Publication date: 11/29/2021

Use the Continuous Fit or Discrete Fit options to fit a distribution to a continuous variable.

Note: Some features of distribution fitting have been updated in JMP 15. This section contains details of the older features from previous JMP releases that have been retained for compatibility purposes. These features are available by selecting Continuous Fit > Enable Legacy Fitters in the red triangle menu for a variable.

A curve is overlaid on the histogram, and a Parameter Estimates report is added to the report window. A red triangle menu contains additional options. See Fit Distribution Options (Legacy).

Note: The Life Distribution platform also contains options for distribution fitting that might use different parameterizations and allow for censoring. See Life Distribution in Reliability and Survival Methods.

This section describes the distributions in the Legacy Fitters submenu that differ from the corresponding distributions in the updated Continuous Fit options.

• The Weibull distribution, Weibull with threshold distribution, and Extreme Value distribution often provide a good model for estimating the length of life, especially for mechanical devices and in biology.

• The Gamma distribution is bound by zero and has a flexible shape.

• The Beta distribution is useful for modeling the behavior of random variables that are constrained to fall in the interval 0,1. For example, proportions always fall between 0 and 1.

• The Smooth Curve distribution fits a smooth curve using nonparametric density estimation (kernel density estimation). The smooth curve is overlaid on the histogram and a slider appears beneath the plot. Control the amount of smoothing by changing the kernel standard deviation with the slider. The initial Kernel Std estimate is calculated from the standard deviation of the data.

• The Johnson Su, Johnson Sb, and Johnson Sl Distributions are useful for its data-fitting capabilities because it supports every possible combination of skewness and kurtosis.

• The Generalized Log (Glog) distribution is useful for fitting data that are rarely normally distributed and often have non-constant variance, like biological assay data.

The All option fits all applicable continuous distributions to a variable. The Compare Distributions report contains statistics about each fitted distribution. Use the check boxes to show or hide a fit report and overlay curve for the selected distribution. By default, the best fit distribution is selected.

The Show Distribution list is sorted by AICc in ascending order.

If your variable contains negative values, the Show Distribution list does not include those distributions that require data with positive values. Only continuous distributions are fitted by this command. Distributions with threshold parameters, like Beta and Johnson Sb, are not included in the list of possible distributions.

For statistical details, see the following sections:

• Statistical Details for Continuous Fit Distributions (Legacy)

• Statistical Details for Fitted Quantiles (Legacy)

• Fit Distribution Options (Legacy)

The Discrete Fit option is available when all data values are integers. Use the Discrete Fit options to fit a distribution (such as Poisson or Binomial) to a discrete variable. The following distributions are available:

• Poisson

• Gamma Poisson

• Binomial

• Beta Binomial

For statistical details, see the following sections:

• Statistical Details for Discrete Fit Distributions (Legacy)

• Statistical Details for Fitted Quantiles (Legacy)

• Fit Distribution Options (Legacy)

Each fitted distribution report has a red triangle menu that contains additional options.

Diagnostic Plot

Creates a quantile or a probability plot. See Diagnostic Plot.

Density Curve

Uses the estimated parameters of the distribution to overlay a density curve on the histogram.

Goodness of Fit

Computes the goodness of fit test for the fitted distribution. See Goodness of Fit.

Fix Parameters

Enables you to fix parameters and re-estimate the non-fixed parameters. An Adequacy LR (likelihood ratio) Test report also appears, which tests your new parameters to determine whether they fit the data.

Quantiles

Returns the unscaled and uncentered quantiles for the specific lower probability values that you specify.

Set Spec Limits for K Sigma

Use this option when you do not know the specification limits for a process and you want to use its distribution as a guideline for setting specification limits.

Usually, specification limits are derived using engineering considerations. If there are no engineering considerations, and if the data are from a well behaved process, then quantiles from a fitted distribution are often used to help set specification limits. See Set Spec Limits for K Sigma.

Spec Limits

Computes generalizations of the standard capability indices, based on the specification limits and target you specify. See Spec Limits.

Save Fitted Quantiles

Saves the fitted quantile values as a new column in the current data table. See Statistical Details for Fitted Quantiles (Legacy).

Save Density Formula

Creates a new column in the current data table that contains fitted values that have been computed by the density formula. The density formula uses the estimated parameter values.

Save Spec Limits

Saves the specification limits as a column property.

Save Transformed

Creates a new column and saves a formula. The formula can transform the column to normality using the fitted distribution. This option is available only when one of the Johnson distributions, the Glog distribution, or the SHASH distribution is fit.

Remove Fit

Removes the distribution fit from the report window.

The Diagnostic Plot option creates a quantile or a probability plot. Depending on the fitted distribution, the plot is in one of the following four formats.

• Weibull with threshold

• Gamma

• Beta

• Poisson

• GammaPoisson

• Binomial

• BetaBinomial

• Normal

• Normal Mixtures

• Exponential

• Weibull

• LogNormal

• Extreme Value

• SHASH

• Johnson Sl

• Johnson Sb

• Johnson Su

• Glog

The following options are available in the Diagnostic Plot red triangle menu:

Rotate

Reverses the x- and y-axes.

Confidence Limits

Draws Lilliefors 95% confidence limits for the Normal Quantile plot, and 95% equal precision bands with a = 0.001 and b = 0.99 for all other quantile plots (Meeker and Escobar 1998).

Line of Fit

Draws the straight diagonal reference line. If a variable fits the selected distribution, the values fall approximately on the reference line.

Median Reference Line

Draws a horizontal line at the median of the response.

The Goodness of Fit option computes the goodness of fit test for the fitted distribution. The goodness of fit tests are not Chi-square tests, but are EDF (Empirical Distribution Function) tests. EDF tests offer advantages over the Chi-square tests, including improved power and invariance with respect to histogram midpoints.

• For Normal distributions, the Shapiro-Wilk test for normality is reported when the sample size is less than or equal to 2000. The KSL test is computed for samples that are greater than 2000.

• For discrete distributions that have sample sizes less than or equal to 30, the Goodness of Fit test is formed using two one-sided exact Kolmogorov tests combined to form a near-exact test. See Conover (1972). For sample sizes greater than 30, a Pearson Chi-squared goodness of fit test is performed.

• For statistical details, see Fit Distribution Options (Legacy).

The Spec Limits option opens a window that enables you to enter specification limits and a target. Then generalizations of the standard capability indices are computed. Note that for the normal distribution, 3σ is both the distance from the lower 0.135 percentile to median (or mean) and the distance from the median (or mean) to the upper 99.865 percentile. These percentiles are estimated from the fitted distribution, and the appropriate percentile-to-median distances are substituted for 3σ in the standard formulas.

• For statistical details, see Fit Distribution Options (Legacy).

Want more information? Have questions? Get answers in the JMP User Community (community.jmp.com).