Chapter 11 Survival Analysis Part 2. 2 Survival Analysis and Regression Combine lots of information Combine lots of information Look at several variables.

Chapter 11Chapter 11Survival AnalysisSurvival Analysis

Part 2Part 2

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Survival Analysis and Survival Analysis and RegressionRegression

Combine lots of informationCombine lots of information Look at several variables simultaneouslyLook at several variables simultaneously

Explore interactionsExplore interactions model interaction directlymodel interaction directly

Control (adjust) for confoundingControl (adjust) for confounding

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Proportional hazards regressionProportional hazards regression(Cox Regression)(Cox Regression)

Can we relate predictors to survival time?Can we relate predictors to survival time?

We would like something like linear regressionWe would like something like linear regression

Can we incorporate censoring too?Can we incorporate censoring too?

Use the hazard functionUse the hazard function

...22110 XBXBBt

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Hazard functionHazard function

Given patient survived to time t, what is the Given patient survived to time t, what is the probability they develop outcome very soon? probability they develop outcome very soon?

(t + small amount of time)(t + small amount of time)

Approximates proportion of patients having Approximates proportion of patients having event around time tevent around time t

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Hazard functionHazard function

) (Prob

)(TttTt

t

Hazard less intuitive than survival curve

Conditional probability the event will occur between t and t+ given it has not previously occurred

Rate per unit of time, as goes to 0 get instant rate

Tells us where the greatest risk is given survival up to that time (risk of the event at that time for an individual)

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Possible Hazard of Death from BirthPossible Hazard of Death from BirthProbability of dying in next year as function of ageProbability of dying in next year as function of age

0 6 17 23 80

t)

At which age would the hazard be greatest?

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Possible Hazard of Divorce Possible Hazard of Divorce

0 2 10 25 35 50

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Why “Why “proportional hazardsproportional hazards”?”?

Ratio of hazards measures relative riskRatio of hazards measures relative risk

If we If we assumeassume relative risk is constant over time…relative risk is constant over time…

The hazards are proportional!The hazards are proportional!

RR(t) (t) for exposed

(t) for unexposed

ct

t

unexposedfor )(

exposedfor )(

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Proportional HazardProportional Hazard of Death from Birth of Death from BirthProbability of dying in next year as function of age Probability of dying in next year as function of age

for two groups (women, men)for two groups (women, men)

0 6 17 23 80

t)

At which age would the hazard be greatest?

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Proportional Hazards and Proportional Hazards and Survival CurvesSurvival Curves

If we assume proportional hazards then If we assume proportional hazards then

The curves should not cross.The curves should not cross.

cba tsts )]([)(

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Proportional hazards regression modelProportional hazards regression modelone covariateone covariate

)exp()()( 110 Xtt

0(t) - unspecified baseline hazard (constant)

(t) the hazard for subject with X=0 (cannot be negative)

1 = regression coefficient associated with the predictor (X)

1 positive indicates larger X increases the hazard

Can include more than one predictor

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Interpretation of Regression ParametersInterpretation of Regression Parameters

)....exp()()( 3322110 ppXXXXtt

For a binary predictor; X1 = 1 if exposed and 0 if unexposed,

exp(1) is the relative hazard for exposed versus unexposed

(1 is the log of the relative hazard)

exp(1) can be interpreted as relative risk or relative rate with all other covariates held fixed.

