six sigma healthcare spc infection control papertttt

7/28/2019 Six Sigma Healthcare SPC Infection Control Papertttt

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Int. J. Six Sigma and Competitive Advantage, Vol. 4, No. 3, 2008 209

The design, selection, and performance of statisticalcontrol charts for healthcare process improvement

James C. Benneyan

Healthcare Systems Engineering Laboratory

Northeastern University

334 Snell Engineering Center

Boston MA 02115, USA

Fax: 6173732921

E-mail: [email protected]

Abstract: Although the value of Statistical Process Control (SPC) charts tomonitor and improve healthcare processes has received growing interest,several important implementation and theoretical issues are largely notunderstood. SPC charts are chronological graphs of process data that are usedin many other industries to help control and improve quality and, while basedin statistical theory, are easy for practitioners to use and interpret. This articleprovides an overview of the use of SPC in healthcare process improvementand Six Sigma activities, including the process of selecting and using theappropriate type of control chart, their statistical performance, and theguidelines for determining the appropriate sample sizes. The operatingcharacteristics are computed and tabulated for a wide range of scenariosin order to aid practitioners in designing control charts with the desiredstatistical properties.

Keywords: statistical process control; SPC; control charts; healthcare;quality improvement; adverse events; patient safety; OC curves; average runlengths; ARLs.

Reference to this paper should be made as follows: Benneyan, J.C. (2008) Thedesign, selection, and performance of statistical control charts for healthcareprocess improvement, Int. J. Six Sigma and Competitive Advantage, Vol. 4,No. 3, pp.209239.

Biographical notes: James C. Benneyan, PhD, is an Associate Professor ofIndustrial Engineering and Operations Research and the Director of the Qualityand Productivity Laboratory of Northeastern University, USA, a faculty forthe Institute for Healthcare Improvement, and an advisor to severalnational healthcare improvement projects. Previously, he was a Senior SystemsEngineer at Harvard Community Health Plan and is a past President and

Fellow of the Society for Health Systems. His research areas include thestatistical methods for quality improvement, healthcare systems engineering,and operations research in nanotechnology and biomanufacturing.

Copyright 2008 Inderscience Enterprises Ltd.


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210 J.C. Benneyan

1 Introduction

This article provides an overview of the use of Statistical Process Control (SPC) charts

within process improvement and Six Sigma activities, the different types and uses of

control charts, and guidelines for selecting an appropriate type of chart and sample size to

achieve desirable properties. The intent is to provide an introduction to these methods and

further insight into their selection, design, and performance beyond what typically exists

in the literature. The utility of control charts to help improve clinical and administrative

processes has received growing interest in the healthcare community. For example,

see Nelson et al. (2004), Sellick (1993), Curran et al. (2008), Benneyan (1998c), Burnett

and Chesher (1995), and a comprehensive review in a series in Infection Control

and Hospital Epidemiology (Benneyan, 1998a; Benneyan, 1998b). These methods are

well-established in other industries and have a long history of use for measuring process

performance and improving outcome quality. Important healthcare applications includeadverse drug events, surgical site infections, patient falls, central line infections, surgical

complications, and many other iatrogenic injuries and adverse events.

The estimated total annual national costs in the USA of such process defects are

staggering, including 770 000 to 2 million patient injuries, 8.7 million hospital days,

44 000 to 180 000 deaths, and $8.8 billion in healthcare costs (Bates, 1997; Bedell et al.,

1991; Bogner, 1994; Brennan et al., 1991; Leape, 1994; Cullen et al., 1997). Studies

summarised in the recent National Academy of Sciences Institute of Medicine (1999)

report, To Err is Human, estimated that between 45 000 to 98 000 patients die each year

in US hospitals from medical errors, more than the annual number of deaths in the USA

from traffic accidents, breast cancer, or AIDS.

It is not surprising that many accrediting and regulatory bodies therefore encourage

healthcare organisations to apply continuous quality improvement methodologies to these

process concerns, including the use of statistical methods such as SPC. For example,

the Joint Commission on Accreditation of Healthcare Organizations (1997) stated their

position on the use of SPC as follows:

An understanding of statistical quality control, including SPC, and variationis essential for an effective assessment process... Statistical tools such as runcharts, control charts, and histograms are especially helpful in comparingperformance with historical patterns and assessing variation and stability.

Several authors from the US Center for Disease Control (Martone et al., 1991) similarly

stated that:

Many of the leading approaches to directing quality improvement in hospitalsare based on the principles of W. E. Deming. These principles include use ofstatistical measures designed to determine whether improvement in quality hasbeen achieved. These measures should include nosocomial infection rates.

Many epidemiologists also have proposed monitoring infection and adverse event rates

continuously over time in manners that are quite similar to SPC (Birnbaum, 1984;

Mylotte, 1996; Childress and Childress, 1981; Mylotte et al., 1989), including the

increasing important problem of antibiotic resistance (Curran et al., 2008). In

conventional epidemiology, in fact, the identification of epidemic and endemic events

are related in SPC terminology to the detection of unnatural and natural variability,


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The design, selection and performance of statistical control charts 211

respectively. It also is interesting that as early as 1942, the late quality pioneer Deming

(1942) advocated the application of SPC to disease surveillance and adverse healthcare

events in theJournal of the American Statistical Association.

As evident in throughout the above publications, SPC is as much a process

improvement philosophy as it is a data analysis method, with several important

implications discussed below. These include the use of data for learning and

improvement (rather than for assigning blame), the frequency of data collection, the type

of data and format in which they should be collected, and the actions taken based on

the results. If control charts are to be used, moreover, it is equally important that they

be used correctly, including the selection of the right chart and sample size. While

details on the mathematics of control charts appear in many publications, the above topics

tend to receive less attention but are equally important to the successful use of SPC for

process improvement.

2 Quality control charts

2.1 Overview and interpretation

Statistical process control charts are chronological graphs of process data that are used to

help understand, control, and improve processes such as infection control or adverse

event processes and that, although based in statistical theory, are easy for practitioners

to use and interpret. While there are several different types of control charts, the general

format and interpretation of the most common and simplest type, called a Shewhart

control chart, are illustrated in Figure 1. As shown, some statistic of interest, such as the

number of cases of ventilator-associated pneumonia per 100 device days, is plotted on the

chart over time and interpreted on a monthly or weekly basis.

