Specialized Capability Indices :: SAS/QC(R) 14.1 User's Guide

The Index k

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

$k = \frac{2 |m - \bar{X}|}{\mi{USL} - \mi{LSL}}$

where $m = \frac{1}{2}(\mi{USL} + \mi{LSL})$ is the midpoint of the specification limits, $\bar{X}$ 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 $m - \bar{X}$ 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 $C_{pm}^{+}$

Boyles (1992) proposed the process capability index $C_{pm}^{+}$ which is defined as

$C_{pm}^{+} = \frac{1}{3} { \left[ \frac{ E_{X<T} \left[ (X-T)^{2} \right] }{ (T - \mr{LSL})^{2} } + \frac{ E_{X>T} \left[ (X-T)^{2} \right] }{ (\mr{USL} - T)^{2} } \right] }^{-1/2}$

He proposed this index as a modification of $C_{pm}$ for use when $\mu \neq T$ . The quantities

$E_{X<T} \left[ (X-T)^{2} \right] = E \left[ (X-T)^2 | X < T \right] Pr \left[ X < T \right]$

and

$E_{X>T} \left[ (X-T)^{2} \right] = E \left[ (X-T)^2 | X > T \right] Pr \left[ X > T \right]$

are referred to as semivariances. Kotz and Johnson (1993) point out that if $T = (\mr{LSL} + \mr{USL} ) / 2$ , then $C_{pm}^{+} = C_{pm}$ .

Kotz and Johnson (1993) suggest that a natural estimator for $C_{pm}^{+}$ is

$\widehat{C}_{pm}^{+} = \frac{1}{3} \left[ \frac{1}{n} \left\{ \frac{ \sum _{X_{i} < T} (X_{i} - T)^{2} }{ (T - \mbox{LSL})^{2} } + \frac{ \sum _{X_{i} > T} (X_{i} - T)^{2} }{ (\mbox{USL} - T)^{2} } \right\} ^{-1/2} \right]$

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 $C_{jkp}$

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

$C_{jkp} = \frac{1}{3 \sqrt {2}} \min \left( \frac{\mbox{USL} - T}{ \sqrt { E_{X>T}[(X-T)^2] } } , \frac{T - \mbox{LSL}}{ \sqrt { E_{X<T}[(X-T)^2] } } \right)$

where $d = ( \mbox{USL} - \mbox{LSL} ) / 2$ .

A natural estimator of this index is

$\widehat{C}_{jkp} = \frac{1}{3 \sqrt {2}} \min \left( \frac{ \mbox{USL} - T }{ \sqrt { \sum _{X_{i} > T} (X_{i} - T)^{2} / n } } , \frac{ T - \mbox{LSL} }{ \sqrt { \sum _{X_{i} < T} (X_{i} - T)^{2} / n } } \right)$

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

The Indices $C_{pm}(a)$

The class of capability indices $C_{pm}(a)$ , 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

$C_{pm}(a) = ( 1 - a \zeta ^2 ) C_{p}$

where $\zeta = (\mu - T) / \sigma$ . The motivation for this definition is that if $|\zeta |$ is small, then

$C_{pm} \approx (1 - \frac{1}{2} \zeta ^{2} ) C_ p$

A natural estimator of $C_{pm}(a)$ is

$\frac{d}{3s} \widehat{C}_{pm}(a) = \left\{ 1 - a \left( \frac{\bar{X}-T}{s} \right) ^2 \right\}$

where $d = ( \mr{USL} - \mr{LSL} ) / 2$ . 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 $C_{p(5.15)}$

