PATH HERE') plot(sort(gamma.quantiles), sort(ozone), xlab = 'Theoretical Quantiles from Gamma Distribution', ylab = 'Sample Quantiles of Ozone', main = 'Gamma Quantile-Quantile Plot of Ozone')

abline(0,1) dev.off()

R has functions for quickly producing Q-Q plots; they are qqnorm(), qqline(), and qqplot(). These are good functions to use.
I built the above Q-Q plots using more rudimentary functions because

•I wanted to use R’s rudimentary functions to illustrate the 5 steps of creating a Q-Q plot •My method ensures that the sample quantiles and the theoretical quantiles are on the same scale.

qqnorm(ozone) qqline(ozone)

boxplot(ozone)

ZellW/LearnPlotting documentation built on May 10, 2019, 1:57 a.m.