risk-adjusted control charts for non- homogeneous dichotomous events qprc annual conference may 26,...

Risk-adjusted Control Charts for Non-Risk-adjusted Control Charts for Non-homogeneous Dichotomous Eventshomogeneous Dichotomous Events

QPRC Annual ConferenceQPRC Annual Conference

May 26, 2005May 26, 2005

James Benneyan, Ph.D.Quality and Productivity Lab, Director

Senior Fellow, Institute for Healthcare Improvement334 Snell Engineering Center

Northeastern University, Boston MA

www.coe.neu.edu/Research/[email protected]

http://www.coe.neu.edu/Research/QPL

http://www.coe.neu.edu/Research/QPL

www.coe.neu.edu/Research/QPL

Outline

1. Preliminary comments on healthcare SPC

2. Non-homogeneity- Types of applications- Non-homogeneous probability model- Variance inequality proof & other results

3. Risk-adjusted control charts- Shewhart, standardized, EWMA- k-sigma & probability limits- Examples (Infection control, mortality, reliability composite measures,

pain management, hand-washing compliance, others)- SPRT-Cusum

4. Performance & comparisons


Typical healthcare applications of SPC

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep Oct

Nov

Dec Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep Oct

Nov

Dec Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Month

Per

cent of Pat

ients

Rep

lyin

g "

Ver

y Sat

isfied

" or

"Exc

elle

nt"

Multiple changes tested

Subgroup Number

Mor

taliti

es /

1000

Disc

harg

es

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Surgical Site Infections

Subgroup (Month) Number

Avera

ge T

ime (

Min

s)

An

tib

ioti

c is

Ad

min

iste

red

Befo

re 1

st

Incis

ion

-200

-100

0

100

200

300

4/9

3

5/9

3

6/9

3

7/9

3

8/9

3

9/9

3

10/9

3

11/9

3

12/9

3

1/9

4

2/9

4

3/9

4

4/9

4

5/9

4

6/9

4

7/9

4

8/9

4

9/9

4

10/9

4

11/9

4

12/9

4

1/9

5

2/9

5

3/9

5

4/9

5

5/9

5

6/9

5

7/9

5

8/9

5

9/9

5

UCL

CL

LCL

Trial X-bar Control ChartPerioperative Antibiotic Timing

X-bar Chart

Patient Satisfaction

Fall Rate

0

0.5

1

1.5

2

2.5

3

3.5

11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9

Fiscal Period

Fal

ls/1

000

pat

ien

t d

ays

Falls and Slips


Public health applications

Scotland Surveillance for Regional MRSA Regional Respiratory Illness (USAF)

Infectious Disease SurveillanceBinary Cusum

Fraction Reporting Used Opioids in Last 30 DaysStandardized p EWMA control chart: Region 5

-1.25

-1.00-.75

-.50

-.25

.00

.25

.50

.75

1.001.25

1.50

24 34 44 54 64 74 84 94

Week

EW

MA

Sta

tist

ic


0

25

50

75

100

125

150

175

5 10 15 20 25 30 35 40 45

Procedure Number

EW

MA

of

Pro

ced

ure

s B

etw

een

C

om

pli

cati

on

s

EWMA: Exponentially weighted moving average

SPC 102: 80/20 rule… (95/5?)

Quarter

Adv

erse

Eve

nt R

ate

UCL = ???

CL = ???

LCL = ???

START-UP SPC METHODSAdverse Event Rate

RARE EVENTSEWMA g chart of Cases Between Complications

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

z

LCL

CL

UCL

"Incorrect" z

RISK-ADJUSTED SPC

Standardized jb EWMA Chart Incorrect

Correct

www.coe.neu.edu/Research/QPLSample

Tsquare

d

7654321

5

4

3

2

1

0

Median=4.652

UCL=5.143

LCL=0.555

181 - Tsquared Chart of C1, ..., C5

Patient monitoring and control

Observation

Indiv

idual V

alu

e

87654321

10

8

6

4

2

0

_X=4.13

UCL=9.44

LCL=-1.19

Pain

ObservationIn

div

idual V

alu

e

87654321

10

8

6

4

2

0

-2

_X=3.38

UCL=9.07

LCL=-2.32

Fatigue

Observation

Indiv

idual V

alu

e

87654321

7

6

5

4

3

2

_X=4.125

UCL=6.405

LCL=1.845

Nausea

Observation

Indiv

idual V

alu

e

87654321

8

6

4

2

0

-2

-4

_X=2.63

UCL=7.94

LCL=-2.69

Sleep

MCEWMA Chart for Systolic Blood Pressure Data

95

105

115

125

135

145

0 50 100 150 200 250

Pain Fatigue Nausea Sleep Distress

Pain 1.00

Fatigue 0.75 1.00

Nausea 0.60 0.82 1.00

Sleep 0.50 0.56 0.52 1.00

Distress 0.76 0.80 0.77 0.60 1.00

Bounded Adjustment Chart with IMA(1,1) Data

-4

0

4

8

12

0 100 200 300 400 500

z(t)

e(t)

