### QQPLOT Statement

Subsections:

QQPLOT <variables> < / options> ;

The QQPLOT statement creates quantile-quantile plots (Q-Q plots) and compares ordered variable values with quantiles of a specified theoretical distribution. If the data distribution matches the theoretical distribution, the points on the plot form a linear pattern. Thus, you can use a Q-Q plot to determine how well a theoretical distribution models a set of measurements.

Q-Q plots are similar to probability plots, which you can create with the PROBPLOT statement. Q-Q plots are preferable for graphical estimation of distribution parameters, whereas probability plots are preferable for graphical estimation of percentiles.

You can use any number of QQPLOT statements in the UNIVARIATE procedure. The components of the QQPLOT statement are as follows.

variables

are the variables for which Q-Q plots are created. If you specify a VAR statement, the variables must also be listed in the VAR statement. Otherwise, the variables can be any numeric variables in the input data set. If you do not specify a list of variables, then by default the procedure creates a Q-Q plot for each variable listed in the VAR statement, or for each numeric variable in the DATA= data set if you do not specify a VAR statement. For example, each of the following QQPLOT statements produces two Q-Q plots, one for Length and one for Width:

proc univariate data=Measures;
var Length Width;
qqplot;

proc univariate data=Measures;
qqplot Length Width;
run;

options

specify the theoretical distribution for the plot or add features to the plot. If you specify more than one variable, the options apply equally to each variable. Specify all options after the slash (/) in the QQPLOT statement. You can specify only one option that names the distribution in each QQPLOT statement, but you can specify any number of other options. The distributions available are the beta, exponential, gamma, lognormal, normal, two-parameter Weibull, and three-parameter Weibull. By default, the procedure produces a plot for the normal distribution.

In the following example, the NORMAL option requests a normal Q-Q plot for each variable. The MU= and SIGMA= normal-options request a distribution reference line with intercept 10 and slope 0.3 for each plot, corresponding to a normal distribution with mean and standard deviation . The SQUARE option displays the plot in a square frame, and the CTEXT= option specifies the text color.

proc univariate data=measures;
qqplot length1 length2 / normal(mu=10 sigma=0.3)
square ctext=blue;
run;


Table 4.22 through Table 4.24 list the QQPLOT options by function. For complete descriptions, see the sections Dictionary of Options and Dictionary of Common Options.

Options can be any of the following:

• primary options

• secondary options

• general options

#### Distribution Options

Table 4.22 lists primary options for requesting a theoretical distribution. See the section Distributions for Probability and Q-Q Plots for detailed descriptions of these distributions.

Table 4.22: Primary Options for Theoretical Distributions

Option

Description

specifies beta Q-Q plot for shape parameters and specified with mandatory ALPHA= and BETA= beta-options

specifies exponential Q-Q plot

specifies gamma Q-Q plot for shape parameter specified with mandatory ALPHA= gamma-option

specifies gumbel Q-Q plot

specifies lognormal Q-Q plot for shape parameter specified with mandatory SIGMA= lognormal-option

specifies normal Q-Q plot

specifies generalized Pareto Q-Q plot for shape parameter specified with mandatory ALPHA= Pareto-option

specifies power function Q-Q plot for shape parameter specified with mandatory ALPHA= power-option

specifies Rayleigh Q-Q plot

specifies three-parameter Weibull Q-Q plot for shape parameter specified with mandatory C= Weibull-option

specifies two-parameter Weibull Q-Q plot

Table 4.23 lists secondary options that specify distribution parameters and control the display of a distribution reference line. Specify these options in parentheses after the distribution keyword. For example, you can request a normal Q-Q plot with a distribution reference line by specifying the NORMAL option as follows:

proc univariate;
qqplot Length / normal(mu=10 sigma=0.3 color=red);
run;


The MU= and SIGMA= normal-options display a distribution reference line that corresponds to the normal distribution with mean and standard deviation , and the COLOR= normal-option specifies the color for the line.

