Dictionary of Options

QQPLOT Statement

Dictionary of Options

The following entries provide detailed descriptions of options for the QQPLOT statement.

ALPHA=value-list|EST

specifies values for a mandatory shape parameter $\alpha$ $(\alpha\gt)$ for Q-Q plots requested with the BETA and GAMMA options. A plot is created for each value specified. For examples, see the entries for the BETA and GAMMA options. If you specify ALPHA=EST, a maximum likelihood estimate is computed for $\alpha$ .

ANNOTATE=SAS-data-set

ANNO=SAS-data-set

[Graphics]
specifies an input data set containing annotate variables as described in SAS/GRAPH Software: Reference. You can use this data set to add features to the plot. The ANNOTATE= data set specified in the QQPLOT statement is used for all plots created by the statement. You can also specify an ANNOTATE= data set in the PROC CAPABILITY statement to enhance all plots created by the procedure; for more information, see "ANNOTATE= Data Sets".

BETA(ALPHA=value-list|EST BETA=value-list|EST <beta-options >)

creates a beta Q-Q plot for each combination of the shape parameters $\alpha$ and $\beta$ given by the mandatory ALPHA= and BETA= options. If you specify ALPHA=EST and BETA=EST, a plot is created based on maximum likelihood estimates for $\alpha$ and $\beta$ .In the following example, the first QQPLOT statement produces one plot, the second statement produces four plots, the third statement produces six plots, and the fourth statement produces one plot:

   proc capability data=measures;
      qqplot width / beta(alpha=2 beta=2);
      qqplot width / beta(alpha=2 3 beta=1 2);
      qqplot width / beta(alpha=2 to 3 beta=1 to 2 by 0.5);
      qqplot width / beta(alpha=est beta=est);
   run;

To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile $B_{ \alpha \beta }^{-1} ( \frac{ i - 0.375}{ n + 0.25 } )$ ,where $B_{\alpha\beta}^{-1} (\cdot)$ is the inverse normalized incomplete beta function, n is the number of nonmissing observations, and $\alpha$ and $\beta$ are the shape parameters of the beta distribution.

The point pattern on the plot for ALPHA= $\alpha$ and BETA= $\beta$ tends to be linear with intercept $\theta$ and slope $\sigma$ if the data are beta distributed with the specific density function


		  
 
where , and 
 
		  lower threshold parameter
		  scale parameter  
		  first shape parameter  
		  second shape parameter

To obtain graphical estimates of $\alpha$ and $\beta$ ,specify lists of values for the ALPHA= and BETA= options, and select the combination of $\alpha$ and $\beta$ that most nearly linearizes the point pattern. To assess the point pattern, you can add a diagonal distribution reference line with intercept $\theta_0$ and slope $\sigma_0$ with the beta-options THETA= $\theta_0$ and SIGMA= $\sigma_0$ .Alternatively, you can add a line corresponding to estimated values of $\theta_0$ and slope $\sigma_0$ with the beta-options THETA=EST and SIGMA=EST. Specify these options in parentheses, as in the following example:

   proc capability data=measures;
      qqplot width / beta(alpha=2 beta=3 theta=4 sigma=5);
   run;

Agreement between the reference line and the point pattern indicates that the beta distribution with parameters $\alpha$ , $\beta$ , $\theta_0$ , and $\sigma_0$ is a good fit. You can specify the SCALE= option as an alias for the SIGMA= option and the THRESHOLD= option as an alias for the THETA= option.

BETA=value-list|EST

specifies values for the shape parameter $\beta$ $(\beta \gt)$ for Q-Q plots requested with the BETA distribution option. A plot is created for each value specified with the BETA= option. If you specify BETA=EST, a maximum likelihood estimate is computed for $\beta$ .For examples, see the preceding entry for the BETA distribution option.

C=value(-list)|EST

specifies the shape parameter c (c>0) for Q-Q plots requested with the WEIBULL and WEIBULL2 options. You must specify C= as a Weibull-option with the WEIBULL option; in this situation it accepts a list of values, or if you specify C=EST, a maximum likelihood estimate is computed for c. You can optionally specify C=value or C=EST as a Weibull2-option with the WEIBULL2 option to request a distribution reference line; in this situation, you must also specify SIGMA=value or SIGMA=EST. For an example, see Output 10.3.1.