EWMA

+L

-L

Anticoagulation Feedback Adjustment Example


Input Populations

Inputs Risk AdjustmentMethods

(Log-Regression)(others)

X1 ~ Binomial(n1,p1)

X2 ~ Binomial(n2,p2)

Xk~ Binomial(nk,pk)

•

•

•

•

•

(NB: ni = 1 often)

Aggregate Model

T = X1 + X2 + ... + Xk

E(T) nipii1

k

V(T) ni(1 pi)pii1

k

GT (s) (1 pspi)ni

i1

k

j

ij,i nN j

ij,i xT iii /NTF

Fractionx1,i n1,i x2,i n2,i x3,i n3,i x4,i n4,i Ti Ni Fi

1 1 1 1 1 0 3 0 0 2 5 0.400002 0 6 0 14 1 5 0 2 1 27 0.037043 1 11 0 4 0 0 0 1 1 16 0.062504 0 1 0 0 0 4 1 1 1 6 0.166675 1 11 1 9 1 2 0 0 3 22 0.136366 0 1 1 12 1 7 1 5 3 25 0.120007 1 2 0 3 1 1 0 1 2 7 0.28571. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .

m x1,m n1,m x2,m n2,m x3,m n3,m x4,m n4,m Tm Nm Fm

Totals

Week

Process 1 (j=1) Process 2 (j=2) Process 3 (j=3) Process 4 (j=4)

J Non-Homogeneous Sub-Populations, ni ≥ 1 i

i weekin SSI of number Expected E ii weekin SSI of number of deviation Standard S i

1 0.10732 0.06603 0.0349

. . . . . .

. . . . . .

. . . . . .1 0.10732 0.02043 0.05134 0.09805 0.0955

1

m

Risk-Adjusted SSI LikelihoodWeek Patient

Actual No. SSI Ei Si

0.2082 0.1911

0.3725 0.3392

3

4

Unique likelihoods, ni = 1 i

Two general types of non-homogeneous data


Healthcare examples of this type of data

SSI Stratified by NNIS Category

Time Period

x1,t n1,t … xJ,t nJ,t

1 37 49 18 222 22 32 6 113 37 43 15 194 41 43 16 195 46 76 13 206 55 72 19 227 32 39 19 338 48 53 20 219 35 37 29 3710 56 68 12 2111 57 58 19 2412 18 22 1 213 6 11 5 514 15 19 6 815 16 19 8 816 13 20 3 417 19 22 3 418 19 33 12 1419 20 21 19 1920 29 37 15 1521 12 21 6 622 19 24 19 19l

l

l

T 57 58 19 24

1Hospital / Provider

J

Hand-washing Compliance

Subgroup n1,t x1,t n2,t x2,t n3,t x3,t n4,t x4,t nt xt

1 41 2 38 2 17 5 1 0 97 92 60 5 34 0 17 6 1 1 112 123 50 1 46 3 15 5 2 2 113 114 47 1 32 4 20 4 4 2 103 115 48 0 36 5 10 2 1 1 95 86 36 0 41 4 12 2 2 1 91 77 64 0 25 0 10 4 3 0 102 48 44 1 33 2 20 3 3 1 100 79 45 4 32 2 11 3 4 0 92 9

10 57 2 24 1 17 3 3 2 101 811 52 1 28 4 15 4 4 3 99 1212 54 0 32 1 16 5 0 0 102 613 38 2 32 2 8 1 0 0 78 514 25 3 16 3 9 3 0 0 50 915 20 1 19 6 5 2 0 0 44 916 19 0 18 1 7 3 0 0 44 417 2 0 4 0 4 2 0 0 10 2

Total 702 23 490 40 213 57 28 13 1433 133p1 0.0328 p2 0.0816 p3 0.2676 p4 0.4643 p 0.8463

TotalCategory 1 Category 2 Category 3 Category 4Week

Note: Sample sizes change week-to-weekNote: Sample sizes change week-to-week


Industry examples of this type of data

Time Period

Sample Size

Number Defects

Sample Size

Number Defects

…Sample

SizeNumber Defects

Total sample

Total Defects

1 10 3 15 3 25 6

2 12 2 23 2 35 4

3

:

etc

Process / Product J TotalProcess / Product 1 Process / Product 2

Other applications

• Manufacturing defects of different lines, products, etc.