Table 4.23: Secondary Distribution Reference Line Options

Option

Description

Options Used with All Distributions

specifies color of distribution reference line

specifies line type of distribution reference line

specifies width of distribution reference line

Beta-Options

specifies mandatory shape parameter

specifies mandatory shape parameter

specifies for distribution reference line

specifies for distribution reference line

Exponential-Options

specifies for distribution reference line

specifies for distribution reference line

Gamma-Options

specifies mandatory shape parameter

specifies change in successive estimates of at which the Newton-Raphson approximation of terminates

specifies initial value for in the Newton-Raphson approximation of

specifies maximum number of iterations in the Newton-Raphson approximation of

specifies for distribution reference line

specifies for distribution reference line

Gumbel-Options

specifies for distribution reference line

specifies for distribution reference line

Lognormal-Options

specifies mandatory shape parameter

specifies slope of distribution reference line

specifies for distribution reference line

specifies for distribution reference line (slope is )

Normal-Options

specifies for distribution reference line

specifies for distribution reference line

Pareto-Options

specifies mandatory shape parameter

specifies for distribution reference line

specifies for distribution reference line

Power-Options

specifies mandatory shape parameter

specifies for distribution reference line

specifies for distribution reference line

Rayleigh-Options

specifies for distribution reference line

specifies for distribution reference line

Weibull-Options

specifies mandatory shape parameter c

specifies for distribution reference line

specifies for distribution reference line

Weibull2-Options

specifies for distribution reference line (slope is )

specifies for distribution reference line (intercept is )

specifies slope of distribution reference line

specifies known lower threshold

Weibull-Options

specifies mandatory shape parameter c

requests table of iteration history and optimizer details

specifies maximum number of iterations in the Newton-Raphson approximation of

specifies for distribution reference line

specifies for distribution reference line

Weibull2-Options

specifies for distribution reference line (slope is )

requests table of iteration history and optimizer details

specifies maximum number of iterations in the Newton-Raphson approximation of

specifies for distribution reference line (intercept is )

specifies slope of distribution reference line

specifies known lower threshold

#### General Options

Table 4.24 summarizes general options for enhancing Q-Q plots.

Table 4.24: General Graphics Options

Option

Description

General Graphics Options

creates a grid

specifies reference lines perpendicular to the horizontal axis

specifies labels for HREF= lines

specifies vertical position of labels for HREF= lines

suppresses label for horizontal axis

suppresses label for vertical axis

suppresses tick marks and tick mark labels for vertical axis

displays a nonlinear percentile axis

replaces theoretical quantiles with percentiles

switches horizontal and vertical axes

displays plot in square format

specifies label for vertical axis

specifies reference lines perpendicular to the vertical axis

specifies labels for VREF= lines

specifies horizontal position of labels for VREF= lines

specifies annotate data set

specifies color for axis

specifies color for frame

specifies color for grid lines

specifies colors for HREF= lines

specifies colors for STATREF= lines

specifies color for text

specifies colors for VREF= lines

specifies description for plot in graphics catalog

specifies software font for text

specifies height of text used outside framed areas

specifies number of horizontal minor tick marks

specifies software font for text inside framed areas

specifies height of text inside framed areas

specifies a line type for grid lines

specifies line types for HREF= lines

specifies line types for STATREF= lines

specifies line types for VREF= lines

specifies name for plot in graphics catalog

suppresses frame around plotting area

requests minor tick marks for percentile axis

specifies AXIS statement for vertical axis

specifies number of vertical minor tick marks

specifies line thickness for axes and frame

specifies line thickness for grid

Options for ODS Graphics Output

specifies footnote displayed on plot

specifies secondary footnote displayed on plot

specifies title displayed on plot

specifies secondary title displayed on plot

Options for Comparative Plots

applies annotation requested in ANNOTATE= data set to key cell only

specifies color for filling frame for row labels

specifies color for filling frame for column labels

specifies color for proportion of frequency bar

specifies distance between tiles

specifies number of columns in comparative Q-Q plot

specifies number of rows in comparative Q-Q plot

Miscellaneous Options

adjusts sample size when computing percentiles

#### Dictionary of Options

The following entries provide detailed descriptions of options in the QQPLOT statement. Options marked with † are applicable only when traditional graphics are produced. See the section Dictionary of Common Options for detailed descriptions of options common to all plot statements.

