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To Appear in Health Care Management Science, 2001

Number-Between g-type Statistical QualityControl Charts for Monitoring Adverse Events

Last Revised: August 17, 2000

James C. Benneyan *

334 Snell Engineering CenterNortheastern University

Boston, MA 02115

tel: 617-373-2975fax: 617-373-2921

email: [email protected]

Running Title: Number-Between g Control Charts

____________________

* Please address correspondence to Professor James C. Benneyan, Ph.D., MIME Department, 334 SnellEngineering Center, Northeastern University, Boston MA 02115; tel: 617-373-2975; fax: 617-373-2921;e-mail: [email protected].

Benneyan: Number-Between g Control Charts page i

Number-Between g-type Statistical QualityControl Charts for Monitoring Adverse Events

Abstract

Alternate Shewhart-type statistical control charts, called “g” and “h” charts, are developed

and evaluated for monitoring the number of cases between hospital-acquired infections and

other adverse events, such as heart surgery complications, catheter-related infections, surgi-

cal site infections, contaminated needle sticks, and other iatrically induced outcomes. These

new charts, based on inverse sampling from geometric and negative binomial distributions,

are simple to use and can exhibit significantly greater detection power over conventional bi-

nomial-based approaches, particularly for infrequent events and low “defect” rates. A com-

panion article illustrates several interesting properties of these charts and design modifica-

tions that significantly can improve their statistical properties, operating characteristics, and

sensitivity.

Running Title: Number-Between g Control Charts

Key Words: SPC, Control charts, Healthcare, Adverse events, Geometric distribution,g charts.

Benneyan: Number-Between g Control Charts page 1

Introduction

Overview of Article

This article illustrates a new type of statistical process control (SPC) chart for monitoring the

number of cases between hospital-acquired infections or other healthcare adverse events,

such as heart surgery complications, catheter-related infections, contaminated needle sticks,

medication errors, and other iatrogenic events. These new charts, called “g” and “h” control

charts, are based on inverse sampling from underlying geometric and negative binomial dis-

tributions and can exhibit improved shift-detection sensitivity over conventional approaches,

particularly when dealing with infrequent events or low “defect” rates. The application and

interpretation of these charts for detecting rate changes are illustrated by several examples

involving cardiac bypass surgical-site infections, Clostridium difficile infections, needle stick

exposures, and related concerns.

In a companion paper [5], the specificity and sensitivity of these new charts are investigated

and contrasted with more conventional methods, with several simple design considerations --

including standard within-limit rules, redefined Bernoulli trials, a new in-control rule, and

probability-based control limits -- shown to significantly improve the chart’s power to detect

true process changes. These charts also are shown in some cases to exhibit better statistical

operating characteristics over traditional binomial-based np and p control charts, especially

when the rate of occurrence (i.e., the Bernoulli parameter p) is sufficiently low. In summary,

these charts are found to be relatively simple to use and interpret, to exhibit comparable or

superior performance to more traditional or more complicated methods, and to be a useful

complement to conventional hospital epidemiology and infection control methods.


Hospital Epidemiology and Infection Control

Epidemiology in the broadest context is concerned with the study, identification, and pre-

vention of adverse healthcare events, disease transmission, and contagious outbreaks, with

particular focus within hospitals on nosocomial infections and infection control. Nosocomial

infections essentially are any infections that are acquired or spread as a direct result of a pa-

tient’s hospital stay (rather than being pre-existent as an admitting condition), with a few ex-

amples including surgical site infections, catheter infections, pneumonia, bacteremia, urinary

tract infections, cutaneous wound infections, bloodstream infections, gastrointestinal infec-

tions, and others.

With estimates of the national costs of nosocomial infections ranging from approximately 8.7

million additional hospital days and 20,000 deaths per year [21] to 2 million infections and

80,000 deaths per year [30], it is clear that these problems represent quite considerable health

and cost concerns. Additionally, the number of U.S. hospital patients injured due to medical

errors and adverse events has been estimated between 770,000 and 2 million per year, with

the national cost of adverse drug events estimated at $4.2 billion annually and an estimated

180,000 deaths caused partly by iatrogenic injury nationwide per year [31, 13, 4, 18, 2, 15].

The costs of a single nosocomial infection or adverse event have been estimated both to av-

erage between $2,000 and $3,000 per episode. The National Academy of Sciences' Institute

of Medicine recently estimated that more Americans die each year from medical mistakes

than from traffic accidents, breast cancer, or AIDS, with $8.8 billion spent annually as a re-

sult of medical mistakes [24].