Figure 1 General format and interpretation of a statistical control chart

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Sample Number

Measurement

of

Interest

Upper Control Limit (UCL)

Lower Control Limit (LCL)

Center Line (CL)

Low probability event

(Do not expect)

Low probability event

(Do not expect)

Natural Range

of Variation

(Expect)

Time

(Upper Warning Limit)

(Lower Warning Limit)


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212 J.C. Benneyan

The three horizontal lines called the Centre Line (CL), the Upper Control Limit (UCL),

and the Lower Control Limit (LCL) define the central tendency and the range of naturalvariation of the plotted values, assuming that the long-term pneumonia rate, in the above

example, does not change. The control limits are computed statistically based on

probability models such as the Gaussian (normal), Poisson, or binomial distributions.

As shown in Figure 2 for the bell-shaped normal distribution, values that fall outside the

upper and lower three standard deviation control limits exceed the range within which

almost all of the values (99.73% under the normal assumption) are expected to lie ifthe

process remains unchanged and produces statistically consistent results.

Figure 2 Relation of control limits to underlying probability theory (normal distribution)

Clinical or Administrative Value of Interest

Mean

0.9973

(very likely)

Mean minus

3 standard

deviations

Mean plus

3 standard

deviations

.00135

(unlikely).00135(unlikely)

Interpretation of values in the tails or outside the control limits is similar to that of

conventional hypothesis tests, namely that these values are statistically significant

indications that the process is producing different results or is not producing outcomes

from only one consistent and homogeneous process. Under the philosophy of quality

improvement, the cause(s) of the process change or inconsistency should be identified

and removed in order to achieve a single, stable, and predictable process (i.e., a state of

statistical control in SPC language). While several different types of charts exist (seeSection 3 below), all are interpreted in essentially this same manner.

In addition to values outside the control limits, there should be no evidence of

non-random behaviour between the limits, such as trends, cycles, and shifts above or

beneath the CL. Various between-limit rules have been defined to aid in the objective

interpretation of such data patterns, such as those summarised in Grant and Leavenworth

(1988), Benneyan (1998b; 1998c), and Section 5.5 below. Most of these supplementary

rules are based on binomial probability calculations of runs of consecutive values in

various zones of the control chart, thereby improving sensitivity but reducing specificity


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(see Section 5.5). See Grant and Leavenworth (1988), Duncan (1986), and Montgomery

(2009) for further information on the mathematical details of SPC. Note that control

limits should not be confused with confidence intervals that contain the process mean

with some probability, but rather are more analogous to prediction or probability intervals

that will contain almost all data (with the specified probability).

2.2 Uses of control charts

It is important to emphasise that control charts have several important and somewhat

sequential roles in quality improvement work. These uses are discussed in greater detail

elsewhere (Benneyan, 1998b) and include (see Figure 3):

understanding current and past process performance and its degree of consistency

and predictability

establishing a state of statistical control by identifying and removing current

existing causes of unnatural (or special cause) variation so as to achieve a

consistent and predictable level of process quality over time

improving a process by identifying and removing causes of natural (or common

cause) variation and by testing whether interventions result in an improvement

monitoring for process deterioration and holding the gains by identifying new

special causes of unnatural variation if and when they arise in the future.

Figure 3 The four phases and uses of control charts

Understand Past PerformanceIs process unstable or in statistical control?

Begin trial control charting

Understand Past PerformanceIs process unstable or in statistical control?

Begin trial control charting

Establish Consistent Process

Iterative process using trial control charts

Bring process into statistical control

Remove existingsources of unnatural variability until in-control

Establish Consistent Process

Iterative process using trial control charts

Bring process into statistical control

Remove existingsources of unnatural variability until in-control

Improve Stable Process

Remove sources of natural variability

Position the average & reduce variability until acceptable

Test and verify impact of interventions and improvements

Improve Stable Process

Remove sources of natural variability

Position the average & reduce variability until acceptable

Test and verify impact of interventions and improvements

Monitor Process for Instability

Maintain a state of statistical control

Hold the gains

Identify & remove newspecial causes of unnatural variability

Monitor Process for Instability

Maintain a state of statistical control

Hold the gains

Identify & remove newspecial causes of unnatural variability


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214 J.C. Benneyan

2.3 Iteratively establishing a state of statistical control

While the latter two uses of control charts testing and holding the gains tend to be the

most well-known, the first two activities are extremely important in process improvement

but often overlooked or misunderstood. In many applications, considerable value exists in

simply achieving a state of statistical control. As in other industries, many healthcare

processes will not be stable and consistent when first examined and will require

significant effort to bring them into a state of statistically consistent behaviour (i.e.,

statistical control).

This activity often is referred to as trial control charting because of the focus on

testing whether the process is consistent, removing found causes of unnatural variability,

retesting for statistical control, and attempting to bring the process into a state of

operation such that it produces consistent and predictable results. This iterative process

occurs over a period of time and consists of:

constructing an initial trial control chart to test for statistical control

searching for and removing any identified assignable causes of unnatural variability

removing all affected data and recalculating the CL and control limits from the

remaining data (with the addition of new data if available or necessary)

searching a second time for additional causes of unnatural variability

removing these data and reconstructing the control chart a second time as above

repeating this process as many times as is necessary until a state of statistical control

is reached.

As an analogy, this iterative improvement process is akin to removing rocks above thesurface of a pond (i.e., the upper limit), thereby lowering the water level only to identify a

next layer of rocks now exposed above the surface, and repeating this process until a

smooth consistent water horizon is achieved. Note that the trial control charting process

typically is conducted retrospectively on historical data with new data added as they

become available, in essence investigating if the process has been in statistical control in

the past and up to the present time. In some cases, after causes of inconsistent variation

are removed, an entirely new data set may need to be collected, although this can greatly

lengthen the time to achieve statistical control. In other cases, however, the affected

time periods of data can be determined and removed from the working data set, which

then can be re-used to re-compute the control limits; if possible, this latter approach is

preferred since it is not necessary to discard all historical data and collect an entire new

set of data.

Either way, practitioners sometimes are discouraged in this first phase of qualityimprovement by the amount of time and effort often required to achieve a predictable

process. It is important to emphasise, however, that this is a critical necessary first step

and that only when a state of statistical control has been established can it be stated that

a single process even exists. This is because without statistical control, there is no

consistent process producing the outcomes; if there were, by definition the process would

exhibit a state of statistical control. Until this distributional stability exists, statements

about quality levels, projections of occurrence rates, assertions about standardised

processes, and predictions about future outcomes all are invalid and misleading, despite


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any amount of wishful thinking to the contrary. Moreover, the widespread interest in

process standardisation implies, by definition, that processes can be shown to exhibit

statistical control.