• Automobile accidents across different driver types

• Number on-time shipments for different vendors

• Airline no-shows for different types of passengers

• k of n system reliability

• Basketball team’s successful free throws


Let: T = X1 + X2 + … + XJ and F = T/N, where Xj ~ binomial(nj,pj)

All (Xi, Xj) independent but not identically distributed

E(Xj) = njpj, V(Xj) = njpj(1 - pj), GXj(s) = (1 – pj + spj)nj

Then:

(MGF, skewness, kurtosis: See paper)

Note: GT(s) is not of binomial form (except for trivial cases J = 1 or pi = pj (i, j)

E(T) has binomial form but V(T) does not have binomial form (see below)

Binomial Approximation

Assume T’ binomial(N,P), with

Then:

But:

In fact V(T) < V(T’) (see below)

J-binomial random variable

J

i iiJJ pnpnpnpnTE12211 ... )(

J

i iiiJJJ ppnppnppnppnTV1222111 )1( )1(...)1()1( )(

J

i

nii

nJJ

nnT

iJ sppsppsppsppsG12211 )1( )1(...)1()1( )( 21

211

2

11)1( /)( )(, )/( )(

J

i i

J

i iii

J

i i

J

i ii nppnNTVFVnpnNTEFE

iJiii

Jiii

Jii

Ji npnNnpPnN 1111 and approx

~

)()'(1111

TEpnnpnnNPTEJ

i ii

J

i i

J

i ii

J

i i

)()1(1)1()'(111111

TVppnnpnnpnnPNPTVJ

i iii

J

i i

J

i ii

J

i i

J

i ii

J

i i


Mixed-risk probability modelJ-binomial probability distribution3

nconvolutio recursive , )()(0 1

11

1

1

t

x

J

xii

ii xtXPxXPtTP

t

x

xt

x

xt

xkk

xxt

x

k

i i

k

xXPxXPxXPxXP0 0 00

332211

1

1

2

121

3

)...( ... ...)()()(

.........)()( ... )...( ... 1

10

110

2

1

1

1

Ji iJ

xt

xJJ

xt

xkk xtXPxXPxXP

J

i i

J

k

i i

k

yrecursivel ,

)1(

)1(1

0 , 11where

1

i

i

i

iii x

k i

ini

in

i-xn

ix

ii

iii

kp

knp)-p(

x)-p( )-p(p

x

n)x P(X

P(F = f) = P(T = t), where f = t/N

cases lbut trivia all in eintractabl very although ,)()(or 0

s

Tt

t

sGds

dtTP MLE) (MOM, ˆ

1 ,1 ,

m

j ji

m

j jii nxp


0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t

P(T

=t)

0.00

0.05

0.10

0.15

0.20

0.25

J-BINOMIAL

BINOMIAL

NORMAL-BIN

NORMAL-JB

TAD ValuesJ-Bin vs Bin: 0.1931J-Bin vs J-Bin Norm al: 0.0074

n0=2 p0=0.01n1=2 p1=0.1n2=2 p2=0.2n3=3 p3=0.3n4=3 p4=0.4n5=3 p5=0.5n6=3 p6=0.6n7=3 p7=0.7n8=3 p8=0.8n9=3 p9=0.9

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

t

P(T

=t)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40J-BINOMIAL

BINOMIAL

NORMAL-BIN

NORMAL-JB


n0=10 p0=0.99n1=100 p1=0.01

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

t

P(T

=t)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

J-BINOMIAL

BINOMIAL

NORMAL-BIN

NORMAL-JB


n0=10 p0=0.99n1=10 p1=0.01

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t

P(T

=t)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20J-BINOMIAL

BINOMIAL

NORMAL-BIN

NORMAL-JB


ni=2 pi=0.5(i=0,1,…10)

Why Does/Might This Matter?

Note: JB ≤ B

J

j jjjN

J

j jjN ppnTVpnTE1

2

1)1()(,)( :ionapproximat Normal


Shewhart Control Charts(same examples)

Example4, Shewhart

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

p

p

UCL-bin

UCL

CL

LCL

LCL-bin

Example1, Shewhart

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

p

p

UCL-bin

UCL

CL

LCL

LCL-bin

Error

Error

Example2, Shewhart

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

p

p

UCL-bin

UCL

CL

LCL

LCL-bin

Error

Error

Example3, Shewhart

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

p

p

UCL-bin

UCL

CL

LCL

LCL-bin

Error

Error

3 standard deviation control limits, Lower power (1 – ), lower false alarm rate (),


Comparison of VariancesJ-binomial vs. Binomial

Example 1

J = 2 n1 = 10 p1 = .99

n2 = 10 p2 = .01

n = 10, sn = 0, p = .50, sp = .693

V(T) =

V(T’) = NP(1-P) = 5.0

V(T)/V(T’) = .0396

KL =

TAD = 1.3001

198.)1(1

J

i iii ppn

Example 3

J = 10 n1 = 2 p1 = .01 n6 = 3 p6 = .5

n2 = 2 p2 = .1 p7 = 3 p7 = .6

n3 = 2 p3 = .2 p8 = 3 p8 = .7

n4 = 3 p4 = .3 p9 = 3 p9 = .8

n5 = 3 p5 = .4 p10 = 3 p10 = .9

n = 2.7, sn = .483, p = .451, sp = .301

V(T) = 4.720 V(T’) = 6.797

V(T)/V(T’) = .700 KL = TAD = .1931

Example 4 (binomial)

