Computes a two-sided confidence interval for properties of a control chart.

Description

Computes a two-sided confidence interval for properties of a control chart.

Usage

SPC2sidedconfint(covprob = 0.9, ...)

Arguments

Value

The desired confidence interval, a vector of length 2.

See Also

SPCproperty

Examples

# Compute 2-sided CI for the ARL of a CUSUM control chart assuming normality.
 X <-  rnorm(100) #observed data
 chart <- new("SPCCUSUM",model=SPCModelNormal(Delta=1)) # CUSUM chart with normal observations
 SPC2sidedconfint(data=X,nrep=100,covprob=0.95,
            property="ARL",chart=chart,params=list(threshold=4))

CUSUM Charts

Description

Class extending SPCChart with a basic CUSUM charts implementation.

Details

The only slot this class contains is the data model. This data model should already incorporate the negative mean for in-control updates that is typical for CUSUM charts.

Let U_t, t=1,2,\dots be the updates from the data model. Then the CUSUM chart is given by S_0=0 and

S_t=max(S_{t-1}+U_t,0)

Examples

X <-  rnorm(1000)
chart <- new("SPCCUSUM",model=SPCModelNormal(Delta=1))
## Not run: 
SPCproperty(data=X,nrep=10,chart=chart,
            property="calARL",params=list(target=100))
SPCproperty(data=X,nrep=10,chart=chart,
            property="calhitprob",params=list(target=0.05,nsteps=1e3))
SPCproperty(data=X,nrep=10,chart=chart,
            property="ARL",params=list(threshold=3))

## End(Not run)
SPCproperty(data=X,nrep=10,chart=chart,
            property="hitprob",params=list(threshold=3,nsteps=1e3))
#increase the number of repetitions nrep for real applications.

Data Model for SPC charts

Description

This is the basic structure for converting observations (data) into updates for control charts. Classes of this type also have the ability to generate new data sets (resampling).

Details

Every element of this class has to consist of a list of the following functions: updates, getcdfupdates, Pofdata, resample, xiofP, which have to be of a specific form. The arguments generally have the following meaning: xi denotes the parameter vector needed to create updates for running the chart from observed data, data is observed data, P is a data model.

• updates(xi,data): Returns updates for the chart using the parameter xi and the observed data data.

• Pofdata(data): Estimates a probability model from the data.

• xiofP(P): Computes the parameter xi needed to compute updates from an (estimated) probability model P.

• resample(P): Generates a new data set from the probability model P.

• getcdfupdates(P,xi,cadlag=TRUE): Returns the cumulative distribution function (CDF) of updates of data generated from the probability model P and computed using the parameter xi. The CDF has to be a function of one argument that also accepts vectors. If cadlag is TRUE then the CDF is right-continuous (i.e. F(x)=P(X\le x)). If cadlag is FALSE then the CDF is left-continuous (i.e. F(x) = P(X<x)).

See Also

SPCModelNormal, SPCModelNonpar, SPCModelNonparCenterScale

EWMA Charts

Description

Class extending SPCChart with a basic EWMA charts implementation.

Details

Let Y_t, t=1,2,\dots be the updates from the data model. Then the EWMA chart is given by Q_0=0 and

Q_t=lambda Y_t+(1-lambda) Q_{t-1}

Slots

model

The data model. The data model should center the in-control updates such that they have mean 0.

lambda

The smoothing constant, 0<lambda<=1.

Examples

X <-  rnorm(1000)
chart <- new("SPCEWMA",model=SPCModelNormal(Delta=0),lambda=0.8)
## Not run: 
SPCproperty(data=X,nrep=10,chart=chart,
            property="calARL",params=list(target=100))
SPCproperty(data=X,nrep=10,chart=chart,
            property="calhitprob",params=list(target=0.05,nsteps=1e3))

## End(Not run)
SPCproperty(data=X,nrep=10,chart=chart,
            property="ARL",params=list(threshold=3))
SPCproperty(data=X,nrep=10,chart=chart,
            property="hitprob",params=list(threshold=3,nsteps=1e3))
#increase the number of repetitions nrep for real applications.

Generic Model for Nonparametric Resampling

Description

Generic model that allows nonparametric resampling (with replacement) of the data. The transformation of data into updates needs to be defined via the arguments.

Usage

SPCModelNonpar(updates, xiofP)

Arguments

Details

The parameters to the functions being returned have the following meaning.

• data: a numeric vector or a matrix where the rows contain the observations.

• xi: depends on the parameter xiofP.

• P: The data with no modifications (thus either a numeric vector or a matrix).

