# PROC CAPABILITY and General Statements

#### Specialized Capability Indices

This section describes a number of specialized capability indices which you can request with the SPECIALINDICES option in the PROC CAPABILITY statement.

##### The Index k

The process capability index k (also denoted by K) is computed as

where is the midpoint of the specification limits, is the sample mean, USL is the upper specification limit, and LSL is the lower specification limit.

The formula for k used here is given by Kane (1986). Note that k is sometimes computed without taking the absolute value of in the numerator. See Wadsworth, Stephens, and Godfrey (1986).

If you do not specify the upper and lower limits in the SPEC statement or the SPEC= data set, then k is assigned a missing value.

##### Boyles’ Index

Boyles (1992) proposed the process capability index which is defined as

He proposed this index as a modification of for use when . The quantities

and

are referred to as semivariances. Kotz and Johnson (1993) point out that if , then .

Kotz and Johnson (1993) suggest that a natural estimator for is

Note that this index is not defined when either of the specification limits is equal to the target T. Refer to Section 3.5 of Kotz and Johnson (1993) for further details.

##### The Index

Johnson, Kotz, and Pearn (1994) introduced a so-called flexible process capability index which takes into account possible differences in variability above and below the target T. They defined this index as

where .

A natural estimator of this index is

For further details, refer to Section 4.4 of Kotz and Johnson (1993).

##### The Indices

The class of capability indices , indexed by the parameter a (a > 0) allows flexibility in choosing between the relative importance of variability and deviation of the mean from the target value T.

The class defined as

where . The motivation for this definition is that if is small, then

A natural estimator of is

where . You can specify the value of a with the SPECIALINDICES(CPMA=) option in the PROC CAPABILITY statement. By default, a = 0.5.

This index is not recommended for situation in which the target T is not equal to the midpoint of the specification limits.

For additional details, refer to Section 3.7 of Kotz and Johnson (1993).

##### The Index

Johnson et al. (1992) suggest the class of process capability indices defined as

where is chosen so that the proportion of conforming items is robust with respect to the shape of the process distribution. In particular, Kotz and Johnson (1993) recommend use of

which is estimated as

For details, refer to Section 4.3.2 of Kotz and Johnson (1993).

##### The Index

Similarly, Kotz and Johnson (1993) recommend use of the robust capability index

where . This index is estimated as

For details, refer to Section 4.3.2 of Kotz and Johnson (1993).

##### The Index

Pearn, Kotz, and Johnson (1992) proposed the index

where . A natural estimator for is

where .

For further details, refer to Section 3.6 of Kotz and Johnson (1993).

##### Wright’s Index

Wright (1995) defines the capability index

where .

A natural estimator of is

where is an unbiasing constant for the sample standard deviation, and is a measure of skewness. Wright (1995) shows that compares favorably with even when skewness is not present, and he advocates the use of for monitoring near-normal processes when loss of capability typically leads to asymmetry.

Chen and Kotz (1996) proposed a modification to Wright’s index which introduces a multiplier, , and is estimated as

If you specify a value for with the SPECIALINDICES(CSGAMMA=) option, the index is computed with this modification. Otherwise it is computed using Wright’s original definition.

##### The Index

Boyles (1994) proposed a smooth version of defined as

The CAPABILITY procedure estimates as

where .

##### The Index

Chen (1998) devised a process incapability index based on the index. The first term measures inaccuracy and the second measures imprecision. The index is estimated as

where .

##### The Index

The index does not handle asymmetric tolerances well, as discussed by Kotz and Lovelace (1998). To address that shortcoming, Chen (1998) defined the index , which is estimated by

where

and .

##### The Index

Marcucci and Beazley (1988) defined the index

which is estimated as

##### The Index

Gupta and Kotz (1997) introduced the index , which is estimated by

##### The Index

Bai and Choi (1997) defined the index

where . It is estimated by

where is the fraction of observations less than or equal to . For more information about , see Kotz and Lovelace (1998).

##### The Index

Bai and Choi (1997) also proposed the index

It is estimated by

where is the fraction of observations less than or equal to . For more information about , see Kotz and Lovelace (1998).

##### The Index

The index , also introduced by Bai and Choi (1997), is defined as

where . It is estimated by

where is the fraction of observations less than or equal to T. For more information about , see Kotz and Lovelace (1998).

##### The Index

Luceño (1996) proposed the index

where . It is estimated by

where

##### Vännmann’s Index

Vännmann (1995) introduced the generalized index , which reduces to the following capability indices given appropriate choices of u and v:

is defined as

and estimated by

You can specify u with the SPECIALINDICES(CPU=) option and v with the SPECIALINDICES(CPV=) option. By default, u = 0 and v = 4.

##### Vännmann’s Index

Vännmann (1997) also proposed the index , which is equivalent to with u = 1. It is estimated as

You can specify v with the SPECIALINDICES(CPV=) option. By default, v = 4.