)(...)()())(( 2211 ppo xxxtLog

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Example - risk of outcome forExample - risk of outcome forwomen vs. menwomen vs. men

For males;For males;

For females;For females;

)exp()(

)exp()(

malesfor hazard

femalesfor hazardhazard Relative 1

0

10

t

t

)exp()()( 110 Xtt Suppose X1=1 for females, 0 for males

)()0*exp()()( 010 ttt

)exp()()1*exp()()( 1010 ttt

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Example - Risk of outcome forExample - Risk of outcome for1 unit change in blood pressure1 unit change in blood pressure

For person with SBP = 114 For person with SBP = 114

)exp(

)113114exp(

)*113exp()(

)*114exp()(

1

11

10

10

t

t

)exp()()( 110 Xtt Suppose X1= systolic bloodpressure (mm Hg)

)114*exp()()( 10 tt

)113*exp()()( 10 tt

Relative risk of 1 unitincrease in SBP:

For person with SBP = 113For person with SBP = 113

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Example - Risk of outcome forExample - Risk of outcome for10 unit change in blood pressure10 unit change in blood pressure

For person with SBP = 110 For person with SBP = 110

)10exp(

)100110exp(

)*100exp()(

)*110exp()(

1

11

10

10

t

t

)exp()()( 110 Xtt Suppose X= systolic bloodpressure (mmHg)

)110*exp()()( 10 tt

)100*exp()()( 10 tt

Relative risk of 10 unitincrease in SBP:

For person with SBP = 100For person with SBP = 100

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Parameter estimationParameter estimation

How do we come up with estimates for How do we come up with estimates for ii??

Can’t use least squares since outcome is not Can’t use least squares since outcome is not continuouscontinuous

Maximum partial-likelihood Maximum partial-likelihood (beyond the scope of this (beyond the scope of this class)class) Given our data, what are the values of Given our data, what are the values of ii that are that are

most likely?most likely?

See page 392 of Le for detailsSee page 392 of Le for details

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Inference for proportional hazards regressionInference for proportional hazards regression

Collect data, choose model, estimate Collect data, choose model, estimate iiss

Describe hazard ratios, exp(Describe hazard ratios, exp(ii), in statistical ), in statistical

terms.terms. How confident are we of our estimate?How confident are we of our estimate? Is the hazard ratio is different from one due to Is the hazard ratio is different from one due to

chance?chance?

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95% Confidence Intervals for the relative 95% Confidence Intervals for the relative risk (hazard ratio)risk (hazard ratio)

Based on transforming the 95% CI for the hazard ratioBased on transforming the 95% CI for the hazard ratio

Supplied automatically by SASSupplied automatically by SAS

““We have a statistically significant association between the predictor We have a statistically significant association between the predictor and the outcome controlling for all other covariates”and the outcome controlling for all other covariates”

Equivalent to a hypothesis test; reject HEquivalent to a hypothesis test; reject Hoo: RR = 1 at alpha = 0.05 : RR = 1 at alpha = 0.05 (H(Haa: RR: RR1)1)

),( 96.196.1 SEiSE ee i

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Hypothesis test for individual PH Hypothesis test for individual PH regression coefficientregression coefficient

Null and alternative hypothesesNull and alternative hypotheses

Ho : BHo : Bi i = 0, Ha: B= 0, Ha: Bii 0 0

Test statistic and p-values supplied by SASTest statistic and p-values supplied by SAS

If p<0.05, “there is a statistically significant association If p<0.05, “there is a statistically significant association between the predictor and outcome variable controlling between the predictor and outcome variable controlling for all other covariates” at alpha = 0.05for all other covariates” at alpha = 0.05

When X is binary, identical results as log-rank testWhen X is binary, identical results as log-rank test

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Hypothesis test for all coefficientsHypothesis test for all coefficients

Null and alternative hypothesesNull and alternative hypotheses

Ho : all BHo : all Bi i = 0, Ha: not all B= 0, Ha: not all Bii 0 0

Several test statistics, each supplied by SASSeveral test statistics, each supplied by SAS Likelihood ratio, score, WaldLikelihood ratio, score, Wald

p-values are p-values are supplied by SASsupplied by SAS

If p<0.05, “there is a statistically significant association If p<0.05, “there is a statistically significant association between the predictors and outcome at alpha = 0.05”between the predictors and outcome at alpha = 0.05”

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Example Example MyelomatosisMyelomatosis: Tumors throughout the body composed of cells : Tumors throughout the body composed of cells

derived from hemopoietic(blood) tissues of the bone marrow. derived from hemopoietic(blood) tissues of the bone marrow.

