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Parameters

This section lists the parameters for the fitted curve as well as the estimated mean and estimated standard deviation. See Formulas for Fitted Curves.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit statistic for a fitted parametric distribution is computed as follows:

where

observed value in th histogram interval
expected value in th histogram interval
number of histogram intervals
number of estimated parameters

The degrees of freedom for the chi-square test is equal to . You can save the observed and expected interval values in the OUTFIT= data set discussed in Output Data Sets.

Note that empty intervals are not combined, and the range of intervals used to compute begins with the first interval containing observations and ends with the final interval containing observations.

EDF Goodness-of-Fit Tests

When you fit a parametric distribution, the HISTOGRAM statement provides a series of goodness-of-fit tests based on the empirical distribution function (EDF). The EDF tests offer advantages over the chi-square goodness-of-fit test, including improved power and invariance with respect to the histogram midpoints. For a thorough discussion, refer to D’Agostino and Stephens (1986).

The empirical distribution function is defined for a set of independent observations with a common distribution function . Denote the observations ordered from smallest to largest as . The empirical distribution function, , is defined as

Note that is a step function that takes a step of height at each observation. This function estimates the distribution function . At any value , is the proportion of observations less than or equal to , while is the probability of an observation less than or equal to . EDF statistics measure the discrepancy between and .

The computational formulas for the EDF statistics make use of the probability integral transformation . If is the distribution function of , the random variable is uniformly distributed between 0 and 1.

Given observations , the values are computed by applying the transformation, as shown in the following sections.

The HISTOGRAM statement provides three EDF tests:

• Kolmogorov-Smirnov

• Anderson-Darling

• Cramér-von Mises

These tests are based on various measures of the discrepancy between the empirical distribution function and the proposed parametric cumulative distribution function .

The following sections provide formal definitions of the EDF statistics.

Kolmogorov-Smirnov Statistic

The Kolmogorov-Smirnov statistic (D) is defined as

The Kolmogorov-Smirnov statistic belongs to the supremum class of EDF statistics. This class of statistics is based on the largest vertical difference between and .

The Kolmogorov-Smirnov statistic is computed as the maximum of and , where is the largest vertical distance between the EDF and the distribution function when the EDF is greater than the distribution function, and is the largest vertical distance when the EDF is less than the distribution function.

Anderson-Darling Statistic

The Anderson-Darling statistic and the Cramér-von Mises statistic belong to the quadratic class of EDF statistics. This class of statistics is based on the squared difference . Quadratic statistics have the following general form:

The function weights the squared difference .

The Anderson-Darling statistic () is defined as

Here the weight function is .

The Anderson-Darling statistic is computed as

Cramér-von Mises Statistic

The Cramér-von Mises statistic () is defined as

Here the weight function is .

The Cramér-von Mises statistic is computed as

Specifications

This section is included in the summary only if you provide specification limits, and it tabulates the limits as well as the observed percentages and estimated percentages outside the limits.

The estimated percentages are computed only if fitted distributions are requested and are based on the probability that an observed value exceeds the specification limits, assuming the fitted distribution. The observed percentages are the percents of observations outside the specification limits.

Indices Using Fitted Curves

This section is included in the summary only if you specify the INDICES option in parentheses after a distribution option, as in the statements that produce Figure 5.7.10. Standard process capability indices, such as and , are not appropriate if the data are not normally distributed. The INDICES option computes generalizations of the standard indices by using the fact that for the normal distribution, is both the distance from the lower 0.135 percentile to the 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 in the standard formulas.

Writing for the target, LSL and USL for the lower and upper specification limits, and for the th percentile, the generalized capability indices are as follows:

If the data are normally distributed, these formulas reduce to the formulas for the standard capability indices, which are given in the section Standard Capability Indices.

The following guidelines apply to the use of generalized capability indices requested with the INDICES option:

• When you choose the family of parametric distributions for the fitted curve, consider whether an appropriate family can be derived from assumptions about the process.

• Whenever possible, examine the data distribution with a histogram, probability plot, or quantile-quantile plot.

• Apply goodness-of-fit tests to assess how well the parametric distribution models the data.

• Consider whether a generalized index has a meaningful practical interpretation in your application.

At the time of this writing, there is ongoing research concerning the application of generalized capability indices, and it is important to note that other approaches can be used with nonnormal data:

• Transform the data to normality, then compute and report standard capability indices on the transformed scale.

• Report the proportion of nonconforming output estimated from the fitted distribution.

• If it is not possible to adequately model the data distribution with a parametric density, smooth the data distribution with a kernel density estimate and simply report the proportion of nonconforming output.

Refer to Rodriguez and Bynum (1992) for additional discussion.

Histogram Intervals

This section is included in the summary only if you specify the MIDPERCENTS option in parentheses after the distribution option, as in the statements that produce Figure 5.7.10. This table lists the interval midpoints along with the observed and estimated percentages of the observations that lie in the interval. The estimated percentages are based on the fitted distribution.

In addition, you can specify the MIDPERCENTS option to request a table of interval midpoints with the observed percent of observations that lie in the interval. See the entry for the MIDPERCENTS option .

Quantiles

This table lists observed and estimated quantiles. You can use the PERCENTS= option to specify the list of quantiles to appear in this list. The list in Figure 5.7.10 is the default list. See the entry for the PERCENTS= option .

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