HSRP 734: Advanced Statistical Methods July 17, 2008

HSRP 734: HSRP 734: Advanced Statistical Advanced Statistical

MethodsMethodsJuly 17, 2008July 17, 2008

ObjectivesObjectives

Describe and use the Cox Describe and use the Cox proportional hazards model to proportional hazards model to describe and compare survival describe and compare survival experiencesexperiences

Use SAS to implement Use SAS to implement

From Stratification to From Stratification to ModelingModeling

What have we done so far?What have we done so far? Estimated the survival function with the Estimated the survival function with the

minimum of assumptionsminimum of assumptions Compared the survival function of Compared the survival function of

various groups using nonparametric various groups using nonparametric teststests

Similar to a contingency table Similar to a contingency table analysis, the above tests are analysis, the above tests are somewhat limited to simple somewhat limited to simple stratificationsstratifications

From Stratification to From Stratification to ModelingModeling

Goal: extend survival analysis to an Goal: extend survival analysis to an approach that allows for multiple approach that allows for multiple covariates of mixed forms (i.e., covariates of mixed forms (i.e., continuous, ordinal and nominal continuous, ordinal and nominal categorical)categorical)

We have two options for our expansionWe have two options for our expansion Model the survival function or timeModel the survival function or time Model the hazard function (between 0 to Model the hazard function (between 0 to

∞)∞)

Cox Proportional Hazards Cox Proportional Hazards ModelModel

We will model the hazard functionWe will model the hazard function

In the Cox proportional hazards model, In the Cox proportional hazards model, we have a regression-based approach to we have a regression-based approach to survival analysis.survival analysis.

What are Proportional HazardsWhat are Proportional Hazards

The constant C does not depend on timeThe constant C does not depend on time

1

2

C1 2

( | )

( | )

( | ) ( | )

h tC

h t

S t S t

x

x

x x


Cox assumed this proportionality constant Cox assumed this proportionality constant and proposed the following model.and proposed the following model.

wherewhere h h00(t)(t) is the baseline hazard; involves is the baseline hazard; involves tt but not X,but not X,

is the exponential function; is the exponential function;

involves X’s but not involves X’s but not tt ( as long as ( as long as the X’s the X’s are are time independent). time independent).

p

i

ii x

o ethXth 1)();(

p

i

ii x

e 1


Hazard rate = baseline hazard rate Hazard rate = baseline hazard rate x positive term that depends on a x positive term that depends on a “score”“score”

Score = linear function of Score = linear function of explanatory factorsexplanatory factors

Note: Baseline hazard rate is the Note: Baseline hazard rate is the same for everyonesame for everyone

““Score” may be negativeScore” may be negative


The Cox proportional hazards (PH) model The Cox proportional hazards (PH) model assumes one of many possible forms.assumes one of many possible forms.

We could use any function g(X) > 0. such We could use any function g(X) > 0. such that that

0

)();(

1

1

p

iii

p

iiio

xg

xgthXth

and


In the Cox PH model, we do not include In the Cox PH model, we do not include an intercept term. This is because any an intercept term. This is because any intercept term could be incorporated into intercept term could be incorporated into the baseline hazard.the baseline hazard.


The regression model for the hazard function The regression model for the hazard function (instantaneous incidence rate) as a function (instantaneous incidence rate) as a function of of p p explanatory (explanatory (XX) variables is specified as ) variables is specified as follows:follows:

log hazard:log hazard:

log h(t; log h(t; XX) = log h) = log h00(t) + (t) + 11XX11 + + 22XX22 + + … … + + ppXXpp

hazard:hazard:

ppXXX

o eeethXth ...)();( 2211


Interpretation of Interpretation of hh00(t)(t)::

Baseline hazard (incidence) rate as a Baseline hazard (incidence) rate as a function of timefunction of time

Baseline can be interpreted as when Baseline can be interpreted as when all all XX’s are zero – often must center ’s are zero – often must center continuous variables to make continuous variables to make hh00(t) (t) interpretableinterpretable


