Health Care Management Science 4, 305-318. 2001© 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

Number-Between ^-Type Statistical Quality Control Charts forMonitoring Adverse Events

JAMES C. BENNEYAN. Ph.D.334 Snetl Engineering Center. Northeastern University. Boston, MA 02115, USA

E-mail: benneyan@coe.neu.edu

Received 21 January 2000; Revised 10 April 2001

Abstract. Alternate Shewhart-type statistical control charts, called "g" and "ft" charts, are developed and evaluated for monitoring thenumber of cases between hospital-acquired infections and other adverse events, such as heart surgery complications, catheter-relatedinfections, surgical site infections, contaminated needle sticks, and other iatrically induced outcomes. These new charts, based on inversesampling from geometric and negative binomial distributions, are simple to use and can exhibit significantly greater detection powerover conventional binomial-based approaches, particularly for infrequent events and low "defect" rates, A companion article illustratesseveral interesting properties of these charts and design modifications that significantly can improve their statistical properties, operatingcharacteristics, and sensitivity.

Keywords: SPC, control charts, healthcare, adverse events, geometric distribution, g charts

1. Introduction

7.7. Oveniew of article

This article illustrates a tiew type of statistical process con-trol (SPC) chart for tnotiitoring the number of cases be-tween hospital-acquired infections or other healthcare ad-verse events, such as heart surgery complications, catheter-relaled infections, contatninated needle sticks, medicationerrors, and other iatrogenic events. These new charts, called"g" and " / j " control charts, are based on inverse samplingfrom underlying geometric and negative binomial distri-butions and can exhibit improved shift-detection sensitiv-ity over conventional approaches, particularly when dealingwith infrequent events or low "defect" rates. The applicationand interpretation of these charts for detecting rate changesare illustrated by several examples involving cardiac bypasssurgical-site infections. Clostridium difficile infections, nee-dle stick exposures, and related concerns.

In a companion paper [5], the specificity and sensitiv-ity of these new charts are investigated and contrasted withmore conventional methods, with several simple design con-siderations - including standard within-limit rules, redefinedBernoulli trials, a new in-control rule, and probability-basedcontrol limits - shown to significantly improve the chart'spower to detect true process changes. These charts also areshown in some cases to exhibit better statistical operatingcharacteristics over traditional binomial-based np and p con-trol charts, especially when the rate of occurrence (i.e., theBernoulli parameter p) is sufficiently low. In summary, thesecharts are found to be relatively simple to use and interpret,to exhibit comparable or superior performance to more tra-ditional or more complicated methods, and to be a useful

complement to conventional hospital epidemiology and in-fection control methods.

7.2. Hospital epidemiology and infection control

Epidemiology in the broadest context is concerned with thestudy, identification, and prevention of adverse healthcareevents, disease transmission, and contagious outbreaks, withparticular focus within hospitals on nosocomial infectionsand infection control. Nosocomial infections essentially areany infections that are acquired or spread as a direct result ofa patient's hospital stay {rather than being pre-existent as anadmitting condition), with a few examples including surgicalsite infections, catheter infections, pneumonia, bacteremia,urinary tract infections, cutaneous wound infections, blood-stream infections, gastrointestinal infections, and others.

With estimates of the national costs of nosocomial in-fections ranging from approximately 8.7 million additionalhospital days and 20,000 deaths per year [21] to 2 millioninfections and 80.000 deaths per year [30]. it is clear thatthese problems represent quite considerable health and costconcerns. Additionally, the number of U.S. hospital patientsinjured due to medical errors and adverse events has beenestimated between 770.000 and 2 million per year, with thenational cost of adverse drug events estimated at $4.2 bil-lion annually and an estimated 180,000 deaths caused partlyby iatrogenic injury nationwide per year (2,4,13,15,18.31].The costs of a single nosocomial infection or adverse eventhave been estimated both to average between $2,000 and$3,000 per episode. The National Academy of Sciences'Institute of Medicine recently estimated that more Ameri-cans die each year from medical mistakes than from traffic

306 J.C, BENNEYAN

accidents, breast cancer, or AIDS, with $8.8 billion spentannually as a result of medical mistakes [24].