CAXIS=color

CAXES=color

[Graphics]
specifies the color for the axes. This option overrides any COLOR= specifications in an AXIS statement. The default is the first color in the device color list.

CFRAME=color

CFR=color

[Graphics]
specifies the color for shading the area enclosed by the axes and frame. This area is not shaded by default.

CHREF=color

CH=color

[Graphics]
specifies the color for reference lines requested with the option. The default is the first color in the device color list.

COLOR=color

[Graphics]
specifies the color for a distribution reference line. Specify the COLOR= option in parentheses following a distribution option keyword. For an example, see Figure 10.3. The default is the fourth color in the device color list.

CPKREF

[Graphics]
draws reference lines extending from the intersections of the specification limits with the distribution reference line to the quantile axis in plots requested with the NORMAL option. Specify CPKREF in parentheses after the NORMAL option. You can use the CPKREF option with the CPKSCALE option for graphical estimation of the capability indices CPU, CPL, and C_pk, as illustrated in Output 10.4.1.

CPKSCALE

rescales the quantile axis in C_pk units for plots requested with the NORMAL option. Specify CPKSCALE in parentheses after the NORMAL option. You can use the CPKSCALE option with the CPKREF option for graphical estimation of the capability indices CPU, CPL, and C_pk, as illustrated in Output 10.4.1.

CTEXT=color

[Graphics]
specifies the color for tick mark values and axis labels. The default is the color specified for the CTEXT= option in the most recent GOPTIONS statement. In the absence of a GOPTIONS statement, the default color is the first color in the device color list.

CVREF=color

CV=color

[Graphics]
specifies the color for reference lines requested by the VREF= option. The default is the first color in the device color list.

DESCRIPTION='string'

DES='string'

[Graphics]
specifies a description, up to 40 characters, that appears in the PROC GREPLAY master menu. The default string is the variable name.

EXPONENTIAL(<(exponential-options)>

EXP<(exponential-options)>)

creates an exponential Q-Q plot. To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile -log( 1 - [( i - 0.375)/( n + 0.25 )] ), where n is the number of nonmissing observations.

The pattern on the plot tends to be linear with intercept $\theta$ and slope $\sigma$ if the data are exponentially distributed with the specific density function

$p( x )= \{ \frac{ 1 }{ \sigma } \exp ( - \frac{ x - \theta }{ \sigma } ) & { for x \geq \theta \space } \ 0 & { for x \lt \theta \space } .$

where $\theta$ is the threshold parameter, and $\sigma$ is the scale parameter $(\sigma \gt)$ .

To assess the point pattern, you can add a diagonal distribution reference line with intercept $\theta_0$ and slope $\sigma_0$ with the exponential-options THETA= $\theta_0$ and SIGMA= $\sigma_0$ .Alternatively, you can add a line corresponding to estimated values of $\theta_0$ and slope $\sigma_0$ with the exponential-options THETA=EST and SIGMA=EST. Specify these options in parentheses, as in the following example: as in the following example:

   proc capability data=measures;
      qqplot width / exponential(theta=4 sigma=5);
   run;

Agreement between the reference line and the point pattern indicates that the exponential distribution with parameters $\theta_0$ and $\sigma_0$ is a good fit. You can specify the SCALE= option as an alias for the SIGMA= option and the THRESHOLD= option as an alias for the THETA= option.

FONT=font

[Graphics]
specifies a software font for horizontal and vertical reference line labels and axis labels. You can also specify fonts for axis labels in an AXIS statement. The FONT= font takes precedence over the FTEXT= font you specify in the GOPTIONS statement. Hardware characters are used by default.

GAMMA(ALPHA=value-list|EST <gamma-options> )

creates a gamma Q-Q plot for each value of the shape parameter $\alpha$ given by the mandatory ALPHA= option or its alias, the SHAPE= option. The following example produces three probability plots:

   proc capability data=measures;
      qqplot width / gamma(alpha=0.4 to 0.6 by 0.1);
   run;

To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile $G_{\alpha}^{-1} ( \frac{i- 0.375 }{n+0.25} )$ ,where $G_{\alpha}^{-1}(\cdot)$ is the inverse normalized incomplete gamma function, n is the number of nonmissing observations, and $\alpha$ is the shape parameter of the gamma distribution.

The pattern on the plot for ALPHA= $\alpha$ tends to be linear with intercept $\theta$ and slope $\sigma$ if the data are gamma distributed with the specific density function


		 
 
where  
 
		  threshold parameter
		  scale parameter  
		  shape parameter

To obtain a graphical estimate of $\alpha$ ,specify a list of values for the ALPHA= option, and select the value that most nearly linearizes the point pattern.

