Hazard Functions for Combination of Causes Farrokh Alemi Ph.D. Professor of Health Administration...

Hazard Functions for Hazard Functions for Combination of CausesCombination of Causes

Farrokh Alemi Ph.D.Farrokh Alemi Ph.D.Professor of Health Administration and PolicyProfessor of Health Administration and Policy

College of Health and Human Services, George Mason UniversityCollege of Health and Human Services, George Mason University4400 University Drive, Fairfax, Virginia 22030 4400 University Drive, Fairfax, Virginia 22030

703 993 1929 703 993 1929 [email protected]@gmu.edu

mailto:[email protected]

PurposePurpose

Risk faced from one cause over Risk faced from one cause over timetime

Relative contribution of various Relative contribution of various causescauses

Joint effect of causesJoint effect of causes

DefinitionsDefinitions Function

NameFormula Definition

Related Terms

Probability Distribution

Functionp(X=t)

Probability of event occurring at time

"t."

Cumulative Distribution

Functionp((X≤t)

Probability of the event occurring

prior to or at time "t."

Sum of PDF for ≤ t

Survival Function

p(X>t)Probability of the

event not occurring prior or at time "t."

1-CDF

Hazard Function

p(X=t|X≥t)

Probability of the event occurring at time t given that it has not occurred prior to this time

PDF/SF

An ExampleAn Example

Year

Probability Distribution

Function

Cumulative Distribution

FunctionSurvival Function

Hazard Function

1 0.20 0.20 1.00 0.20

2 0.20 0.40 0.80 0.25

3 0.20 0.60 0.60 0.33

4 0.20 0.80 0.40 0.50

5 0.20 1.00 0.20 1.00

5+ 0

What Do You Know?What Do You Know?

““Suppose that a cancer patient Suppose that a cancer patient is equally likely to die at any is equally likely to die at any time between 0 to 3 years from time between 0 to 3 years from now. What is the hazard now. What is the hazard function for this person?”function for this person?”

From Cox LA. Risk Analysis, From Cox LA. Risk Analysis, Foundations, Models and MethodsFoundations, Models and Methods

Poisson Process of ArrivalPoisson Process of Arrival

Constant hazard rates, hConstant hazard rates, h Poisson DistributionPoisson Distribution

Arrival of sentinel eventArrival of sentinel event After “t” periods have passed After “t” periods have passed

without the eventwithout the event


h

periodnexteventSentinel

h

periodnextsurviving

ep

ep

1

Constant

hazard

function


)1log(periodnextoccurringeventSentinel

ph Hazard function

Hazard Rate & Hazard Rate & Probability Probability

When the sentinel event is rare, the When the sentinel event is rare, the hazard function is essentially the same hazard function is essentially the same as the probability distribution functionas the probability distribution function

Hazard Function from Hazard Function from Different SourcesDifferent Sources

)(...)()()(21

ththththn

Haza

rd

function

from all

sources Hazard

function from first source

Hazard function

from source n

Hazard Function from Hazard Function from Different SourcesDifferent Sources

)(...)()()(21

ththththn

)()()( thththAsbestosSmokingsourcesBoth

Attributable RiskAttributable Risk

)()(thth

AR i

iAttributable

Risk to Source i


)()(thth

AR i

i

Hazard function

for source i

Attributable Risk to

Source i


)()(thth

AR i

i

Hazard function from all sources

Hazard function

for source i


Source i

ExampleExample

If the hazard function for If the hazard function for medication error caused by a medication error caused by a fatigued nurse is 1 in 1000 and fatigued nurse is 1 in 1000 and the hazard function for the hazard function for medication error caused by medication error caused by illegible prescription order is 2 in illegible prescription order is 2 in 1000, what is the attributable 1000, what is the attributable risk to fatigued nurse?risk to fatigued nurse?

Attributable Risk & Probable Attributable Risk & Probable CauseCause

%33002.001.

001.)()(

)(

thth

thAR

MDRN

RN

RN

Hazard function from all sources

Hazard function for

fatigued nurse


fatigued nurse

Complete Specification of Complete Specification of Combination of Hazard FunctionsCombination of Hazard Functions

Combinatorial combinationCombinatorial combination Not enough dataNot enough data Estimate missing effectsEstimate missing effects

Estimating Missing Hazard Estimating Missing Hazard FunctionsFunctions

1.1. Unknown combination of Unknown combination of cause X and a binding cause X and a binding constraint Y:constraint Y:

0&

YX

hFor example, a cause of medication error is

miscalculation by the nurse of the necessary dose. A constraint for this is verification by an independent observer. If a nurse miscalculates the dose but the

error is found in verification, then it is unlikely to have a medication error from this cause.


2.2. Unknown hazard of several Unknown hazard of several causes (X, Y, and Z) required causes (X, Y, and Z) required to be simultaneously present:to be simultaneously present:

min,0

min,0

min,0

)1log(&&

Z

Y

X

ZYX

h

h

h

ph


2.2. Unknown hazard of several Unknown hazard of several causes (X, Y, and Z) required causes (X, Y, and Z) required to be simultaneously present:to be simultaneously present:

min,0

min,0

min,0

)1log(&&

Z

Y

X

ZYX

h

h

h

phFor example, for a patient to fall, there must be a slippery floor and some cognitive impairment. The hazard of slippery floor or cognitive impairment by itself is minimal but the combination of these two

causes makes falls much more likely.


3.3. Unknown combination of non-Unknown combination of non-interacting, independent interacting, independent causes:causes:

ZYXZYXhhhh

&&


3.3. Unknown combination of non-Unknown combination of non-interacting, independent interacting, independent causes:causes:

ZYXZYXhhhh

&&

For example, wrong side surgery might be due to erroneous marking, not following the nurse’s marking or wrong information from the patient. The missing hazard of the combination is the sum of the hazard

associated with each cause.


4.4. Unknown combination of Unknown combination of interacting, dependent causes:interacting, dependent causes:

ZYXZYX

YZXZYX

ZYXZYX

hhh

hhh

hhh

ofMax

&&&

&&&

&&&

:


4.4. Unknown combination of Unknown combination of interacting, dependent causes:interacting, dependent causes:

ZYXZYX

YZXZYX

ZYXZYX

hhh

hhh

hhh

ofMax

&&&

&&&

&&&

:

For example, consider the effect of poor training, fatigue and similar bottles on medication error rates. If fatigue and poor training interact to make things worse, then the hazard of combined poor training and fatigue should be added to the hazard associated with similar bottles.


5.5. Unknown, non-interacting and Unknown, non-interacting and independent causesindependent causes

ZZX

ZYXX

hhh

hh

&&3

1

Take Home LessonTake Home Lesson

Hazard Functions Are Key to Hazard Functions Are Key to Understanding Relative Contribution of Understanding Relative Contribution of

Multiple Causes to Sentinel EventsMultiple Causes to Sentinel Events

Minute EvaluationsMinute Evaluations

Please use the course web site Please use the course web site to ask a question and rate this to ask a question and rate this lecturelecture

Hazard Functions for Combination of Causes Farrokh Alemi Ph.D. Professor of Health Administration...

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