Given the above figures, it is not surprising that manyfederal,, regulatory, and healthcare accrediting bodies - suchas the Joint Commission on Accreditation of Healthcare Or-ganizations (JCAHO), the Nationa] Committee for QualityAssurance (NCQA), the U.S. Center for Disease Control(CDC), the Health Care Financing Administration (HCFA),and others - either require or strongly encourage hospitalsand HMO's to apply both classical epidemiology atid moremodem continuous quality improvement (CQI) methodolo-gies to these significant process concerns, including theuse of statistical methods such as statistical process control(SPC). For example, the Joint Commission on Accreditationof Healthcare Organizations recently stated their position onthe use of SPC as follows [27]:

"An understanding of statistical quality control, includ-ing SPC, and variation is essential for an effective as-sessment process... Statistical tools such as run charts,control charts, and histograms are especially helpful incomparing performance with historical patterns and as-sessing variation and stability,"

Similarly, a recent position paper by several epidemiol-ogists from the U.S. Center for Disease Control [33] staledthat

"Many of the leading approaches to directing quality im-provement in hospitals are based on the principles ofW.E. Deming. These principles include use of statisticalmeasures designed to determine whether improvement inquality has been achieved. These measures should in-clude nosocomial infection rates."

Conventional epidemiology methods, in fact, includeboth various statistical and graphical tools for retrospectiveanalysis, such as described by Mausner and Kramer [34] andGordis [22], and several more surveillance-oriented meth-ods, such as reviewed by Larson [29] (also see Becker [3]).It is worth noting that collectively these methods tend to beconcerned with both epidemic (i.e,, outbreaks) and endemic(i.e., systemic) events, which in SPC terminology equate tounnatural and natural variability, respectively, and thereforeare candidates for effective study via control charts. Sev-eral epidemiologists (for example, see Bimbaum [12], My-lotte [35], Burnett and Chesher [16], Childress and Chil-dress [17], and Mylotteetal. [36]) have proposed monitoringcertain infection and adverse event rates more dynamicallyover time, rather than "time-statically", in manners that arequite similar in nature and philosophy to SPC. It also is inter-esting that as early as 1942, Dr. Deming advocated the im-portant potential of SPC in disease surveillance and to rareevents [19].

1.3. Use of statistical process control (SPC)

The application of standard SPC methods to healthcareprocesses, infection control, and hospital epidemiology bas

been discussed by several authors, including a comprehen-sive review in a recent series in Infection Control and Hos-pital Epidemiology [6]. Example applications include med-ication errors, patient falls and slips, central line infections,surgical complications, and other adverse events. In somecases, however, none of the most common types of controlcharts will be appropriate, for example due to the mannerin which data are collected, pre-established measuring andreporting metrics, or low occurrence rates and infrequentdata. One example of particular note is the use of events-between, number-between, days-between. or time-betweentype of data that occasionally are used by convention in somehealthcare and other settings. Several important clinical ap-plications of such measures are described below, and thisarticle therefore derives and illustrates appropriate controlcharts for these cases, such as for the number of procedures,opportunities, or days between infrequent adverse events.(Note that these same methods, of course, are equally ap-plicable for monitoring other types of low defect processes,such as in manufacturing and service settings.)

As some general background, statistical process controlcharts are chronological displays of process data used to helpstatistically understand, control, and improve a system, herean infection control or adverse event process. The generalformat of a Shewhart-type control chart is shown in figure 1.Observed process data, such as the monthly rate of infec-tion or the number of procedures between infections, areplotted on the chart and interpreted, ideally, soon after theybecome available. Three horizontal lines also are plotted,called the center line (CL), the upper control limit (UCL).and the lower control limit (LCD. which are calculated sta-tistically and help define the central tendency and the rangeof natural variation of the plotted values, assuming that therate of occurrence does not change. By observing and inter-preting the behavior of these process data, plotted over timeon an appropriate control chart, a determination can be madeabout the stability (i.e., the "state of statistical control") ofthe process according to the following criteria.

Values that fall outside the control limits exceed theirprobabilistic range and therefore are strong indications thatnon-systemic causes almost certainly exist that should beidentified and removed in order to achieve a single, stable,and predictable process. There also should be no evidenceof non-random behavior between the limits, such as trends,cycles, and shifts above or beneath the center line. To aidin the objective interpretation of such data patterns, various"within-limit" rules have been defined, described in greaterdetail elsewhere. See Duncan [20] and Grant and Leaven-worth [23] for further information about statistical processcontrol in general and Benneyan [6-8] for discussion andapplication of SPC to healthcare processes.

J.4. Limitations of standard types of control charts

Several approaches to applying SPC to hospital infectionsor other adverse events are possible, dependent on the sit-uation and ranging in complexity and data required. For

NUMBER-BETWEEN g CONTROL CHARTS 307

SubgroupStatistic

(e.g.. average,standard deviation,

rate, number,proportion)

Upper Control Limit (UCL)

Center Line (CL) = Central Tendency

Lower Control Limit (LCL)

I • I . I . I . I r • r . r . I • I . r

3 t;

Is^ Q.