Given the above figures, it is not surprising that many federal, regulatory, and healthcare ac-

crediting bodies -- such as the Joint Commission on Accreditation of Healthcare Organiza-

tions (JCAHO), the National Committee for Quality Assurance (NCQA), the U.S. Center for

Disease Control (CDC), the Health Care Financing Administration (HCFA), and others --


either require or strongly encourage hospitals and HMO’s to apply both classical epidemiol-

ogy and more modern continuous quality improvement (CQI) methodologies to these signifi-

cant process concerns, including the use of statistical methods such as statistical process

control (SPC). For example, the Joint Commission on Accreditation of Healthcare Organi-

zations recently stated their position on the use of SPC as follows [27]:

“An understanding of statistical quality control, including SPC, and variation is es-sential for an effective assessment process... Statistical tools such as run charts,control charts, and histograms are especially helpful in comparing performance withhistorical patterns and assessing variation and stability.”

Similarly, a recent position paper by several epidemiologists from the U.S. Center for Dis-

ease Control [33] stated that

“Many of the leading approaches to directing quality improvement in hospitals arebased on the principles of W. E. Deming. These principles include use of statisticalmeasures designed to determine whether improvement in quality has been achieved.These measures should include nosocomial infection rates.”

Conventional epidemiology methods, in fact, include both various statistical and graphical

tools for retrospective analysis, such as described by Mausner and Kramer [34] and Gordis

[22], and several more surveillance-oriented methods, such as reviewed by Larson [29] (also

see Becker [3]). It is worth noting that collectively these methods tend to be concerned with

both epidemic (i.e., outbreaks) and endemic (i.e., systemic) events, which in SPC terminol-

ogy equate to unnatural and natural variability, respectively, and therefore are candidates for

effective study via control charts. Several epidemiologists (for example, see Birnbaum [12],

Mylotte [35], Burnett and Chesher [16], Childress and Childress [17], and Mylotte et al [36])

have proposed monitoring certain infection and adverse event rates more dynamically over

time, rather than "time-statically", in manners that are quite similar in nature and philosophy

to SPC. It also is interesting that as early as 1942, Dr. Deming advocated the important po-

tential of SPC in disease surveillance and to rare events [19].


Use of Statistical Process Control (SPC)

The application of standard SPC methods to healthcare processes, infection control, and hos-

pital epidemiology has been discussed by several authors, including a comprehensive review

in a recent series in Infection Control and Hospital Epidemiology [6]. Example applications

include medication errors, patient falls and slips, central line infections, surgical complica-

tions, and other adverse events. In some cases, however, none of the most common types of

control charts will be appropriate, for example due to the manner in which data are collected,

pre-established measuring and reporting metrics, or low occurrence rates and infrequent data.

One example of particular note is the use of events-between, number-between, days-

between, or time-between type of data that occasionally are used by convention in some

healthcare and other settings. Several important clinical applications of such measures are

described below, and this article therefore derives and illustrates appropriate control charts

for these cases, such as for the number of procedures, opportunities, or days between infre-

quent adverse events. (Note that these same methods, of course, are equally applicable for

monitoring other types of low defect processes, such as in manufacturing and service set-

tings.)

As some general background, statistical process control charts are chronological displays of

process data used to help statistically understand, control, and improve a system, here an in-

fection control or adverse event process. The general format of a Shewhart-type control

chart is shown in Figure 1. Observed process data, such as the monthly rate of infection or

the number of procedures between infections, are plotted on the chart and interpreted, ide-

ally, soon after they become available. Three horizontal lines also are plotted, called the

center line (CL), the upper control limit (UCL), and the lower control limit (LCL), which are

calculated statistically and help define the central tendency and the range of natural variation

of the plotted values, assuming that the rate of occurrence does not change. By observing

and interpreting the behavior of these process data, plotted over time on an appropriate con-


trol chart, a determination can be made about the stability (i.e., the "state of statistical con-

trol") of the process according to the following criteria.

Figure 1: General Format of a Statistical Control Chart

Values that fall outside the control limits exceed their probabilistic range and therefore are

strong indications that non-systemic causes almost certainly exist that should be identified

and removed in order to achieve a single, stable, and predictable process. There also should

be no evidence of non-random behavior between the limits, such as trends, cycles, and shifts

above or beneath the center line. To aid in the objective interpretation of such data patterns,

various “within-limit” rules have been defined, described in greater detail elsewhere. See

Duncan [20] and Grant and Leavenworth [23] for further information about statistical pro-

cess control in general and Benneyan [6-8] for discussion and application of SPC to

healthcare processes.

Limitations of Standard Types of Control Charts

Several approaches to applying SPC to hospital infections or other adverse events are possi-

ble, dependent on the situation and ranging in complexity and data required. For example,

two standard approaches are to use u or p Shewhart control charts, such as for the number or

the fraction of patients, respectively, per time period who acquire a particular type of infec-

tion. As an example, Figure 2 illustrates a recent u chart for the quarterly number of infec-

tions per 100 patient days, although it should be noted that this example ideally should contain

many more subgroups (time periods) of data. In terms of proper control chart selection, note

that each type of chart is based on a particular underlying probability model and is appropriate

in different types of settings. In this particular example, a Poisson-based u chart would be

appropriate if each patient day is considered as an “area of opportunity” in which one or more

infections theoretically could occur. Alternatively, a binomial-based p chart would be more


appropriate if the data were recorded as the number of patient days with one or more infec-

tions (i.e., Bernoulli trials). Further information on different types of control charts and sce-

narios in which each is appropriate can be found in the above references [20, 23, 6].