Interpretation of Interpretation of is the relative hazard associated with a is the relative hazard associated with a 1 1 unit unit

change in change in X1 X1 (i.e.(i.e., X, X11+1 vs. X+1 vs. X11), holding other ), holding other XXs constant, independent of times constant, independent of time

or, in relative risk terms,or, in relative risk terms, is the relative risk for is the relative risk for XX11+1 vs. X+1 vs. X11, , holding holding

other other XXs constant, independent of times constant, independent of time Other Other s have similar interpretationss have similar interpretations

1e


Note:Note:

““multiplies” the baseline hazard multiplies” the baseline hazard hh00(t) (t) by the same amount regardless of by the same amount regardless of the time t. This is therefore a the time t. This is therefore a “proportional hazards” model – the effect “proportional hazards” model – the effect of any (fixed) of any (fixed) X X is the same at any time is the same at any time during follow-upduring follow-up

1e


Applying the formula relating Applying the formula relating S(t) S(t) to the to the cumulative hazard to the proportional cumulative hazard to the proportional hazards model,hazards model,

pXpXXpXpXX

t

tpXpXX

t

eeduuh

dueuh

duuh

tSe

eXtS

etS

...2211

...2211

0

0

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);(

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gives,


is the focus whereas is the focus whereas hh00(t) (t) is a nuisance is a nuisance variablevariable

David Cox (1972) showed how to estimate David Cox (1972) showed how to estimate without having to assume a model for without having to assume a model for hh00(t) (t)

““Semi-parametric”Semi-parametric” hh00(t) (t) is the baseline hazard - “non-parametric” is the baseline hazard - “non-parametric”

part of the modelpart of the model 11, , 22, …, , …, pp are the regression coefficients - are the regression coefficients -

“parametric” part of the model“parametric” part of the model Think of estimating Think of estimating hh00(t) (t) with a step functionwith a step function Let # steps get large Let # steps get large —— “partial likelihood” “partial likelihood”

for for depends on depends on , not , not hh00(t) (t)

Partial likelihoodPartial likelihood

The likelihood function used in Cox The likelihood function used in Cox PH models is called a partial PH models is called a partial likelihood likelihood

We use only the part of the We use only the part of the likelihood function that contains the likelihood function that contains the ’s’s

It depends only on the ranks of the It depends only on the ranks of the data and not the actual time values.data and not the actual time values.

Partial likelihoodPartial likelihood Let the survival times (times to failure) be:Let the survival times (times to failure) be:

tt11 < t < t22 < ... < t < ... < tkk

And let the “risk sets” corresponding to these times be:And let the “risk sets” corresponding to these times be:

RR11, R, R22, ..., R, ..., Rkk

RRjj = list of persons at risk just before = list of persons at risk just before ttjj

Then, the “partial likelihood” for Then, the “partial likelihood” for isis

(Assumes no ties in event times)(Assumes no ties in event times) To estimate To estimate , find the values of , find the values of s that maximize s that maximize L(L() above.) above.

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Rj

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pjpjj

pipii

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2211

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Partial likelihoodPartial likelihood

Why does the partial likelihood make Why does the partial likelihood make sense?sense?

Choose Choose so that the one who failed at so that the one who failed at each time was most likely - relative to each time was most likely - relative to others who might have failed!others who might have failed!

it at failed have could whoones of hazards

person failed of hazard

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pipii

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Some General Comments Some General Comments ThoughtsThoughts

Similar to logistic regression, a simple Similar to logistic regression, a simple function of function of

the has a particularly nice the has a particularly nice interpretationinterpretation

can be interpreted as a can be interpreted as a relative relative risk (risk ratio) for a one unit change risk (risk ratio) for a one unit change in the predictorin the predictor

e

β

0.60

0.60

ˆ 0.60 0.55 (protective effect)

ˆ 0.60 1.82 (increased risk)

e

e


Using the common methods of Using the common methods of estimation, it can be shown that estimation, it can be shown that estimated regression parameters estimated regression parameters have an asymptotically normal have an asymptotically normal distribution with mean distribution with mean and finite and finite variancevariance