5 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Subgroup Number

Figure 1. GeDeral format of a statistical control chart.

UCL

CL

LCL

Quarter

Figure 2. Example of u control chart with loo few values: number infections per quarter.

example, two standard approaches are to use u OT p She-whan control charts, such as for the number or the fractionof patients, respectively, per time period who acquire a par-ticular type of infection. As an example, figure 2 illustratesa recent u chart for the quarterly number of infections per100 patient days, although it should be noted that this ex-ample ideally should contain many more subgroups (timeperiods) of data. In terms of proper control chart selection,note that each type of chart is based on a particular underly-ing probability model and is appropriate in different types ofsettings. In this particular example, a Poisson-based u chart

would be appropriate if each patient day is considered asan "area of opportunity" in which one or more infectionstheoretically could occur. Alternatively, a binomial-hasedp chart would be more appropriate if the data were recordedas the number of patient days with one or more infections(i.e., Bernoulli trials). Further information on different typesof control charts and scenarios in which each is appropriatecan be found in the above references [6,20,23].

In either of the above cases, note that some knowledgeof the denominator is required to construct the correspond-ing chart, such as the number of patient days, discharges.

J.C. BENNEYAN

or surgeries per time period. A more troubling problem isthat use of these charts in some cases can result in an inad-equate number of points to plot or in data becoming avail-able too infrequently to be able to make rational decisions ina timely manner (especially, ironically, for better processeswith lower infection rates). For example, recall that a mini-mum of at least 25-35 subgroups are recommended to con-fidently determine whether a process is in statistical controland that, in the case of p and np charts., a conventional rule-of-thumb for forming binomial subgroups is to pick a sub-group size, n. at least large enough so that np ^ 5.

For processes with low infection rates, say /J < O.Ol, theconsequence of the above comments can be a significant in-crease in total sample size across all subgroups and in theaverage run length until a process change is detected. Forexample, even for p = 0.01, this translates to 500 data persubgroup X 25 subgroups = 12,500 total data. Addition-ally, this can necessitate waiting until the end of longer timeperiods (e.g.. week, month, or perhaps quarter) to calculate,plot, and interpret each subgroup value, perhaps being toolate to react to important changes in a critical process andno longer in best alignment with the philosophy of processmonitoring in as real-time as possible. Note that similar sub-group size rules-of-thumb exist for c and u charts and leadto the same general dilemma as for np and p charts, in partbecause all conventional control charts consider the numberof infections or adverse events either at the end of some timeperiod or after some pre-determined number of cases.

1.5. The number of procedures, events, or days betweeninfections

As an alternate measure, the number of procedures, events,or days between infections has been proposed due to easeof use, more timely feedback, near immediate availability ofeach individual observation, low infection rates, and the sim-phcity with which non-technical personnel can implementsuch measures. Being easy to calculate, a control chart canbe updated immediately in real-time, on the hospital floor,without knowing the census or other base denominator (al-though assumed for now to be reasonably constant). In somesettings, additionally, "number-between" or "time-between"types of process measures simply may be preferred as a stan-dard or traditional manner in which to report outcomes.

As a few recent examples of the traditional use of suchmeasures. Nelson et al. [37] examined the number of openheart surgeries between post-operative sternal wound infec-tions, Plourde et al. [40] analyzed the number of infection-free coronary artery bypass graft (CABG) procedures be-tween adverse events, Finison et al. [21] considered thenumber of days between Clostridium difficile colitis posi-tive stool assays, Jacquez et al. [26] analyzed the numberof time intervals between occurrences of infectious diseases,Nugent et al. [38] described the use of time between adverseevents as a key hospital-wide metric. Pohlen [40] monitoredthe time between insulin reactions, and Benneyan [6] dis-cussed the number of cases between needle sticks.

In order to plot any of these types of data on controlcharts, however, note that none of the standard charts are ap-propriate. For example, standard c and u control charts arebased on discrete Poisson distributions, np and p charts arebased on discrete binomial distributions, and X and S chartsare based on continuous Gaussian distributions. By descrip-tion, conversely, the number of cases between infectionsmost closely fits the classic definition of a geometric randomvariable, as discussed below and illustrated by ihe histogramin figure 3 of surgeries-between-infections empirical data.Note that if the infection rate is unchanged (i.e.. in statisti-cal control), then the probability, p, of an infection occurringwill be reasonably constant for each case (with patients strat-ified into reasonably homogenous groups if necessary), andthis scenario therefore satisfies the definition of a geometricrandom variable.