Example of u Control Chart with Too Few Values: Number Infections per Quarter

Figure 2

In either of the above cases, note that some knowledge of the denominator is required to con-

struct the corresponding chart, such as the number of patient days, discharges, or surgeries

per time period. A more troubling problem is that use of these charts in some cases can re-

sult in an inadequate number of points to plot or in data becoming available too infrequently

to be able to make rational decisions in a timely manner (especially, ironically, for better

processes with lower infection rates). For example, recall that a minimum of at least 25 to 35

subgroups are recommended to confidently determine whether a process is in statistical con-

trol and that, in the case of p and np charts, a conventional rule-of-thumb for forming bino-

mial subgroups is to pick a subgroup size, n, at least large enough so that np ≥ 5.

For processes with low infection rates, say p < 0.01, the consequence of the above comments

can be a significant increase in total sample size across all subgroups and in the average run

length until a process change is detected. For example, even for p = .01, this translates to

500 data per subgroup x 25 subgroups = 12,500 total data. Additionally, this can necessitate

waiting until the end of longer time periods (e.g., week, month, or perhaps quarter) to calcu-

late, plot, and interpret each subgroup value, perhaps being too late to react to important

changes in a critical process and no longer in best alignment with the philosophy of process

monitoring in as real-time as possible. Note that similar subgroup size rules-of-thumb exist

for c and u charts and lead to the same general dilemma as for np and p charts, in part be-

cause all conventional control charts consider the number of infections or adverse events ei-

ther at the end of some time period or after some pre-determined number of cases.


The Number of Procedures, Events, or Days Between Infections

As an alternate measure, the number of procedures, events, or days between infections has

been proposed due to ease of use, more timely feedback, near immediate availability of each

individual observation, low infection rates, and the simplicity with which non-technical per-

sonnel can implement such measures. Being easy to calculate, a control chart can be updated

immediately in real-time, on the hospital floor, without knowing the census or other base de-

nominator (although assumed for now to be reasonably constant). In some settings, addi-

tionally, “number-between” or “time-between” types of process measures simply may be

preferred as a standard or traditional manner in which to report outcomes.

As a few recent examples of the traditional use of such measures, Nelson et al [37] examined

the number of open heart surgeries between post-operative sternal wound infections, Plourde

et al [40] analyzed the number of infection-free coronary artery bypass graft (CABG) proce-

dures between adverse events, Finison et al [21] considered the number of days between

Clostridium difficile colitis positive stool assays, Jacquez et al [26] analyzed the number of

time intervals between occurrences of infectious diseases, Nugent et al [38] described the use

of time between adverse events as a key hospital-wide metric, Pohlen [40] monitored the

time between insulin reactions, and Benneyan [6] discussed the number of cases between

needle sticks.

In order to plot any of these types of data on control charts, however, note that none of the

standard charts are appropriate. For example, standard c and u control charts are based on

discrete Poisson distributions, np and p charts are based on discrete binomial distributions,

and X and S charts are based on continuous Gaussian distributions. By description, con-

versely, the number of cases between infections most closely fits the classic definition of a

geometric random variable, as discussed below and illustrated by the histogram in Figure 3


of surgeries-between-infections empirical data. Note that if the infection rate is unchanged

(i.e., in statistical control), then the probability, p, of an infection occurring will be rea-

sonably constant for each case (with patients stratified into reasonably homogenous groups if

necessary), and this scenario therefore satisfies the definition of a geometric random vari-

able.

Comparison of Empirical Heart Surgery Infection Data with Geometric Distribution

Figure 3

As can be seen by comparison with the corresponding geometric distribution (with the infec-

tion probability p estimated via the method of maximum likelihood), these data exhibit a

geometrically decaying shape that is very close to the theoretic model. (More rigorously, a

chi-square goodness-of-fit test indicates close statistical agreement, with an effective signifi-

cance value of 0.693.) Appropriate control charts for these data therefore should be based on

underlying geometric distributions, as developed and illustrated below, rather than using any

of the above traditional charts. Use of inappropriate discrete distributions and control charts,

in fact, can lead to erroneous conclusions about the variability and state of statistical control

of the infection rate, a situation which has been described previously [25, 28, 8] and is shown

in some of the examples below.

Events-Between g and h Control Charts

Three General Application Scenarios

Alternate control charts for such situations, called g and h charts, were developed and inves-

tigated by Benneyan [11] and Kaminsky et al [28] based on underlying geometric and nega-

tive binomial distributions. With respect to the above motivation, these new charts were de-

veloped specifically to be appropriate for the following three general scenarios:


1. For situations dealing with classic geometric random variables, as discussed in

greater detail below, such as for the various “number-between” types of appli-

cations described above;

2. For more powerful control (i.e., greater sensitivity) of low frequency process

data than if using traditional np and p charts; and

3. For cases in which a geometrically decaying shape simply appears naturally,

such as periodically observed in histograms of empirical data, even though the

existence of a classic geometric scenario may not be apparent.

