XCHART Statement: CUSUM Procedure

Dictionary of Special Options

The notes Traditional Graphics and Line Printer identify options that apply only to traditional graphics and line printer charts, respectively.

ALPHA=value

specifies the probability $\alpha$ of incorrectly deciding that a shift has occurred when the process mean is equal to the target mean. This is known as the probability of a Type 1 error. The value must be between zero and one, and it is typically set at 0.05 or 0.10. If you specify the ALPHA= option, the error probability approach is used to determine the V-mask. For details, see Defining the V-Mask for a Two-Sided Cusum Scheme.

The ALPHA= option is applicable only with two-sided cusum schemes. As an alternative to the ALPHA= value, you can specify the percentile $z_{1-\alpha /2}$ from a standard normal distribution with the SIGMAS= option. As a second alternative, you can specify the geometric parameter h for the V-mask (in standard error units) with the H= option.

In addition to the ALPHA= option, you can optionally specify the probability of a Type 2 error with the BETA= option.

BETA=value

specifies the probability $\beta$ of failing to discover that the specified shift has occurred. This is known as the probability of a Type 2 error. The value must be between zero and one. The BETA= option is used in conjunction with either the ALPHA= option or the SIGMAS= option.

The interpretation of $\beta$ is based on the analogy between cusum charts and sequential probability ratio tests, and it is inexact since the cusum chart does not provide an acceptance region. Refer to Johnson (1961) and van Dobben de Bruyn (1968) for further details.

CINFILL=color

Traditional Graphics specifies the color for the area under the decision interval or inside the V-mask arms. See also the COUTFILL= option.

CLIMITS=color

Traditional Graphics specifies the color for the decision interval line.

CMASK=color

Traditional Graphics specifies the color for the V-mask arms.

DATAUNITS

computes cumulative sums without standardizing the subgroup means or individual measurements. As a result, the vertical axis of the cusum chart is scaled in the same units as the data.

The DATAUNITS option requires constant subgroup sample sizes. If your data do not have constant subgroup sample sizes, you need to specify a constant nominal sample size n for the V-mask or decision interval with the LIMITN= option or with the variable _LIMITN_ in the LIMITS= data set.

DELTA=value

specifies the absolute value of the smallest shift to be detected as a multiple $\delta$ of the process standard deviation $\sigma$ or the standard error $\sigma _{\bar{X}}$ , depending on whether $\delta$ is viewed as a shift in the population mean or a shift in the sampling distribution of the subgroup mean $\bar{X}$ , respectively.

If you specify SCHEME=ONESIDED (see the SCHEME= option later in this list) and the value is positive, a shift above the process mean is to be detected, whereas if the value is negative, a shift below the process mean is to be detected.

As an alternative to specifying the DELTA= option, you can specify the shift in the same units as the data with the SHIFT= option.

H=value

specifies the decision interval h for a one-sided cusum scheme. This type of scheme is completely specified by the parameters h and k (see the K= option later in this list). You can also specify the H= option as an alternative to the ALPHA= or SIGMAS= options for a two-sided cusum scheme with a V-mask. In this case, the H= option specifies the vertical distance h between the origin for the V-mask and the upper or lower arm of the V-mask. In either case, the H=value must be positive and must be expressed as a multiple of standard error.

You can use a table of average run lengths to choose h (this is typically between zero and 10). See Table 6.3 and Table 6.4

HEADSTART=value HSTART=value

specifies a headstart value $S_{0}$ for a one-sided cusum scheme. The value must be expressed as a multiple of standard error. See the section Headstart Values,and refer to Lucas and Crosier (1982), Ryan (1989), and Montgomery (1996).

K=value

specifies the reference value k for a one-sided (decision interval) cusum scheme. This type of scheme is completely specified by the parameters k and h (see the H= option earlier in this list). You can also specify the K= and H= options as geometric parameters for a two-sided cusum scheme with a V-mask . In this case, the K= option specifies the slope of the lower arm of the V-mask, and the K= and H= options together are alternatives to the error probability options ALPHA=, SIGMAS=, and BETA=. In either case, the K= value must be positive and must be expressed as a multiple of standard error.

You can use a table of average run lengths to choose k and h (k is typically between zero and two). See Table 6.3 and Table 6.4.