The main operations are defined as follows:

• resample(P): generates a new data set of the same size by either resampling the values (if the data is a vector) or by resampling the rows of the data matrix (both use resampling with replacement).

Value

An object of class SPCDataModel.

Nonparametric Resampling with Centering and Scaling

Description

Nonparametric resampling of univariate observations. Updates are centered and scaled transformations of the data (with a constant potentially being subtracted).

Usage

SPCModelNonparCenterScale(Delta = 0)

Arguments

Details

Calls SPCModelNonpar to generate the data object, so it only needs to specify the meaning of xi, the parameter needed to compute updates and the definition of the updates.

• xi: a list with two elements:

  • mu: the mean.

  • sd: the standard deviation.

• updates(xi,data): returns the centered and scale version of the data from which Delta/2 has been subtracted, i.e.

  \frac{data-mu-Delta/2}{sd}

Value

An object of class SPCDataModel.

An object of class SPCDataModel.

See Also

SPCModelNonpar

Examples

X <- rnorm(1000)

#CUSUM chart
model <- SPCModelNonparCenterScale(1)
chart <- new("SPCCUSUM",model=model)
SPCproperty(data=X,nrep=10,property="calARL",
            chart=chart,params=list(target=100))

#Shewhart chart
model <- SPCModelNonparCenterScale(0)
chart <- new("SPCCUSUM",model=model)
SPCproperty(data=X,nrep=10,property="calARL",
            chart=chart,params=list(target=100))

Parametric Model for Normally Distributed Data with Centering and Scaling

Description

Returns a data model for univariate observations using normality assumptions with updates that center and scale the observations and potentially subtract half of a constant Delta. Subtracting Delta/2 is useful for CUSUM charts.

Usage

SPCModelNormal(Delta = 0)

Arguments

Details

The parameters to the function have the following meaning.

• data: a numeric vector.

• xi: a list with two elements:

  • mu: the mean.

  • sd: the standard deviation.

• P: a list with three elements:

  • mu: the mean.

  • sd: the standard deviation.

  • m: the number of data points to resample.

The main operations are defined as follows:

• updates(xi,data): returns the centered and scale version of the data from which Delta/2 has been subtracted, i.e.

  \frac{data-mu-Delta/2}{sd}

  .

• resample(P): resamples m new data points from a normal distribution if mean mu and standard deviation sd.

Value

An object of class SPCDataModel.

Data model based on a linear model

Description

The parameters needed for running the chart are the fitted linear model. Resampled data sets are created by resampling cases with replacement (i.e. keeping observations together).

Usage

SPCModellm(formula, Delta = 0)

Arguments

Examples

n <- 1000
Xlinreg <- data.frame(x1= rbinom(n,1,0.4), x2= runif(n,0,1), x3= rnorm(n))
Xlinreg$y <- 2 + Xlinreg$x1 + Xlinreg$x2 + Xlinreg$x3 + rnorm(n)
## Not run: 
chartlinregCUSUM <- new("SPCCUSUM", model=SPCModellm(Delta=1,formula="y~x1+x2+x3"))
SPCproperty(data=Xlinreg,nrep=10,property="calARL",
            chart=chartlinregCUSUM,params=list(target=100))
#increase nrep in real applications.
#' chartlinregCUSUM2 <- new("SPCCUSUM",model=SPCModellm(Delta=1,formula="y~x1"))
SPCproperty(data=Xlinreg,nrep=10,property="calARL",
            chart=chartlinregCUSUM2,params=list(target=100))
#increase nrep in real applications.

chartlinregEWMA <- new("SPCEWMA", model=SPCModellm(Delta=0,formula="y~x1+x2+x3"),lambda=0.8)
SPCproperty(data=Xlinreg,nrep=10,property="calARL",
            chart=chartlinregEWMA,params=list(target=100))
#increase nrep in real applications.

chartlinregEWMA2 <- new("SPCEWMA",model=SPCModellm(Delta=0,formula="y~x1"),lambda=0.8)
SPCproperty(data=Xlinreg,nrep=10,property="calARL",
            chart=chartlinregEWMA2,params=list(target=100))

## End(Not run)
#increase nrep in real applications.

Data Model for Binary Responses using a logarithmic model and likelihood ratio updates.

Description

Data Model for Binary Responses using a logarithmic model and likelihood ratio updates.

