Chapter Contents |
Previous |
Next |

QQPLOT Statement |

See CAPQQ2 in the SAS/QC Sample Library |

The three-parameter lognormal distribution depends on a threshold parameter , a scale parameter , and a shape parameter .You can estimate from a series of lognormal Q-Q plots with different values of .The estimate is the value of that linearizes the point pattern. You can then estimate the threshold and scale parameters from the intercept and slope of the point pattern. The following statements create the series of plots in Output 10.2.1 through Output 10.2.3 for values of 0.2, 0.5, and 0.8:

title 'Lognormal Q-Q Plot for Diameters'; proc capability data=measures noprint; qqplot diameter / lognormal(sigma=0.2 0.5 0.8) square cframe = ligr; run;

The plot in Output 10.2.2 displays the most linear point pattern, indicating that the lognormal distribution with provides a reasonable fit for the data distribution.

Data with this particular lognormal distribution have the density function

The points in the plot fall on or near the line with intercept and slope .Based on Output 10.2.2, and , giving .

See CAPQQ2 in the SAS/QC Sample Library |

You can use a Q-Q plot to estimate percentiles such as
the 95^{ th}
percentile of the lognormal distribution.^{*}

The point pattern in Output 10.2.2 has a slope of
approximately 0.39 and an intercept of 5.
The following statements
reproduce this plot, adding a lognormal
reference line with this slope and intercept.
The result is shown in Output 10.2.4.

title 'Lognormal Q-Q Plot for Diameters'; legend1 frame cframe=ligr cborder=black position=center; proc capability data=measures noprint; qqplot diameter / lognormal(sigma=0.5 theta=5 slope=0.39 color=yellow) pctlaxis(grid lgrid=35) vref = 5.8 5.9 6.0 cvref = blue cframe = ligr legend=legend1; run;

The PCTLAXIS option labels the major percentiles, and the GRID option draws percentile axis reference lines. The 95

Alternatively, you can compute this percentile from
the estimated lognormal parameters.
The percentile of the lognormal distribution is

where is the inverse cumulative standard normal distribution. Consequently,

See CAPQQ2 in the SAS/QC Sample Library |

If a known threshold parameter is available, you can construct a two-parameter lognormal Q-Q plot by subtracting the threshold from the data and requesting a normal Q-Q plot. The following statements create this plot for DIAMETER, assuming a known threshold of five:

data measures; set measures; label logdiam = 'log(Diameter-5)'; logdiam = log( diameter - 5 );

title 'Two-Parameter Lognormal Q-Q Plot for Diameters'; legend1 frame cframe=ligr cborder=black position=center; proc capability data=measures noprint; qqplot logdiam / normal(mu=est sigma=est color=yellow) vaxis=axis1 square cframe=ligr legend=legend1; axis1 label=(a=90 r=0); run;

Because the point pattern in Output 10.2.5 is linear, you can estimate the lognormal parameters and as the normal plot estimates of and ,which are -0.99 and 0.51. These values correspond to the previous estimates of -0.92 for and 0.5 for .

Chapter Contents |
Previous |
Next |
Top |

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.