The first two scenarios are of particular interest in this article given the above discussion,

with a few examples of the third type of application briefly also illustrated. As shown below,

in cases with low infection rates and with immediate availability of each observation, consid-

ering Bernoulli processes with respect to geometric rather than traditional binomial probabil-

ity distributions can produce more plotted subgroup data and greater ability to more quickly

detect process changes.

Theoretic Motivation

To provide a context for comments to follow, and because some healthcare practitioners and

other readers might not be familiar with the underlying mathematical development, a brief

motivation of these charts is given here; further detail can be found in the cited references

[10, 11, 28]. Recall that if each case (e.g., procedure, catheter, day, et cetera) is considered

as an independent Bernoulli trial with reasonably the same probability of resulting in a “fail-

ure” (e.g., an infection or other adverse event), then the number of Bernoulli trials until the

first failure and the number of Bernoulli trials before the first failure are random variables

with type I and type II geometric distributions, respectively. For example, the probabilities


of the next infection occurring on the xth case or immediately after the xth case are well-

known to be

P X x p p x a a ax a( ) ( ) , , , ,...= = − = + +−1 1 2for ,

where the minimum possible value a = 1 for type I (number-until) and a = 0 for type II

(number-before) geometric data, with the sum of n independent geometric random variables,

T = X1 + X2 + ... + Xn,

being negative binomial with probability function

P T t p p t na na nan a t

nn t na( ) ( ) , , , ,...

( )= =

− = + +− + −

−−1 1

11 1 2for .

It follows that the expected values, variances, and standard deviations of the total number of

cases, T, and the average number of cases, X , before (a = 0) or until (a = 1) the next n infec-

tions or adverse events occur then are

E T n ap

p( ) = +

−1, V T

n p

p( )

( )= −12

, σ T

n p

p

( )= −12 ,

and

E X ap

p( ) = +−1

, V Xp

np( ) = −1

2, σ

X

p

np = −1

2 ,

where n = 1 and n ≥ 1 for the geometric and negative binomial cases, respectively. In cases

where the Bernoulli parameter p is not known, it can be estimated in several manners, with

the conventional method of maximum likelihood and method of moment estimators both

producing

ˆ pX a

=− +

1

1 ,


where X = the average of all the individual data in all subgroups or samples. (Note that a

few alternate approaches to the “parameters-unknown” or “parameters-estimated” case are

possible, as mentioned briefly later in this paper.)

Also note that in the number-until and number-before Bernoulli cases, the uniform minimum

variance unbiased estimator (MVUE) is recommended, especially when dealing with very

few data, because the above tends to slightly over-estimate p. Although not the focus here

and slightly more work for practitioners, using the above notation this can be shown to be

ˆ pN

t X a

N

N= =−

− − +⋅

−1

1

1

1

1 ,

where t is the sum of all the individual data adjusted for the shift parameter a to equate to the

type I case,

t = ( ) ( ) ( )+ + − − + + + − − + + + + − − , , , , , ,... ( ) ... ( ) ... ... ( )x x a n x x a n x x a nn n m m n mm11 1 1 2 1 2 2 11 21 1 1

= − −−=∑∑ ( ),x a Nj ii

n

j

m m

111

,

and N is the total number of trials in all subgroups such that N = njjm=∑ 1 in general or N = nm

if all subgroups are of the same size n.

The above results then can be used to develop appropriate k-sigma or probability-based con-

trol charts for the cases where the parameter p is known and the (more common) case where

this probability must be estimated from historical data [11]. These calculations, for conven-

tional k-sigma type g and h charts, are summarized in Table 1. In the simplest and most

likely case where the subgroup size n = 1, such as here for the number-between individual

occurrences, note that these calculations conveniently simplify for the practitioner to those

summarized in Table 2, where:


x = the average number of procedures, events, or days between infections,

n = the subgroup size (if other than 1),

m = the number of subgroups used to calculate the center line and control limits,

N = the total number of data used to calculate the center line and control limits,

p = the infection or adverse event rate (if known), and

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

Parameters-Known and Parameters-Estimated g and h Control Chart Calculations

Table 1

Standard Calculations for Adverse Event g Control Charts (n = 1)(Appropriate for the Number-Before Type Scenarios, a = 0)

Table 2

Alternate Calculations for Adverse Event g Control Charts (n = 1)(Appropriate for the Number-Until Type Scenarios, a = 1)

Table 3

Note that a negative lower control limit by convention usually is rounded up to a = 0, as

plotting observed values beneath this is not possible given the non-negative nature of “num-

ber-between” data. As a slight variation, if the number-until the next occurrence are plotted,

such as for the number of catheters until the next infection (i.e., up to and including the first

infected catheter), then the alternate calculations shown in Table 3 should be used, with the

minimum possible value for the LCL now being a = 1 (i.e., on the first Bernoulli trial). Note

that both approaches always will yield precisely identical results and statistical properties

(e.g., the geometric type I event of exactly X cases until the next infection is identical to the

geometric type II event of exactly X - 1 cases before the next infection), and thus the choice

is strictly a matter of user preference or reporting conventions.

