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Control Charts: Theory and Use
Heather
Disclosures
I have no conflicts of interest to disclose or resolve
Objectives
Examine the “anatomy,” structure and statistical basis of a control chart
Review the basic “types” of control charts.
Using examples, apply the rules for detecting special cause variation in control charts
Control Charts
The Shewhart chart (a.k.a. control chart) is a statistical tool used to distinguish between common cause and special cause variation
Chart Title
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Run Order
Mea
sure Center Line
Upper Limit
Lower Limit
Provost, LP and Murray S. The Health Care Data Guide. 2011
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Measure
A control chart is a run chart with some differences
Run chart: Center line is the median.
Control chart: Center line is often the mean.
Add control limits that reflect variability in data or the extent of common cause variation KEY
Mean
Upper Control Limit (UCL)
Lower Control Limit (LCL)
From Run Charts to Control ChartsVa
lue
Time
±2
SD
95.
4%±
3 S
D 9
9.7%
Relationship to Probability Theory
± 2 SD 95.4%
± 3 SD 99.7%
Constructing a Control Chart
Underlying data distribution dictates population parameters. Parameters dictate:
• Measure of central tendency (the “centerline”)• Measure of variability standard deviation values for
the upper and lower control limits.
Underlying distribution depends on type of data being observed (e.g., normal/Gaussian, Poisson, binomial, geometric)
Need to know what type of data you have to construct the proper type of control chart!
Continuous Data1. Numerical value for each unit in a group
Discrete (Integer) Data2. Classification: Presence or not of an attribute3. Count: How many attributes occur in sample
Type of data
Sample Size
Type of Chart
Math(software)
Constructing Control Charts
Types of Data & Control Charts
Healthcare Systems Engineering Institute
Common cause probability model Example
Disc
rete
Classification: Binomial
Parameter: p
Patient develops an SSI (Y/N)
Count: Poisson
Parameter: λ
Number of catheter-associated HAIs
Cont
inuo
us Normal
Parameters: m, s
Time to deliver thrombolytics
Type of data
Sample Size
Type of Chart
Math(software)
Constructing Control Charts
Single Observation
Multiple Observations• Equal Sample Size or Area of Opportunity• Unequal Sample Size or Area of Opportunity
Which Control Chart To Use
Type of Data
Discrete / Attribute(data is counted or classified)
Continuous / Variable(data is measured on a scale)
Count(events/errors are counted; numerator can be greater
than denominator)
Classification(each item is classified; numerator cannot be
greater than denominator)
Equal or fixed area of
opportunity
Unequal or variable area of opportunity
Equal or unequal
subgroup size
Subgroup size = 1(each subgroup is single observation)
Subgroup size > 1(each subgroup has multiple observations)
C chartCount of events
U chartEvents per unit
P chartPercent classified
X and MR chartsIndividual measures and
moving range
X-bar and S chartsAverage and standard
deviation
Gupta M and Kaplan HC, Clinics in Perinatology, 2017.
Quiz: Determine the Right Chart
Type of Data
Discrete / Attribute(data is counted or classified)
Continuous / Variable(data is measured on a scale)
Count(events/errors are counted; numerator can be greater
than denominator)
Classification(each item is classified; numerator cannot be
greater than denominator)
Equal or fixed area of
opportunity
Unequal or variable area of opportunity
Equal or unequal
subgroup size
Subgroup size = 1(each subgroup is single observation)
Subgroup size > 1(each subgroup has multiple observations)
C chartCount of events
U chartEvents per unit
P chartPercent classified
X and MR chartsIndividual measures and
moving range
X-bar and S chartsAverage and standard
deviation
A surgical service tracks a sample of 10 patients each week and records whether or not each patient received antibiotics on time.
Gupta M and Kaplan HC, Clinics in Perinatology, 2017.
