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HRP223-2008

Topic 11- Survival Analysis

HRP223 2008

What is survival analysis? They are statistical methods to study not only if an an

event happens but also when it happens. Time is measured for each person from the first

observation until when an event happens or when time is censored. Time typically stops when:– An event occurs (aka a failure).– The study is out of time/money.– A person is lost to follow-up or dies from other causes.

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Examples How long a person lives after surgery How long a person is in remission from leukemia until

relapse How long a woman undergoing fertility treatments

takes to get pregnant

All would be called time to event or time until failure. Events are categorical changes in states.

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Why special methods?

Logistic regression can predict the presence or absence of events but not time until events and it can not handle time dependent covariates.

Linear regression can not handle censoring well or time dependent covariates or the fact that time can only be positive.

Time starts when the risk for the possible outcome starts or alternatively, when treatment starts.

Right Censoring

Interval Censoring

Worry about bias if you have censoring before the end of the study. Is the censoring related to the outcome or the risk of the outcome (informative censoring)?

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Survival Analysis Steps

Get some data and make sure it is valid. Estimate the survival/hazard functions. Compare the functions between groups. Assess the impact of predictors on survival

rates.

Survival function– Probability of an event happening some time after this

time point

Hazard function– Instantaneous risk of having an event given the subject

is still at risk

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Methods Kaplan-Meier method (product limit method)

– Most commonly done method to get the survival function– The math is easy.

Life-table method Parametric methods Cox regression (proportional hazards regression)

S(t) [1# dead at time j

# at risk at time jj:t j t

]

Group 1 = Radiation + ChemotherapyGroup 2 = Radiation only

Censor 0 = DeathCensor 1 = Censored

12

3

4

56

7

8

9

1

2

3

4

Reality check the data first.

Don’t use the mean if there are censored people.

Two died at 301 days.

Lots of subjects are censored at the end.

Time to percent failed.

Failure Rates

These CLs are too narrow! Hall-Wellner CLs are better but require some code.

Late differences

Early differences

Different Tests None are powerful when the curves cross. The -2 log LR (likelihood ratio) test assumes the

distribution is exponential (risk doesn’t change over time). The likelihood ratio is a parametric test that requires the distribution to be exponential. When true, it is the most powerful test. Wilcoxon is more powerful than Log Rank when the data follows a log normal distribution. Do the diagnostic plots.

You need straight lines starting at 0,0 for the -2 log LR (likelihood ratio) to be valid. Here the radiation/chemo line flattening suggests decreasing risk over time. Curving up suggests increasing risk over time.

You want straight lines in the log normal plots for the Wilcoxon to be powerful in detecting the group differences. Differing rates of censoring will cause bias in the Wilcoxon.

Parallel lines suggest a ratio of hazards that is constant over time and therefore, log rank is

powerful.

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Differences

Wilcoxon puts more weight on the early parts of the curves whereas the log rank puts equal weight on all observations.

Wilcoxon is more sensitive to early differences in the curves and Log Rank is more sensitive to late differences in the curves.

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Other Comparisons

There are other methods to compare the curves that I expect to see in the literature in upcoming years. They just use different weights at each time point. Watch for Harrington-Fleming and the modified Peto-Peto statistics.

These should only be used with large samples with many events.

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Hazard Plot

Paul Allison wrote the best book on survival analysis. In it you will find code to produce hazard plots when you do KM analysis. If you ever do survival analysis, get a copy!

www.sas.com/apps/pubscat/bookdetails.jsp?catid=1&pc=55233

You can download the smooth macro from the website. Get the macro and mend lines and everything between.

Run it first then add the call to the smooth macro in your code.

Comment out the one in green below (at the end of the program) to be able to double check the groups in the dataset.

Group 1 is radiation only.

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Life Table Analysis (Actuarial Method)

If you have a lot of data or if you have measures on everyone at precise times, you can do life table analyses.

You can choose the interval widths but be sure to include some events.

The analyses are done on the ungrouped data so the statistics to compare strata will be the same as in the KM analyses.

You can get rough estimated hazard functions.

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Comparing Multiple Groups If you have 3 or more groups, you can compare them

using KM or life table analyses to find out if there are overall differences between the groups. SAS 9.1.3 does not have a built-in method to do the post hoc comparisons between all the levels. Survival Analysis Using the Proportional Hazards Model by SAS Press has a macro do it (but it costs $100 for the paperback book). Come see me.

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Impact of Covariates

You may have statistically significant differences between groups, but what causes it? In 1972, (Sir) David Cox published a paper “Regression Models and Life Tables” that revolutionized how we assess the impact of predictors on survival.

Cox’s paper is the most referenced article in the history of statistics and it is in the top 100 for all of science.

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Non- and Parametric Modeling

There are non-parametric methods for assessing survival (KM) curves which are not particularly powerful for assessing relationships. There are also parametric models, where you mathematically specify the shape of the hazard functions. These are very powerful if you specify the correct shape of the hazard but often the hazard functions are unknown or extremely complex.

What does it do? You say everyone has a baseline hazard (which

you don’t need to specify!) and the covariates push the hazard up or down. It works using the rank order of events and censoring and gives you hazard ratios.

The method assumes that the hazards are proportional but it can deal with them when they are not.

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Hazard Ratios

Interpret them like odds ratios.

HR(isFemale) Hazard for female

Hazard for male2

HR(age) Age + 1

Age1.75

Hazard of event increases by 75% every year older someone was at the start of the study.

Double the hazard if the subject was female.

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Weaknesses

You don’t get equations that can be used to make predictions.

You don’t have group specific hazard rates. Does the HR of 2 mean that a subject goes from .25 hazard to .5 or .001 to .002?

Clinical significance can’t be judged.

How it works… It uses partial likelihood to assess hazards at the

times when there are events. Because of this, it needs to deal with tied times. Use Efron’s method.

Like logistic regression, it can fail to find an answer because of 0 cell counts. If you see huge standard errors relative to the size of the parameter estimates, you have problems.

Make sure you have the correct number censored.

See if it notices that it failed to converge.

These measure overall goodness of the model. AIC or Schwartz Bayesian Criteria include penalizations to -2 log likelihood for less parsimonious models.Does the model do any good? Use the likelihood ratio if the sample is small.

For every year increase in baseline age, the model gives a .035 increase in the log of the hazard of death or a 3.6% increase in the hazard.

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Dealing with Censoring

Do additional analysis where you:– set all censored people to having events immediately after

they were censored.– set all censored people to having events after the longest

follow-up of anyone in the study. This sensitivity analysis give you worst case scenario

information as well as your best guess given the actual data that you already ran.