J = 10 n1 = 2 p1 = .5 n6 = 2 p6 = .5

n2 = 2 p2 = .5 p7 = 2 p7 = .5

n3 = 2 p3 = .5 p8 = 2 p8 = .5

n4 = 2 p4 = .5 p9 = 2 p9 = .5

n5 = 2 p5 = .5 p10 = 2 p10 = .5

n = 2, sn = 0, p = .5, sp = 0

V(T) = 5 V(T’) = 5

V(T)/V(T’) = 1.000 KL = TAD = 0.000

Example 2

J = 2 n1 = 10 p1 = .99

n2 = 100 p2 = .01

n = 55, sn = 63.64, p = .50, sp = .693

V(T) = 1.089

V(T’) = 9.820

V(T)/V(T’) = .111

KL =

TAD = 1.0485


Let: T ~ J-binomial(p1, p2, …, pJ, n1, n2, …, nJ) with pi ≠ pk for at least some (i, k)

T’~ binomial(N,P) with

Assume: p1 ≤ p2 ≤ … ≤ pJ without loss of generality such that p1 = p(1) (min), …, pJ = p(J) (max)

pi ≠ pk for at least one (i, k)

Define: 1 ≥ 2 ≥ … ≥ J-1 such that i = pJ - pi

i.e., p1 = pJ – 1, p2 = pJ – 2, …, pJ-1 = pJ – J-1

Then: since i k k i and i > k for at least one (i, k)

After some algebra it can be shown that

and then after additional work that

V(T) < V(T’)

(Also: Skewness of T < skewness of T’)

Under-dispersion: V(T) ≤ V(T’)Outline of Proof3

iJiii

Jii

Ji npnPnN 111 and

0)( 2211 kikiik

Ji nn

211

221 ii

Jii

Jiii

Ji pnnpn

J

i i

J

i ji

J

i i

J

i jiJ

i ijj

J

i in

pn

n

pnnPNPppn

1

1

1

111

1)1()1(


Statistical jb Control Charts2,4 k-sigma Shewhart Limits

LCLt

UCLt

Shewhart np-type jb chart

CLt

Plotted Statistict

Shewhart p-type jb chart

where xj,t ~ binomial(nj,t,pj)

nj,t = sub-subgroup size of sub-population j at time t

xj,t = number of adverse events in sub-population j at time t

J = number of sub-populations, t = time period

J

j jtjt pnTE1 , )(

J

j jjtj

J

j jtj ppnkpn1 ,1 , )1(

J

j tjt xT1 ,

J

j tj

J

j jtjttt npnNTEFE1 ,1 , )/( )(

21 ,1 ,1 ,1 , )1(

J

j tj

J

j jjtj

J

j tj

J

j jtj nppnknpn

21 ,1 ,1 ,1 , )1(

J

j tj

J

j jjtj

J

j tj

J

j jtj nppnknpn

J

j tj

J

j tjt nxF1 ,1 ,

J

j jjtj

J

j itj ppnkpn1 ,1 , )1(


Probability jb Limits

CLi

UCLi

np-type jb chart

LCLi

Plotted Statistict

p-type jb chart

Note: CL (mean and median) of all non-standardized charts (including p-type) vary longitudinally unless all ni’s remain constant

See papers2,4 for standardized cases and examples

Find limits numerically, possibly starting with LCLT’, UCLT’, and CLT’ approximations

LCLi tTPTtLCL

)(argmax,

)(

)()(

5.0,5.0,

1 ,

2

1medianTT

meanpnTE

ii

J

j jjii

J

j jtt xT1 ,

J

j jt

J

j jtt nxF1 ,1 ,

UCLi tTPTtUCL

)(argmin,

J

j jii nTLCL 1 ,,

J

j jii nTUCL 1 ,,

)(

)()(

1 ,5.0,5.0,

1 ,1 ,

2

1mediannTT

meannpnFE

J

j jiii

J

j ji

J

j jjii


J-Binomial UCL:Standard UCL:

J-Binomial CL:Standard CL:

J-Binomial LCL:Standard LCL:

iti

t iti

t iti

t iti

t iti

t iti

t iti

n

n

x

n

x

n

x

,

,

,

,

,

,

,

1

3

t iti

t iti

n

x

,

,

iti

t iti

t iti

t iti

t iti

t iti

t iti

n

n

x

n

x

n

x

,

,

,

,

,

,

,

1

3

iti

it

ti

tti

tti

tti

ti

iti

it

ti

tti

ti

n

n

x

n

xn

n

n

xn

,

,

,

,

,

,

,

,

,

,1

3

iti

it

ti

tti

ti

n

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xn

,

,

,

,

iti

it

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tti

tti

ti

iti

it

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ti

n

n

x

n

xn

n

n

xn

,

,

,

,

,

,

,

,

,

,1

3

Parameters Estimated jb Chartsnp-type jb Shewhart case

periods timeofnumber where, ...,2,1, ˆ :binomial

ˆ :Binomial

1 ,1 ,

1 1 ,1 1 ,1 ,1 ,

mJjnxpJ

nxNtp

m

i ji

m

i jij

m

i

J

j ji

m

i

J

j ji

m

i ji

m

i ji


-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

26 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 44

Week

Zt

Zt

Zt bin

UCL

CL

LCL

-1.5

-1

-0.5

0

0.5

1

1.5

2

Week

SS

I Ra

te

p-fract

UCL-bin

UCL

CL

CL-bin

LCL

LCL-bin

Surgical Site Infection Examples: Moving CL (Non-constant mean)