Also note that an alternate approach to constructing these charts, especially given the skew-

ness of geometric data, could be to use probability-based limits and a center line equal to the


median rather than the traditional arithmetic mean. Additionally, although traditional 3-

sigma control limits almost always should be used, the k-sigma notation is meant to recog-

nize that in some special cases k can be set to different values (hopefully only when based on

sound analysis) in order to achieve the most preferable tradeoff between the false alarm rate,

power to detect an infection rate change, and all associated costs and consequences. These

concepts and probability-based control limits are explored in greater detail in a companion

article [5].

Some Examples

Example 1

To illustrate the construction and interpretation of g-type control charts, note that the average

of the 75 number-between-infections heart surgery data shown previously in Figure 3 (out of

a total of 3,090 cases) is

x .= =3090

7541 2,

resulting in an infection rate estimate of

ˆ ..

p = ≈− +1

41 2 0 10 0237

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

CL = 41.2,

UCL = 41.2 3 41.2(41.2 1) 41.2 125.09 166.29+ + ≈ + ≈ , and

LCL = max{a, 41.2 - 125.09} = max{0, -83.89} = 0.


The MVUE results in this case are very similar, with

ˆ ..

p = ≈− +

⋅−1

41 2 0 1

75 1

750 0234,

CL =75

75 1

1

7541 2 41 7703

−+

≈. . ,

UCL =75

75 -11

7541.7703 3 (41.2 +1) 41.2+ +( )

≈ + ≈ 1.0135[41.7703 125.1112] 168.5722 , and

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

The corresponding infection control g chart for these data is shown in Figure 4, with the

control limits for a conventional c chart also added for comparison. In terms of visual inter-

pretation of this type of chart, note that a decrease in the number of cases between infections

corresponds to an increase in the infection rate and, unlike the case for more familiar types of

charts, to values now closer to, rather than further from, the horizontal x-axis. In this par-

ticular example, the process appears to exhibit a state of statistical control throughout the en-

tire time period examined (in contrast with previous suggestions that an infection rate in-

crease was followed by a subsequent reduction due to procedural interventions [37]). As an

aside, note that if a traditional c chart had been incorrectly used for these “count” data, an

entirely different and erroneous conclusion would have been drawn about the consistency of

this process, due to a grossly inflated false alarm α probability, with approximately 72% of

the in-control values incorrectly being interpreted as out-of-control.

Heart Surgery Infection Control g Chart for Earlier Data

Figure 4


Example 2

As a second example for which g charts are applicable, Figure 5 compares a histogram of the

number of days between positive Clostridium difficile colitis infected stool assays with the

appropriate theoretic geometric probability distribution. Treating the number of days as dis-

crete data, by definition a geometric distribution and g control chart are appropriate for these

data, as this histogram illustrates visually (assuming for now a reasonably constant infection

probability from day-to-day; see Discussion section). Note that while ideally in such situa-

tions it would be preferable to know the exact number of cases, rather than the number of

days, between positive specimens in order to have a more precise infection rate measure, in

many cases such detailed data may not be available easily. Related examples for which the

true underlying sample size typically may not be easy to obtain include the number of cathe-

ters used between catheter-associated infections, the number of needle handlings between

accidental sticks, the number of medications administered between adverse drug events

(ADE’s), and so on.

In such cases, the number of days or other time periods often can serve as a reasonable sur-

rogate, especially given the important considerations of feasibility of use and implementation

by practitioners. For example, the corresponding g control chart of days-between-infections

for the above Clostridium difficile data is shown in Figure 6. Note that although all points

are contained within the control limits and the rate of infection by this criterion thus appears

to be in a state of statistical control (i.e., unchanged), several within-limit signals indicate a

rate increase between observations 34 to 55. Under the philosophy of statistical process

control, therefore, a first step in reducing the infection rate would be to bring this process

into a state of statistical control so that it is operating with only natural variability. An epi-

demiologic investigation thus might be conducted in an attempt to determine and remove the

cause(s) of this increase. (As previously, again note the significant error in the UCL if a c


control chart incorrectly had been used based on the reasoning that these are integer count

data, which is not appropriate as by definition this process is not Poisson.)