Control Charts for Discrete Data (1)
Classification data
• P chart: Percent of observations with a given attribute• Measure of variability comes from binomial distribution
Centerline = p-bar = Average of the Statistic
UCL = CL + 3 σs
LCL = CL - 3 σs
100×=∑ ndp
( )in
pppUCL −+=
1003
( )in
pppLCL −−=
1003
σs from the binomial
distribution
Provost, LP and Murray S. The Health Care Data Guide. 2011 Slide courtesy of Terri Byczkowski, PhD, CCHMC
P-chart Calculations
Example: Late-Onset Sepsis
Performance Metric: Percent of infants discharged with Late-Onset Nosocomial Sepsis
Subgroup: variable number of infants discharged in a given month
MonthInfants with
Late InfectionPatients
Discharged4/1/2006 10 615/1/2006 13 816/1/2006 19 947/1/2006 20 788/1/2006 7 779/1/2006 18 77
10/1/2006 16 8411/1/2006 12 8312/1/2006 15 76
1/1/2007 17 902/1/2007 16 733/1/2007 16 1004/1/2007 13 755/1/2007 16 996/1/2007 12 887/1/2007 22 1058/1/2007 16 919/1/2007 19 93
Example: Late-Onset Sepsis
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/1/0
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/1/0
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9
Month
Y Axis: Proportion of Infants D/C with Late-Onset
Infection
Centerline: Average Proportion of Infants with Late-Onset Infection (over 45 months)
Control Limits
P-chart: % of VLBW infants with Late-Onset Infection
Quiz: Determine the Right Chart
Type of Data
Discrete / Attribute(data is counted or classified)
Continuous / Variable(data is measured on a scale)
Count(events/errors are counted; numerator can be greater
than denominator)
Classification(each item is classified; numerator cannot be
greater than denominator)
Equal or fixed area of
opportunity
Unequal or variable area of opportunity
Equal or unequal
subgroup size
Subgroup size = 1(each subgroup is single observation)
Subgroup size > 1(each subgroup has multiple observations)
C chartCount of events
U chartEvents per unit
P chartPercent classified
X and MR chartsIndividual measures and
moving range
X-bar and S chartsAverage and standard
deviation
The NICU is tracking the number of unplanned extubations each month as compared to total ventilator days.
Gupta M and Kaplan HC, Clinics in Perinatology, 2017.
Control Charts for Discrete Data (2)
Count Data• “C” Chart (as in count), plots the raw number of instances• “U” Chart (as in unit) plots the number of instances per
opportunities to observe
• Measure of variability comes from Poisson distribution
Example: Catheter-Associated InfectionsMetric: Catheter-Associated Infection Rate
Data obtained from infection control as reported to CDC. Each day, number of catheters is counted. This is used to obtain catheter days each month. Number of infections (catheter-associated) occurring each month is also reported.
Subgroup: Monthly
Unit Count: number of infections per opportunity (catheter day)
Month # InfectionsCatheter
Days10/1/2008 8 221211/1/2008 15 306412/1/2008 15 30071/1/2009 6 27832/1/2009 14 24993/1/2009 4 26924/1/2009 8 27845/1/2009 14 27726/1/2009 9 26907/1/2009 10 31458/1/2009 16 31719/1/2009 12 3209
10/1/2009 11 307611/1/2009 17 274912/1/2009 7 2759
Example: Catheter-Associated InfectionsY Axis: CA-Infection Rate
per 1000 line days
Centerline: Average CA-Infection rate (over 24 months)
Control Limits
U-Chart
Quiz: Determine the Right Chart
Type of Data
Discrete / Attribute(data is counted or classified)
Continuous / Variable(data is measured on a scale)
Count(events/errors are counted; numerator can be greater
than denominator)
Classification(each item is classified; numerator cannot be
greater than denominator)
Equal or fixed area of
opportunity
Unequal or variable area of opportunity
Equal or unequal
subgroup size
Subgroup size = 1(each subgroup is single observation)
Subgroup size > 1(each subgroup has multiple observations)
C chartCount of events
U chartEvents per unit
P chartPercent classified
X and MR chartsIndividual measures and
moving range
X-bar and S chartsAverage and standard
deviation
A call center samples 20 calls each day & records the average amount of time it takes for the scheduler to handle the call.
Gupta M and Kaplan HC, Clinics in Perinatology, 2017.
Control Charts for Continuous Data
Two charts Charts of Value or Sample Mean
• “X”Chart plots an individual value • “Xbar” Chart plots the sample average
Charts of Variation-S (average & standard deviation)• “MR” chart plots the moving range of the individual values• “S” chart plots the standard deviation of the sample
Measure of variability comes from normal (Gaussian) distribution
Example: C-section Decision to IncisionPerformance Metric: Time to incision following decision
for emergent c-section
What it means operationally: Minutes between decision to do c-section and time of incision
How it is observed: 10 charts sampled per week
Subgroup: The 10 charts sampled each week
Summary stats for the subgroups:
X-bar: The average decision to incision time for the 10 charts sampled each week
S: The standard deviation of decision to incision times for the 10 charts sampled each week.
