Analysis of Complex Survey Data

Day 4: Survival analysis and Cox proportional hazards models

Nonparametric Survival Analysis

• Kaplan-Meier Method (also called Product-Limit Method)

• Life Table Method (also called Actuarial Method)

A statistical method to study time to an event

1) Divide risk period into many small time intervals

2) Treat each interval as a small cohort analysis

3) Combine the results for the intervals

Nonparametric Survival Analysis

Basic Concepts of Survival Analysis

1. Censoring

2. Time to an event

3. Survival Function

Censoring

• At the end of study, subjects did not experience the event (outcome). Or subjects withdrew from a study (lost to follow up or died from other diseases).

• Survival analysis assumes LTF and competing cause censoring is random (independent of exposure and outcome)

• When using longitudinal complex surveys (e.g., PSID, AddHealth), survival analysis is most useful

• We can also use it in cross-sectional studies when incorporating retrospective age of onset information.

Cohort Size at Start : 1,000 for 1 year

Number with disease : 28

Number LTF: 15

If assume all dropped out on 1st day of study, rate of disease/y

Censoring

= 28

1,000 - 15=

28

985= .0284

If assume all dropped out on last day of study, probability of disease =

28

1,000= .0280

If drop out rate is constant over the period best estimate of when dropped out is midpoint : probability of disease then is

= 28

1,000 – 7.5=

28

992.5= .0282

Example:

Survival Function

The probability of surviving beyond a specific time [i.e., S(t) = 1 – F(t)]

F(t) = cumulative probability distribution for endpoint (e.g., death)

Probability for survival at each new time period =Probability at that time period conditioned “surviving” to that interval

n

o

p

Probability survival to S4 =

n * o * p * q

S1

S2

q

F

S3

S4

F

F

FFailures (F) = deaths or cases or losses to follow up

Life Table Method

• Time is partitioned into a fixed sequence of intervals (not necessarily of equal lengths)

A classical method of estimating the survival function in epidemiology and actuarial science

Useful for large samples

Interval lengths (arbitrary)Larger the interval, larger the bias

• The LIFETEST Procedure• Stratum 1: platelet = 0• Life Table Survival Estimates

• Conditional• Effective Conditional Probability• Interval Number Number Sample Probability Standard• [Lower, Upper) Failed Censored Size of Failure Error Survival Failure

• 0 10 4 0 9.0 0.4444 0.1656 1.0000 0• 10 20 2 1 4.5 0.4444 0.2342 0.5556 0.4444• 20 30 0 0 2.0 0 0 0.3086 0.6914• 30 40 1 0 2.0 0.5000 0.3536 0.3086 0.6914• 40 50 0 0 1.0 0 0 0.1543 0.8457• 50 60 1 0 1.0 1.0000 0 0.1543 0.8457

Effective sample size: whenever there is censoring (withdrawal or loss), we assume that, on average, those individuals who became lost or withdrawn during the interval were at risk for half the interval.

Thus, effective sample size (n*)= n – ½ (censoring #)E.g., effective sample size (1st interval) = 9 – ½ (0) = 9

N*

E.g., effective sample size (2nd interval) = 5 – ½ (1) = 4.5

• The LIFETEST Procedure• Stratum 1: platelet = 0• Life Table Survival Estimates

• Conditional• Effective Conditional Probability• Interval Number Number Sample Probability Standard• [Lower, Upper) Failed Censored Size of Failure Error Survival Failure

• 0 10 4 0 9.0 0.4444 0.1656 1.0000 0• 10 20 2 1 4.5 0.4444 0.2342 0.5556 0.4444• 20 30 0 0 2.0 0 0 0.3086 0.6914• 30 40 1 0 2.0 0.5000 0.3536 0.3086 0.6914• 40 50 0 0 1.0 0 0 0.1543 0.8457• 50 60 1 0 1.0 1.0000 0 0.1543 0.8457

Conditional Probability of Failure: Number failed / Effective Sample Sizee.g., P(F) (1st interval) = 4/9 = .44

P(F)

e.g., P(F) (2nd interval) = 2/4.5 = .44Survival probability (in each interval) = 1- failure probability (in each interval)Cum Survival Prob (S(t)) = S (t-1) * S(t)

e.g., S(1) = 1 * (1-.4444) = 1* 0.5556 =.5556e.g., S(2) = S(0)* S(1) * S(2)

