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## Chapter 11 Survival Analysis Part 2

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Survival Analysis and Regression

- Combine lots of information
- Look at several variables simultaneously
- Explore interactions
- model interaction directly
- Control (adjust) for confounding

Proportional hazards regression(Cox Regression)

- Can we relate predictors to survival time?
- We would like something like linear regression
- Can we incorporate censoring too?
- Use the hazard function

Hazard function

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

(t + small amount of time)

- Approximates proportion of patients having event around time t

Hazard function

Hazard less intuitive than survival curve

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

Rate per unit of time, as d 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)

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

l (t)

0 6 17 23 80

At which age would the hazard be greatest?

Possible Hazard of Divorce

0 2 10 25 35 50

Why “proportional hazards”?

Ratio of hazards measures relative risk

If we assumerelative risk is constant over time…

The hazards are proportional!

Proportional Hazard of Death from BirthProbability of dying in next year as function of age for two groups (women, men)

l (t)

0 6 17 23 80

At which age would the hazard be greatest?

Proportional Hazards and Survival Curves

- If we assume proportional hazards then
- The curves should not cross.

Proportional hazards regression modelone covariate

0(t)- unspecified baseline hazard (constant)

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

b1 = regression coefficient associated with the predictor (X)

b1 positive indicates larger X increases the hazard

Can include more than one predictor

Interpretation of Regression Parameters

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

exp(b1) is the relative hazard for exposed versus unexposed

(b1 is the log of the relative hazard)

exp(b1) can be interpreted as relative risk or relative rate with all other covariates held fixed.

Example - risk of outcome forwomen vs. men

For males;

Suppose X1=1 for females, 0 for males

For females;

Example - Risk of outcome for1 unit change in blood pressure

Suppose X1= systolic blood

pressure (mm Hg)

For person with SBP = 114

For person with SBP = 113

Relative risk of 1 unit

increase in SBP:

Example - Risk of outcome for10 unit change in blood pressure

Suppose X= systolic blood

pressure (mmHg)

For person with SBP = 110

For person with SBP = 100

Relative risk of 10 unit

increase in SBP:

Parameter estimation

- How do we come up with estimates for bi?
- Can’t use least squares since outcome is not continuous
- Maximum partial-likelihood (beyond the scope of this class)
- Given our data, what are the values of bi that are most likely?
- See page 392 of Le for details

Inference for proportional hazards regression

- Collect data, choose model, estimate bis
- Describe hazard ratios, exp(bi), in statistical terms.
- How confident are we of our estimate?
- Is the hazard ratio is different from one due to chance?

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

- Based on transforming the 95% CI for the hazard ratio
- Supplied automatically by SAS

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

- Equivalent to a hypothesis test; reject Ho: RR = 1 at alpha = 0.05 (Ha: RR1)

Hypothesis test for individual PH regression coefficient

- Null and alternative hypotheses
- Ho : Bi = 0, Ha: Bi 0
- Test statistic and p-values supplied by SAS
- If p<0.05, “there is a statistically significant association between the predictor and outcome variable controlling for all other covariates” at alpha = 0.05
- When X is binary, identical results as log-rank test

Hypothesis test for all coefficients

- Null and alternative hypotheses
- Ho : all Bi = 0, Ha: not all Bi 0
- Several test statistics, each supplied by SAS
- Likelihood ratio, score, Wald
- p-values are supplied by SAS
- If p<0.05, “there is a statistically significant association between the predictors and outcome at alpha = 0.05”

Example

Myelomatosis: Tumors throughout the body composed of cells derived from hemopoietic(blood) tissues of the bone marrow.

N=25

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

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

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

Renal=>has a value of 0 if renal functioning was normal at the time 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 269

SAS- PHREG

PROCPHREGDATA = myel;

MODEL dur*status(0) =treat;

RUN;

- Fit proportional hazards model with time to death as outcome
- “ status(0)”; observations with status variable = 0 are censored
- status= 1 means an event occurred
- Look at effect of Treatment 2 vs. Treatment 1 on mortality.

Same as LIFETEST

PROC PHREG Output

Analysis of Maximum Likelihood Estimates

Parameter Standard Hazard

Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio

treat 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?

Complications

- Complications
- competing risks (high death rate)– RENAL FUNCTION
- Non proportional hazards -time dependent covariates (will show you later)
- Extreme censoring in one group

SAS- PHREG

PROCPHREGDATA = myel;

MODEL dur*status(0) = renal treat;

RUN;

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

Same as LIFETEST

Output with adjusted treatment effect

Analysis of Maximum Likelihood Estimates

Parameter Standard Hazard

Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio

renal 1 4.10540 1.16451 12.4286 0.0004 60.667

treat 1 1.24308 0.59932 4.3021 0.0381 3.466

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