Russian SSI’s (j: NNIS scores)

0

0.05

0.1

0.15

0.2

0.25

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Subgroup

Fra

cti

on

No

nc

on

form

ing

p

LCL

CL

UCL

Std LCL

Std CL

Std UCL

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

z

LCL

CL

UCL

"Incorrect" z

Georgia Hospital SSI’s (j: hospitals)

Not standardized Not standardized

Standardized Standardized

Incorrect

Correct


Standardized jbin Charts

LCLt

UCLt

Standardized Shewhart*

CLt

Plotted Statistict

Standardized EWMA *

where E(Z) = 0, V(Z) = 1, f(z) unknown (j-binomial)

= EWMA weight

* Standardized charts are the same for fraction (F) or total (T)

tt kLCL 2)1(1)2(

tt kUCL 2)1(1)2(

0tCL

12

1 ,1 ,

1 ,1 ,)1(

)1(

tJ

j tj

J

j jjtj

J

j tj

J

j jtjt

t ynppn

npnFy

3 kLCL t

3kUCLt

0tCL

2

1 ,1 ,

1 ,1 ,

)1(

J

j tj

J

j jjtj

J

j tj

J

j jtjt

t

nppn

npnFz

2

1 ,1 ,

1 ,1 ,

1 ,

1 ,

)1(

)(

)(

)1(

)(

)(

J

j tj

J

j jjtj

J

j tj

J

j jtjt

t

tt

J

j jjtj

J

j jtjt

t

ttt

nppn

npnF

FV

FEF

ppn

pnT

TV

TETZ


Standardized EWMA Charts(Same Examples)

Example2, Standardized EWMA

-1.5

-1

-0.5

0

0.5

1

1.5

Zt

Zt

Zt bin

UCL

CL

LCL


-1.5

-1

-0.5

0

0.5

1

1.5

Zt

Zt

Zt bin

UCL

CL

LCL


-1.5

-1

-0.5

0

0.5

1

1.5

Zt

Zt

Zt bin

UCL

CL

LCL


-1.5

-1

-0.5

0

0.5

1

1.5

Zt

Zt

Zt bin

UCL

CL

LCL

2

1 ,1 ,

1 ,1 ,

)1(

J

i ti

J

i iiti

J

i ti

J

i itit

t

nppn

npnFz

Note: Difference now is in plotted points (limits are the same)


Interpretation errorsOther applications

Pain Management Bundle

Percent Patients with Pain Assessment and Plan of Care

-5

-4

-3

-2

-1

0

1

2

3

4

Jun-00

Jul-00

Aug-00

Sep-00

Oct-00

Nov-00

Dec-00

Jan-01

z*

z*bin

UCL

CL

LCL

Percent Patients with Pain Assessment and Plan of Care

-1.5

-1

-0.5

0

0.5

1

1.5

Jun

-00

Jul-0

0

Au

g-0

0

Se

p-0

0

Oct-0

0

No

v-00

De

c-00

Jan

-01

Zt

Zt bin

LCL

CL

UCL

Handwashing Compliance forj = 6 MD’s)

Standardized

Standardized-6

-5

-4

-3

-2

-1

0

1

2

3

4

11

/27

/20

00

12

/20

/20

00

12

/26

/20

00

12

/28

/20

00

1/3

/20

01

1/1

0/2

00

1

1/2

0/2

00

1

2/2

0/2

00

1

3/2

1/2

00

1

4/1

9/2

00

1

5/3

1/2

00

1

6/3

0/2

00

1

7/3

0/2

00

1

z*

z*bin

UCL

CL

LCL

-2

-1.5

-1

-0.5

0

0.5

1

1.5

11

/27

/20

00

12

/20

/20

00

12

/26

/20

00

12

/28

/20

00

1/3

/20

01

1/1

0/2

00

1

1/2

0/2

00

1

2/2

0/2

00

1

3/2

1/2

00

1

4/1

9/2

00

1

5/3

1/2

00

1

6/3

0/2

00

1

7/3

0/2

00

1

Zt

Zt bin

LCL

CL

UCL

Standardized

Standardized


60

70

80

90

100

110

Subgroup Number

Incorrect limits Correct limits

CL

105.93

98.18

71.88

64.13

Error

Error

n2 = 103p2 = .06

k = 2n1 = 95p1 = .83

Example


.00

.20

.40

.60

.80

1.00

.45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95

Shifted Value of p1

Pro

ba

bil

ity

of

Sig

na

l Correct limits Incorrect limits

Example of performance

Example

2 sub-populations (J = 2)

n1 = 95 p1 = .83

n2 = 103 p2 = .06

Correct variance = 77.87 (jb)

Incorrect variance = 48.51 (bin)

• Correct limits roughly 1/3 closer to CL

• Significantly faster change detection

• Average time-to-detect (pop 1):

Shift to Correct Conventional.75 10 points 300 points.70 2-3 points 20 points.65 1-2 points 5 points


Relative performanceAverage run lengths. More detailed results in appendix & paper.