2.4 Monitoring and improving a stable process

Once a stable process exists (i.e., a state of statistical control has been established), the

control chart is used to monitor for signals that a change has occurred (special cause

of unnatural variability in SPC terminology) points outside the control limits or

violations of any of the within-limit rules. If these are changes for the worse, such as an

increase in the ventilator-associated pneumonia rate, then an effort should be made to

discover the causes so that they can be removed and prevented in the future. Causes of

changes for the better also should be investigated and understood so that they can be

implemented on a regular basis.While this monitoring activity tends to be the most familiar use associated with

control charts, it also is the most passive use from a process improvement perspective

as it is focused primarily on maintaining the status quo. It also is important to note

that being in a state of statistical control must not necessarily imply that the process

is performing at an acceptable level, nor that the outcome rate is good, and that

either an increase or decrease (i.e., an improvement) in the outcome rate represents an

out-of-control process. Statistical control is defined as all data being produced by the

same constant process and probability model, which may or may not have an acceptable

mean or variance. For example, hypothetically it is possible to have a stable and

consistent adverse drug event rate of 50% (1 of every 2 medications on average),

although this obviously would be far from acceptable.

The improvement focus at this stage now is on the difficult task of identifyingchanges to the existent (and consistent) process that will result in improvements. Changes

to the existent standardised process will be necessary in order to improve outcomes. The

role of control charts here is two-fold to test whether changes actually result in the

hypothesised benefits and whether improvement goals have been reached. The first case

would be evidenced by (desirable) unnatural variability and out-of-control improvement

signals based on control limits computed from pre-change data, whereas the second case

would be demonstrated by post-change data exhibiting in-control behaviour and natural

variability with respect to control limits computed from the targets.

3 Types of control charts

The most familiar types of control charts, called Shewhart control charts, originally weredeveloped by Shewhart in 1924, one for each of several types of data that are commonly

encountered in practice. Each of these types of data can be described by a statistical

distribution that is used to determine the expected value, theoretical standard deviation,

and natural variation of the data (i.e., the CL and control limits). Examples of the most

common types of data distributions the normal, binomial, Poisson, and geometric are

shown in Figure 4. While many other types of data exist, these distributions are very

common and appropriate in many applications.


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216 J.C. Benneyan

Figure 4 Four common types of data distributions

Figure 4b: Binomialigure 4a: Normal

Figure 4c: Poisson Figure 4d: Geometric

One of the most common difficulties that practitioners have in using SPC is determiningwhich type of control chart they should construct. As shown in Table 1, the chart type

to use in any particular situation is based on identifying which type of data is most

appropriate. For example, the three most common types of control charts should be used

in the following situations:

Either an np or ap control chart should be used when analysing discrete data that

should be distributed according to a binomial distribution

Either a c or u control chart should be used when analysing count data that should be

distributed according to a Poisson distribution

Both anX-barand an Schart should be used together for continuous data that should

be distributed according to a normal distribution.

Many standard statistical packages of SPC software will construct these types of control

charts, as well as plot histograms or provide mathematical goodness-of-fit tests to help

verify the type of distribution that fits a particular data set. See Benneyan (1998a; 1998c)

for further discussion on the distinctions between each type of data and examples of each

type of control chart.


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Table 1 Common types of control charts and examples of applications

Type ofcontrol chart

Probabilitydistribution When appropriate to use Examples

Xbarand S

(Plot samplemean andstandarddeviation)

Normal(Gaussian)

Continuous measurements withbell shape

Note:XbarandR sometimesused as an alternative, althoughstatistical properties are not asgood. (Individuals chartshould be used only as a lastresort for same reason.)

Length of patient waits

Procedure durations

Timing of perioperativeantibiotics

Physiologic data

Time from decision tofirst incision for emergentCesarean deliveries

np

(Plot sampletotal)

Binomial Total number of dichotomous

cases generated by a processthat result in a certain outcome

Note: Sample size assumedconstant for each sample

Number of surgeries that

develop a surgical site infectionNumber of patients who receivean antibiotic on time

Number of patients readmitted

p

(Plot samplefraction)

Binomial Fraction of dichotomous casesgenerated by a process thatresult in a certain outcome

Note: Sample size can changefrom sample to sample

Fraction of surgeries thatdevelop a surgical site infection

Fraction of patients who receivean antibiotic on time

Fraction of patients readmitted

c

(Plot sample

rate)

Poisson Total number of some event,where no exact upper bound,can be more than one eventper patient or sampling unit

Note: Assumes constantopportunity or sampling area ineach time period

Number of patient falls

Number of centralline infections

Number of ventilatorassociated pneumonias

Number of needle sticks

u

(Plot samplerate adjustedper commonbase)

Poisson Rate of some event, where noexact upper bound, can bemore than one event per patientor sampling unit

Note: Rate is adjusted toaverage per some commonsampling denominator size

Average number of patient fallsper 100 patient days

Number of central lineinfections per 100 line-days

Number of ventilatorassociated pneumonias per100 ventilator days

g

(Plot countbetweenevents)

Geometric Number of cases or amount oftime between occurrences

Note: Particularly useful forrare events or when rate is low(e.g., rate < .01)

Number of surgeriesbetween infections

Number patientsbetween complications

Number days betweenadverse drug events

Number days betweenneedle sticks

Figure 5 illustrates Xbarand Scharts for the mean and standard deviation, respectively,

of the time from decision to the first incision for emergent Cesarean deliveries. Note that

this process appears fairly consistent with the exception of the three weeks when either


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218 J.C. Benneyan

theXbaror the Schart is out-of-control. The appropriate first action in this case therefore

would be to investigate what occurred during these time periods to cause thesestatistically significant differences in outcomes. Both charts are necessary, only for

normal data, because (1) the estimation of the Xbar control limits assumes that the

standard deviation is in statistical control and homogeneous and (2) either the mean or

standard deviation can become out-of-control independently of the other.

Figure 5 Example ofXbarand Scontrol chart for emergent Cesarean delivery decision toincision delays

Average Decision-to-Incision Time(Xbar Chart)

35

40

45

50

55

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2 0 21 22 23 24 25 26 27 28 29 30

Week Number

AaTme(mine

Standard Deviations of Decision-to-Incision Time(S Chart)

0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Week Number

Sa

dDao(mine

Note: n = sample of ten charts per week.


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For example, Benneyan (1998a) describes an experience in which the average timing of

perioperative prophylaxis was equal to the target of 60 min prior to the first incision and

appeared to be in statistical control, but the standard deviation was excessively large and

not in-control. Out of control values on theXbaror Schart indicate that the process mean

or standard deviation is not in statistical control, respectively, somewhat analogous to T

and Ftests for statistically significant differences in means and variances. (In fact, similar

to Fand Ttests, technically interpretation of anXbarchart is valid only if the Schart first

is found to be in-control.)