Number of Days Between Clostridium difficile colitis Toxin Positive Stool Assays

Figure 5

Clostridium difficile colitis Infection g-type Control Chart

Figure 6

Note that if days-between and other time-between measures were recorded as continuous

data then a slight variation of the g chart, now based on a negative exponential distribution,

would be used. (The formulae for the "rate estimated" case will be almost identical, simply

omitting the a, na, and ±1 terms under the square root in the control limits.) For practical

purposes, however, this alternative would only be appropriate if the specific time of day were

recorded, whereas otherwise a g chart should be used to produce a more accurate approxi-

mation. In many cases with reasonably low infection rates, moreover, the difference is neg-

ligible, as the geometric distribution can be shown to essentially become continuous as p ap-

proaches zero and to converge to its continuous exponential analogue.

Other Healthcare Applications

In addition to the above examples, similar g charts also may be applicable to other types of

adverse healthcare events and medical concerns, especially in cases for which occurrence

rates are low, data are scarce or infrequent, and immediate interpretation of each data value is

of interest, such as for needle stick staff exposures, medication errors, and other types of pa-

tient complications. As three examples, Figures 7-9 illustrate g charts for the number of pro-

cedures between surgical site infections, the number of time intervals between infectious dis-

eases [26], and the number of days between needle sticks [6], respectively. Again, in the last

two cases note that ideally a better basis for comparison might be the number of "potential


sticks" (procedures, injections, et cetera) between actual sticks and the number of patients

between diagnoses (especially if the assumptions of a relatively constant probability day-to-

day or patient-to-patient are not reasonable), although these data are extremely unlikely to be

available easily. Thus in such cases the number of days or time periods again may suffice as

a very reasonable surrogate.

Note that while the needle stick rate shown in Figure 9 exhibits a state of statistical control

(with a slight improving trend), Figures 7 and 8 conversely exhibit several signals that these

processes are not stable over time. For example, Figure 7 indicates that the surgical site in-

fection rate has increased, with the 8th plotted point being above the upper control limit and

an evident downward trend in the data thereafter. Figure 8, conversely, exhibits several in-

stances of unnatural variability in the form of rate decreases that represent important oppor-

tunities for improvement. (Namely, statistically longer sojourns are evident between occur-

rences 5 and 6, between occurrences 15 and 16, and between occurrences 20 and 21 than the

higher disease rate associated with the remainder of this period.)

g Control Chart of Number of Procedures Between Surgical Site Infections

Figure 7

Time Between Infectious Diseases g Control Chart

Figure 8

g Control Chart of Number of Days Between Needle Sticks

Figure 9

Other recent “number-between” and "time-between" applications in which g control charts

have been useful include:

• the number of days between gram stain errors,

• the number of patients between catheter-associated infections,


• the number of blood cultures taken from patients with pyrexia between blood stream

infections (BSI),

• the number of days between preventable adverse drug events, and

• the number of medical intensive care unit patients colonized with Staphylococcus

aureus between methicillin-resistant (MRSA) cases.

Additionally, situations for which geometric distributions and g charts have been found to be

appropriate simply as a "state of nature", although sometimes counter-intuitive, have in-

cluded certain patient length of stay (LOS) data, recidivism (number of revisits per patient),

the number of re-worked welds per manufactured item, the number of detected software

bugs, the number of items on delivery trucks, and the number of invoices received per day

[9, 10]. It is important to note that these applications will not always be appropriate and, just

as is the case for other control charts, g charts should not be blindly applied without investi-

gating and verifying the underlying statistical distribution.

As one example of such an arisal, the histogram shown in Figure 10 of patient lengths-of-

stay in a particular seniorcare facility exhibits a geometric shape, with the corresponding g

control chart of individual LOS’s shown in Figure 11. Alternately, Figure 12 illustrates the

use of an h control chart for the average lengths-of-stay per week, subgrouped by admit date,

of all patients admitted in each week. As is evident in these two control charts, early efforts

to standardize procedures and reduce LOS’s have been effective. As previously, Figure 12

also illustrates the significant possible consequence of an incorrect u chart leading to errone-

ous and potentially costly conclusions. In the bloodstream infection application mentioned

above, for example, reacting to perceived (but in fact non-existent) increases in hospital-wide

BSI's (i.e., “false alarms”) can be expensive and potentially dangerous to patients, frequently

resulting in the medical staff changing all intravenous lines or in inappropriate blanket ad-

ministration of prophylaxis antibiotics.


Histogram of Seniorcare Lengths-of-Stay (LOS), in Days

Figure 10

Seniorcare Length-of-Stay g Control Chart

Figure 11

h-type Control Chart of Average Seniorcare LOS per Week

Figure 12

Discussion

This article developed and illustrated the use geometric-based Shewhart-type SPC charts to

help study and control adverse events as processes over time. Several empirical examples

demonstrated that statistical process control, applied correctly, is an effective technique that

can complement traditional hospital epidemiology methods. Because the costs of nosoco-

mial infection and other adverse events can be quite high, rapid detection of an increase in a

clinical unit is of obvious interest (as well as detection of rate decreases so that root causes

can be investigated and standardized).