As can be seen by comparison with the correspond-ing geometric distribution (with the infection probability pestimated via the method of maximum likelihood), thesedata exhibit a geometrically decaying shape that is veryclose to the theoretic model. (More rigorously, a chi-square goodness-of-fit test indicates close statistical agree-ment, with an effective significance value of 0.693,) Ap-propriate control charts for these data therefore should bebased on underlying geometric distributions, as developedand illustrated below, rather than using any of the above tra-ditional charts. Use of inappropriate discrete distributionsand control charts, in fact, can lead to erroneous conclu-sions about the variability and state of statistical control ofthe infection rate, a situation which has been described pre-viously [9,25,28] and is shown in some of the examples be-low.

2. Events-between g and h cuntrul chartsI

2.1. Three general application scenarios

Alternate control charts for such situations, called g and hcharts, were developed and investigated by Benneyan [11]and Kaminsky et al. [28] based on underlying geometric andnegative binomial distributions. With respect to the abovemotivation, these new charts were developed specifically tobe appropriate for the following three general scenarios:

1. for situations dealing with classic geometric random vari-ables, as discussed in greater detail below, such as forthe various "number-between" types of applications de-scribed above;

2. for more powerful control (i.e.. greater sensitivity) of lowfrequency process data than if using traditional np andp charts; and

3. for cases in which a geometrically decaying shape simplyappears naturally, such as periodically observed in his-tograms of empirical data, even though the existence of aclassic geometric scenario may not be apparent.

NUMBER-BETWEEN g CONTROL CHARTS

0.14 1

0.12 -

0.1 -

BU-

0.08 -

£ 0.06 -Ica

0.04 -I

0.02 -

I Empirical Waiting Times

I Theoretical (Geometric) Distribution

Mill I I I Iinin

i nCO

i n i no

i n.,—

i n inCO

Number of Cases Between Infections

Figure 3. Comparison of empirical heart surgery infection data with geometric distribution.

The first Iwo scenarios are of particular interest in this ar-ticle given the above discussion, with a few examples of thethird type of application briefly also illustrated. As shownbelow, in cases with low infection rates and with immedi-ate availability of each observation, considering Bernoulliprocesses with respect to geometric rather than traditionalbinomial probability distributions can produce more plot-ted subgroup data and greater ability to more quickly detectprocess changes.

2.2. Theoretic motivation

To provide a context for comments to follow, and becausesome healthcare practitioners and other readers might not befamiliar with the underlying mathematical development, abrief motivation of these charts is given here; further detailcan be found in the cited references [10.11.28]. Recall thatif each case (e.g.. procedure, catheter, day, etc.) is consid-ered as an independent Bernoulli trial with reasonably thesame probability of resulting in a "failure" (e.g., an infec-tion or other adverse event), ihen the number of Bernoullitrials until the first failure and the number of Bernoulli trialsbefore the first failure are random variables with type I andtype II geometric distrihutions, respectively. For example.

the probabilities of the next infection occurring on the xthcase or immediately after the A th case are well-known to be

for .X = a,a + \.a +2

where the minimum possible value a = 1 for type I (number-until) and a = 0 for type II (number-be fore) geometric data,with the sum of n independent geometric random variables,

T = Xi+X2 + --- + Xn,

being negative binomial with probability function

for / = na. na + \.na +2,

It follows that the expected values, variances, and standarddeviations of the total number of cases, T, and the averagenumber of cases, X, before (a = 0) or until (a = 1) the nextn infections or adverse events occur then are

E(T) = n ViT) =

n(\-p)

310 I.e. BENNEYAN

Table 1Parameters-known and parameters-estimated g and h control chart calculations.

Rate known Rate estimated MVUE

g chart: Total number events per subgroup

, I- pUpper control limit (UCL) n

N -

Center line (CL)

Lower control limit (LCL) ti

P

l-p

N

\-a\ - k - a)(x - a-i- ]} CL -

h chart: Average number events per subgroup

Upper control limit (UCL)

Center line (CL)

ix-a)ix-a+ I)

+ a

CLH-

N

N

. \ — p \ /I — P - I(x ~ a)ix - (JLower control limit (LCL) | + a 1 - kj ^ x-k.