ββ


Two important implications of Two important implications of asymptotic normalityasymptotic normality We can use the likelihood ratio, score, and We can use the likelihood ratio, score, and

Wald tests to make inference about our data Wald tests to make inference about our data We can use the usual method to construct a We can use the usual method to construct a

95% confidence interval95% confidence interval

ˆ ˆ1.96 ( )SEe

Confidence IntervalsConfidence Intervals

Instead of comparing a 49 year old Instead of comparing a 49 year old to a 50 year old (a one unit to a 50 year old (a one unit difference in age), what if we want difference in age), what if we want the hazard ratio and confidence the hazard ratio and confidence interval comparing a 49 year old to a interval comparing a 49 year old to a 59 year old?59 year old?


The Cox PH model is a regression The Cox PH model is a regression model and we can use the usual model and we can use the usual tools for model building (e.g., tools for model building (e.g., stepwise methods or linearity of stepwise methods or linearity of predictor via higher order terms)predictor via higher order terms)

Two ExamplesTwo Examples

AML AML —— one covariate one covariate UIS UIS —— more than one covariate more than one covariate

Example 1: Cox PH model Example 1: Cox PH model for AML datafor AML data

Semi-parametric model for the hazard Semi-parametric model for the hazard (incidence) rate for the AML data(incidence) rate for the AML data

where where hhii(t) (t) is the hazard for person is the hazard for person i i at at week week tt, , hh00(t) (t) is the hazard if is the hazard if XXii = 0 = 0 (not (not maintained group), and is the maintained group), and is the multiplicative effect of multiplicative effect of XXii=1 =1 (maintained (maintained group)group)

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oi ethth )()(

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Cox PH Model using SAS Cox PH Model using SAS —— AMLAML

Example 1: Cox PH model Example 1: Cox PH model for AML data (cont’d)for AML data (cont’d)

= 0.444 – relative rate of AML relapse = 0.444 – relative rate of AML relapse maintained vs. not maintainedmaintained vs. not maintained

95% CI : (0.16, 1.23)95% CI : (0.16, 1.23) 1/0.444 = 2.25 – relative rate of AML 1/0.444 = 2.25 – relative rate of AML

relapse not maintained vs. maintainedrelapse not maintained vs. maintained

95% CI : (1/1.23, 1/0.16) = (0.81, 6.26)95% CI : (1/1.23, 1/0.16) = (0.81, 6.26)

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Example 2: Cox PH model Example 2: Cox PH model for UIS datafor UIS data

Description of the variables from the Description of the variables from the UIS study in Table 1.3 of Hosmer, D.W. UIS study in Table 1.3 of Hosmer, D.W. and Lemeshow, S. (1998) Applied and Lemeshow, S. (1998) Applied Survival Analysis: Regression Modeling Survival Analysis: Regression Modeling of Time to Event Data, John Wiley and of Time to Event Data, John Wiley and Sons Inc., New York, NYSons Inc., New York, NY

This data set is available atThis data set is available at

http://www-unix.oit.umass.edu/~statdata select “datasets” and then “survival select “datasets” and then “survival analysis”analysis”

Example 2: Cox PH model Example 2: Cox PH model for UIS data (cont’d)for UIS data (cont’d)

We use Cox PH model to compare two We use Cox PH model to compare two treatment randomization assignments, treatment randomization assignments, controlling for several covariatescontrolling for several covariates Compare long treatment randomization Compare long treatment randomization

assignment with short treatment randomization assignment with short treatment randomization assignmentassignment

Use time to drug relapse as the response Use time to drug relapse as the response variablevariable

Time variable is time from admission date to Time variable is time from admission date to drug relapse or censoring due to the end of the drug relapse or censoring due to the end of the study or lost to follow-up (the definition for study or lost to follow-up (the definition for variable CENSOR is questionable in the data set; variable CENSOR is questionable in the data set; however, we still use it as a demonstration.)however, we still use it as a demonstration.)