1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p

Percent Change in p(i)

Ave

rage

Run

Len

gth

X1

X0

X1

X0

Avg n = 200, SD n = 14.1, Avg p = .5, SD p = .566, J = 2SR = 0.6005, T AD = 0.4850, KL = 0.0828

Risk-Adjusted Not Risk-Adjusted

1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

B X1B X0

JB X1

JB X0



1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X0X0

X1

X1

Avg n = 200, SD n = 141.4, Avg p = .0505, SD p = .07, J = 2SR = 0.9263, T AD = 0.03704, KL = 6.16E-4


1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X1X0

Avg n = 200, SD n = 141.4, Avg p = .5, SD p = .028, J = 2SR = 0.9994, T AD = 5.816E-4, KL = 1.57E-7



Factors affecting difference(Preliminary – more than just p)

25 factorial design: 5 main effects (J, n, sn, p, sp) and all interactions

Responses:

1. Kullback-Leibler information:

2. Total Absolute Deviation:

3. Standard Deviation Ratio:

Y = 0 + 1J + 2 n + 3ssn + 4 p + 5sp + 12J n + 13J sn + … + 123J snsp + …

J

i in

tBJB tTPtTPTAD

1

0

)()(

N

t B

JBJB tTP

tTPtTPKL

0 )(

)(log)('

J

i i

J

i ii

J

i i

J

i ii

J

i i

J

i iii

B

JB

npnnpnn

ppnSR

11111

12

22

1

)1(

Parameters Interactions Response

p J

Ave p (pbar)

Ave n (nbar)

n

p x J p x pbar

p x pbar x nbar

p x nbar

x J

p x pbar

n x nbar All others

SR 0 * * * * 0 * .046 * .001 *

TAD * * * * * 0 * 0 .008 * *

KL * * * * * 0 0 * .004 * *

* Not statistically significant (.05)


Other applications: Dr. Harold Shipman (GP)

• Britain’s most prolific serial killer

• Murdered 236 (215-289) elderly patients between 1971(75)-1998

by lethal injections of heroin (diamorphine)

• > 189 females, > 55 males

• Postulated related to mother’s death from cancer when Shipman was young

• National inquiry, media, analysis

• 15 life sentences, 1/31/00 Hangs himself in jail, 1/13/04


Risk-adjusted mortality rates

Male Standardized Risk-Adjusted Mortality Rate

-6

-3

0

3

6

9

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Year

Mo

rtal

ity

Rat

e (S

tan

dar

diz

ed)

LCL

UCL

CL

Female Standardized Risk-Adjusted Mortality Rate

-6

0

6

12

18

24

30

36

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Year

Mo

rta

lity

Ra

te (

Sta

nd

ard

ize

d)

LCL

UCL

CL

• Adjusted for patient acuity using comparison groups

• Standardized p charts

• Higher than expected mortality, increasing trends for both males and females

• Noticeable difference between male and female populations

• ARL = 1.7 years to detect this magnitude of rate difference (pre-trend, binary sprt)5


100K Lives Campaign^

• Save 100,000 lives by 6/14/06 via ‘bundles’ of proven interventions in six areas:

• Adverse drug events (ADE)• Surgical site infection (SSI)• Myocardial infarction (AMI)• Ventilator pneumonia (VAP)• Central line infection (CLI)• Rapid response teams

• More than 2,000 US hospitals participating

• Accomplish by consistent ‘bundle’ implementation using a reliability model

^ www.ihi.org/ihi/programs

Institute for Healthcare Improvement (IHI)

Background

• National leader of healthcare system improvement

• Safety, medical error, waits & delays, appointment access, specific clinical diagnoses, …

• Large-scale multi-hospital (15-70) problem area projects

• National network of leading hospitals, faculty, researchers

• Ties to Institute of Medicine, JCAHO, CMS, NAE, VA, BMJ…

• Modern Healthcare: 3rd most powerful person in healthcare


IHI growth & impact: 1986 - present

• Active in over 50 countries, internationally, and in developing countries

• Roughly 14,000 organizations involved

• Annual forum conference > 5,000 attendees worldwide

• > 200 nationally-recognized faculty, researchers, experts

1986

15,000

50


Examples of composite bundles^

AMI CHF CAP CABG TJ VAP

ASA (aspirin) within 24 hrs

BETA blocker within 24 hrs

Thrombolytics within 30 min

PCI within 120 minutes

ACE Inhibitor for LVSD

Smoking counseling

ASA at d/c

BETA blocker at discharge

LVEF assessed

ACE Inhibitor for LVSD

Detailed discharge instructions

Smoking cessation counseling

O2 assessed within 24 hours

Blood culture prior to abx

First dose of antibiotic within 4 hours

Influenza screening / vaccination

Pneumococ. vaccination

Smoking cessation counseling

First dose antibioitc 1 hour prior to first incision

Appropriate antibiotic selection

Discontinued antibiotics within 24 hours

Use internal mammary artery (IMA)