N~\

C L -N

N ~

NCL-I--T -k nlx-a + -

1 -fl

N -k

(.X - (

[IN ^

\ ^

/- a + 1)

kl \ .

k\

l \ . .

and

P

where n = 1 and n ^ 1 for the geometric and negative bi-nomial cases, respectively. In cases where the Bemoulli pa-rameter p is not known, it can be estimated in several man-ners, with the conventional method of maximum likelihoodand method of moment estimators both producing

1P==

where X = the average of all the individual data in all sub-groups or samples. (Note that a few alternate approaches tothe "parameters-unknown" or "parameters-estimated" caseare possible, as mentioned briefly later in this paper.)

Also note that in the number-until and number-beforeBemoulli cases, the uniform minimum variance unbiased es-timator (MVUE) is recommended, especially when dealingwith very few data, because the above tends to slightly over-estimate p. Although not the focus here and slightly morework for practitioners, using the above notation this can beshown to he

N - 1 1 N -

,«i - {a ~

X2,n-, - { a - l)/72) +

^^m.n,,, - {a - Drim)

P = N

where / is the sum of all the individual data adjusted for theshift parameter a to equate to the type I case.

H

H

j=\ /-I

and N is the total number of trials in all subgroups such thatN = l]7=i "J ^^ general or N = nm if all subgroups are ofthe same size n.

The above results then can be used to develop appropri-ate )fc-sigma or probability-based control charts for the caseswhere the parameter p is known and the (more common)case where this probability must be estimated from historicaldata 111]. These calculations, for conventional/:-sigma typeg and h charts, are summarized in table 1. In the simplestand most likely case where the subgroup size n = 1, suchas here for the number-between individual occurrences, notethat these calculations conveniently simplify for the practi-tioner to those summarized in table 2, where:

X - the average number of procedures, events, or daysbetween infections;

n - the subgroup size (if other than 1);tn - thenumberof subgroups used to calculate the cen-

ter line and control limits;N - the total number of data used to calculate the cen-

ter line and control limits;p - the infection or adverse event rate (if known); and

NUMBER-BETWEEN j CONTROL CHARTS 31t

Table 2Standard calculations for adverse event g conirol charts (n = 1). (Appropriate for number-before type scenarios, a = 0.)

Rate known Rale estimated MVUE

Upper control limit (UCL) -\-kJ

Center line (CL)

P m- 1

m- J

m - 1

1-i- —

- \ + k m

Table 3Altemate calculations for adverse evctit g control charts (n = I). (Appropriate ior number-uniil type scenarios,

a= 1.)

Rale known Rate cstimaicd MVUE

1 / I — p - /=—; III

Upper control hmit (UCL) - + k —r— Jt -t- kJx(x - I)P \ p^ "I - 1

-V - 1 + -m

Cetiter line (CL)m - 1

Lower control hmil (LCD *: X -k

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

Note that a negative lower control limit by conventionusually is rounded up to i? = 0, as plotting observed val-ues beneath this is not possible given the non-negative na-ture of "number-between" data. As a slight variation, if thenutnber-until the next occurrence are plotted, such as for thenumber of catheters tmtil the next infection (i.e., up to andincluding the first infected catheter), then the alternate calcu-lations shown in table 3 should be used, with the minimumpossible value for the LCL now being a = 1 {i.e., on the firstBernoulli trial). Note that both approaches always will yieldprecisely identical results and statistical properties (e.g., thegeometric type I event of exactly X cases until the next in-fection is identical to the geometric type II event of exactlyX— I cases/jf/ore the next infection), and thus the choice isstrictly a matter of user preference or reporting conventions.

Also note that an alternate approach to constructing thesecharts, especially given the skewness of geometric data,could be to use probability-based limits and a center lineequal to the median rather than the traditional arithmeticmean. Additionally, although traditional 3-sigma controllimits almost always should be used, the t-sigma notation ismeant to recognize that in some special cases k can be set todifferent values (hopefully only when based on sound analy-sis) in order lo achieve the most preferable tradeoff betweenthe false alarm rate, power to detect an infection rate change,and all associated costs and consequences- These conceptsand probability-based control limits are explored in greaterdetail in a companion article [5].

3. Some examples

3.}. Example 1

To illustrate the construction and interpretation of ,?-typecontrol charts, note that the average of the 75 number-between-infections heart surgery data shown previously infigure 3 (out of a total of 3,090 cases) is

- 309041275

resulting in an infection rate estimate of

1P = 4 1 . 2 - 0 - K l

0.0237

and a center line, upper control limit, and lower control limitof

CL = 41.2.