Control for other risk factors in making the Control for other risk factors in making the comparisoncomparison

Cox PH Model using SAS Cox PH Model using SAS —— UISUIS

The Description of UIS The Description of UIS datadata

Data are in the file uissurv.datData are in the file uissurv.dat

n = 628n = 628

VariableVariable DescriptionDescription Codes/ValuesCodes/Values

IDID Identification CodeIdentification Code 1 - 6281 - 628

AGEAGE Age at EnrollmentAge at Enrollment YearsYears

BECKTOTABECKTOTA Beck Depression ScoreBeck Depression Score 0.000 - 54.0000.000 - 54.000

HERCOCHERCOC Heroin/Cocaine Use DuringHeroin/Cocaine Use During 1 = Heroin & Cocaine1 = Heroin & Cocaine3 Months Prior to Admission3 Months Prior to Admission 2 = Heroin Only2 = Heroin Only

3 = Cocaine Only3 = Cocaine Only4 = Neither Heroin 4 = Neither Heroin nor Cocaine nor Cocaine

IVHXIVHX History of IV Drug UseHistory of IV Drug Use 1 = Never1 = Never2 = Previous2 = Previous3 = Recent3 = Recent

The Description of UIS data The Description of UIS data (cont’d)(cont’d)

VariableVariable DescriptionDescription Codes/ValuesCodes/Values

NDRUGTXNDRUGTX Number of Prior Drug TreatmentsNumber of Prior Drug Treatments 0 - 400 - 40

RACERACE Subject's RaceSubject's Race 0 = White0 = White1 = Non-White1 = Non-White

TREATTREAT Treatment RandomizationTreatment Randomization 0 = Short0 = ShortAssignmentAssignment 1 = Long1 = Long

SITESITE Treatment SiteTreatment Site 0 = A0 = A1 = B1 = B

LOSLOS Length of Stay in TreatmentLength of Stay in Treatment DaysDays(Admission Date to Exit Date)(Admission Date to Exit Date)

TIMETIME Time to Drug RelapseTime to Drug Relapse DaysDays(Measured from Admission Date)(Measured from Admission Date)

CENSORCENSOR Event for Treating Lost toEvent for Treating Lost to 1 = Returned to Drugs or 1 = Returned to Drugs or Follow-Up as Returned to Drugs Follow-Up as Returned to Drugs Lost to Follow-Up Lost to Follow-Up

0 = Otherwise 0 = Otherwise


Model 1:Model 1:

log h(t) = log hlog h(t) = log h00(t) + (t) + 11TREAT TREAT Model 2:Model 2:

log h(t) = log hlog h(t) = log h00(t) + (t) + 11TREAT + TREAT + 22AGE + AGE + 33RACE + RACE + 44BECKTOTA + BECKTOTA + 55HERCOC.1 + HERCOC.1 + 66HERCOC.2 + HERCOC.2 + 77HERCOC.3HERCOC.3

where where HERCOC.1 = 1 if HERCOC = 1; = 0 otherwise,HERCOC.1 = 1 if HERCOC = 1; = 0 otherwise,

HERCOC.2 = 1 if HERCOC = 2; = 0 otherwise,HERCOC.2 = 1 if HERCOC = 2; = 0 otherwise,

HERCOC.3 = 1 if HERCOC = 3; = 0 otherwise,HERCOC.3 = 1 if HERCOC = 3; = 0 otherwise,


What is the relative risk of drug relapse What is the relative risk of drug relapse for the long treatment group compared for the long treatment group compared to the short treatment group, adjusting to the short treatment group, adjusting for age and other risk factors?for age and other risk factors?

ee-0.2273-0.2273 = 0.797 – about 20% reduction in = 0.797 – about 20% reduction in the risk of drug relapse for the patients the risk of drug relapse for the patients in the long treatment randomization in the long treatment randomization assignment compared with patients in assignment compared with patients in the short treatment randomization the short treatment randomization assignment.assignment.