Usage

SPCModellogregLikRatio(formula, Delta = 1)

Arguments

Examples

n <- 1000
Xlogreg <- data.frame(x1=rbinom(n,1,0.4), x2=runif(n,0,1), x3=rnorm(n))
xbeta <- -1+Xlogreg$x1*100+Xlogreg$x2+Xlogreg$x3
Xlogreg$y <- rbinom(n,1,exp(xbeta)/(1+exp(xbeta)))
chartlogreg <- new("SPCCUSUM",
                   model=SPCModellogregLikRatio(Delta= 1, formula="y~x1+x2+x3"))
SPCproperty(data=Xlogreg,nrep=10,property="calARL",
            chart=chartlogreg,params=list(target=100))
#increase nrep for real applications.

Data Model for Binary Responses using a Logarithmic Model and observed minus expected updates.

Description

Data Model for Binary Responses using a Logarithmic Model and observed minus expected updates.

Usage

SPCModellogregOE(formula, Delta = 0)

Arguments

Examples

n <- 1000
Xlogreg <- data.frame(x1=rbinom(n,1,0.4), x2=runif(n,0,1), x3=rnorm(n))
xbeta <- -1+Xlogreg$x1*100+Xlogreg$x2+Xlogreg$x3
Xlogreg$y <- rbinom(n,1,exp(xbeta)/(1+exp(xbeta)))
chartlogreg <- new("SPCEWMA",
                   model=SPCModellogregOE(Delta= 0, formula="y~x1+x2+x3"), lambda=0.8)
SPCproperty(data=Xlogreg,nrep=10,property="calARL",
            chart=chartlogreg,params=list(target=100))
#increase nrep for real applications.

Shewhart charts.

Description

Shewhart charts.

Slots

twosided

TRUE if a two-sided chart should be used. Default FALSE.

Examples

X<-rnorm(100);
##calibrate to ARL 100
chartShew <- new("SPCShew",model=SPCModelNormal(),twosided=TRUE)
## Not run: 
SPCproperty(data=X,nrep=500,
            property="calARL",chart=chartShew,params=list(target=100),
            covprob=c(0.7,0.9))

chartShewOneSided <- new("SPCShew",model=SPCModelNormal(),twosided=FALSE)
SPCproperty(data=X,nrep=500,
            property="calARL",chart=chartShewOneSided,
            params=list(target=100),covprob=c(0.7,0.9))

##calibrate to a hitting probability of 0.01 in 100 steps
SPCproperty(data=X,nrep=500,
            property="calhitprob",
            chart=chartShew,params=list(target=0.01,nsteps=100))
SPCproperty(data=X,nrep=500,
            property="calhitprob",chart=chartShewOneSided,params=list(target=0.01,nsteps=100))

## work out  for ARL for a fixed threshold of 4
SPCproperty(data=X,nrep=500,
            property="ARL",chart=chartShew,params=list(threshold=4))
SPCproperty(data=X,nrep=500,
            property="ARL",chart=chartShewOneSided,
            params=list(threshold=4))

SPCproperty(data=X,nrep=500,
            property="hitprob",chart=chartShew,params=list(nsteps=100,threshold=4))

SPCproperty(data=X,nrep=500,
            property="hitprob",chart=chartShewOneSided,params=list(nsteps=100,threshold=4))

## End(Not run)

X<-rnorm(100)
chartShew <- new("SPCShew",model=SPCModelNormal())
## Not run: 
SPCproperty(data=X,nrep=500,
            property="calARL", chart=chartShew,
            params=list(target=1000))
SPCproperty(data=X,nrep=500,
            property="calhitprob",chart=chartShew,
            params=list(target=0.01,nsteps=100))
SPCproperty(data=X,nrep=10,chart=chartShew,
            property="ARL",params=list(threshold=3))
SPCproperty(data=X,nrep=500,
            property="hitprob",
            chart=chartShew,params=list(nsteps=100,threshold=4))

## End(Not run)

Virtual Base Class for Control Charts

Description

Virtual S4 base class for Control Charts.

Slots

model

The data model to be used in the chart. Must be of type SPCDataModel.

Computes bootstrap adjusted properties for control charts

Description

Computes bootstrap adjusted properties for control charts.

Usage

SPCproperty(data, nrep = 500, chart, property, params, covprob = 0.9,
  quiet = FALSE, reportdistr = FALSE, parallel = 1)

Arguments

Value

An object of type SPCpropertyres.

See Also

SPC2sidedconfint

Examples

 # calibrate CUSUM chart to an in-control ARL of 100.
 # run with a larger number of replications in real examples!
 X <-  rnorm(100) #observed data
 chart <- new("SPCCUSUM",model=SPCModelNormal(Delta=1)) # CUSUM chart with normal observations
SPCproperty(data=X,nrep=15,chart=chart,property="calARL", params=list(target=100))

Results of SPCproperty.

Description

Results of SPCproperty.