S(2) =1*(1-.4444)* (1-.4444) =1 * .5556 * .5556 = .3086

Cumulative Survival

Kaplan-Meier (Product-limit) Method

• Time is partitioned into variable intervals

Whenever a case arises, set up a time interval. Use the actual censored and event times

If censored times > last event time, then the average duration will be underestimated using KM method

Kaplan-Meier Method

Patient 1

Patient 2

Patient 3

Patient 4

Patient 5

Patient 6

died

died

died

died

Lost to follow-up

Lost to follow-up

Months Since Enrollment

4 10 14 24

Kaplan-Meier Method(1)

Times to death from

starting treatment (Months)

(2)Number alive at

each time

(3)Number who died at each

time

(4)HAZARDProportion who died at that time:(3)/(2)

(5)Proportion

who survived at that time:

1.00-(4)

(6)Cumulative Survival

4 6 1 .167 .833 .833

10 4 1 .250 .750 .625

14 3 1 .333 .667 .417

24 1 1 1.00 .000 .000

Kaplan-Meier Plot (N=6)

0 4 10 14 24

Months After Enrollment

% Surviving

0

20

40

80

100

60

.833

.625

.417

.0

Kaplan-Meier Curve (N = 5,398).

Days to Settlement Censored at Nov 1, 1997

1400120010008006004002000

Pro

po

rtio

n o

f C

laim

an

ts

1.0

.8

.6

.4

.2

0.0

Tort

No Fault 1

No Fault 2

“Effect of eliminating compensation for pain and suffering on the outcome of insurance claims for whiplash injury” Cassidy JD et al., N Engl J Med 2000;342:1179-1186

Median Survival Time

Days to Settlement Censored at Nov 1, 1997

1400120010008006004002000

Pro

po

rtio

n o

f C

laim

an

ts

1.0

.8

.6

.4

.2

0.0

Tort

No Fault 1

No Fault 2

Semi-Parametric Methods

• Not required to choose some particular probability distribution to represent survival time

• Incorporate time-dependent covariates

Example: exposure increases over time as with drug dosage or with workers in hazardous occupations

Cox Proportional Hazards Model

• 1. Proportional Hazards Model

Basic Model of the hazard for individual i at time t

hi(t) = 0(t) exp{β1xi1 + ….. + βkxik}Baseline hazard

functionNon-negative

Linear function of fixed covariates

Take the logarithm of both sides, log hi(t) = (t) +β1xi1 + ….. + βkxik

log 0(t) No need to specify the

functional form of baseline hazard function

Cox Proportional Hazards Model• 1. Proportional Hazards Model

Consider the hazard ratio of two individuals i and j hi(t) = 0(t) exp{β1xi1 + ….. +

βkxik}hi(t) = 0(t) exp{β1xj1 + ….. + βkxjk}

Hazard ratio = exp{β1(xi1 -xj1) ….. + βk(xik-xjk)} Hazard functions are multiplicatively

related, hazard ratio is constant over survival time.

Hazards of any two individuals are proportional.

Cox Proportional Hazards Model• 2. Partial Likelihood Estimation

Estimate the β coefficients of the Cox model without having to specify the baseline hazard function 0(t)

Partial likelihood depends only on the order in which events occur, not on the exact times of occurrence.

Partial likelihood estimates are not fully efficient because of loss of information about exact times of event occurrence

• No intercept h0(t): an arbitrary function of time. Cancel out of the estimating equations

Interpretation of Coefficients

eβ: Hazard ratioIndicator variables (coded as 0 and 1)

Hazard ratio of the estimated hazard for those with a value of 1 in X to the estimated hazard for those with a value of 0 in X (controlling for other covariates)

Quantitative (Continuous) variablesEstimated percent change in the hazard for each one-unit increase in X. For example, variable AGE, eβ=1.5, which yields 100(1.5 - 1) =50. For each one-year increase in the age at diagnosis, the hazard of death goes up by an estimated 50 percent, controlling for other covariates.

Lab 4: estimating survival curves and Cox models in SUDAAN