To assess the point pattern, you can add a diagonal distribution reference line with intercept $\theta_0$ and slope $\sigma_0$ with the gamma-options THETA= $\theta_0$ and SIGMA= $\sigma_0$ .Alternatively, you can add a line corresponding to estimated values of $\theta_0$ and $\sigma_0$ with the gamma-options THETA=EST and SIGMA=EST. Specify these options in parentheses, as in the following example:

   proc capability data=measures;
      qqplot width / gamma(alpha=2 theta=3 sigma=4);
   run;

Agreement between the reference line and the point pattern indicates that the gamma distribution with parameters $\alpha$ , $\theta_0$ ,and $\sigma_0$ is a good fit. You can specify the SCALE= option as an alias for the SIGMA= option and the THRESHOLD= option as an alias for the THETA= option.

HAXIS=name

[Graphics]
specifies the name of an AXIS statement describing the horizontal axis.

HMINOR=n

HM=n

[Graphics]
specifies the number of minor tick marks between each major tick mark on the horizontal axis. Minor tick marks are not labeled. The default is 0.

HREF=value-list

draws reference lines perpendicular to the horizontal axis at the values specified. See Example 10.3 for illustrations. Related options include the HREFCHAR=, CHREF=, and LHREF=options.

HREFCHAR='character'

[Line Printer]
specifies the character used to form the reference lines requested by the HREF=option for a line printer. The default is the vertical bar (|).

HREFLABELS='label1' ... 'labeln'

HREFLABEL='label1' ... 'labeln'

HREFLAB='label1' ... 'labeln'

specifies labels for the reference lines requested by the HREF=option. The number of labels must equal the number of lines. Enclose each label in quotes. Labels can be up to 16 characters.

L=linetype

[Graphics]
specifies the line type for a distribution reference line. Specify the L= option in parentheses following a distribution option keyword. The default is 1, which produces a solid line.

LEGEND=name | NONE

specifies the name of a LEGEND statement describing the legend for specification limit reference lines and fitted curves. Specifying LEGEND=NONE is equivalent to specifying the NOLEGEND option.

LHREF=linetype

LH=linetype

[Graphics]
specifies the line type for reference lines requested by the HREF=option. The default is 2, which produces a dashed line.

LOGNORMAL(SIGMA=value-list|EST <lognormal-options >)

LNORM(SIGMA=value-list|EST <lognormal-options >)

creates a lognormal Q-Q plot for each value of the shape parameter $\sigma$ given by the mandatory SIGMA= option or its alias, the SHAPE= option. For example,

   proc capability data=measures;
      qqplot width/ lognormal(shape=1.5 2.5);
   run;

To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile $\exp( \sigma\Phi^{-1} ( \frac{i- 0.375}{n+ 0.25 } ) )$ ,where $\Phi^{-1}(\cdot)$ is the inverse cumulative standard normal distribution, n is the number of nonmissing observations, and $\sigma$ is the shape parameter of the lognormal distribution.

The pattern on the plot for SIGMA= $\sigma$ tends to be linear with intercept $\theta$ and slope $\exp(\zeta)$ if the data are lognormally distributed with the specific density function


		 
 
where
		  threshold parameter
		  scale parameter
		  shape parameter

To obtain a graphical estimate of $\sigma$ ,specify a list of values for the SIGMA= option, and select the value that most nearly linearizes the point pattern. For an illustration, see Example 10.2.

To assess the point pattern, you can add a diagonal distribution reference line corresponding to the threshold parameter $\theta_0$ and the scale parameter $\zeta_0$ with the lognormal-options THETA= $\theta_0$ and ZETA= $\zeta_0$ .Alternatively, you can add a line corresponding to estimated values of $\theta_0$ and $\zeta_0$ with the lognormal-options THETA=EST and ZETA=EST. This line has intercept $\theta_0$ and slope $\exp(\zeta_0)$ . Agreement between the line and the point pattern indicates that the lognormal distribution with parameters $\sigma$ , $\theta_0$ , and $\zeta_0$ is a good fit. See Output 10.2.4 for an example. You can specify the THRESHOLD= option as an alias for the THETA= option and the SCALE= option as an alias for the ZETA= option. You can also display the reference line by specifying THETA= $\theta_0$ , and you can specify the slope with the SLOPE= option. For example, the following two QQPLOT statements produce charts with identical reference lines:

   proc capability data=measures;
      qqplot width / lognormal(sigma=2 theta=3 zeta=1);
      qqplot width / lognormal(sigma=2 theta=3 slope=2.718);
   run;

LVREF=linetype

LV=linetype

[Graphics]
specifies the line type for reference lines requested by the VREF= option. The default is 2, which produces a dashed line.