For a one-sided scheme, the default K= value is $\delta /2$ , which is referred to as the central reference value. For a two-sided scheme where the V-mask is specified geometrically with the H= option, the default K= value is $\delta /2$ . If, however, the V-mask is specified by an error probability with the ALPHA= option, then the K= option should not be specified.

Caution: The interpretation of the K= value depends on the subgroup-variable and the interval between subgroups that is specified with the INTERVAL= option. For a two-sided scheme, the value is the increase in the lower V-mask arm per unit change on the subgroup axis, so the value depends on how the subgroup-variable is scaled.

If integer values are assigned to the subgroup-variable, then a unit change is defined as one.
If the subgroup-variable has character values, then a unit change is defined as the increment between adjacent values of the subgroup-variable.
If the subgroup-variable is numeric and is formatted with a SAS date or time format, then a unit change is defined as the default value for the INTERVAL= option. For example, if a DATE7. format is associated with the subgroup-variable, then a unit change is defined as one day.

You can use the INTERVAL= option to modify the definition of a unit change. For example, if a DATE7. format is associated with the subgroup-variable but subgroups are collected hourly, then INTERVAL=HOUR defines a unit change as one hour rather than one day.

LIMITN=n LIMITN=VARYING

specifies either a fixed or varying nominal sample size for the control limits. If you specify LIMITN=n, cusums are calculated and displayed only for those subgroups with a sample size equal to n, although you can specify the ALLN option to force all cusums to be plotted. If you specify LIMITN=VARYING, cusums are calculated and displayed for all subgroups, regardless of sample size.

LLIMITS=linetype

Traditional Graphicsspecifies the line type for the decision interval.

LMASK=linetype

Traditional Graphics specifies the line type for the V-mask arms.

MU0=value

specifies the target mean $\mu _{0}$ for the process. The target mean must be scaled in the same units as the data.

NOARL

suppresses calculation of average run lengths. By default, this calculation is performed if you specify the TABLESUMMARY option or an OUTLIMITS= data set.

NOMASK

suppresses the display of the V-mask on charts for two-sided schemes. This option does not affect computations of cusums or V-mask parameters.

NOREADLIMITS

specifies that the cusum scheme parameters for each process listed in the chart statement are not to be read from the LIMITS= data set specified in the PROC CUSUM statement. The NOREADLIMITS option is available only in SAS 6.10 and later releases. See the READLIMITS option later in this list.

ORIGIN=value

specifies the origin of the V-mask, which is defined as the horizontal coordinate of the right edge of the V-mask. If a date, time, or datetime format is associated with the subgroup-variable, you must specify the value as a date, time, or datetime constant, respectively. If the subgroup variable is character, you must specify the value as a quoted string. The default value is the last (most recent) value of the subgroup-variable.

Note that estimates for the process mean and standard deviation are calculated only from subgroups up to and including the origin subgroup.

READINDEX=’value’

reads cusum scheme parameters from a LIMITS= data set (specified in the PROC CUSUM statement) for each process listed in the chart statement. The ith set of control limits for a particular process is read from the first observation in the LIMITS= data set for which

the value of _VAR_ matches process
the value of _SUBGRP_ matches the subgroup-variable
the value of _INDEX_ matches value

The value can be up to 16 characters and must be enclosed in quotes.

READLIMITS

specifies that cusum scheme parameters are to be read from a LIMITS= data set specified in the PROC CUSUM statement. The parameters for a particular process are read from the first observation in the LIMITS= data set for which

the value of _VAR_ matches process
the value of _SUBGRP_ matches the subgroup variable

The use of the READLIMITS option depends on which release of SAS/QC software you are using.

In SAS 6.10 and later releases, the READLIMITS option is not necessary. To read cusum scheme parameters as described previously, you simply specify a LIMITS= data set. However, even though the READLIMITS option is redundant, it continues to function as in earlier releases.
In SAS 6.09 and earlier releases, you must specify the READLIMITS option to read cusum scheme parameters as described previously. If you specify a LIMITS= data set without specifying the READLIMITS option (or the READINDEX= option), the cusum scheme parameters are computed from the data.

READSIGMAS

specifies that the variable _SIGMAS_ (instead of _ALPHA_) is to be read from a LIMITS= data set that contains both variables. The variables _SIGMAS_ and _ALPHA_ provide the same parameters as the SIGMAS= and ALPHA= options. By default, _ALPHA_ is read from the LIMITS= data set.

