Survival Analysis Final

Question 1

Cox Model Assumptions

Answer 1

1. Proportional Hazards
2. Linear functional form of covariates
3. Well fit to data
4. No influencing points

Question 2

Res. For PH Assumption

Answer 2

Scheonfeld

Score

Question 3

Linearity of Covariates

Answer 3

Martingale

Question 4

Overall Fit

Answer 4

Cox-Snell

Question 5

Outliers

Answer 5

Score

Deviance

Question 6

Influential Points

Answer 6

Score

Deviance

Question 7

Stratification Assumption

Answer 7

That there is a different baseline hazard rate for each strata
That the effect of the covariate is the same
That the effect of the stratified variable doesnt need to be estimated.

Used for matched pair data, differing clinical trial data, and PH violators

Question 8

Proportional Hazard AFT Models

Answer 8

Weibull and Exponential

Question 9

Proportional Odds AFT Models

Answer 9

Log Logistic

Question 10

AFT Interpretation

Answer 10

exp(Beta) is the multiplicative factor of TIME

Question 11

Cox Model Building

Answer 11

1. Start with global null test
2. Find what confounders are significant and lower the AIC
3. Add these one at a time to the model
4. Continue till there are no more significant factors

Question 12

AIC

Answer 12

AIC=-2LogL+pk

k=2, p=number of parameters, L=likelihood eqn.

Question 13

Local Test

Answer 13

Tests a subset of q parameters of the full (p) model
b1=qx1 vector, b2=(p-q)x1 vector

When using the c vector you are using a Wald test with a chi square distribution with q DF

Question 14

Piecewise Exponential Parametric Model

Answer 14

Separate the model so there are different models for different chunks of time
There will be different coefficients and baseline rates for each chunk

Question 15

Accelerated Failure Time Models

Answer 15

Y=log(t)=bX+s*e

b is parameters, X is the covariate, s is the unknown scale, and e is the error from known distribution

Question 16

Parametric Model Difference

Answer 16

Gives shape to the hazard curve where cox doesn’t. Treats the baseline hazard as a non-needed parameter.

Use MLE instead of partial likelihood since all data goes into the model and not just the ones with observed event times

PH doesn’t hold true for the majority of the models

Question 17

Using Parametric and NP Models (Which one)

Answer 17

Check fit (CS Res), outliers (score), AIC

Bias due to omitted due to variables is more important than the model fit. Most will give roughly the same

NP is 95% as efficient as Parametric

Question 18

95% CI on HR in Cox

Answer 18

exp(B+/-1.96*SE)

Question 19

Partial Likelihood with Ties

Answer 19

1. Breslow (use with few ties/default in SAS)
   Adds each of their likelihood to eqn. at time point
2. Efron
   Closer to the actual partial likelihood
   Acts the same as Breslow with small ties
3. Cox
   Assumes a logistic model for h(t)

Question 20

Likelihood Ratio Test

Answer 20

Must estimate the Betas

Chi Square with p Df (q DF for local test)

Question 21

Wald Test

Answer 21

Must estimate the Betas

Chi Square with p Df (q DF for local test)

Question 22

Score Test

Answer 22

Don’t need to estimate Betas with the global test
Do need to estimate with local test

Chi Square with p DF (q DF for local test)

With no ties the score test becomes a log rank test

Question 23

Partial Likelihood for Cox

Answer 23

It is the product of:
hazard of the individual that died at that time point/(sum of the hazard for all individuals at risk just prior to the event time)

Take the partial derivative of log likelihood with respect to each beta and set it equal to zero. Then solve with an iterative method

Question 24

Martingale Residual

Answer 24

Difference over time of observed number of events minus the expected under the model
Calculated off individual
Checks for the functional form fit of each covariate
Transformation of the CS residuals

Question 25

Deviance Residual

Answer 25

Normalized transformation of the Martingale residuals Should be symmetric and random around zero Can find outliers and leverage Calculated off of the individual

Question 26

Cox-Snell Residuals

Answer 26

Pro: Checks the fit of the overall model Calculated off of the individual ~exp(1) with good fit so a plot of H(Rcs) vs. Rcs should be a linear line with slope=1 and intercept=0 Con: Doesn't say what type of departure there is. Betas in this case must be well estimate (PH, linearity, no outliers)

Question 27

Score Residuals

Answer 27

Based on the score eqn of partial likelihood Able to assess PH and find outliers in the data Finds the outliers by finding the abnormal covariate profiles Calculated for each covariate

Question 28

Covariate Specific Residuals

Answer 28

Score | Schoenfeld

Question 29

Individual Specific Residuals

Answer 29

Martingale Deviance Cox-Snell

Question 30

Schoenfeld Residuals

Answer 30

Partial Score residuals based off of individuals with observed times Assess PH for each covariate Plot of time vs. scaled schoenfeld residuals should be a random scatter around zero with no time dependency Will give information on location and type of violation of PH (time where there is a switch, points of leverage)

Question 31

PH Assumption Test

Answer 31

Residual Plots (Schoenfeld, Score) Significance test (z2*ln(t)) H0 vs. Time plot Anderson Plot

Question 32

Fixing PH

Answer 32

Stratify by the covariate Use a different model like AFT that doesnt require PH assumption Partition time and fit multiple models to the data *After PH fails check other covariates to see if they are still an issue. A wrong functional form can appear like a non-PH model

Question 33

Finding Outliers

Answer 33

A plot of the deviance vs. the risk score should be a flat line around zero Plot of score vs. ID

Question 34

Cox Model Formula

Answer 34

h(t|Z)=h0(t)exp(B'Z)

Question 35

Semiparametric Explanation of Cox

Answer 35

H0(t) is a nonparametric formula for hazard and c() is a know parametric function

Question 36

Log Likelihood Equation

Answer 36

Chi Square=2[LL(b)-LL(B0)]