UCL = 41.2 + 3 / 4 ^ 2 ( 4 ^ 2 + 1 )

fti 41.2-h 125.09 =« 166.29,

and

LCL = max{a, 4 1 . 2 - 125.09} = maxjO,-83.89) = 0.

The MVUE results in this case are very similar, with

1 7 5 - 1P =

CL =

4 1 . 2 - 0 + 175

75

.41.2+ —75 - 1 V 75

0.0234.

41.7703.

312 J.C. BENNEYAN

5 ^COS

.>"ra• 5 EO 0)

175 T

150 --

125

100 --

75 -- ;

50 - •

25 -•

UCL(g)

; f

• • •

4 •

; UCL(c}

1 • I H h H H

CL

LCL(c)

LCL(g)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Infection Number

Figure 4. Heart surgery infection control g chart for earlier data.

75UCL -: ——1-[41.7703 + 37(41.2 +1)(41.2+1/75)1

=« 1.0135[41.7703+ 125.1112] ^ 168.5722,

and

= maxjO. 1.0135(41.7703-125.11))= max{0,-85.03} = 0.

The corresponding infection control g chart for these datais shown in figure 4, with the control limits for a conven-tional c chart also added for comparison. In terms of visualinterpretation of this type of chart, note that a decrease inthe number of cases between infections corresponds to anincrease in the infection rate and, unlike the case for morefamiliar types of charts, to values now closer to, rather thanfurther from, the horizontal .r-axis. In this particular exam-ple, the process appears to exhibit a state of statistical con-trol throughout the entire time period examined (in contrastwith previous suggestions that an infection rate increase wasfollowed by a subsequent reduction due to procedural inter-ventions [37]). As an aside, note that if a traditional c charthad been incorrectly used for these "count" data, an entirelydifferent and erroneous conclusion would have been drawnabout the consistency of this process, due to a grossly in-flated false alarm a probability, with approximately 72% ofthe in-control values incorrectly being interpreted as out-of-control.

3.2. Example 2

As a second example for which g charts are applicable,figure 5 compares a histogram of the number of days be-tween positive Clostridium difficile colitis infected stool as-says with the appropriate theoretic geometric probability dis-tribution. Treating the number of days as discrete data, bydefinition a geometric distribution and g control chart are ap-propriate for these data, as this histogram illustrates visually(assuming for now a reasonably constant infection probabil-ity from day-to-day; see section 4). Note that while ideallyin such situations it would be preferable to know the exactnumber of cases, rather than the number of days, betweenpositive specimens in order to have a more precise infectionrate measure, in many cases such detailed data may not beavailable easily. Related examples for which the true under-lying sample size typically may not be easy to obtain includethe number of catheters used between catheter-associated in-fections, the number of needle handlings between accidentalsticks, the number of medications administered between ad-verse drug events (ADE's), and so on.

In such cases, the number of days or other time periodsoften can serve as a reasonable surrogate, especially giventhe important considerations of feasibility of use and imple-mentation by practitioners. For example, the correspond-ing g control chart of days-between-infectionsfor the aboveClostridium difficile data is shown in figure 6. Note that al-though all points are contained within the contro] limits and

NUMBER-BETWEEN g CONTROL CHARTS 313

>c0crCD

Days Between InfectionsGeometric Distribution

.

1 2 3 4 5 6 7 8 9

Days Between Nosocomial Infections

Figure 5. Number of days between Clostridium difficile colitis toxin positive slool assays.

10 11 12

10 T.

8 - • • •

4 +CO

Q 2 - • * - * -

! • • • • •

• • !• i •( H 1- H h15 30 45

Occurrence of Infection

Figure 6. Clostridium difficile colitis infection g-type control chart.

60 75

the rate of infection by this criterion thus appears to be ina state of statistical control (i.e., unchanged), several within-limit signals indicate a rate increase between observations 34and 55. Under the philosophy of statistical process control,therefore, a first step in reducing the infection rate wouldbe to bring this process into a state of statistical control sothat it is operating with only natural variability. An epidemi-ologic investigation thus might be conducted in an attemptlo determine and remove the cause(s) of this increase. (Aspreviously, again note the significant error in the UCL if a ccontrol chart incorrectly had been used based on the reason-ing that these are integer count data, which is not appropriateas by definition this process is not Poisson also see figures 8and 12.)