NN=25=25

durdur=>is time in days from the point of randomization to either =>is time in days from the point of randomization to either death or censoring (which could occur either by loss to follow-up death or censoring (which could occur either by loss to follow-up or termination of the observation). or termination of the observation).

StatusStatus=>has a value of 1 if dead; it has a value of 0 if censored.=>has a value of 1 if dead; it has a value of 0 if censored.

TreatTreat=>specifies a value of 1 or 2 to correspond to two treatments.=>specifies a value of 1 or 2 to correspond to two treatments.

RenalRenal=>has a value of 0 if renal functioning was normal at the time =>has a value of 0 if renal functioning was normal at the time of randomization; it has a value of 1 for impaired functioning.of randomization; it has a value of 1 for impaired functioning.

The MYEL Data set take from: Survival Analysis Using SAS, A Practical Guide by Paul D. Allison - page 269The MYEL Data set take from: Survival Analysis Using SAS, A Practical Guide by Paul D. Allison - page 269

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SAS- PHREGSAS- PHREG

PROCPROC PHREGPHREG DATA DATA = myel= myel;; MODELMODEL dur*status(0) =treat; dur*status(0) =treat; RUNRUN;;

Fit proportional hazards model with time to death as outcomeFit proportional hazards model with time to death as outcome

“ “ status(0)”; observations with status variable = 0 are censoredstatus(0)”; observations with status variable = 0 are censored

status= 1 means an event occurredstatus= 1 means an event occurred

Look at effect of Treatment 2 vs. Treatment 1 on mortality.Look at effect of Treatment 2 vs. Treatment 1 on mortality.

Same as LIFETEST

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PROC PHREG OutputPROC PHREG Output

Analysis of Maximum Likelihood EstimatesAnalysis of Maximum Likelihood Estimates

Parameter Standard HazardParameter Standard Hazard

Variable DF Estimate Error Chi-Square Pr > ChiSq RatioVariable DF Estimate Error Chi-Square Pr > ChiSq Ratio

treat 1 0.57276 0.50960 1.2633 0.2610 1.773treat 1 0.57276 0.50960 1.2633 0.2610 1.773

77% increased risk of death for treatment 2 vs. treatment 1, But it is not significant? Why?

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Complications Complications

ComplicationsComplications competing risks (high death rate)– RENAL competing risks (high death rate)– RENAL

FUNCTIONFUNCTION Non proportional hazards -time dependent Non proportional hazards -time dependent

covariates (will show you later)covariates (will show you later) Extreme censoring in one group Extreme censoring in one group

SAS- PHREGSAS- PHREG

PROCPROC PHREGPHREG DATA DATA = myel= myel;; MODELMODEL dur*status(0) = renal treat; dur*status(0) = renal treat; RUNRUN;;

Look at effect of Treatment 2 vs. Treatment 1 on mortality Look at effect of Treatment 2 vs. Treatment 1 on mortality adjusted for renal functioning at baseline.adjusted for renal functioning at baseline.

Same as LIFETEST

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Output with adjusted Output with adjusted treatment effecttreatment effect

Analysis of Maximum Likelihood EstimatesAnalysis of Maximum Likelihood Estimates

Parameter Standard Parameter Standard Hazard Hazard

Variable DF Estimate Error Chi-Square Pr > ChiSq RatioVariable DF Estimate Error Chi-Square Pr > ChiSq Ratio

    renal 1 4.10540 1.16451 12.4286 0.0004 60.667renal 1 4.10540 1.16451 12.4286 0.0004 60.667

treat 1 1.24308 0.59932 4.3021 0.0381 3.466treat 1 1.24308 0.59932 4.3021 0.0381 3.466

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