By contrast, thep control chart in Figure 6 illustrates a surgical site infection rate that

appears consistent and in-control but also is higher than is acceptable. Whereas this p

chart is for the fraction of surgeries that develop infections, an np chart alternatively

could be used for the total numberof surgeries to develop infections. The appropriate

action in this case would be to brainstorm and test process changes that might reduce

the infection rate, using the control chart to verify whether improvement occurs. The ucontrol chart in Figure 7, conversely, illustrates a ventilator-associated pneumonia rate

that is both in-control and most likely acceptably low. While improvement efforts

certainly should continue, this chart might be used primarily to detect increases and hold

the gains.

For both types of control charts (i.e., Poisson and binomial data), because the number

of surgeries, device-days, and so on typically is not constant in each sample, p and u

charts tend to be used more often than np and c charts, respectively, with the differing

sample sizes resulting in control limits that vary over time (as shown in Figures 6 and

7). Interpretation of the points between or outside these limits however is the same as

for all control charts (if the limits vary widely, if desirable for aesthetics and easier

interpretation the plotted points and limits can be standardised so as to now have straight

limits, analogous to aZtransformation, although this will not change results in any way).

Figure 6 Example ofp control chart of surgical site infection rate

.00

.05

.10

.15

.20

.25

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Month


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220 J.C. Benneyan

Figure 7 Example ofu control chart for ventilator associated pneumonia rate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Jan-98

Feb-98

Mar-98

Apr-98

May-98

Jun-98

Jul-98

Aug-98

Sep-98

Oct-98

Nov-98

Dec-98

Jan-99

Feb-99

Mar-99

Apr-99

May-99

Jun-99

Jul-99

Aug-99

Sep-99

Oct-99

Nov-99

Dec-99

Jan-00

Feb-00

Mar-00

Month

Note that if the data do not exhibit the appropriate shape and distribution for one of

the above control charts, then an alternate chart should be constructed (Benneyan, 1994;

Jackson, 1972). As a general rule of thumb, if a histogram does not exhibit a fairly

symmetric bell-shape, such as those shown in Figures 4(a) through 4(c), then none of

the conventional charts may be appropriate or sample sizes may be too small, and further

advice should be sought. As one example, a fourth type of control chart called g and h

charts should be used for count data that have a geometric distribution, which can be

useful when dealing with rare events (see Section 6.1 below).

4 Data and statistical issues

4.1 Distinctions between traditional measurement and SPC

As the above examples illustrate, several important distinctions exist between traditional

measurement practices for quality assurance and the role of measurement when using

SPC for process improvement. Most importantly, data on almost all key indicators should

be collected and evaluated much more frequently via control charts and closer to the

continuous manner in which they actually are produced, rather than retrospectively in

large aggregate quantities. For example, a typical current manner of reporting key

performance data is to summarise several aggregate values annually, semiannually, or

quarterly such as shown in Table 2.

Instead, process data now should be collected longitudinally in real-time and in much

smaller samples in the format necessary for control charts shown in Figure 8. These

samples are called subgroups in SPC terminology in order to distinguish them from

the total group of all data in all subgroups together. Note that this format and use of data

for the purpose of process improvement is a fundamental change from the format used


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for the traditional purpose of internal or external reports and documentation. In many

settings, the implication is that current data collection systems will need to be revised

significantly in order to provide the necessary data in the necessary format. The increased

use of simple run charts within hospitals to track key performance data over time is a

positive move in this direction. Subgroup size guidelines in this context also differ from

traditional sample (total) size calculation methods, as discussed in Section 5 below.

Table 2 Example of traditional aggregate data format for reporting and quality assurance

Measure 1998 1999 2000

Cesarean section births 5.6 / 100 21 / 100 19 / 100

Average maternity length-of-stay 66 h 58 h 61 h

Needle sticks 98 120 113

Surgical nosocomial infections 6.9 / 100 patients 3.4 / 100 patients 4 / 100 patients

Breast wound infections 12% 3% 6%

Adverse drug events 105 68 83

Methicillin-resistantStaphylococcus Aureus (MRSA)

325 / 1677 (19%) 525 / 1629 (32%) 694 / 1735 (40%)

Ventilator-associated pneumonia 21.7 / 1000ventilator days

14.3 / 1000ventilator days

16.5 / 1000ventilator days

CR BSI 1.7 1.4 3.1

ICU lengths-of-stay 4.9 days 5.3 days 5.1 days

Figure 8 Longitudinal data collection format for SPC and process improvement

Individual Subgroup Observations

1 2 3 4 . . . n

Time (Within Subgroups)

(e.g., Counts,

Proportions,

Averages,

St. Devs.

1

2

3

.

.

.

k

Su

bgroupNumber

Time(Between

Subgroups)

Xk1

Xk2

Xk4

Xkn

Xk3 . . .

X11

X12

X14

X1n

X13 . . .

X21

X22

X24

X2n

X23 . . .

X31

X32

X34

X3n

X33 . . .

SubgroupStatistics

Notes: Samples are called subgroups in SPC terminology. This example contains ksamples of data recorded over a significant period of time, with each samplecontaining n values taken over a shorter period of time. The rate or otherstatistic of interest is calculated for each small subgroup sample and plotted ona control chart in real time.


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222 J.C. Benneyan

Also note that a minimum of at least 25 to 35 subgroups of data are necessary in order to

conclude reliably that a process has been in statistical control over time (many more arenecessary for more accurate limits), a requirement that has radical implications on how

data should be gathered and analysed within many organisations and on the frequency of

data collection. Using only aggregate quarterly data, for example, can take six to nine

years to verify a state of statistical control exists. Many current data systems therefore are

woefully insufficient for the purpose of SPC and improvement, whereas plotting smaller

amounts of data on control charts more frequently monthly, weekly, or even daily if

at all possible is a much better approach. In manufacturing and service applications,

data often are plotted hourly or at least daily so that the process can be monitored and

controlled in real time.

4.2 Chart calculations and use of three standard deviation limits

The conventional formulae for all Shewhart control charts are based on the expected

value (i.e., the theoretic mean) and the theoretic standard deviation (sigma) of the plotted

data, which are computed differently for each type of data distribution. For example, the

CL for np charts is set equal to the mean of the binomial distribution (np) with the control

limits equal to the CL plus and minus three binomial standard deviations ([np(1 p)]^0.5),

where n is the sample size and p is either the historical, baseline, current, or target rate.

An alternate approach is to use probability-based control limits calculated so that the

probability of falling between them is some desired value, typically somewhere in the

vicinity of 99.73% (equating to three standard deviations for normally distributed data).