ASA at discharge

First dose antibioitc 1 hour prior to first incision

Appropriate antibiotic selection

Discontinued antibiotics within 24 hours

Head of bed elevation

DVT prophylaxis

Sedation vacation

PUD prophylaxis

Others for SSI, other disease diagnoses, rapid response teams, pain mgmt (see JCAHO, CMS, & IHI websites)

^ Center for Medicare & Medicaid Services (CMS), “Clinical conditions and measures for reporting and incentives” 2003; AHA; Leapfrog Group; National Quality Forum; AHQR; JCAHO Core Measures


Composite measures

Approach Explanation Example

Individual measures

% patients meeting that measure

- Fraction TJ patients receiving prophylactic antibiotic received one hour prior to surgical incision

CompositeTotal measures not met

Total # opportunities

(across all patients)

Total not met

3 measures x Number patients

“All-or nothing”

(Aggregate bundle)

Fraction patients meeting all measures (total bundle met,

‘perfect encounter’)

Fraction TJ patients with ALL 3:

- Prophylactic antibiotic 1 hour prior

- Prophylactic antibiotic selection for surgical patients

- Antibiotics discontinued within 24 hours of surgery end time

Front-line improvement and analysis

Ultimate goal

Used to measure

improvement


Composite example

Patient Measure 1 Measure 2 Measure 3 Measure 4 # Met All measures?

1 Yes Yes Yes Yes 4 Yes

2 Yes No Yes Yes 3 No

3 Yes Yes No No 2 No


5 Yes Yes Yes No 3 No

6 Yes No Yes No 2 No

7 No Yes Yes No 2 No


9 Yes No No Yes 2 No


Reliability 9/10 (.9) 7/10 (.70) 8/10 (.8) 6/10 (.6) 30/40 4/10

Composite measure = (# met) / (# opportunities) = (9+7+8+6) / (10x4) = .75


jb Composite Control Chart

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17 19 21 23 25

Week

Co

mp

osi

te S

core

Correct limits

Incorrect limits

Application to clinical evidence ‘bundles’

Individual measure Historical rate

1. Head of bed elevation 61%

2. DVT prophylaxis 75%

3. Sedation protocol 43%

4. PUD prophylaxis 88%

Ventilator Associated Pneumonia Bundle

Suppose tests 10% improvement in each measure

Week X1 X2 X3 X4

1 10 6 8 2 6 40 22 55%

2 10 4 9 5 10 40 28 70%

3 10 7 5 4 8 40 24 60%

4 10 5 7 4 9 40 25 62.5%

5 10 6 8 3 7 40 24 60%

etc etc etc etc etc etc etc etc etc

Number Patients

# Satisfying Measure Xi TotalOps

TotalMet

CompositeScore

Data Format

Undetectedimprovement

Undetected negative spike


Other composite reliability control charts

Columbia ADE Reliability(High hazard and regular risk populations (4) combined)

0

0.1

0.2

0.3

0.4

0.5

1 3 5 7 9 11 13 15 17 19 21 23

Correct

Incorrect

CAMC CAP jb Composite Reliability

0.70

0.80

0.90

1.00

12/2

7/04

1/3/

05

1/10

/05

1/17

/05

1/24

/05

1/31

/05

2/7/

05

2/14

/05

2/21

/05

2/28

/05

3/7/

05

3/14

/05

3/21

/05

3/28

/05

Correct LCL

Correct UCL

Baptist HF Composite Reliability

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Correct

Columbia ADE Reliability(High hazard and regular risk populations (4) combined)

-1.5

-1

-0.5

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19 21 23

Correct

Incorrect


• If at least k of the n measures are met for a patient, the likelihood of the adverse outcome significantly decreases

• Non-linear “bundle effect” (observation)

• Safety culture and human factors phenomenon

• “Tipping point” of patient-centered care

• All-or-nothing: k = n

• n measures are not i.i.d.

• Usual k of n calculation is not binomial

“Bundle effect” and k of n reliability

Pro

babili

ty N

o A

E

.