An appealing surveillance feature of these new charts is that they can take immediate ad-

vantage of each observed adverse event, rather than waiting until the end of a pre-specified

time period, increasing the likelihood of identifying root causes soon after detecting rate

changes. The ability of these charts to detect rate changes of different magnitudes and sev-

eral variations are explored extensively in a companion article [5], including comparison to

conventional np charts and several simple ways to improve performance.

In addition to the Shewhart-type charts discussed in this article, more sophisticated types of g

charts also can be applied to number-between type data, such as exponentially weighted


moving average (EWMA) and cumulative sum (Cusum) g control charts [10, 14, 31], here

based on geometric and negative binomial distributions. Although slightly more complicated

to construct and interpret, these charts tend to detect smaller process shifts more quickly

while still maintaining low false alarm rates. Additionally, like the Shewhart-type g chart,

these charts also can take immediate advantage of each observed adverse event, rather than

waiting until the end of a specified time period. Cusum charts also tend to be particularly

effective with samples of size n = 1, as often will be the case in these applications.

In terms of chart administration, note that because significant differences typically exist be-

tween service-specific infection rates, such as for adult and pediatric intensive care units,

surgical patients, and high-risk nursery patients, separate control charts might be applied to

each of these categories. Additionally, infection rates generally are more representative if

based on the number or duration at-risk when these are known, such as the number of patient

days, surgeries, and device-use/device-days, rather than simply on the number of admissions,

discharges, et cetera. [32]. Of course, to study each category separately and adjusted in an

appropriate manner requires more detailed data availability and additional calculations.

Conversely, note that when combining process data with significantly unequal rates - such as

various types of infections or adverse events for different departments or nonhomogeneous

patients - into one overall rate, standard charts will be incorrect and an alternative should be

used [9]. Also see Alemi et al [1].

Another important assumption in many applications, of course, is that the probability of an

infection or other adverse event remains fairly constant for each time period or device used,

such as due to a fairly constant census or duration in use. (This assumption also is inherent

in most transformation approaches, so the applicability and performance of g charts are no

worse in this respect.) In the use of other types of charts (such as for p, u, and X charts),

however, the impact of a small amount of variation in the denominator usually is considered


negligible and, therefore, frequent advice is that it is fairly safe to ignore if it does not vary

from its average by more than around 10%.

In other cases, although beyond the present scope, number-between approaches also could be

developed for situations in which the probability is not constant from case-to-case. Addi-

tionally, if the census varies quite considerably, approaches to the number of events between

occurrences might be developed by adjusting on the area of opportunity in some manner

analogous to u and p charts or by using a prior distribution on the infection rate, although

these concepts are not yet well-developed. Although beyond the present scope here, boot-

strap, non-parametric, or robust approaches also might be explored in exact results prove

mathematically intractable or if distributional assumptions are not reasonably satisfied; see

[5] for further discussion.


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Table and Figure Legends

Table 1: Parameters-Known and Parameters-Estimated g and h Control Chart Calculations

Table 2: Standard Calculations for Adverse Event g Control Charts (n = 1).(Appropriate for the Number-Before Type Scenarios)

Table 3: Alternate Calculations for Adverse Event g Control Charts (n = 1).(Appropriate for the Number-Until Type Scenarios)

Figure 1: General Format of a Statistical Control Chart

Figure 2: Example of a u Control Chart with Too Few Values: Number Infections per Quar-ter

Figure 3: Comparison of Empirical Heart Surgery Infection Data with Geometric Distribu-tion

Figure 4: Heart Surgery Infection Control g Chart for Earlier Data

Figure 5: Number of Days Between Clostridium difficile colitis Toxin Positive Stool Assays

Figure 6: Clostridium difficile colitis Infection g-type Control Chart

Figure 7: g Control Chart of Number of Procedures Between Surgical Site Infections

Figure 8: Time Between Infectious Diseases g Control Chart

Figure 9: g Control Chart of Number of Days Between Needle Sticks

Figure 10: Histogram of Seniorcare Lengths-of-Stay (LOS), in Days

Figure 11: Seniorcare Length-of-Stay g Control Chart

Figure 12: h-type Control Chart of Average Seniorcare LOS per Week


0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4

Subgroup Number

SubgroupStatistic

(e.g., average,standard deviation,

rate, number,proportion)

Upper Control Limit (UCL)

Lower Control Limit (LCL)

Center Line (CL) = Central Tendency

General Format of a Statistical Control Chart

Figure 1

Quarter

Infe

ctio

n R

ate

(Num

ber

per

100

Pat

ient

Day

s)

UCL

CL

LCL

Example of a u Control Chart with Too Few Values:Number of Nosocomial Infections per Quarter