Usage

## S4 method for signature 'SPCpropertyres'
show(object)

Arguments

Methods (by generic)

• show: Prints the object nicely.

Slots

nrep

number of repetitions used in the simulation.

chart

the chart used.

property

the property of interest, ARL, calARL, calhitprob or hitprob.

covprob

the probability of the guaranteed conditional performance.

res

the guaranteed conditional performance.

raw

the unadjusted result.

params

additional parameters used for computing this property.

restext

a readable version of the result.

Cardiac surgery data

Description

A dataset describing the results of cardiac surgery. The data give information about the date of surgery, surgeon, Parsonnet score and outcome up to 90 days after surgery.

Usage

data(cardiacsurgery)

Format

A data frame with 5595 rows and 5 variables:

Source

Based on the data described in Steiner et al (2000). A subset of the data has been selected, some noise has been introduced and the follow-up was censored at 90 days.

References

Steiner SH, Cook RJ, Farewell VT, Treasure T (2000). Monitoring surgical performance using risk-adjusted cumulative sum charts. Biostat 1(4) 441-452.

CDF of Updates of a Control Chart

Description

Consider running a control chart with given parameters with data coming from a given probability model. This function computes the cumulative distribution function (CDF) of the updates of the control charts as they would be computed by the method updates.

Usage

getcdfupdates(chart, P, xi)

## S4 method for signature 'SPCchart'
getcdfupdates(chart, P, xi)

## S4 method for signature 'SPCShew'
getcdfupdates(chart, P, xi)

Arguments

Value

A function mapping one-dimensional numerical values into the interval [0,1], having all properties of a cumulative distribution function.

Methods (by class)

• SPCchart: Standard method which simply first applies getcdfresiduals from the data model.

• SPCShew: Computes the CDF of the updates taking into account if the chart is one-sided or two-sided.

Returns a List to Compute Properties of a chart

Description

Returns functions to compute desired properties of a given control chart.

Usage

getq(chart, property, params)

## S4 method for signature 'SPCCUSUM'
getq(chart, property, params)

## S4 method for signature 'SPCEWMA'
getq(chart, property, params)

## S4 method for signature 'SPCShew'
getq(chart, property, params)

Arguments

Value

A list with the elements q, trafo, lowerconf, format.

• q(P,xi): The transformed property of interest. To improve the bootstrap a log transform is used for calARL,calhitprob and ARL, and a logit transform for hitprob. This function depends on the distribution of updates P and the chart parameters xi.

• trafo(x): The inverse of the transformation of the property used in the bootstrap. Needed to back-transform the result to the correct scale.

• lowerconf: Logical value. TRUE if a lower confidence interval should be reported, FALSE otherwise. Default is TRUE for properties calARL, calhitprob and hitprob and FALSE for ARL.

• format(res): Output summary given as a text string.

Methods (by class)

• SPCCUSUM: Implements the properties ARL, calARL, hitprob and calhitprob.

• SPCEWMA: Implements the properties ARL, calARL, hitprob and calhitprob.

• SPCShew: Implements the properties ARL, calARL, hitprob and calhitprob.

Runs a chart

Description

Generic method for running a chart on new data using given parameters xi.

Usage

runchart(chart, newdata, xi)

## S4 method for signature 'SPCCUSUM'
runchart(chart, newdata, xi)

## S4 method for signature 'SPCEWMA'
runchart(chart, newdata, xi)

## S4 method for signature 'SPCShew'
runchart(chart, newdata, xi)

Arguments

Value

The path of the chart over time.

Methods (by class)

• SPCCUSUM: Generic function for running CUSUM charts. Relies on updates being implemented for the chart.

• SPCEWMA: Generic function for running EWMA charts. Relies on updates being implemented for the chart.

• SPCShew: Simply computes the updates appropriate for the Shewhart chart and returns them.

Updates of a Control Chart

Description

Computes updates of a control chart using the given parameters and the given data.

Usage

updates(chart, xi, data)

## S4 method for signature 'SPCchart'
updates(chart, xi, data)

## S4 method for signature 'SPCShew'
updates(chart, xi, data)

Arguments

Value

A vector of the same length as data.

Methods (by class)

• SPCchart: Standard method which simply first applies getresiduals from the data model.

• SPCShew: Computes the updates taking into account if the chart is one-sided or two-sided.

Estimate Chart Parameters

Description

Estimates the parameters used to run a control chart from given data.

Usage

xiofdata(chart, data)

## S4 method for signature 'SPCchart'
xiofdata(chart, data)

Arguments

Value

The parameter values for running the control chart chart.

Methods (by class)

• SPCchart: Standard method which simply first applies PofData to get a model and then xiofP to get the parameters.