Note that if days-between and other time-between mea-sures were recorded as continuous data then a slight vari-ation of the g chart, now based on a negative exponentialdistribution, would be used. (The formulae for the "rate es-timated" case will be almost identical, simply omitting thea. na, and ±1 terms under the square root in the control lim-its.) For practical purposes, however, this alternative wouldonly be appropriate if the specific time of day were recorded,whereas otherwise a g chart should be used to produce amore accurate approximation. In many cases with reason-ably low infection rates, moreover, the difference is negligi-ble, as the geometric distribution can be shown to essentiallybecome continuous as p approaches zero and to converge toits continuotis exponential analogue.

314 J.C. BENNEYAN

3.3. Other healthcare applications

In addition to the above examples, similar g charts also maybe applicable to other types of adverse healthcare eventsand medical concerns, especially in cases for which occur-rence rates are low, data are scarce or infrequent, and im-mediate interpretation of each data value is of interest, suchas for needle stick staff exposures, medication errors, andother types of patient complications. As three examples, fig-ures 7-9 illustrate g charts for the number of procedures be-

CQw0)CO(0

O(D

nEz

tween surgical site infections, the number of time intervalsbetween infectious diseases [26], and the number of days be-tween needle sticks |6]. respectively. Again, in the last twocases note that ideally a better basis for comparison mightbe the number of "potential sticks" (procedures, injectiotis,et cetera) between actual sticks and the number of patientsbetween diagnoses (especially if the assumptions of a rel-atively constant probability day-to-day or patient-to-patientare not reasonable), although these data are extremely un-

10 15 20 25 30 35 40 45 50

Surgical Site Infection Number

Figure 7, g control chart of number of procedures between surgical sile infections.

55 60

Q

cn

(0

co

5"SCOCO

1 6 T

14 +

12 --

10 --

6 -

4 . .

2 - •

UCL(g)

UCL(c)

CL

LCL

10 12 14 16 IB 20 22 24 26 28 30

Infectious Disease Occurrence

Figure 8. Time between infections diseases g control chart.

NUMBER-BETWEEN g CONTROL CHARTS 313

likely to be available easily. Thus in such cases the numberof days or lime periods again may suffice as a very reason-able surrogate.

Note that while the needle stick rate shown in figure 9exhibits a state of statistical control (with a slight improv-ing trend), figures 7 and 8 conversely exhibit several sig-nals that these processes are not stable over time. For ex-ample, figure 7 indicates that tbe surgical site infection ratehas increased, with the 8th plotted point being above the up-per control limit and an evident downward trend in the datathereafter. Figure 8, conversely, exhibits several instances ofunnatural variability in the form of rate decreases that repre-

oT3

sent important opportunities for improvement. (Namely, sta-tistically longer sojourns are evident between occurrences 5and 6. between occurrences 15 and 16.. and between occur-rences 20 and 21 than the higher disease rate associated withthe remainder of this period.)

Other recent "number-between" and "time-between" ap-plications in which g control charts have been useful in-clude:

• the number of days between gram stain errors;

• the number of patients betv^een catheter-associated infec-tions;

c0)0)

0}CO

to

nE

H h -1 1- H h H \-

Date of Exposure

Figure 9. g control chart of number of days between needle sticks.

03LL03

cc

nLength of Stay (Number of Days)

Figure 10. Histogram of seniorcare lengths-of-stay (LOS), in days.

316 J.C. BENNEYAN

COO

CO

o

c

Figure 11. Seniorcare length-of-stay g control chart.

>toQ

t o•-<CO

enc

cn

2

UCL(g)

20 22 24 26 28 30 32 34 368 10 12 14 16 1

Subgroup (Week) Number

Figure 12. /i-type control chart of average seniorcare LOS per week.

• the number of blood cultures taken from patients withpyrexia between blood stream infections (BSI);

• the number of days between preventable adverse drugevents; and

• the number of medical intensive care unit patients col-onized with Staphylococcus aureus between methicillin-resistant (MRSA) cases.

Additionally, health care and other situations for whichgeometric distributions and g charts have been found to beappropriate simply as a "state of nature", although some-times counter-intuitive, have included certain patient lengthof stay (LOS) data, recidivism (number of revisits per pa-tient), the number of re-worked welds per manufactureditem, the number of detected software bugs, the number of

items on delivery trucks, and the number of invoices receivedper day [9,10]. It is important to note that these applicationswill not always be appropriate and. just as is the case forother control charts, g charts should not be blindly appliedwithout investigating and verifying the underlying statisticaldistribution.