SCHEME=ONESIDED SCHEME=TWOSIDED

indicates whether the cusum scheme is a one-sided (decision interval) scheme or a two-sided scheme with a V-mask. By default, SCHEME=TWOSIDED.

SHIFT=value

specifies the shift to be detected in the same units as the data. The value is interpreted as the shift in the mean of the sampling distribution of the subgroup mean. The SHIFT= option is an alternative to the DELTA= option. To specify the SHIFT= option, one of the following must be true:

The subgroup sample sizes are constant.
A constant nominal sample size n is provided for the cusum scheme with the LIMITN= option or the _LIMITN_ variable in a LIMITS= data set.

The relationship between the SHIFT= value (denoted by $\Delta$ ) and the DELTA= value (denoted by $\delta$ ) is $\delta = \Delta / (\sigma / \sqrt {n} )$ , where $\sigma$ is the process standard deviation.

SIGMA0=value

specifies a known standard deviation $\sigma _{0}$ for the process standard deviation $\sigma$ . The value must be positive. By default, PROC CUSUM estimates $\sigma$ from the data using the formulas given in Methods for Estimating the Standard Deviation. You can use the variable _STDDEV_ in a LIMITS= data set as an alternative to the SIGMA0= option.

SIGMAS=value

specifies the probability $\alpha$ of false detection for a two-sided cusum scheme with a V-mask as the probability that the absolute value of a standard normally distributed variable is greater than the value. For example, SIGMAS=3 corresponds to the probability $\alpha$ =0.0027. The value must be positive. The SIGMAS= option is an alternative to the ALPHA= and H= options, and only one of these three options can be specified.

The SIGMAS= option is useful for defining cusum charts that correspond to Shewhart charts whose control limits are defined with the same value as the multiple of $\sigma$ . Refer to Johnson and Leone (1962, 1974).

SMETHOD=NOWEIGHT | MVLUE | RMSDF

specifies a method for estimating the process standard deviation from subgroup observations, $\sigma$ , as summarized by the following table.

Keyword	Method for Estimating Standard Deviation
NOWEIGHT	estimates $\sigma$ as an unweighted average of unbiased subgroup estimates of $\sigma$
MVLUE	calculates a minimum variance linear unbiased estimate for $\sigma$
RMSDF	calculates a root-mean square estimate for $\sigma$

For formulas, see Methods for Estimating the Standard Deviation.

TABLEALL

requests all the tables specified by the options TABLECHART, TABLECOMP, TABLEID, TABLEOUT, and TABLESUMMARY.

TABLECHART <(EXCEPTIONS)>

creates a table of the subgroup variable, the subgroup sample sizes, the subgroup means, the cumulative sums, and the decision interval or V-mask limits. A table is produced for each process specified in the XCHART statement. The keyword EXCEPTIONS (enclosed in parentheses) is optional and restricts the tabulation to those subgroups for which the decision interval or V-mask values are exceeded.

TABLECOMP

tabulates the computational form of the cusum scheme as described by Lucas (1976) and Lucas and Crosier (1982). Upper or lower cumulative sums (or both) are tabulated for each process given in the XCHART statement. See Formulas for Cumulative Sums for more information.

TABLEID

augments the tables specified by the TABLECHART and TABLECOMP options with a column for each of the ID variables.

TABLEOUT

augments the table specified by the TABLECHART option with a column indicating whether the decision interval or V-mask values are exceeded.

TABLESUMMARY

produces a table that summarizes the cusum scheme. The table lists the parameters of the scheme and the average run lengths corresponding to shifts of zero and $\delta$ . The average run lengths are computed using the method of Goel and Wu (1971). A table is produced for each process. You can save the summary in a data set by specifying the OUTLIMITS= option. See OUTLIMITS= Data Set for details.

TYPE=ESTIMATE TYPE=STANDARD

specifies the value of _TYPE_ in an OUTLIMITS= data set. The variable _TYPE_ indicates whether the variable _STDDEV_ in the OUTLIMITS= data set represents an estimate or a standard (known) value. The default is 'STANDARD' if the SIGMA0= option is specified; otherwise, the default is 'ESTIMATE'.

WLIMITS=linetype

Traditional Graphics specifies the width (in pixels) of the decision interval line.

WMASK=linetype

Traditional Graphics specifies the width (in pixels) of the V-mask arms.