Math 141. Quantile-Quantile Plots. Albyn Jones 1. jones/courses/ Library 304. Albyn Jones Math 141

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1 Math 141 Quantile-Quantile Plots Albyn Jones 1 1 Library 304 jones@reed.edu jones/courses/141

2 Outline: Quantile-Quantile Plots Quantiles and Order Statistics Quantile-Quantile Plots Normal Quantile Plots

3 Quantiles and Order Statistics Definition: p-th quantile q p Earlier we defined the p-th quantile of the distribution of a RV X as any number q p satisfying P(X q p ) p and P(X q p ) (1 p).

4 Order Statistics Sample quantiles are based on Order Statistics: Let X 1, X 2,..., X n be a sample of size n. The order statistics X (1), X (2),..., X (n) are just the observations sorted into ascending order. # The data > x [1] # The order statistics > sort(x) [1]

5 Order Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen to perform well under various conditions. All involve interpolation between neighboring order statistics.

6 Order Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen to perform well under various conditions. All involve interpolation between neighboring order statistics. Suppose that we want the pth quantile, where p lies between k/n and (k + 1)/n. There are variations, but all are, for some choice of a [0, 1]: ˆq p = ax (k) + (1 a)x (k+1)

7 Order Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen to perform well under various conditions. All involve interpolation between neighboring order statistics. Suppose that we want the pth quantile, where p lies between k/n and (k + 1)/n. There are variations, but all are, for some choice of a [0, 1]: ˆq p = ax (k) + (1 a)x (k+1) Example, when n is even, the sample median q.5 is usually taken to be the average of the two middle observations.

8 Order Statistics as Sample Quantiles Let s turn it around, and ask what sample quantiles correspond to the order statistics? Consider 4 observations from a population. On the average, we expect them to at least approximately divide the population into equal chunks corresponding to equally spaced percentiles: 1 5, 2 5, 3 5, 4 5 In other words, they correspond to the sample quantiles q.2, q.4, q.6, q.8

9 Comparing Two Samples Suppose we have two samples of size n, X 1, X 2,... X n and Y 1, Y 2,... Y n.

10 Comparing Two Samples Suppose we have two samples of size n, X 1, X 2,... X n and Y 1, Y 2,... Y n. If they were samples from the same distribution, then the order statistics X (1), X (2),... X (n) and Y (1), Y (2),... Y (n) would be estimates of the same quantiles.

11 Comparing Two Samples Suppose we have two samples of size n, X 1, X 2,... X n and Y 1, Y 2,... Y n. If they were samples from the same distribution, then the order statistics X (1), X (2),... X (n) and Y (1), Y (2),... Y (n) would be estimates of the same quantiles. Thus we expect that X (1) Y (1), X (2) Y (2), etc.

12 The QQ plot The quantile quantile plot, or QQplot, is a simple graphical method for comparing two sets of sample quantiles. Plot the pairs of order statistics (X (k), Y (k) ). If the two datasets come from the same distribution, the points should lie roughly on a line through the origin with slope 1.

13 QQ plot example QQplot of two N(0,1) samples of size 200 Y X

14 QQ plot example, small sample! Alert!! With small samples, expect variation! QQplot of two N(0,1) samples of size 20 Y X

15 QQ plot example: location shift Two samples from similar distributions which differ only in location: the green reference line is y = x. QQplot of two samples of size 200 Y X

16 QQ plot example: different spread Two samples from similar distributions which differ only in spread, with reference line. QQplot of two samples of size 200 Y X

17 QQ plot example: different shape Two samples from distributions which differ in shape, as well as location and spread, with reference line. QQplot of two samples of size 200 Y X

18 QQ plot example: Anorexia data The Family Therapy group had 17 subjects, the Control Therapy 26. qqplot() uses estimated quantiles for the larger dataset. QQplot of Family Therapy vs Control Family Control

19 Normal Quantile Plots Often we wish to compare a dataset to the Normal distribution, a theoretical population, rather than to a second dataset. R has a function that plots the order statistics of a sample against the corresponding quaintiles of the standard normal distribution: qqnorm(x) If the plot is roughly linear, then our data are approximately normally distributed.

20 Normal Quantile Plot: Normal data Don t expect a perfectly straight plot even with normal data! Normal Q Q Plot Theoretical Quantiles Sample Quantiles

21 Normal Quantile Plot: Mean and SD We can estimate the mean and SD from a Normal quantile plot: the mean is roughly equal to the median (plotted above 0), and the slope is roughly the SD. Normal Q Q Plot Sample Quantiles rise: Theoretical Quantiles

22 Normal Quantile Plot: Short Tails Normal Q Q Plot Theoretical Quantiles Sample Quantiles

23 Normal Quantile Plot: Short Tails Short tails are hard to diagnose from a density plot! Short Tails Density N = 50 Bandwidth = 2.436

24 Normal Quantile Plot: Long Tails Normal Q Q Plot Theoretical Quantiles Sample Quantiles

25 Density Plot: Long Tails Long Tails Density N = 50 Bandwidth =

26 Normal Quantile Plot: Positive Skewness Normal Q Q Plot Theoretical Quantiles Sample Quantiles

27 Density Plot: Positive Skewness Positive Skewness Density N = 50 Bandwidth =

28 Normal Quantile Plot: Negative Skewness Normal Q Q Plot Theoretical Quantiles Sample Quantiles

29 Normal Quantile Plot: Bimodal Data Normal Q Q Plot Sample Quantiles Theoretical Quantiles

30 Density Plot: Bimodal Data Bimodal data Density N = 50 Bandwidth = 1.372

31 Other distributions Suppose we would like to make a theoretical quantile plot for a dataset X to compare to some other distribution, say a Chisquared distribution with 5 degrees of freedom. Easy:

32 Other distributions Suppose we would like to make a theoretical quantile plot for a dataset X to compare to some other distribution, say a Chisquared distribution with 5 degrees of freedom. Easy: Sort your dataset: SampleQuantiles <- sort(x)

33 Other distributions Suppose we would like to make a theoretical quantile plot for a dataset X to compare to some other distribution, say a Chisquared distribution with 5 degrees of freedom. Easy: Sort your dataset: SampleQuantiles <- sort(x) Compute the theoretical quantiles: ChiSqQuantiles <- qchisq(ppoints(x),5)

34 Other distributions Suppose we would like to make a theoretical quantile plot for a dataset X to compare to some other distribution, say a Chisquared distribution with 5 degrees of freedom. Easy: Sort your dataset: SampleQuantiles <- sort(x) Compute the theoretical quantiles: Plot: ChiSqQuantiles <- qchisq(ppoints(x),5) plot(chisqquantiles,samplequantiles, pch=19)

35 Gamma Quantile Plot Gamma Quantile Plot sort(x) Gq

36 Summary QQplots are an excellent graphical tool for comparing two samples to each other, or one sample to a theoretical distribution like the Normal. They reveal differences in location, spread and shape more clearly than do density plots or histograms.