Construction and Interpretation of P-P Plots :: SAS/QC(R) 12.3 User's Guide

Construction and Interpretation of P-P Plots

A P-P plot compares the empirical cumulative distribution function (ecdf) of a variable with a specified theoretical cumulative distribution function $F(\cdot )$ . The ecdf, denoted by , is defined as the proportion of nonmissing observations less than or equal to x, so that $F_ n (x_{(i)}) = \frac{i}{n}$ .

To construct a P-P plot, the n nonmissing values are first sorted in increasing order:

$x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$

Then the ith ordered value $x_{(i)}$ is represented on the plot by the point whose x-coordinate is $F(x_{(i)})$ and whose y-coordinate is $\frac{i}{n}$ .

Like Q-Q plots and probability plots, P-P plots can be used to determine how well a theoretical distribution models a data distribution. If the theoretical cdf reasonably models the ecdf in all respects, including location and scale, the point pattern on the P-P plot is linear through the origin and has unit slope.

Unlike Q-Q and probability plots, P-P plots are not invariant to changes in location and scale. See CAPPP2 in the SAS/QC Sample LibraryFor example, the data in the section Getting Started: PPPLOT Statement are reasonably described by a normal distribution with mean 10 and standard deviation 0.3. It is instructive to display these data on normal P-P plots with a different mean and standard deviation, as created by the following statements:

ods graphics on;
proc capability data=Sheets noprint;
   ppplot Distance / normal(mu=9.5 sigma=0.3) square;
   ppplot Distance / normal(mu=10 sigma=0.5) square;
run;

The ODS GRAPHICS ON statement specified before the PROC CAPABILITY statement enables ODS Graphics, so the P-P plots are created using ODS Graphics instead of traditional graphics. The resulting plots are shown in Figure 5.32 and Figure 5.33.

Figure 5.32: Normal P-P Plot with Mean Specified Incorrectly

Figure 5.33: Normal P-P Plot with Standard Deviation Specified Incorrectly

Specifying a mean of 9.5 instead of 10 results in the plot shown in Figure 5.32, while specifying a standard deviation of 0.5 instead of 0.3 results in the plot shown in Figure 5.33. Both plots clearly reveal the model misspecification.