What we want to see: process behavior over 30 weeks
WeekDecision to
Incision Time (minutes)
X-bar Standard Deviation
404536495035345039474952364950423850445044523649464136524351
1
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3
42.5 6.4
46 5.6
45 6.1
Example from Benneyan, Int J Six Sigma and Competitive Advantage, 2008
C-section Incision: X-Bar & S Charts
S chart looks at the variation withinsubgroups. High variation within subgroups it makes it difficult to interpret variation between subgroups.
X-bar chart looks at the variation betweensubgroups.
Example from Benneyan, Int J Six Sigma and Competitive Advantage, 2008
Why two charts (Xbar & S)?
2 types of possible process changes (unnatural variation)
Mean or standard deviation
Either can change without the other
One chart to detect each type of change
Change in mean
Change in SD
How to Interpret a Control Chart
Goal to identify common or special cause variation and take appropriate action
Probability-based rules
Rules designed to balance Type I (alpha error, p<0.05) and Type II errors
Rules for Identifying Special Cause
Rules for Identifying Special Cause
Gupta M and Kaplan HC, Clinics in Perinatology, 2017.
Quiz: Interpretation
Points outside control limits?
Runs of 8 or more consecutive points on one side of the centerline?
Trends of 6 or more consecutive points increasing or decreasing?
Two of three consecutive points near the outer control limits?
Yes
Yes
No
Benneyan JC, et al. Qual Saf Health Care. 2003;12:458-464.
LCL
UCL
Yes
Quiz: Interpretation
This process appears to be in control, i.e. no special cause variation, only common cause variation.
Points outside control limits?
Runs of 8 or more consecutive points on one side of the centerline?
Trends of 6 or more consecutive points increasing or decreasing?
Two of three consecutive points near the outer control limits?
No
No
No
Benneyan JC, et al. Qual Saf Health Care. 2003;12:458-464.
No
Stable Process Process defined and predictable Range of variation (performance)
intrinsic to the process• Common Cause variation:
sampling error, noise; no signals
Changing results from a stable process requires a new process… current process designed to get what it gets
noise
Unstable Process Process not defined, unpredictable Range of variation not intrinsic–
influenced by external factors• Special cause: outside chance
variation – signals Changing results achieved by an unstable
process begins with removing the special causes to establish core process Learning from experience with unstable
processes is limited… all one can say is that no one knows what will happen next! signal
The Goal: Standardize then Improve
Standard process
Improved process
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1 2 3Smal
ler i
s bet
ter
Unstableprocess
c/o J. Benneyan
Why Not Just Use a Run Chart?
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Appointment AccessNumber pts appt < 30 days of request
Pure chaos!No standard process at all
Why Not Just Use a Run Chart?
Why Control Charts Over Run Charts?
Allow you to distinguish between common cause and special cause variation
More sensitive / more powerful in detecting changes
Estimate capability of a stable process more accurately predict performance
But… more difficult to generate
Take Home Points1. Control charts allow you to distinguish common cause and
special cause variation.
2. Key to control chart use is statistically-derived control limits to assess variation.
3. The type of data determines the type of control chart.
4. Probability-based rules should be used to identify special cause variation and interpret control charts.
5. Control charts are generally more powerful than run charts if you have enough data, but run charts can still be very useful.
References Benneyan, J.C., R.C. Lloyd, and P.E. Plsek, Statistical process control as a tool
for research and healthcare improvement. Qual Saf Health Care, 2003. 12(6): p. 458-64.
Benneyan, J.C., The design, selection, and performance of statistical control charts for healthcare process improvement. Int J Six Sigma and Competitive Advantage, 2008. 4(3):p.209-239.
Gupta, M, and H Kaplan, Using Statistical Process Control to Drive Improvement in Neonatal Care: A Practical Introduction to Control Charts. Clinics in Perinatology, 2017. 44:627-644.
Provost, L.P. and S.K. Murray, The Health Care Data Guide: Learning From Data for Improvement. 1st ed. 2011, San Francisco, CA: Jossey-Bass. 445 p.