MU=value|EST

specifies a value for the mean $\mu$ for a normal Q-Q plot requested with the NORMAL option. Specify MU= $\mu_0$ and SIGMA= $\sigma_0$ to request a distribution reference line with intercept $\mu_0$ and slope $\sigma_0$ .Specify MU=EST to request a distribution reference line with intercept equal to the sample mean, as illustrated in Figure 10.3.

NADJ=value

specifies the adjustment value added to the sample size in the calculation of theoretical quantiles. The default is (1/4), as described by Blom (1958). Also refer to Chambers and others (1983) for additional information.

NAME='string '

[Graphics]
specifies a name for the plot, up to eight characters, that appears in the PROC GREPLAY master menu. The default name is 'CAPABILI'.

NOFRAME

suppresses the frame around the area bounded by the axes.

NOLEGEND

LEGEND=NONE

suppresses legends for specification limits, fitted curves, distribution lines, and hidden observations. For an example, see Output 10.4.1.

NOLINELEGEND

NOLINEL

suppresses the legend for the optional distribution reference line.

NOOBSLEGEND

NOOBSL

[Line Printer]
suppresses the legend that indicates the number of hidden observations.

NORMAL<(normal-options)>

NORM<(normal-options)>

creates a normal Q-Q plot. This is the default if you do not specify a distribution option. To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile $\Phi^{-1} ( \frac{i- 0.375}{n+ 0.25} )$ ,where $\Phi^{-1}(\cdot)$ is the inverse cumulative standard normal distribution, and n is the number of nonmissing observations.

The pattern on the plot tends to be linear with intercept $\mu$ and slope $\sigma$ if the data are normally distributed with the specific density function

$p(x) = \frac{1}{\sigma \sqrt{2 \pi} } \exp ( -\frac{(x - \mu)^2}{2 \sigma^2} ) & {for all x} \$

where $\mu$ is the mean, and $\sigma$ is the standard deviation $(\sigma \gt)$ .

To assess the point pattern, you can add a diagonal distribution reference line with intercept $\mu_0$ and slope $\sigma_0$ with the normal-options MU= $\mu_0$ and SIGMA= $\sigma_0$ .Alternatively, you can add a line corresponding to estimated values of $\mu_0$ and $\sigma_0$ with the normal-options THETA=EST and SIGMA=EST; the estimates of $\mu_0$ and ]sigma₀ are the sample mean and sample standard deviation. Specify these options in parentheses, as in the following example:

   proc capability data=measures;
      qqplot length / normal(mu=10 sigma=0.3);
   run;

For an example, see "Adding a Distribution Reference Line". Agreement between the reference line and the point pattern indicates that the normal distribution with parameters $\mu_0$ and $\sigma_0$ is a good fit. You can specify MU=EST and SIGMA=EST to request a distribution reference line with the sample mean and sample standard deviation as the intercept and slope.

Other normal-options include CPKREF and CPKSCALE. The CPKREF option draws reference lines extending from the intersections of specification limits with the distribution reference line to the theoretical quantile axis. The CPKSCALE option rescales the theoretical quantile axis in C_pk units. You can use the CPKREF option with the CPKSCALE option for graphical estimation of the capability indices CPU, CPL, and C_pk, as illustrated in Output 10.4.1.

NOSPECLEGEND

NOSPECL

suppresses the legend for specification limit reference lines. For an example, see Figure 10.3.

