The UNIVARIATE Procedure

specifies the shape parameter ( > 0) for P-P plots that are requested by the BETA, GAMMA, PARETO, and POWER options.

creates a beta P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical beta CDF value

where is the normalized incomplete beta function, , and

You can specify , , , and in the ALPHA=, BETA=, SIGMA=, and THETA= beta-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / beta(theta=1 sigma=2 alpha=3 beta=4);
run;

If you do not specify values for these parameters, then by default, , , and maximum likelihood estimates are calculated for and .

IMPORTANT: If the default unit interval (0,1) does not adequately describe the range of your data, then you should specify THETA= and SIGMA= so that your data fall in the interval .

If the data are beta-distributed with parameters , , , and , then the points on the plot for ALPHA=, BETA=, SIGMA=, and THETA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified beta distribution 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.

specifies the shape parameter ( > 0) for P-P plots that are requested by the BETA distribution option. For an example, see the preceding entry for the BETA distribution option.

specifies the shape parameter c for P-P plots that are requested by the WEIBULL option. For examples, see the entry for the WEIBULL option.

creates an exponential P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical exponential CDF value

where

You can specify and in the SIGMA= and THETA= exponential-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / exponential(theta=1 sigma=2);
run;

If you do not specify values for these parameters, then by default, and a maximum likelihood estimate is calculated for .

IMPORTANT: Your data must be greater than or equal to the lower threshold . If the default is not an adequate lower bound for your data, specify in the THETA= option.

If the data are exponentially distributed with parameters and , the points on the plot for SIGMA= and THETA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified exponential distribution 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.

creates a gamma P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical gamma CDF value

where is the normalized incomplete gamma function and

You can specify , , and in the ALPHA=, SIGMA=, and THETA= gamma-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / gamma(alpha=1 sigma=2 theta=3);
run;

If you do not specify values for these parameters, then by default, and maximum likelihood estimates are calculated for and .

IMPORTANT: Your data must be greater than or equal to the lower threshold . If the default is not an adequate lower bound for your data, specify in the THETA= option.

If the data are gamma distributed with parameters , , and , the points on the plot for ALPHA=, SIGMA=, and THETA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified gamma distribution is a good fit. You can specify the SHAPE= option as an alias for the ALPHA= option, the SCALE= option as an alias for the SIGMA= option, and the THRESHOLD= option as an alias for the THETA= option.

creates a Gumbel P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical Gumbel CDF value

where

You can specify and in the MU= and SIGMA= Gumbel-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / gumbel(mu=1 sigma=2);
run;

If you do not specify values for these parameters, then by default, the maximum likelihood estimates are calculated for and .

If the data are Gumbel distributed with parameters and , the points on the plot for MU= and SIGMA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified Gumbel distribution is a good fit.

creates an inverse Gaussian P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical inverse Gaussian CDF value

where is the standard normal distribution function and

You can specify and in the LAMBDA= and MU= IGauss-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / igauss(lambda=1 mu=2);
run;

If you do not specify values for these parameters, then by default, the maximum likelihood estimates are calculated for and .

If the data have an inverse Gaussian distribution with parameters and , the points on the plot for LAMBDA= and MU= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified inverse Gaussian distribution is a good fit.

specifies the shape parameter for fitted curves that are requested by the IGAUSS option. Enclose the LAMBDA= option in parentheses after the IGAUSS distribution keyword. If you do not specify this option, the procedure calculates a maximum likelihood estimate for .

creates a lognormal P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical lognormal CDF value

where is the cumulative standard normal distribution function and

You can specify , , and with the THETA= ZETA=, and SIGMA= lognormal-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / lognormal(theta=1 zeta=2);
run;

If you do not specify values for these parameters, then by default, and estimates of and are computed as described in the section Lognormal Distribution.

IMPORTANT: Your data must be greater than the lower threshold . If the default is not an adequate lower bound for your data, specify in the THETA= option.