0 k n

0

1.0

"tipping point"


Bundles of bundles (aggregate bundle)

Bundle # Measures

VAP 4

AMI 7

CHF 6

SSI 4

Total 21

Week VAP CHF AMI SSI

1 10 6, 8, 2, 6 210

2 10 4, 9, 5, 10 210

3 10 7, 5, 4, 8 210

4 10 5, 7, 4, 9 210

5 10 6, 8, 3, 7 210

etc etc etc etc etc etc etc etc etc

# Pat-ients

TotalOps

TotalMet

CompositeScore

Data Format

• Two aggregation trends

• Meet all individual measures across all campaign bundles

• Meet at least k of the 6 all-or-nothing campaign bundles

# Satisfying Measure Xi


SPRT Cusums

Standard homogeneous Bernoulli SPRT-Cusum. Given xi = m of ni = n failures where Xi=1 w.p. p0 or p1 i.i.d. under H0 or H1, respectively,

^ Sequential probability ratio tests for non-homogeneous Bernoulli and binomial events”, working paper. So=0, bo=B, ao=A

App

pp

pXXP

pXXPB

mnm

mnm

n

n

1log

)1(

)1(log

)0|,...(

)|,...(log

1log

00

11

1

11

)1(

)1(log

1

1log

)1(

)1(log

1log

)1(

)1(log

1

1log

)1(

)1(log

1log

10

01

0

1

10

01

10

01

0

1

10

01

pp

pp

p

pn

pp

ppx

pp

pp

p

pn

pp

ppi

Scenario Example SPRT-Cusum

One item at a time

(J=1, n=1)

Operate on one patient at a time. Interpret after each individual case.

J Bernoulli items at a time

(J>1, nj=1, j=1,…J)

Operate on J patients per day. Interpret at end of each day.a. Know which T of J items failedb. Don’t know (true j-Bernoulli)

J binomial samples at a time

(J>1, nj>1, j=1,…J)

Operate on nj patients in each of J risk categories.a. Know Xi’s in sub-samplesb. Don’t know (true j-binomial)

0

1

110

01

10

1

1

00

00

11

1

1log

)1(

)1(log

1

1log

)1)(1(

)1)(1(log

i

ii

ii

iiii

i

ii

ii

iiii

p

pa

pp

ppXS

p

pb

Apxpx

pxpxB

1log

),|(

),|(log

1log

),|()...,|(),|(

),|()...,|(),|(log

00

11

1

0002

0222

01

0111

111

12

1222

11

1111

iiiiJB

iiiiJBii

iiiiJBJBJB

iiiiJBJBJB

nptTP

nptTPXS

AnptTPnptTPnptTP

nptTPnptTPnptTPB

iii J

j ji

jijii

J

j jiji

jijijii

J

j ji

jijii

jijiJiiJJ

jijiJiiJJ

p

pna

pp

ppXS

p

pnb

AXXXXXXP

XXXXXXPB

10,

1,

,11

1,

0,

0,

1,

,11

0,

1,

,1

,0,,1,,21,2,11,1

,1,,1,,21,2,11,1

1

1log

)1(

)1(log

1

1log

,|),...),...(,...)(,...(

,|),...),...(,...)(,...(log

np

np

Following similar logic:

a.

b.


PerformanceAverage Run Lengths (ARL’s)

HHHHL ARLProbability Limits

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p

B X1B X0

JB X1

JB X0


Comments

• Excessively low in-control false alarm rate (standard binomial chart)

• Detection power is significantly better (J-bin risk-adjusted chart)

Example: J = (≤) .001 probability limits


Effects and Interactions(p, p)

Low p High p

Lowpbar

Highpbar

1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X0

X1


Risk-Adjus ted Not Risk-Adjus ted

1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X0X0

X1

X1



1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X1X0



1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X1X0

X1

X0




Variation in p’s (p, n )

Low p High p

Lownbar

Highnbar

1

10

100

1000

10000

100000

1000000

10000000

-0.9

p

-0.8

p

-0.7

p

-0.6

p

-0.5

p

-0.4

p

-0.3

p

-0.2

p

-0.1

p p

0.1p

0.2p

0.3p

0.4p

0.5p

0.6p

0.7p

0.8p

0.9p


Ave

rage

Run

Len

gth

X1

X0

Avg n = 10, SD n = 11.3, Avg p = .5, SD p = .028, J = 2SR = 0.9997, T AD = 0.0282, KL = 3.80E08


1

10

100

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Avg n = 10, SD n = 11.3, Avg p = .5, SD p = .566, J = 2SR = 0.7809, T AD = 2.915E-4, KL = 0.0281


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Avg n = 10, SD n = 11.3, Avg p = .5, SD p = .566, J = 2SR = 0.7809, T AD = 2.915E-4, KL = 0.0281


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Avg n = 200, SD n = 141, Avg p = .5005, SD p = .006, J = 20SR = 0.9999, T AD = 0.0252, KL = 2.17E-4


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Avg n = 200, SD n = 141, Avg p = .494, SD p = .351, J = 20SR = 0.8102, T AD = 0.2045, KL = 1.70E-2


Number of Strata (J)

risk-adjusted control charts for non- homogeneous dichotomous events qprc annual conference may 26,...

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