Note that in the case of normal data both these methods yield exactly the same (and good)

specificity of 0.9973 for any single plotted value (i.e., the probability that any single

in-control value correctly will be determined to be in-control) or equivalently a false

alarm probability of 0.0027.

There also are several more technical rationales for using 3-sigma limits. From a

multiple comparison Bonferroni-type of perspective, if a retrospective 3-sigma trialXbar

control chart contains 25 samples of historical data from an in-control process, then the

overall probability of at least one false alarm is = 1 (0.9973)25 0.0654 (similar tothe false over-signalling from using too many between-limit rules mentioned earlier and

in Section 5.5 below). That is, the specificity of the overall chart is 1 = 0.9346(implying, if anything, that perhaps something greaterthan 3-sigma limits might be used

for retrospective data). Additionally, a mathematical technique developed to determine

the optimal control limits for monitoring prospectively typically yields results between

2.5 and 3.5 sigma from the CL across a wide range of applications in many industries,

lending the convention of three standard deviations further justification (Benneyan,

1998b; Montgomery, 2009; Duncan, 1956; Montgomery, 1980; Duncan, 1978;Montgomery et al., 1975). Further research could determine the extent to which these

conclusions extend broadly to healthcare applications, although it is likely that they do

(Benneyan, 1998b).

In other applications, it periodically is suggested that narrower limits should be used

in order to obtain a different tradeoff between sensitivity and specificity. In many

industries, therefore, an additional pair of lines called warning limits sometimes are

plotted at two standard deviations above and below the CL in order to provide earlier but

less definite warnings of possible problems (i.e., greater sensitivity but lower specificity).

A typical response to values falling between the warning and control limits is to start


15/31


investigating and searching for assignable causes on a smaller scale and with less urgency

than if the control limits themselves had been exceeded. See Benneyan (1998b) for

further discussion of these topics.

5 Chart performance and subgroup size selection

Like almost any other statistical method, sample sizes and sensitivity are intrinsically

related to one another, with larger subgroups producing control charts with greater power

to detect process changes but at the expense of greater sampling cost (or less frequent

subgroups). Several guidelines therefore exist to help determine an appropriate minimum

subgroup size that will produce decent statistical properties due to:

a reasonably symmetric sampling distribution

good sensitivity to detect process changes

a non-zero LCL.

Each of these rules-of-thumb are described below.

5.1 p andnp charts

For np and p charts, the two most common rules-of-thumb are that the minimum

subgroup size, n, should be large enough so that (Rule 1) both:

np 5 and n(1 p) 5 (1)

are satisfied or equivalently:

5 5 5max , ,

1 min( ,1n

)p p p

= p(2)

or that (Rule 2) both:

ln(.05)

ln(1 )n

p

and

ln(.05)

ln( )n

p (3)

or equivalently:

ln(.05),

ln[max( ,1 )]n

p p

(4)

where p is the expected proportion of cases resulting in the medication error, surgicalsite infection, or other adverse event being studied. As illustrated in Figure 9, the first

rule (based on the binomial approximation to the normal) ensures that a reasonably

symmetric bell-shaped distribution exists, thereby yielding statistical properties and

within-limit rules performance similar to those for a normal distribution. The second

rule similarly avoids significant skewness and poor statistical properties, now by ensuring

the control chart will not have a large number (no more than 5% on average) of

plotted points clustered either at zero or the upper possible bound (1 and n forp and np

charts, respectively), solving the expressions P(X= 0) = (1 p)n 0.05 and P(X= n)=pn 0.05 for n in the binomial case and P(X= 0) = en in the Poisson case.


16/31

224 J.C. Benneyan

Figure 9 Illustration of subgroup size rules of thumb and impact on normal approximation to

binomial and Poisson distributions (see online version for colours)

Binomial Case

np= 1

np= 3

n(1-p)=1

n(1-p)=3

Poisson Case

n = 3

n = 1

np= 5 np= 10n(1-p)=10

(n= 20)

n(1-p) = 5 n = 5

n = 10

Sometimes if data are scarce the first rule is relaxed to np 3 and n(1 p) 3 andthe second to n ln(.25)/ln(1 p) and n ln(.25)/lp(p), although these usually are thelowest subgroup sizes that one should consider. As discussed below, Figure 11 in Section

5.5 illustrates the consequence of smaller subgroup sizes on poorer chart performance. In

all cases, non-integer subgroup sizes should be rounded up to the next whole number.

As an illustration, for the surgical site infection rate in Figure 6 of 0.09 (nine out of

every 100), the minimum subgroup size n using the first rule should be between:

5[55.6] 56

0.09n

+

+ = =

and3

[33.3] 34,0.09

n

+

+ = =

(5)

or using the second rule:

ln(.05) 2.996[31.8] 32,

ln(1 .09) .0943n

+ +

+ = = = (6)

where [ ]+ is the integer round-up function.


17/31


5.2 u andc charts

Similarly, for c and u control charts the two corresponding rules-of-thumb are that

the minimum subgroup size, n, should be (by Rule 1) between n 5 and n 3,or equivalently:

n 5/(best) and n 3/(given scarce data), (7)

or that (by Rule 2):

ln(.05)n

, (8)

where is the average number of occurrences per some common unit of calculation (suchas the ventilator associated pneumonia rate per 100 device days or the needle stick rate

per 100 patient days). To illustrate, for the catheter-associated infection rate in Figure 7of 1.25 per every 100 catheter days, the minimum subgroup size using the first rule

should be between:

5

1.25 /100 daysn = 4.00 100 days = 400 catheter days (9)

and

3

1.25 /100 daysn = 2.40 100 days = 240 catheter days, (10)

or using the second rule:

ln(.05) 2.996

1.25 /100 days 1.25 /100 daysn

= = 2.40 100 days = 240 catheter days, (11)

as above often producing a result equal or similar to the relaxed version of Rule 1.

Again, Figure 9 illustrates the impact of subgroup size selection and these rules on the

normal approximation.

5.3 Non-zero LCL

Note that forp, np, c, and u charts, if the LCL is equal to zero then, unless supplementary

rules are used (see below), detecting or verifying improvements in the form of rate

decreases will not be possible. In order for the LCL to be greater than zero, subgroup

sizes 80% to 200% larger than those that result from the above formulae often arerequired. Setting the equations for the LCL > 0 and solving for n, the minimum subgroup

size now must be large enough (Rule 3) so that forp and np charts:

2 (1 )k pn

p

> (12)

and for u and c charts:


18/31

226 J.C. Benneyan

2

,

k

n > (13)

where k is the number of standard deviations used in the control limits (typically 3).