PCTLAXIS(axis-options)

adds a nonlinear percentile axis along the frame of the Q-Q plot opposite the theoretical quantile axis. The added axis is identical to the axis for probability plots produced with the PROBPLOT statement. When using the PCTLAXIS option, you must specify HREF=values in quantile units, and you cannot use the NOFRAME option. You can specify the following axis-options:

GRID	draws vertical grid lines at major percentiles
GRIDCHAR='character'	specifies grid line plotting character on line printer
LABEL='string'	specifies label for percentile axis
LGRID=linetype	specifies line type for grid

See CAPQQ1 in the SAS/QC Sample Library

For example, the following statements display the plot in Figure 10.4:

   title 'Normal Quantile-Quantile Plot for Hole Distance'; 
   proc capability data=sheets noprint;
      qqplot distance / normal(mu=est sigma=est color=yellow w=2)
                        pctlaxis(grid lgrid=35 label='Normal Percentiles')
                        nolegend
                        cframe = ligr;
   run;

Figure 10.4: Normal Q-Q Plot with Percentile Axis

PCTLMINOR

requests minor tick marks for the percentile axis displayed when you use the PCTLAXIS option. See the entry for the PCTLAXIS option for an example.

PCTLSCALE

requests scale labels for the theoretical quantile axis in percentile units, resulting in a nonlinear axis scale. Tick marks are drawn uniformly across the axis based on the quantile scale. In all other respects, the plot remains the same, and you must specify HREF=values in quantile units. For a true nonlinear axis, use the PCTLAXIS option or use the PROBPLOT statement. For example, the following statements display the plot in Figure 10.5:

See CAPQQ1 in the SAS/QC Sample Library

   title 'Normal Quantile-Quantile Plot for Hole Distance';
   proc capability data=sheets noprint;
      spec  lsl=9.5     usl=10.5
            llsl=2      lusl=20
            clsl=blue  cusl=blue;
      qqplot distance / normal(mu=est sigma=est color=yellow cpkref w=2)
                        pctlscale
                        pctlaxis(grid lgrid=35)
                        nolegend
                        cframe = ligr;
   run;

Figure 10.5: Normal Q-Q Plot for Reading Percentiles of Specification Limits

QQSYMBOL='character'

[Line Printer]
specifies the character used to plot the Q-Q points on a line printer. The default is the plus sign (+).

RANKADJ=value

specifies the adjustment value added to the ranks in the calculation of theoretical quantiles. The default is -(3/8), as described by Blom (1958). Also refer to Chambers and others (1983) for additional information.

ROTATE

[Graphics]
switches the horizontal and vertical axes so that the theoretical percentiles are plotted vertically while the data are plotted horizontally. Regardless of whether the plot has been rotated, horizontal axis options (such as HAXIS=) refer to the horizontal axis, and vertical axis options (such as VAXIS=) refer to the vertical axis. All other options that depend on axis placement adjust to the rotated axes.

SCALE=value|EST

is an alias for the SIGMA= option with the BETA, EXPONENTIAL, GAMMA, WEIBULL, and WEIBULL2 options and for the ZETA= option with the LOGNORMAL option. See the entries for the SIGMA= and ZETA= options.

SHAPE=value-list|EST

is an alias for the ALPHA= option with the GAMMA option, for the SIGMA= option with the LOGNORMAL option, and for the C= option with the WEIBULL and WEIBULL2 options. See the entries for the ALPHA=, C=, and SIGMA= options.

SIGMA=value-list|EST

specifies the value of the distribution parameter $\sigma$ ,where $\sigma\gt$ . Alternatively, you can specify SIGMA=EST to request a maximum likelihood estimate for $\sigma_0$ .The use of the SIGMA= option depends on the distribution option specified, as indicated by the following table:

Distribution Option	Use of the SIGMA= Option
BETA	THETA= $\theta_0$ and SIGMA= $\sigma_0$ request a distribution reference
EXPONENTIAL	line with intercept $\theta_0$ and slope $\sigma_0$ .
GAMMA
WEIBULL
LOGNORMAL	SIGMA= $\sigma_1 ... \sigma_n$ requests n Q-Q plots with shape parameters $\sigma_1 ... \sigma_n$ . The SIGMA= option is mandatory.
NORMAL	MU= $\mu_0$ and SIGMA= $\sigma_0$ request a distribution reference line with intercept $\mu_0$ and slope $\sigma_0$ . SIGMA=EST requests a slope equal to the sample standard deviation.
WEIBULL2	SIGMA= $\sigma_0$ and C=c₀ request a distribution reference line with intercept $\log(\sigma_0)$ and slope [1/(c₀)].

For an example using SIGMA=EST, see Output 10.4.1. For an example of lognormal plots using the SIGMA= option, see Example 10.2.

SLOPE=value|EST

specifies the slope for a distribution reference line requested with the LOGNORMAL and WEIBULL2 options.