ALPHA=value-list | EST

specifies the mandatory shape parameter for quantile plots requested with the BETA, GAMMA, PARETO, and POWER options. Enclose the ALPHA= option in parentheses after the distribution keyword. If you specify ALPHA=EST, a maximum likelihood estimate is computed for .

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

creates a beta quantile plot for each combination of the required shape parameters and specified by the required ALPHA= and BETA= beta-options. If you specify ALPHA=EST and BETA=EST, the procedure creates a plot based on maximum likelihood estimates for and . You can specify the SCALE= beta-option as an alias for the SIGMA= beta-option and the THRESHOLD= beta-option as an alias for the THETA= beta-option. To create a plot that is based on maximum likelihood estimates for and , specify ALPHA=EST and BETA=EST. See the section Beta Distribution for details.

To obtain graphical estimates of and , specify lists of values in the ALPHA= and BETA= beta-options and select the combination of and that most nearly linearizes the point pattern. To assess the point pattern, you can add a diagonal distribution reference line corresponding to lower threshold parameter and scale parameter with the THETA= and SIGMA= beta-options. Alternatively, you can add a line that corresponds to estimated values of and with the beta-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the beta distribution with parameters , , , and is a good fit.

BETA=value-list | EST
B=value | EST

specifies the mandatory shape parameter for quantile plots requested with the BETA option. Enclose the BETA= option in parentheses after the BETA option. If you specify BETA=EST, a maximum likelihood estimate is computed for .

C=value-list | EST

specifies the shape parameter for quantile plots requested with the WEIBULL and WEIBULL2 options. Enclose this option in parentheses after the WEIBULL or WEIBULL2 option. C= is a required Weibull-option in 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 . 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 Weibull2-option SIGMA=value or SIGMA=EST.

† CGRID=color

specifies the color for grid lines when a grid displays on the plot. This option also produces a grid.

EXPONENTIAL<(exponential-options)>
EXP<(exponential-options)>

creates an exponential quantile plot. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= exponential-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the exponential-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the exponential distribution with parameters and is a good fit. You can specify the SCALE= exponential-option as an alias for the SIGMA= exponential-option and the THRESHOLD= exponential-option as an alias for the THETA= exponential-option. See the section Exponential Distribution for details.

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

creates a gamma quantile plot for each value of the shape parameter given by the mandatory ALPHA= gamma-option. If you specify ALPHA=EST, the procedure creates a plot based on a maximum likelihood estimate for . To obtain a graphical estimate of , specify a list of values for the ALPHA= gamma-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= gamma-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the gamma-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the gamma distribution with parameters , , and is a good fit. You can specify the SCALE= gamma-option as an alias for the SIGMA= gamma-option and the THRESHOLD= gamma-option as an alias for the THETA= gamma-option. See the section Gamma Distribution for details.

GRID

displays a grid of horizontal lines positioned at major tick marks on the vertical axis.

GUMBEL<(Gumbel-options)>

creates a Gumbel quantile plot. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the MU= and SIGMA= Gumbel-options. Alternatively, you can add a line corresponding to estimated values of the location parameter and the scale parameter with the Gumbel-options MU=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the exponential distribution with parameters and is a good fit. See the section Gumbel Distribution for details.

† LGRID=linetype

specifies the line type for the grid requested by the GRID option. By default, LGRID=1, which produces a solid line. The LGRID= option also produces a grid.

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

creates a lognormal quantile plot for each value of the shape parameter given by the mandatory SIGMA= lognormal-option. If you specify SIGMA=EST, the procedure creates a plot based on a maximum likelihood estimate for . To obtain a graphical estimate of , specify a list of values for the SIGMA= lognormal-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and ZETA= lognormal-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the lognormal-options THETA=EST and ZETA=EST. Agreement between the reference line and the point pattern indicates that the lognormal distribution with parameters , , and is a good fit. You can specify the THRESHOLD= lognormal-option as an alias for the THETA= lognormal-option and the SCALE= lognormal-option as an alias for the ZETA= lognormal-option. See the section Lognormal Distribution for details, and see Example 4.31 through Example 4.33 for examples that use the LOGNORMAL option.