What is the interpretation of each What is the interpretation of each coefficient?coefficient? AGE AGE —— controlling for treatment assignment and controlling for treatment assignment and

other risk factors, the risk of drug relapse, as other risk factors, the risk of drug relapse, as estimated from a Cox model, is 0.98 times lower estimated from a Cox model, is 0.98 times lower per year of ageper year of age

RACE RACE —— controlling for treatment assignment and controlling for treatment assignment and other risk factors, the risk of drug relapse is 0.78 other risk factors, the risk of drug relapse is 0.78 times lower for non-white compared with whitetimes lower for non-white compared with white

BACKTOTA BACKTOTA —— controlling for treatment controlling for treatment assignment and other risk factors, the risk of drug assignment and other risk factors, the risk of drug relapse is 1.01 times higher per unit difference in relapse is 1.01 times higher per unit difference in Beck Depression scoreBeck Depression score


HERCOC.1 HERCOC.1 —— controlling for treatment controlling for treatment assignment and other risk factors, the assignment and other risk factors, the risk of drug relapse is 1.217 times risk of drug relapse is 1.217 times higher for patients who use Heroin and higher for patients who use Heroin and Cocaine compared with those who use Cocaine compared with those who use neither Heroin nor Cocaine; however, neither Heroin nor Cocaine; however, this risk is not statistically different this risk is not statistically different from 1 from 1

HERCOC.2 HERCOC.2 — you do!— you do! HERCOC.3 HERCOC.3 — you do!— you do!


You must think about another way to You must think about another way to deal with variable HERCOC since deal with variable HERCOC since none of the dummy variables is none of the dummy variables is significant.significant.

How to do it?How to do it? I randomly chose the covariates for I randomly chose the covariates for

the demonstration. To find a best the demonstration. To find a best model seriously, you need to go model seriously, you need to go through the model selection.through the model selection.


What is the relative risk of drug What is the relative risk of drug relapse forrelapse for(A) A short treatment randomization (A) A short treatment randomization

assigned 45-year oldassigned 45-year old

vs.vs.

(B) A long treatment randomization (B) A long treatment randomization assigned 75 -year oldassigned 75 -year old


Log hazard for (A) Log hazard for (A) = const + 0 x (-0.2273) + 45 x (-0.0185)= const + 0 x (-0.2273) + 45 x (-0.0185)= const – 0.8325= const – 0.8325

Log hazard for (B) Log hazard for (B) = const + 1 x (-0.2273) + 75 x (-0.0185)= const + 1 x (-0.2273) + 75 x (-0.0185)= const – 1.6148= const – 1.6148

Difference in log hazards, (A) vs. (B): Difference in log hazards, (A) vs. (B): (const – 0.8325) – (const – 1.6148)(const – 0.8325) – (const – 1.6148)= 0.7823= 0.7823

Relative Risk (A) vs. (B)Relative Risk (A) vs. (B)ee0.78230.7823 = 2.19 – higher risk for younger, short = 2.19 – higher risk for younger, short treatment randomization assigned patient than treatment randomization assigned patient than for older, long treatment randomization assigned for older, long treatment randomization assigned patient.patient.


How much higher is the risk of a 70 years old How much higher is the risk of a 70 years old patient compared with a 60 years old patient, patient compared with a 60 years old patient, assuming treatment and other risk factors are assuming treatment and other risk factors are the same?the same?

The estimated difference in log hazards for two The estimated difference in log hazards for two patients whose ages differ by 10 years, holding patients whose ages differ by 10 years, holding other covariates fixed isother covariates fixed is10 x =10 x (-0.0185) = -0.18510 x =10 x (-0.0185) = -0.185RR = eRR = e-0.185-0.185 = 0.83 – a ten year difference in the = 0.83 – a ten year difference in the age decreases the risk of drug relapse by 20%age decreases the risk of drug relapse by 20%

How would you determine age modifies the risk How would you determine age modifies the risk of drug relapse for long treatment assignment vs. of drug relapse for long treatment assignment vs. short treatment assignment?short treatment assignment?

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HSRP 734: Advanced Statistical Methods July 17, 2008

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Transcript of HSRP 734: Advanced Statistical Methods July 17, 2008