Graphical Estimation

QQPLOT Statement

Graphical Estimation

You can use Q-Q plots to estimate shape, location, and scale parameters and to estimate percentiles. If you are working with a normal Q-Q plot, you can also estimate certain capability indices.

Shape Parameters

Some distribution options in the QQPLOT statement require that you specify one or two shape parameters in parentheses after the distribution keyword. These are summarized in Table 10.15.

You can visually estimate a shape parameter by specifying a list of values for the shape parameter option. A separate plot is displayed for each value, and you can then select the value that linearizes the point pattern. Alternatively, you can request that the plot be created using an estimated shape parameter. See the entries for the distribution options in "Dictionary of Options" for details on specification of shape parameters. Example 10.2 and Example 10.3 illustrate shape parameter estimation with lognormal and Weibull Q-Q plots.

Note that for Q-Q plots requested with the WEIBULL2 option, you can estimate the shape parameter c from a linear pattern using the fact that the slope of the pattern is [1/c]. For an illustration, see Example 10.3.

Table 10.15: Shape Parameter Options for the QQPLOT Statement

Distribution Keyword	Mandatory Shape Parameter Option	Range
BETA	ALPHA= $\alpha$ , BETA= $\beta$	$\alpha\gt$ , $\beta\gt$
EXPONENTIAL	None
GAMMA	ALPHA= $\alpha$	$\alpha\gt$
LOGNORMAL	SIGMA= $\sigma$	$\sigma\gt$
NORMAL	None
WEIBULL	C=c	c>0
WEIBULL2	None

Location and Scale Parameters

When the point pattern on a Q-Q plot is linear, its intercept and slope provide estimates of the location and scale parameters. (An exception to this rule is the two-parameter Weibull distribution, for which the intercept and slope are related to the scale and shape parameters.) Table 10.16 shows how the intercept and slope are related to the parameters for each distribution supported by the QQPLOT statement.

Table 10.16: Intercept and Slope of Linear Q-Q Plots

	Parameters			Linear Pattern
Distribution	Location	Scale	Shape	Intercept	Slope
Beta	$\theta$	$\sigma$	$\alpha$ , $\beta$	$\theta$	$\sigma$
Exponential	$\theta$	$\sigma$		$\theta$	$\sigma$
Gamma	$\theta$	$\sigma$	$\alpha$	$\theta$	$\sigma$
Lognormal	$\theta$	$\zeta$	$\sigma$	$\theta$	$\exp(\zeta)$
Normal	$\mu$	$\sigma$		$\mu$	$\sigma$
Weibull (3-parameter)	$\theta$	$\sigma$	c	$\theta$	$\sigma$
Weibull (2-parameter)	$\theta_0$ (known)	$\sigma$	c	$\log(\sigma)$	[1/c]

You can enhance a Q-Q plot with a diagonal distribution reference line by specifying the parameters that determine the slope and intercept of the line; alternatively, you can request estimates for these parameters. This line is an aid to checking the linearity of the point pattern, and it facilitates parameter estimation. For instance, specifying MU=3 and SIGMA=2 with the NORMAL option requests a line with intercept 3 and slope 2. Specifying SIGMA=1 and C=2 with the WEIBULL2 option requests a line with intercept log(1) = 0 and slope (1/2).

With the LOGNORMAL and WEIBULL2 options, you can specify the slope directly with the SLOPE= option. That is, for the LOGNORMAL option, specifying THETA= $\theta_0$ and SLOPE= $\exp(\zeta_0)$ gives the same reference line as specifying THETA= $\theta_0$ and ZETA= $\zeta_0$ . For the WEIBULL2 option, specifying SIGMA= $\sigma_0$ and SLOPE=[1/(c₀)] gives the same reference line as specifying SIGMA= $\sigma_0$ and C=c₀.

For an example of parameter estimation using a normal Q-Q plot, see "Adding a Distribution Reference Line". Example 10.2 illustrates parameter estimation using a lognormal plot, and Example 10.3 illustrates estimation using two-parameter and three-parameter Weibull plots.

Theoretical Percentiles

There are two ways to estimate percentiles from a Q-Q plot:

Specify the PCTLAXIS option, which adds a percentile axis opposite the theoretical quantile axis. The scale for the percentile axis ranges between 0 and 100 with tick marks at percentile values such as 1, 5, 10, 25, 50, 75, 90, 95, and 99. See Figure 10.4 for an example.
Specify the PCTLSCALE option, which relabels the horizontal axis tick marks with their percentile equivalents but does not alter their spacing. For example, on a normal Q-Q plot, the tick mark labeled "0" is relabeled as "50" since the 50^th percentile corresponds to the zero quantile. See Figure 10.5 for an example.

You can also estimate percentiles using probability plots created with the PROBPLOT statement. See Output 9.2.1 for an example.

Capability Indices

When the point pattern on a normal Q-Q plot is linear, you can estimate the capability indices CPU, CPL, and C_pk from the plot, as explained by Rodriguez (1992). This method exploits the fact that the horizontal axis of a Q-Q plot indicates the distance in standard deviation units (multiple of $\sigma$ ) between a measurement or specification limit and the process average.

In particular, one-third the standardized distance between an upper specification limit and the mean is the one-sided capability index CPU.

${CPU} = \frac{USL - \mu}{3\sigma}$

Likewise, one-third the standardized distance between a lower specification limit and the mean is the one-sided capability index CPL.

${CPL} = \frac{\mu - LSL}{3\sigma}$

Consequently, if you rescale the quantile axis of a normal Q-Q plot by a factor of three, you can read CPU and CPL from the horizontal coordinates of the points at which the upper and lower specification lines intersect the point pattern. Since C_pk is defined as the minimum of CPU and CPL, this method also provides a graphical estimate of C_pk. For an illustration, see Example 10.4.

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