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

$C_{p(\theta )} = \frac{\mbox{USL} - \mbox{LSL}}{\theta \sigma }$

where $\theta$ 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

$C_{p(5.15)} = \frac{\mbox{USL} - \mbox{LSL}}{5.15 \sigma }$

which is estimated as

$\widehat{C}_{p(5.15)} = \frac{\mbox{USL} - \mbox{LSL}}{5.15 s}$

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

The Index $C_{pk(5.15)}$

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

$C_{pk(5.15)} = \frac{d - | \mu - (\mbox{USL} + \mbox{LSL}) / 2 | }{2.575 \sigma }$

where $d = ( \mbox{USL} - \mbox{LSL} ) / 2$ . This index is estimated as

$\widehat{C}_{pk(5.15)} = \frac{d - | \bar{X} - (\mbox{USL} + \mbox{LSL}) / 2 |}{2.575 s}$

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

The Index $C_{pmk}$

Pearn, Kotz, and Johnson (1992) proposed the index $C_{pmk}$

$C_{pmk} = \frac{(\mbox{USL} - \mbox{LSL})/2 - |\mu - m |}{3 \sqrt { \sigma ^2 + (\mu - T)^2}}$

where $m = (\mbox{LCL} + \mbox{UCL}) / 2$ . A natural estimator for $C_{pmk}$ is

$\widehat{C}_{pmk} = \frac{(\mbox{USL} - \mbox{LSL})/2 - |\bar{X} - m |}{3 \sqrt {(\frac{n-1}{n})s^2 + (\bar{X} - T)^2}}$

where $m = ( \mr{USL} + \mr{LSL} ) / 2$ .

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

Wright’s Index $C_{s}$

Wright (1995) defines the capability index

$C_ s = \frac{ \min \left( \mr{USL} - \mu , \mu - \mr{LSL} \right) }{ 3 \sqrt { \sigma ^2 + (\mu - T)^2 + \mu _3 / \sigma } }$

where $\mu _3 = E(X - \mu )^3$ .

A natural estimator of $C_ s$ is

$\widehat{C}_ s = \frac{ ( \mr{USL} - \mr{LSL} ) / 2 - | \bar{X} - m | }{ 3 \sqrt { \left( \frac{n-1}{n} \right) s^2 + (\bar{X} - T)^2 + |c_4 s^2 b_3| } }$

where $c_4$ is an unbiasing constant for the sample standard deviation, and $b_3$ is a measure of skewness. Wright (1995) shows that $C_ s$ compares favorably with $C_{pmk}$ even when skewness is not present, and he advocates the use of $C_ s$ for monitoring near-normal processes when loss of capability typically leads to asymmetry.

Chen and Kotz (1996) proposed a modification to Wright’s $C_ s$ index which introduces a multiplier, $\gamma > 0$ , and is estimated as

$\widehat{C}_ s = \frac{ ( \mr{USL} - \mr{LSL} ) / 2 - | \bar{X} - m | }{ 3 \sqrt { \left( \frac{n-1}{n} \right) s^2 + (\bar{X} - T)^2 + \gamma |c_4 s^2 b_3| } }$

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

The Index $S_{jkp}$

Boyles (1994) proposed a smooth version of $C_{jkp}$ defined as

$S_{jkp} = S \left( \frac{\mbox{USL} - T}{ \sqrt { 2 E_{X>T}[(X-T)^2] } } , \frac{T - \mbox{LSL}}{ \sqrt { 2 E_{X<T}[(X-T)^2] } } \right)$

The CAPABILITY procedure estimates $S_{jkp}$ as

$\widehat{S}_{jkp} = S \left( \frac{ \mbox{USL} - T }{ \sqrt { 2 \sum _{X_{i} > T} (X_{i} - T)^{2} / n } } , \frac{ T - \mbox{LSL} }{ \sqrt { 2 \sum _{X_{i} < T} (X_{i} - T)^{2} / n } } \right)$

where $S(x,y) = \Phi ^{-1}[\{ \Phi (x) + \Phi (y)\} /2]/3$ .

The Index $C_{pp}$

Chen (1998) devised a process incapability index based on the $C_{pm}^{*}$ index. The first term measures inaccuracy and the second measures imprecision. The $C_{pp}$ index is estimated as

$\widehat{C}_{pp} = \left( \frac{~ \bar{X} - T}{d^{*} / 3} \right)^2 + \left( \frac{s}{d^{*} / 3} \right)^2$

where $d^{*} = \min ( \mbox{USL} - T, T - \mbox{LSL} )$ .