Additionally, if the occurrence ratep 0.5 then in order for the UCL ofp and np chartsto be able to detect rate increases the subgroup size must be larger than:

2

1

k pn

p>

(14)

in order for that the UCL to be less than the upper possible bound of 1 (p chart) and n (np

chart), respectively. Note that these rules use a strict inequality so integer results should

be increased to the next whole number. In the above two examples, the minimum

subgroup size so that LCL > 0 (using k= 3) becomes:

23 (1 .09)91.0

.09

92

n

> =

=

(15)

for the surgical site infectionp control chart and:

23720

1.25 /100 days

721

n > =

=

(16)

for the ventilator-associated pneumonia u control chart. Table 3 summarises the

minimum recommended subgroup sizes forp, np, c, and u charts using these rules for a

range of adverse event rates. Note that since the occurrence ratesp and appear in thedenominators, smaller values of these (i.e., higher quality processes) result in larger

required minimum subgroup sizes, presenting the ironic dilemma that better processes

require more data to control them. Performance of these charts to detect rate decreases

using smaller subgroup sizes and supplementary rules is discussed below.

5.4 Xbar charts

The specificity (i.e., probability of a plotted point not falsely signalling if the process

remains in-control) ofXbarcharts with any subgroup size always will be 0.9973 (using

k= 3) if the data are normally distributed. In many healthcare applications, however,

continuous data such as activity durations or delays can be naturally skewed, although

subgroups of size n 10 usually will produce reasonable normality of averages due tothe central limit theorem. The impact on chart performance if subgroup averages are

plotted therefore is minimal (Burr, 1967), especially as n increases, another strength of

Xbar charts. Logarithmic, square root, and other transformations also are possible,

although seeking the best empirical transformation can be misleading (especially for

out-of-control data) since data are not known a priori to be in-control, an inherent but

often not understood Catch-22 in SPC.


19/31


Table 3 Recommended minimum subgroup sizes

r=.25

541

136

61

34

22

16

12

9

7

6

5

4

4

3

r=.5

900

225

100

57

36

25

19

15

12

9

8

7

6

5

Subgroupsize

3

r=.75

1351

338

151

85

55

38

28

22

17

14

12

10

8

7

Xbarcontrolchart

Shiftsiz

e

(sigma

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Rule3

361

181

121

91

37

19

13

10

8

7

6

5

5

4

Rule2

120

60

40

30

12

6

4

3

3

2

2

2

2

2

Subgroupsize

2

Ru

le1

200

100

67

50

20

10

7

5

4

4

3

3

3

2

cor

ucontrolchart

Rateperunit

()

0.0

25

0.0

50

0.0

75

0.1

00

0.2

5

0.5

0

0.7

5

1.0

0

1.2

5

1.5

0

1.7

5

2.0

0

2.2

5

2.5

0

Rule3

3592

1792

1192

892

352

172

112

82

64

52

43

37

32

28

Rule2

1197

598

398

299

119

59

39

29

23

19

16

14

12

11

Subg

roupsize

1

Rule1

2001

1001

667

500

200

100

67

50

40

34

29

25

23

20

nporpcontrolchart

Defectrate

(p)

0.0

025

0.0

050

0.0

075

0.0

100

0.0

25

0.0

50

0.0

75

0.1

00

0.1

25

0.1

50

0.1

75

0.2

00

0.2

25

0.2

50


20/31

228 J.C. Benneyan

Table 3 Recommended minimum subgroup sizes (continued)

r=.25

3 3 2 2 2 2 2 2 2 1

r=.5

4 4 4 3 3 3 3 2 2 2

Subgroupsize

3

r=.75

7 6 5 5 4 4 4 3 3 3

Xbarcontrolchart

Shiftsi

ze

(sigm

a)

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

Rule3

4 4 3 3 3 3 3 3 2 2

Rule2

2 1 1 1 1 1 1 1 1 1

Subgroupsize

2

R

ule1

2 2 2 2 2 2 2 2 2 1

corucontrolchart

Rateperunit

()

2.7

5

3.0

0

3.2

5

3.5

0

3.7

5

4.0

0

4.2

5

4.5

0

4.7

5

5.0

0

Rule3

24

22

19

17

16

14

13

12

10

10

Rule2

10 9 8 7 7 6 6 6 5 5

Sub

groupsize

1

Rule1

19

17

16

15

14

13

12

12

11

10

nporpcontrolchart

Defectrate

(p)

0.2

75

0.3

00

0.3

25

0.3

50

0.3

75

0.4

00

0.4

25

0.4

50

0.4

75

0.5

00

Notes:

1npandpchartrule

s:Rule1:n5

/min(pk2m

ax((1p)/p,p/(1p)).

2canduchartrules:Rule1:n5

/,

Rule2:n

ln(.05)/,R

ule3:nk

2/.

3Xbarchartrule:n(

(kzr)/)2.

k=standarddeviationmultiple,p=binomialrate,

=

Poissonrate,

=meanshiftsizetodetect,Zr=

stdnormalvaluewithupper

tailprobabilityr.


21/31


An appropriate subgroup size for Xbarcharts can be determined by selecting a desired

probability, r, that the next subgroup average will fall outside k-sigma control limits if the

process mean shifts by a certain amount, , which yields the bound:

2

,rk z

n

(17)

wherezr is the standardised normal coordinate corresponding to an upper tail probability

of rand is the number of standard deviations equivalent to the magnitude of a shiftin either direction that is desirable to detect. To illustrate, for the decision-to-incision

Cesarean data in Figure 5, with a mean and standard deviation of 46 and 5 min

respectively, in order to detect a shift with 0.5 probability (r= 0.5, z0.5 = 0) from the

mean of 46 min to 49 min ( = (49 46)/5 = 0.6 standard deviations), the subgroup size

must be:

[ ]2

3 025 25.

0.6n

+

+ =

= (18)

To detect the same shift with 0.25 probability (z0.25 = .6745), the subgroup size must be:

[ ]2

3 .674515.02 16.

0.6n

+

+ =

= (19)

Table 3 summarises the subgroup sizes necessary for various degrees of mean shifts

in order to achieve single-point detection probabilities of .25, .5, and .75. These

probabilities equate to averages of 1/.25 = 4, 1/.5 = 2, and 1/.75 = 2 subgroups plottedfrom the time the process changes until a value exceeds the control limits, any of which

typically are considered to be good chart performance.