MU=value | EST

specifies the mean for a quantile plot requested with the GUMBEL and NORMAL options. Enclose MU= in parentheses after the distribution keyword. You can specify MU=EST to request a distribution reference line with equal to the sample mean with the normal distribution. If you specify MU=EST for the Gumbel distribution, the procedure computes a maximum likelihood estimate.

specifies the adjustment value added to the sample size in the calculation of theoretical percentiles. By default, NADJ=. Refer to Chambers et al. (1983) for additional information.

NORMAL<(normal-options)>

creates a normal quantile plot. This is the default if you omit a distribution option. To assess the point pattern, you can add a diagonal distribution reference line corresponding to and with the MU= and SIGMA= normal-options. Alternatively, you can add a line corresponding to estimated values of and with the normal-options MU=EST and SIGMA=EST; the estimates of the mean and the standard deviation are the sample mean and sample standard deviation. Agreement between the reference line and the point pattern indicates that the normal distribution with parameters and is a good fit. See the section Normal Distribution for details, and see Example 4.28 and Example 4.30 for examples that use the NORMAL option.

PARETO(ALPHA=value | EST <Pareto-options>)

creates a generalized Pareto quantile plot for each value of the shape parameter given by the mandatory ALPHA= Pareto-option. If you specify ALPHA=EST, the procedure creates a plot based on a maximum likelihood estimate for . To obtain a graphical estimate of , specify a list of values for the ALPHA= Pareto-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= Pareto-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the Pareto-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the generalized Pareto distribution with parameters , , and is a good fit. See the section Generalized Pareto Distribution for details.

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:

Table 4.25: PCTLAXIS Axis Options

Option

Description

CGRID=

specifies color for grid lines

GRID

draws grid lines at major percentiles

LABEL='string'

specifies label for percentile axis

LGRID=linetype

specifies line type for grid

WGRID=n

specifies line thickness for grid

† PCTLMINOR

requests minor tick marks for the percentile axis when you specify PCTLAXIS. The HMINOR option overrides the PCTLMINOR option.

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.

POWER(ALPHA=value | EST <power-options>)

creates a power function quantile plot for each value of the shape parameter given by the mandatory ALPHA= power-option. If you specify ALPHA=EST, the procedure creates a plot based on a maximum likelihood estimate for . To obtain a graphical estimate of , specify a list of values for the ALPHA= power-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= power-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the power-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the power function distribution with parameters , , and is a good fit. See the section Power Function Distribution for details.

RAYLEIGH<(Rayleigh-options)>

creates a Rayleigh quantile plot. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= Rayleigh-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter and the scale parameter with the Rayleigh-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the exponential distribution with parameters and is a good fit. See the section Rayleigh Distribution for details.

specifies the adjustment value added to the ranks in the calculation of theoretical percentiles. By default, RANKADJ=, as recommended by Blom (1958). Refer to Chambers et al. (1983) for additional information.

ROTATE

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

SIGMA=value | EST

specifies the parameter , where . Alternatively, you can specify SIGMA=EST to request a maximum likelihood estimate for . The interpretation and use of the SIGMA= option depend on the distribution option with which it is used, as summarized in Table 4.26. Enclose this option in parentheses after the distribution option.

Table 4.26: Uses of the SIGMA= Option

Distribution Option

Use of the SIGMA= Option

BETA EXPONENTIAL GAMMA PARETO POWER RAYLEIGH WEIBULL

THETA= and SIGMA= request a distribution reference line corresponding to and .

GUMBEL

MU= and SIGMA= request a distribution reference line corresponding to and .

LOGNORMAL

SIGMA= requests quantile plots with shape parameters . The SIGMA= option must be specified.

NORMAL

MU= and SIGMA= request a distribution reference line corresponding to and . SIGMA=EST requests a line with equal to the sample standard deviation.