When you use the SLOPE= option with the LOGNORMAL option, you must also specify a threshold parameter value $\theta_0$ with the THETA= option. Specifying the SLOPE= option is an alternative to specifying ZETA= $\zeta_0$ , which requests a slope of $\exp(\zeta_0)$ .See Output 10.2.4 for an example.

When you use the SLOPE= option with the WEIBULL2 option, you must also specify a scale parameter value $\sigma_0$ with the SIGMA= option. Specifying the SLOPE= option is an alternative to specifying C=c₀, which requests a slope of [1/(c₀)].

For example, the first and second QQPLOT statements that follow produce plots identical to those produced by the third and fourth QQPLOT statements:

   proc capability data=measures;
      qqplot width  / lognormal(sigma=2 theta=0 zeta=0);
      qqplot width  / weibull2(sigma=2 theta=0 c=0.25);
      qqplot width  / lognormal(sigma=2 theta=0 slope=1);
      qqplot width  / weibull2(sigma=2 theta=0 slope=4);
   run;

For more information, see "Graphical Estimation".

SQUARE

displays the Q-Q plot in a square frame. Compare Figure 10.1 with Figure 10.3. The default is a rectangular frame.

SYMBOL='character'

[Line Printer]
specifies the character used to plot a distribution reference line when the plot is produced on a line printer. The default character is the first letter of the distribution option keyword.

THETA=value|EST

specifies the lower threshold parameter $\theta$ for Q-Q plots requested with the BETA, EXPONENTIAL, GAMMA, LOGNORMAL, WEIBULL, and WEIBULL2 options.

When used with the WEIBULL2 option, the THETA= option specifies the known lower threshold $\theta_0$ , for which the default is 0. See Output 10.3.2 for an example.

When used with the other distribution options, the THETA= option specifies $\theta_0$ for a distribution reference line; alternatively in this situation, you can specify THETA=EST to request a maximum likelihood estimate for $\theta_0$ .To request the line, you must also specify a scale parameter See Output 10.2.4 for an example of the THETA= option with a lognormal Q-Q plot.

THRESHOLD=value|EST

is an alias for the THETA= option.

VAXIS=name

[Graphics]
specifies the name of an AXIS statement describing the vertical axis. For an example, see Example 10.1.

VMINOR=n

VM=n

[Graphics]
specifies the number of minor tick marks between each major tick mark on the vertical axis. Minor tick marks are not labeled. The default is 0.

VREF=value-list

draws reference lines perpendicular to the vertical axis at the values specified. For illustrations, see Output 10.2.4 or Example 10.3. Related options include the VREFCHAR=, CVREF=, and LVREF= options.

VREFCHAR='character'

[Line Printer]
specifies the character used to form the reference lines requested by the VREF= option for a line printer. The default is the hyphen (-).

VREFLABELS='label1' ... 'labeln'

VREFLABEL='label1' ... 'labeln'

VREFLAB='label1' ... 'labeln'

specifies labels for the reference lines requested by the VREF= option. The number of labels must equal the number of lines. Enclose each label in quotes. Labels can be up to 16 characters.

W=n

[Graphics]
specifies the width in pixels for a distribution reference line, as in the following example. The default is 1.

   proc capability data=measures;
      qqplot length / normal(mu=5 sigma=2 w=2);
   run;

WEIBULL(C=value-list|EST <Weibull-options >)

WEIB(C=value-list <Weibull-options >)

creates a three-parameter Weibull Q-Q plot for each value of the shape parameter c given by the mandatory C= option or its alias, the SHAPE= option. For example,

   proc capability data=measures;
      qqplot width / weibull(c=1.8 to 2.4 by 0.2);
   run;

To create the plot, the observations are ordered from smallest to largest, and the i^th ordered observation is plotted against the quantile ( - log( 1- [( i - 0.375 )/( n + 0.25 )] ) )^[1/c], where n is the number of nonmissing observations, and c is the Weibull distribution shape parameter.

The pattern on the plot for C=c tends to be linear with intercept $\theta$ and slope $\sigma$ if the data are Weibull distributed with the specific density function

$p(x)= \{ \frac{c}{\sigma} ( \frac{x - \theta}{\sigma} )^{c - 1} \exp ( - ( \f... ...sigma} )^c ) & { for x \gt \theta\space } \ 0 & { for x \leq \theta\space } .$

where $\theta$ is the threshold parameter, $\sigma$ is the scale parameter $(\sigma \gt)$ ,and c is the shape parameter ( c > 0 ).