5.5 Chart sensitivity, OC curves, and average run lengths

The sensitivity or power (i.e., the probability that a point will fall outside k-sigma limits if

the process mean shifts by standard deviations) of anXbarchart with a subgroup size ofn to detect other magnitudes of shifts in the process mean can be calculated using the

following formula:

( ( ) 1 ( ) (P X LCL P X UCL k n k n = < + > = + ), (20)

where k and

are defined as previously and

(z) is the standard normal cumulativeprobability evaluated at z, P(Z


22/31

230 J.C. Benneyan

Figure 10 Operating characteristics ofXbarcharts (see online version for colours)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Shift in Mean (number standard deviations)

k = 3

k = 2

1

1

4

4

10

1030

30

Numbers on curves indicate subgroup size

(Italic subgroup sizes are for k = 2 case)

a.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Shift in Mean (number standard deviations)

8 consecutive values on same side of CL

13 of 14 consecutive values on same side of CL

30

30

1

1

4

410

10

Numbers on curves indicate subgroup size

(Italic subgroup sizes are for 13 of 14 rule)(Assumes normality conditions met)

b.

Corresponding power curves for np, p, c, and u control charts will be different for any

particular in-control rate and must be computed numerically, in the binomial (p and np

charts) case as:


23/31


) x[ ]

1 1

[ ]

( ) (1UCL

x n

x LCL

nP LCL X UCL p px

+

= =

(21)

and in the Poisson (u, c) case as:

1[ ]1

[ ]

( )!

xUCL

x LCL

eP LCL X UCL

x

,

+

=

= (22)

where [ ]+ and [ ] denote the integer round down and round up functions, respectively,

the control limits are computed from these in-control values ofp or , and p1 and 1denote the values of the out-of-control rates. (If the subgroup sizes are large enough that

a normal approximation is reasonable, then the OC curves can be approximated by

those for Xbar charts, using 2 = np(1 p) and 2 = in the binomial and Poisson

cases respectively.)Figure 11 illustrates the exact OC curves using Equations (21) and (22) corresponding

to the earlier p (Figure 11a) and u (Figure 11b) control charts in Figures 6 and 7,

respectively, for various sample sizes, including those specified by the above rules.

Again, chart performance for other values of n can be approximated by interpolation.

Note in each case that the OC curves exhibit power to detect rate decreases only when the

sample size equals or exceeds that specified by Rule 3 above. All other OC curves,

plotted in solid lines, converge to 1.0 as the shifted rate converges (improves) to 0. As

would be expected, the specificity for values ofn lower than indicated by Rules 1 and 2

also is less than when these rules are satisfied.

Figure 11 Operating characteristics ofp and u charts

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

p Chart Shifted Rate p 1

10

32

(rule 1)

250

In-control rate p 0 = .09

56(rule 2)

92(rule 3)

1000

a.


24/31

232 J.C. Benneyan

Figure 11 Operating characteristics ofp and u charts (continued)

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

u Chart Shifted Rate 1

50

240(rule 1)

1500

In-control rate 0 = 1.25/100 days

400(rule 2)

721(rule 3)

6000

(NB: Rate in terms of 100 days)

b.

Note: Value below each curve denotes subgroup size.

A related common performance measure is the Average Run Length (ARL), the expected

number of plotted points until a process change is detected, with shorter ARLs beingpreferable for larger process changes. Then using only the control limits to signal a

process change, the ARL is simply the reciprocal of the probability of a point falling

outside the limits, i.e., ARL = 1/. Several examples of ARL curves for healthcarecontrol charts are illustrated by Benneyan (1998b).

Table 4 Examples of between-limit rules for unnatural variability (only the first three rules arerecommended to prevent over false-signalling)

Supplementary rules for statistical control1 Any point outside either control limit

2 8 consecutive points on the same side of the Centre Line (CL)

3 4 of 5 consecutive points outside of CL 1 SD, on the same side

4 13 consecutive values within CL 1 SD, indicating a reduction in variability

5 2 of 3 consecutive pts outside of CL 2 SD, on the same side

6 12 of 14 consecutive points on same side of CL

7 8 consecutive points exhibiting either an increasing or decreasing trend

8 Cyclic or periodic behaviour

The sensitivity of all chart types can be improved using the supplementary

between-limit rules mentioned in Section 2.1, although at the expense of reduced

specificity (Waller et al., 1991). These rules also will add power to detect rate decreases


25/31


for np, p, c, and u control charts even with LCLs of zero. A typical set of such rules is

listed in Table 4, each having low probability of occurring if the process operates with

only natural variability and does not change, although readers should be cautioned to not

use all these rules together in order to avoid significant inflation of the overall false alarm

rate (e.g., see poor simulation results in Benneyan (1998b) when using all rules together

on np charts).

Figure 12 Detection probability of runs-beneath-CL rule ofp and u charts with LCL = 0

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

u Chart Shifted Rate 1

50240(rule 1)

In-control rate 0 = 1.25/100 days

400(rule 2)

1500

(NB: Rate in terms of 100 days)

b.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

p Chart Shifted Rate p 1

32(rule 1)

In-control rate p 0 = .09

250 1056(rule 2)

a.


26/31

234 J.C. Benneyan

For example, one of the most commonly used rules is 8 consecutive values on the same

side of the CL. Figure 12 illustrates the probability when using this rule that a sequenceof eight consecutive subgroups will not generate an out-of-control signal for the earlierp

(Figure 12a) and u (Figure 12b) charts in Figures 6 and 7. Note that small subgroup sizes

with LCL = 0 now yield power to detect rate decreases.

Similarly, Figure 9(b) illustrates the correspondingXbarchart in-control probabilities

for this and a second commonly used rule (13 of 14 consecutive values on the same side

of the CL), if used independently of one another. In general, the more rules that are used

together, the greater the improvement in sensitivity but also the greater the frequency of

false alarms. Although historical advice was to use all rules together, the thinking today

based on extensive simulation analysis is that typically using only the first three rules

(any point outside the limits, eight consecutive points on same side of CL, and four of

five consecutive points above (below) CL plus (minus) 1 standard deviation) gives the

best tradeoff between true and false signals.

6 Other types of control charts

6.1 Control charts for low rates and infrequent data

As shown above in Table 3, when dealing with rare events standard control charts can

be problematic in terms of the amount of data and time required until a subgroup of

sufficient size can be plotted. This can result in feedback becoming available too

infrequently to be able to make rational process decisions in a timely manner. In such

cases, a simple alternative to plotting the number of occurrences per time period on ap or

u chart instead is to use a g-type of control chart for the number of cases or the amount

of time between occurrences. These charts are simple to use, when dealing with rareevents have better statistical properties for detecting rate increases (using probability

limits) or decreases than conventional charts, and are particularly useful for verifying

improvements (Benneyan, 2001a; Plourde et al., 1998; Xie and Goh, 1993).