As one example of such an arisal, the histogram shown infigure 10 of patient lengths-of-stay in a particular seniorcarefacility exhibits a geometric shape, with the corresponding gcontrol chart of individual LOS's shown in figure 11. Alter-nately, figure 12 illustrates the use of an h control chart forthe average lengths-of-stay per week, subgrouped by admitdate, of all patients admitted in each week. As is evident inthese two control charts, early efforts to standardize proce-dures and reduce LOS's have been effective. As previously.

NUMBER-BETWEEN g CONTROL CHARTS 317

figure 12 also illustrates the significant possible consequenceof an incorrect M chart leading to erroneous and potentiallycostly conclusions. In the bloodstream infection applicationmentioned above, for example, reacting to perceived (butin fact non-existent) increases in hospital-wide BSI's (i.e.."false alarms'") can be expensive and potentially dangerousto patients, frequently resulting in the medical staff changingall intravenous lines or in inappropriate blanket administra-tion of prophylaxis antibiotics.

4. Discussion

This article developed and illustrated the use geometric-based Shewhart-type SPC charts to help study and controladverse events as processes over time. Several empiricalexamples demonstrated that statistical process control, ap-plied correctly, is an effective technique that can comple-ment traditional hospital epidemiology methods. Becausethe costs of nosocomial infection and other adverse eventscan be quite high, rapid detection of an increase in a clin-ical unit is of obvious interest (as well as detection of ratedecreases so that root causes can be investigated and stan-dardized).

An appealing surveillance feature of these new charts isthat they can take immediate advantage of each observedadverse event, rather than waiting until the end of a pre-specified time period, increasing the likelihood of identify-ing root causes soon after detecting rate changes. The abil-ity of these charts to detect rate changes of different mag-nitudes and several variations are explored extensively in acompanion article [5], including comparison to conventionalnp charts and several simple ways to improve performance.

In addition to the Shewhart-type charts discussed in thisarticle, more sophisticated types of g charts also can be ap-plied lo number-between type data, such as exponentiallyweighted moving average (EWMA) and cumulative sum(Cusum) g control charts [10,14.32], here based on geomet-ric and negative binomial distributions. Although slightlymore complicated to construct and interpret, these chartstend to detect smaller process shifts more quickly while stillmaintaining low false alarm rates. Additionally, like theShewhart-type g chart, these charts also can take immediateadvantage of each observed adverse event, rather than wait-ing until the end of a specified time period. Cusum chartsalso tend to be particularly effective with samples of sizen = 1, as often will be the case in these applications.

In terms of chart administration, note that because sig-nificant differences typically exist between service-specificinfection rates, such as for aduli and pediatric intensive careunits, surgical patients, and high-risk nursery patients, sep-arate control charts might be applied to each of these cate-gories. Additionally, infection rates generally are more rep-resentative if based on the number or duration at-risk whenthese are known, such as the number of patient days, surg-eries, and device-use/device-days, rather than simply on thenumber of admissions, discharges, et cetera [33]. Of course.

to study each category separately and adjusted in an appro-priate manner requires more detailed data availability andadditional calculations. Conversely, note that when combin-ing process data with significantly unequal rates - such asvarious types of infections or adverse events for differentdepartments or nonhomogeneous patients - into one over-all rate, standard charts will be incorrect and an alternativeshould be used 19]. Also see Alemi et al. (1 ].

Another important assumption in many applications, ofcourse, is that the probability of an infection or other ad-verse event remains fairly constant for each time period ordevice used, such as due to a fairly constant census or dura-tion in use. (This assumption also is inherent in most trans-formation approaches, so the applicability and performanceof g charts are no worse in this respect.) In the use of othertypes of charts (such as for p. u, and X charts), however,the impact of a small amount of variation in the denomina-tor usually is considered negligible and, therefore, frequentadvice is that it is fairly safe to ignore if it does not vary fromits average by more than around 10%.

In other cases, although beyond the present scope,number-between approaches also could be developed for sit-uations in which the probability is not constant from case-to-case. Additionally, if the census varies quite considerably,approaches to the number of events between occurrencesmight be developed by adjusting on the area of opportu-nity in some manner analogous to u and p charts or by us-ing a prior distribution on the infection rate, although theseconcepts are not yet well-developed. Although beyond thepresent scope here, bootstrap, non-parametric, or robust ap-proaches also might be explored if exact results prove math-ematically intractable or if distributional assumptions are notreasonably satisfied; see [6] for further discussion.

Acknowledgement

A portion of this research was conducted while the authorwas supported by the National Science Foundation underGrant No. DMl-0085262.

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