To obtain a graphical estimate of c, specify a list of values for the C= option, and select the value that most nearly linearizes the point pattern. For an illustration, see Example 10.3. To assess the point pattern, you can add a diagonal distribution reference line with intercept $\theta_0$ and slope $\sigma_0$ with the Weibull-options THETA= $\theta_0$ and SIGMA= $\sigma_0$ .Alternatively, you can add a line corresponding to estimated values of $\theta_0$ and $\sigma_0$ with the Weibull-options THETA=EST and SIGMA=EST. Specify these options in parentheses, as in the following example:

   proc capability data=measures;
      qqplot width / weibull(c=2 theta=3 sigma=4);
   run;

Agreement between the reference line and the point pattern indicates that the Weibull distribution with parameters c, $\theta_0$ ,and $\sigma_0$ is a good fit. You can specify the SCALE= option as an alias for the SIGMA= option and the THRESHOLD= option as an alias for the THETA= option.

WEIBULL2<(Weibull2-options)>

W2<(Weibull2-options)>

creates a two-parameter Weibull Q-Q plot. You should use the WEIBULL2 option when your data have a known lower threshold $\theta_0$ . You can specify the threshold value $\theta_0$ with the THETA= option or its alias, the THRESHOLD= option. If you are uncertain of the lower threshold value, you can estimate $\theta_0$ graphically by specifying a list of values for the THETA= option. Select the value that most linearizes the point pattern. The default is $\theta_0 = 0$ .

To create the plot, the observations are ordered from smallest to largest, and the log of the shifted i^th ordered observation x_(i), $\log(x_{(i)}-\theta_0)$ ,is plotted against the quantile log(-log(1-[(i- 0.375)/(n+ 0.25)] ) ), where n is the number of nonmissing observations. Unlike the three-parameter Weibull quantile, the preceding expression is free of distribution parameters. This is why the C= shape parameter option is not mandatory with the WEIBULL2 option.

The pattern on the plot for THETA= $\theta_0$ tends to be linear with intercept $\log(\sigma)$ and slope [1/c] if the data are Weibull distributed with the specific density function

$p(x) = \{ \frac{c}{\sigma} ( \frac{x - \theta_0}{\sigma} )^{c - 1} \exp ( - (... ...ma} )^c ) & { for x \gt \theta_0\space } \ 0 & { for x \leq \theta_0\space } .$

where $\theta_0$ is a known lower threshold parameter, $\sigma$ is a scale parameter $(\sigma \gt)$ ,and c is a shape parameter (c>0).

The advantage of a two-parameter Weibull plot over a three-parameter Weibull plot is that you can visually estimate the shape parameter c and the scale parameter $\sigma$ from the slope and intercept of the point pattern; see Example 10.3 for an illustration of this method. The disadvantage is that the two-parameter Weibull distribution applies only in situations where the threshold parameter is known. See "Graphical Estimation" for more information.

To assess the point pattern, you can add a diagonal distribution reference line corresponding to the scale parameter $\sigma_0$ and shape parameter c₀ with the Weibull2-options SIGMA= $\sigma_0$ and C=c₀. Alternatively, you can add a distribution reference line corresponding to estimated values of $\sigma_0$ and c₀ with the Weibull2-options SIGMA=EST and C=EST. This line has intercept $\log(\sigma_0)$ and slope [1/(c₀)]. Agreement between the line and the point pattern indicates that the Weibull distribution with parameters c₀, $\theta_0$ , and $\sigma_0$ is a good fit. You can specify the SCALE= option as an alias for the SIGMA= option and the SHAPE= option as an alias for the C= option.

You can also display the reference line by specifying SIGMA= $\sigma_0$ , and you can specify the slope with the SLOPE= option. For example, the following QQPLOT statements produce identical plots:

   proc capability data=measures;
      qqplot width / weibull2(theta=3 sigma=4 c=2);
      qqplot width / weibull2(theta=3 sigma=4 slope=0.5);
   run;

ZETA=value|EST

specifies a value for the scale parameter $\zeta$ for lognormal Q-Q plots requested with the LOGNORMAL option. Specify THETA= $\theta_0$ and ZETA= $\zeta_0$ to request a distribution reference line with intercept $\theta_0$ and slope $\exp(\zeta_0)$ .

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