Figure 13 Example ofg control chart of number of procedures between preventable complications

0

50

100

150

200

250

300

350

5 10 15 20 25 30 35 40 4

Procedure Number5


27/31


Figure 13 illustrates a recent g control chart for the number of procedures between

preventable complications. Note that for this type of chart, higher rather than lower

values equate to process improvements and longer times or numbers of cases between

adverse events. The formulae, statistical performance, subgroup size considerations,

ARLs, and impact of between-limit rules for these charts are discussed in detail

by Benneyan (2001a; 2001b). Other examples of this type of chart include the

number of surgeries between surgical site infections, the number of patients between

catheter-associated infections, the number of days between adverse drug events, the

number of days between needle sticks, and so on.

6.2 Non-Shewart control charts

Although beyond the scope of this paper, several other types of control charts also exist,

including Moving Average (MA), Exponentially Weighted Moving Average (EWMA),Cumulative sum (Cusum), Cumulative score (Cuscore) charts, and others. While more

complicated to use, these methods have potential value within healthcare, such as for

detecting or verifying smaller process changes faster, monitoring physiologic, seasonal,

or auto-correlated data, or simultaneous monitoring and feedback adjustment (such as for

managing the blood glucose of a diabetic patient) (Benneyan, 2006). Figures 14 and 15

illustrates an EWMA control chart (using a smoothing weight of 0.2) and a Cusum-type

chart (based on a sequential probability ratio test for a 25% increase from a baseline

rate of .016) for the same complication data as shown in Figure 13, with the small

rate decreases around samples 20 and 36 now being more visually evident as step

and slope changes, respectively. The first type of chart also is related to methods for

handling autocorrelated or seasonally cyclic processes (Hunter, 1986; Vasilopoulas and

Stamboulis, 1978; Montgomery and Mastrangelo, 1991).

Figure 14 Example of EWMA control chart for number procedures between complications

0

25

50

75

100

125

150

175

5 10 15 20 25 30 35 40 4

Procedure Number5

Cmpco

EWMA: Exponentially weighted moving average


28/31

236 J.C. Benneyan

Figure 15 Example of Cusum-SPRT chart for number procedures between complications

0

500

1000

1500

2000

2500

3000

3500

4000

5 10 15 20 25 30 35 40 4

Procedure Number

CmoPo

eBw

Cmpco

Cusum: C umulative sum

5

7 Discussion

As healthcare organisations begin to use control charts in process improvement work,

several implications on process management and the use of data begin to emerge.

Achieving consistent levels of patient care is especially critical in clinical situations

where a lack of statistical control can have direct consequences on risk managementand liability. For many organisations this effort may mean transitioning from

traditional orientations largely focused on external reporting and regulatory adherence

to orientations more focused on continual process analysis, experimentation, and

improvement. This effort also will require a focus as much on processes as on outcomes,

and therefore collecting different data elements in different formats may be necessary.

In many applications, instead of current practices of reporting data in large infrequent

samples (such as quarterly), data should be collected in smaller samples much more

frequently (such as weekly or monthly) and plotted on an appropriate control chart. Much

of the aggregate data currently collected for various report cards should be plotted and

evaluated for process stability on control charts in more frequent and smaller samples.

The meaning and interpretation of standards, benchmarks, and improvement goals also

now is much less clear, such as a Cesarean delivery rate standard of 15%. In such a case,

should an organisations monthly rate never exceed 15%, its long-term CL of a p control

chart equal 15%, the UCL be less than 15%, or something else?

One logical answer is that a process must be in-control (otherwise, of course, there is

no true rate non-compliance to a standard to even consider) and that the CL should equal

the standard rate nor goal, in this example 0.15. This would mean that the long-term rate

(i.e., the CL) is equal to the standard, but also requires management to be statistically

literate and to recognise that approximately half of the plotted subgroup values should

fall above the standard. (This type of control chart, when the CL and limits are calculated

from standards, although less commonly used actually is called a standards given


29/31


control chart (Grant and Leavenworth, 1988; Duncan, 1986; Montgomery, 2009).) When

dealing with low rates, it also can be advantageous to collect data on the number of cases

or the amount of time between adverse events, rather than monthly rates. Perhaps more

generally, it becomes clear that the purpose of data largely is to understand and improve

process performance, rather than to evaluate, reward, or punish individual performance.

As important as it is to use SPC, it is equally important to use it correctly (Humble,

1998; Benneyan, 1999). Use of incorrect charts or formulae in the past has resulted in a

failure to detect infection rate increases and changes in clinical laboratory equipment,

obvious liability concerns. Other common errors include using insufficient amounts

of data when estimating control limits, misuse of short-cut formulas and empirical

transformations, over-use and inappropriate use of individuals charts such as for

discrete or non-normal data (Benneyan and Shenoy, 2008), and using conventional charts

when combining data from non-homogeneous processes or for risk-adjusted outcomes

(Benneyan, 1994). In some situations, it also is important to properly adjust forseasonality, case mix, severity, age, gender, and so on. While beyond the intended scope

here, regression, logistic regression, and risk-adjusted SPC methods also have been

proposed for such scenarios although further research is needed in this area (Taseli and

Benneyan, 2008).

More generally, organisations sometimes get swept up in creating control charts

simply for the sake of creating charts, including widespread software-generated charts of

almost all available data, without much planning and follow-through on how the resultant

information will be used. Using SPC requires time and effort, and there is little point

in investing these resources unless results will be used to inform and improve. Better

success tends to result from focusing on a few key concerns and then expanding ones

use of SPC based on these experiences and the knowledge gained.

References

Bates, D.W. (1997) The costs of adverse drug events in hospitalized patients, Journal of theAmerican Medical Association, Vol. 277, pp.307311.

Bedell, S.E., Deitz, D.C., Leeman, D. and Delbanco, T.L. (1991) Incidence and characteristicsof preventable iatrogenic cardiac arrests, Journal of the American Medical Association,Vol. 265, No. 21, pp.28152820.

Benneyan, J.C. (1994) The importance of modeling discrete data in SPC, Proceedings of theTenth International Conference of the Israel Society for Quality, pp.640646.

Benneyan, J.C. (1998a) Statistical quality control methods in infection control and hospitalepidemiology. Part 1: introduction and basic theory, Infection Control and HospitalEpidemiology, Vol. 19, No. 3, pp.194214.

Benneyan, J.C. (1998b) Statistical quality control methods in infection control and hospitalepidemiology. Part 2: chart use, statistical properties, and research issues, Infection Controland Hospital Epidemiology, Vol. 19, No. 3, pp.265277.

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