The Index $C_{pp}^{''}$

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

$\widehat{C}_{pp}^{''} = \left( \frac{\widehat{A}}{d^{*} / 3} \right)^2 + \left( \frac{s}{d^{*} / 3} \right)$

where

$\widehat{A} = \max \left\{ \frac{(\bar{X} - T)d}{T - \mbox{LSL}} , \frac{(T - \bar{X})d}{\mbox{USL} - T} \right\}$

and $d = (\mi{USL} - \mi{LSL}) / 2$ .

The Index $C_{pg}$

Marcucci and Beazley (1988) defined the index

$C_{pg} = \frac{1}{C_{pm}^2}$

which is estimated as

$\widehat{C}_{pg} = \frac{1}{\widehat{C}_{pm}^2}$

The Index $C_{pq}$

Gupta and Kotz (1997) introduced the index $C_{pq}$ , which is estimated by

$\widehat{C}_{pq} = \widehat{C}_ p \left[ 1 - \frac{1}{2} \left( \frac{\bar{X} - T}{s} \right)^2 \right]$

The Index $C_{p}^{W}$

Bai and Choi (1997) defined the index

$C_ p^ W = \frac{C_ p}{\sqrt { 1 + | 1 - 2 P_ x | }}$

where $P_ x = \mbox{Pr} (X \leq \mu )$ . It is estimated by

$\widehat{C}_ p^ W = \frac{\widehat{C}_ p}{\sqrt { 1 + | 1 - 2 \widehat{P}_ x | }}$

where $\widehat{P}_ x$ is the fraction of observations less than or equal to $\bar{X}$ . For more information about $C_{p}^{W}$ , see Kotz and Lovelace (1998).

The Index $C_{pk}^{W}$

Bai and Choi (1997) also proposed the index

$C_{pk}^ W = \min \left\{ \frac{\mbox{USL} - \mu }{3 \sigma \sqrt {2 P_ x}} , \frac{\mu - \mbox{LSL}}{3 \sigma \sqrt {2 (1 - P_ x)}} \right\}$

It is estimated by

$\widehat{C}_{pk}^ W = \min \left\{ \frac{\mbox{USL} - \bar{X}}{3 s \sqrt {2 \widehat{P}_ x}} , \frac{\bar{X} - \mbox{LSL}}{3 s \sqrt {2 (1 - \widehat{P}_ x)}} \right\}$

where $\widehat{P}_ x$ is the fraction of observations less than or equal to $\bar{X}$ . For more information about $C_{pk}^{W}$ , see Kotz and Lovelace (1998).

The Index $C_{pm}^ W$

The index $C_{pm}^ W$ , also introduced by Bai and Choi (1997), is defined as

$C_{pm}^ W = \frac{C_{pm}}{\sqrt {1 + | 1 - 2P_ T |}}$

where $P_ T = \mbox{Pr}(X \leq T)$ . It is estimated by

$\widehat{C}_{pm}^ W = \frac{\widehat{C}_{pm}}{\sqrt {1 + | 1 - 2 \widehat{P}_ T | }}$

where $\widehat{P}_ T$ is the fraction of observations less than or equal to T. For more information about $C_{pm}^{W}$ , see Kotz and Lovelace (1998).

The Index $C_{pc}$

Luceño (1996) proposed the index

$C_{pc} = \frac{\mbox{USL} - \mbox{LSL}}{6 \sqrt {\frac{\pi }{2} E |X - M|}}$

where $M = (\mbox{USL} + \mbox{LSL}) / 2$ . It is estimated by

$\widehat{C}_{pc} = \frac{\mbox{USL} - \mbox{LSL}}{6 \sqrt {\frac{\pi }{2} c}}$

where

$c = \frac{1}{n} \sum _{i = 1}^ n | X_ i - M |$

Vännmann’s Index $C_ p(u,v)$

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