If the data are lognormally distributed with parameters , , and , the points on the plot for SIGMA=, THETA=, and ZETA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified lognormal distribution is a good fit. You can specify the SHAPE= option as an alias for the SIGMA=option, the SCALE= option as an alias for the ZETA= option, and the THRESHOLD= option as an alias for the THETA= option.

specifies the parameter for P-P plots that are requested by the GUMBEL, IGAUSS, and NORMAL options. By default, the sample mean is used for with inverse Gaussian and normal distributions. A maximum likelihood estimate is computed by default with the Gumbel distribution. See Example 4.36.

suppresses the diagonal reference line.

creates a normal P-P plot. By default, if you do not specify a distribution option, the procedure displays a normal P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical normal CDF value

where is the cumulative standard normal distribution function and

You can specify and in the MU= and SIGMA= normal-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / normal(mu=1 sigma=2);
run;

By default, the sample mean and sample standard deviation are used for and .

If the data are normally distributed with parameters and , the points on the plot for MU= and SIGMA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified normal distribution is a good fit. See Example 4.36.

creates a generalized Pareto P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical generalized Pareto CDF value

where is the threshold parameter is the scale parameter is the shape parameter

The parameter for the generalized Pareto distribution must be less than the minimum data value. You can specify in the THETA= Pareto-option. The default value for is 0. In addition, the generalized Pareto distribution has a shape parameter and a scale parameter . You can specify these parameters in the ALPHA= and SIGMA= Pareto-options. By default, maximum likelihood estimates are computed for and .

If the data have a generalized Pareto distribution with parameters , , and , the points on the plot for THETA=, SIGMA=, and ALPHA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified generalized Pareto distribution is a good fit.

creates a power function P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical power function CDF value

where is the lower threshold parameter (lower endpoint) is the scale parameter is the shape parameter

The power function distribution is bounded below by the parameter and above by the value . You can specify and by using the THETA= and SIGMA= power-options. The default values for and are 0 and 1, respectively.

You can specify a value for the shape parameter, , with the ALPHA= power-option. If you do not specify a value for , the procedure calculates a maximum likelihood estimate.

The power function distribution is a special case of the beta distribution with its second shape parameter, .

If the data have a power function distribution with parameters , , and , the points on the plot for THETA=, SIGMA=, and ALPHA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified power function distribution is a good fit.

creates a Rayleigh P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical Rayleigh CDF value

where is the threshold parameter is the scale parameter

The parameter for the Rayleigh distribution must be less than the minimum data value. You can specify in the THETA= Rayleigh-option. The default value for is 0. You can specify in the SIGMA= Rayleigh-option. By default, a maximum likelihood estimate is computed for .

If the data have a Rayleigh distribution with parameters and , the points on the plot for THETA= and SIGMA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified Rayleigh distribution is a good fit.

specifies the parameter , where . When used with the BETA, EXPONENTIAL, GAMMA, GUMBEL, NORMAL, PARETO, POWER, RAYLEIGH, and WEIBULL options, the SIGMA= option specifies the scale parameter. When used with the LOGNORMAL option, the SIGMA= option specifies the shape parameter. See Example 4.36.

displays the P-P plot in a square frame. (By default, the plot is in a rectangular frame.) See Example 4.36.

specifies the lower threshold parameter for plots that are requested by the BETA, EXPONENTIAL, GAMMA, LOGNORMAL, PARETO, POWER, RAYLEIGH, and WEIBULL options.

creates a Weibull P-P plot. To create the plot, the n nonmissing observations are ordered from smallest to largest:

The y-coordinate of the ith point is the empirical CDF value . The x-coordinate is the theoretical Weibull CDF value

where

You can specify c, , and in the C=, SIGMA= and THETA= Weibull-options, as illustrated in the following example:

proc univariate data=measures;
   ppplot width / weibull(theta=1 sigma=2);
run;

If you do not specify values for these parameters, then by default and maximum likelihood estimates are calculated for and c.

IMPORTANT: Your data must be greater than or equal to the lower threshold . If the default is not an adequate lower bound for your data, you should specify with the THETA= option.

If the data have a Weibull distribution with parameters c, , and , the points on the plot for C=c, SIGMA=, and THETA= tend to fall on or near the diagonal line , which is displayed by default. Agreement between the diagonal line and the point pattern is evidence that the specified Weibull distribution is a good fit. You can specify the SHAPE= option as an alias for the C= option, the SCALE= option as an alias for the SIGMA= option, and the THRESHOLD= option as an alias for the THETA= option.

specifies a value for the scale parameter for lognormal P-P plots that are requested by the LOGNORMAL option.