WEIBULL2

SIGMA= and C= request a distribution reference line corresponding to and .

SLOPE=value | EST

specifies the slope for a distribution reference line requested with the LOGNORMAL and WEIBULL2 options. Enclose the SLOPE= option in parentheses after the distribution option. When you use the SLOPE= lognormal-option with the LOGNORMAL option, you must also specify a threshold parameter value with the THETA= lognormal-option to request the line. The SLOPE= lognormal-option is an alternative to the ZETA= lognormal-option for specifying , because the slope is equal to .

When you use the SLOPE= Weibull2-option with the WEIBULL2 option, you must also specify a scale parameter value with the SIGMA= Weibull2-option to request the line. The SLOPE= Weibull2-option is an alternative to the C= Weibull2-option for specifying , because the slope is equal to .

For example, the first and second QQPLOT statements produce the same quantile plots and the third and fourth QQPLOT statements produce the same quantile plots:

proc univariate data=Measures;
qqplot Width / lognormal(sigma=2 theta=0 zeta=0);
qqplot Width / lognormal(sigma=2 theta=0 slope=1);
qqplot Width / weibull2(sigma=2 theta=0 c=.25);
qqplot Width / weibull2(sigma=2 theta=0 slope=4);


SQUARE

displays the quantile plot in a square frame. By default, the frame is rectangular.

THETA=value | EST
THRESHOLD=value | EST

specifies the lower threshold parameter for plots requested with the BETA, EXPONENTIAL, GAMMA, PARETO, POWER, RAYLEIGH, LOGNORMAL, WEIBULL, and WEIBULL2 options. Enclose the THETA= option in parentheses after a distribution option. When used with the WEIBULL2 option, the THETA= option specifies the known lower threshold , for which the default is 0. When used with the other distribution options, the THETA= option specifies for a distribution reference line; alternatively in this situation, you can specify THETA=EST to request a maximum likelihood estimate for . To request the line, you must also specify a scale parameter.

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

creates a three-parameter Weibull quantile plot for each value of the required shape parameter specified by the mandatory C= Weibull-option. To create a plot that is based on a maximum likelihood estimate for , specify C=EST. To obtain a graphical estimate of , specify a list of values in the C= Weibull-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the THETA= and SIGMA= Weibull-options. Alternatively, you can add a line corresponding to estimated values of and with the Weibull-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the Weibull distribution with parameters , , and is a good fit. You can specify the SCALE= Weibull-option as an alias for the SIGMA= Weibull-option and the THRESHOLD= Weibull-option as an alias for the THETA= Weibull-option. See Example 4.34.

WEIBULL2<(Weibull2-options)>
W2<(Weibull2-options)>

creates a two-parameter Weibull quantile plot. You should use the WEIBULL2 option when your data have a known lower threshold , which is 0 by default. To specify the threshold value , use the THETA= Weibull2-option. By default, THETA=0. An advantage of the two-parameter Weibull plot over the three-parameter Weibull plot is that the parameters and can be estimated from the slope and intercept of the point pattern. A disadvantage is that the two-parameter Weibull distribution applies only in situations where the threshold parameter is known. To obtain a graphical estimate of , specify a list of values for the THETA= Weibull2-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to and with the SIGMA= and C= Weibull2-options. Alternatively, you can add a distribution reference line corresponding to estimated values of and with the Weibull2-options SIGMA=EST and C=EST. Agreement between the reference line and the point pattern indicates that the Weibull distribution with parameters , , and is a good fit. You can specify the SCALE= Weibull2-option as an alias for the SIGMA= Weibull2-option and the SHAPE= Weibull2-option as an alias for the C= Weibull2-option. See Example 4.34.

† WGRID=n

specifies the line thickness for the grid when producing traditional graphics. The option does not apply to ODS Graphics output.

ZETA=value | EST

specifies a value for the scale parameter for the lognormal quantile plots requested with the LOGNORMAL option. Enclose the ZETA= lognormal-option in parentheses after the LOGNORMAL option. To request a distribution reference line with intercept and slope , specify the THETA= and ZETA=.