Figure 2


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0-5

6-1

0

11

-15

16

-20

21

-25

26

-30

31

-35

36

-40

41

-45

46

-50

51

-55

56

-60

61

-65

66

-70

71

-75

76

-80

81

-85

86

-90

91

-95

96

-10

0

10

1-1

05

10

6-1

10

11

1-1

15

11

6-1

20

12

1-1

25

12

6-1

30

13

1-1

35

13

6-1

40

Number of Cases Between Infections

Rel

ativ

e F

requ

ency Empirical Waiting Times

Theoretical (Geometric) Distribution

Comparison of Empirical Heart Surgery Infection Data with Geometric Distribution

Figure 3

0

2 5

5 0

7 5

100

125

150

175

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5

Infection Number

Num

ber

of C

onse

cutiv

e C

ases

Bet

wee

n P

ost-

Ope

rativ

e S

tern

al W

ound

Inf

ectio

ns UCL(g)

LCL(g)

CL

UCL(c)

LCL(c)

Heart Surgery Infection Control g Chart for Earlier Data

Figure 4


0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

Days Between Nosocomial Infections

Fre

qu

en

cy

Days Between Infections

Geometric Distribution

Number of Days Between Clostridium difficile colitis Toxin Positive Stool Assays

Figure 5

0

2

4

6

8

10

0 15 30 45 60 75

Occurrence of Infection

Day

s B

etw

een

Infe

ctio

ns

Clostridium difficile colitis Infection g-type Control Chart

Figure 6


5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0

Surgical Site Infection Number

Num

ber

Cas

es B

etw

een

Infe

ctio

ns

g Control Chart of Number of Procedures Between Surgical Site Infections

Figure 7

0

2

4

6

8

1 0

1 2

1 4

1 6

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0

Infectious Disease Occurrence

Wai

t B

etw

een

Infe

ctio

ns (

e.g.

, D

ays)

UCL(g)

CL

UCL(c)

LCL

Time Between Infectious Diseases g Control Chart

Figure 8


Date of Exposure

Num

ber

of D

ays

Bet

wee

n N

eedl

e S

ticks

g Control Chart of Number of Days Between Needle Sticks

Figure 9

Length of Stay (Number of Days)

Rel

ativ

e F

requ

ency

Histogram of Seniorcare Lengths-of-Stay (LOS), in Days

Figure 10


Leng

th o

f S

tay

(Num

ber

of D

ays)

Seniorcare Lengths-of-Stay g Control Chart

Figure 11

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6

Subgroup (Week) Number

Ave

rage

Len

gth

of S

tay

(Day

s)

UCL(g)

CL(c)

L

CL(c)

CL(g)

h-type Control Chart of Average Seniorcare LOS per Week

Figure 12


g Chart: Total Number Events per Subgroup

Rate Known Rate Estimated MVUE

Upper Control Limit (UCL) np

pa

n pp

k1 1−

+ −

+ ( )

2 nx k n x a x a ( )( )+ − − +1 CLN

Nk n x a x a

N+

−− + − +

1

11

( )

Center Line (CL) np

pa

1 −+

nxN

Nn x

a

N−+

−

1

1

Lower Control Limit (LCL) np

pa

n pp

k1 1−

+ −

− ( )

2 nx k n x a x a ( )( )− − − +1 CLN

Nk n x a x a

N−

−− + − +

1

11

( )

h Chart: Average Number Events per Subgroup


Upper Control Limit (UCL)1 1−

+ −

+p

pa

pnp

k 2 x kx a x a

n

( )( )+

− − +1CL

N

Nk

nx a x a

N+

−− + − +

1

11

1( )

Center Line (CL)1 −

+

p

pa x

N

Nx

a

N−+

−

1

1

Lower Control Limit (LCL)1 1−

+ −

−p

pa

pnp

k 2 x kx a x a

n

( )( )−

− − +1CL

N

Nk

nx a x a

N−

−− + − +

1

11

1( )

Parameters-Known and Parameters-Estimated g and h Control Chart Calculations

Table 1



Upper Control Limit (UCL)1 1− −+p

p

pp

k 2 x k x x ( )+ +1m

mx

mk x

mx

−+ + + +

1

1 11( )

Center Line (CL)1 − p

px

m

m mx

−+

1

1

Lower Control Limit (LCL)1 1− −−p

p

pp

k 2 x k x x ( )− +1m

mx

mk x

mx

−+ − + +

1

1 11( )

Standard Calculations for Adverse Event g Control Charts (n = 1)(Appropriate for the Number-Before Type Scenarios, a = 0)

Table 2


Upper Control Limit (UCL)1 1

p

pp

k + −2 x k x x ( )+ −1

m

mx k x

mx

−+ − +

1

11

Center Line (CL)1

px

m

mx

−1

Lower Control Limit (LCL)1 1

p

pp

k − −2 x k x x ( )− −1

m

mx k x

mx

−− − +

1

11

Alternate Calculations for Adverse Event g Control Charts (n = 1)(Appropriate for the Number-Until Type Scenarios, a = 1)

Table 3

number-between g-type statistical quality control charts ...benneyan/papers/g_chartsb.pdf ·...

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