This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis...
Transcript of This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis...
![Page 1: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/1.jpg)
Copyright 2009, The Johns Hopkins University and John McGready. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site.
![Page 2: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/2.jpg)
Regression for Survival Analysis
John McGready Johns Hopkins University
![Page 3: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/3.jpg)
3
Lecture Topics
Cox proportional hazards (PH) regression
Interpreting coefficients from Cox PH regression
Performing inference on Cox PH regression coefficients
![Page 4: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/4.jpg)
Section A
The Cox Proportional Hazard Regression Model
![Page 5: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/5.jpg)
5 Continued
Multivariate Analysis
In Statistical Reasoning 1 we learned why we need a different approach to analyzing time to event data in the presence of censoring
We learned to estimate the “survival function” via the Kaplan Meier method for a cohort of subjects
We learned to statistically compare the “survival functions” of two or more cohorts via the logrank and Wilcoxon-Gehan tests
![Page 6: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/6.jpg)
6 Continued
Multivariate Analysis
However, those statistical tests did not provide an estimate of the magnitude of the difference in survival for the cohorts being compared
Further, the tests allowed for comparisons based on one grouping factor (predictor) at a time
![Page 7: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/7.jpg)
7
Multivariate Analysis
How can we get an estimate of the magnitude of the survival-predictor relationship of interest?
How can we account for multiple factors simultaneously for each subject in a time to event study?
How can we estimate adjusted survival-predictor relationships in the presence of potential confounding?
![Page 8: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/8.jpg)
8 Continued
Regression Methods for Censored Survival Data
Objective Relate survival times to (potentially multiple)
predictors (the x’s or independent variables)
![Page 9: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/9.jpg)
9
Regression Methods for Censored Survival Data
Problem Can’t use ordinary linear regression because how
do you account for the censored data? Can’t use logistic regression without ignoring the
time component
![Page 10: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/10.jpg)
10 Continued
Proportional Hazards Regression Model
Developed by D.R. Cox (1972)
Relates survival time to predictors
Handles incomplete follow-up Censoring
![Page 11: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/11.jpg)
11
Proportional Hazards Regression Model
It assumes the ratio of time-speci#c outcome (event) risks (hazard) of two groups remains about the same over time
This ratio is called the hazards ratio or the relative risk
![Page 12: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/12.jpg)
12 Continued
Proportional Hazards Assumption
All Cox regression requires is an assumption that ratio of hazards is constant over time across groups
The good news—we don’t need to know anything about overall shape of risk/hazard over time
The bad news—the proportionality assumption can be restrictive
![Page 13: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/13.jpg)
13 Time
Haz
ard
Smokers
Non-Smokers
Constant Ratio
Continued
Proportional Hazards Assumption
![Page 14: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/14.jpg)
14 Time
Haz
ard
Vegetarians
Vegans
Carnivores
Proportional Hazards Assumption
![Page 15: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/15.jpg)
15 Continued
Cox Proportional Hazards Model
Formulation of model: Group hazard = Baseline hazard × “group factor”
![Page 16: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/16.jpg)
16 Continued
Cox Proportional Hazards Model
Formulation of the model:
![Page 17: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/17.jpg)
17 Continued
Cox Proportional Hazards Model
Such that . . .
![Page 18: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/18.jpg)
18
Cox Proportional Hazards Model
312 patients with primary biliary cirrhosis (PBC) studied at the Mayo clinic
Patients were followed from diagnosis until death or censoring
Information available includes sex and age (years) of each patient
Question—how do patient’s age and sex predict survival?
![Page 19: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/19.jpg)
19 Continued
Data as It Appears in Stata
Here is a snippet of the data:
![Page 20: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/20.jpg)
20
Data as It Appears in Stata
The variables:
survyr is a time measurement in years
death is an indicator of death (1) or censoring (0)
sex is an indicator (1 = female, 0 = male)
ageyr is age in years
![Page 21: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/21.jpg)
21 Continued
Letting Stata Know It Is Time to Event Data
The “stset” command tells Stata that we have time to event data—Stata converts it internally and then we have use of a bunch of built in commands
Syntax . . .
stset time_variable, failure(event_variable =1)
So for our data the syntax is . . .
stset survyr, failure(death=1)
![Page 22: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/22.jpg)
22
Letting Stata Know It Is Time to Event Data
Here is what Stata does . . .
![Page 23: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/23.jpg)
23 Continued
Using Stata for Survival Analysis
Let’s look at Kaplan-Meier curves by sex
Command: sts graph, by(sex)
![Page 24: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/24.jpg)
24 Continued
Using Stata for Survival Analysis
To do a log rank test
Command: sts test, by(sex)
![Page 25: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/25.jpg)
25
Using Stata for Survival Analysis
So there is visual evidence that females have longer survival than males
The results of the log-rank test show a signi#cant difference in the survival experience of males and females
However, so far we have no measure of the association between longer survival and being female—how can we get this?
![Page 26: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/26.jpg)
26 Continued
Example
How about a regression model?
(here is sex, 1 for females and 0 for males)
![Page 27: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/27.jpg)
27 Continued
Example
Let’s #gure out how to interpret Model for females
![Page 28: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/28.jpg)
28 Continued
Example
Let’s #gure out how to interpret Model for males
![Page 29: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/29.jpg)
29 Continued
Example
Taking the difference
![Page 30: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/30.jpg)
30 Continued
Example
Invoking our favorite property of logs
So is the log of the hazard ratio (relative risk) of death for females compared to males
![Page 31: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/31.jpg)
31
Such that . . .
So is the hazard ratio (relative risk) of death for males to females
Example
![Page 32: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/32.jpg)
32 Continued
Interpreting Coefficients
Interpretation > 0: Higher hazard (poorer survival) associated
with being female Because > 1
![Page 33: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/33.jpg)
33 Continued
Interpreting Coefficients
Interpretation < 0: Lower hazard (better survival) associated
with being female Because < 1
![Page 34: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/34.jpg)
34
Interpreting Coefficients
Interpretation = 0: No association between hazard (and
survival) and being female Because = 1
![Page 35: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/35.jpg)
35 Continued
Example
With a sample of data, we are only going to be estimating —so the regression equation we estimate from our sample of 312 patients looks like . . .
![Page 36: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/36.jpg)
36
For this dataset, the estimated regression equation is . . .
So
Example
![Page 37: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/37.jpg)
37
Interpretation
So we’ve estimated a negative association between death and being female—the hazard (risk) of death is lower for females in this sample
How to describe this association?
The estimate hazard ratio (relative risk) of death for females relative to males is = .62
Females have .62 the hazard (risk) that males have of death (or females have 38% lower hazard (risk) of death than males)
![Page 38: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/38.jpg)
38
The “stcox” command will estimate the regression
General syntax: stcox
Where is the predictor of interest
Notice we do not need to specify an outcome (Stata already knows it because we have declared the time and censoring variables with the stset command!)
Getting Stata To Do the Work
![Page 39: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/39.jpg)
39 Continued
The Stata Output
Results from the stcox command
![Page 40: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/40.jpg)
40 Continued
The Stata Output
Notice stcox does the conversion to the hazard ratio for us!
If we wanted information about the coefficient, , we could use the “nohr” option
General syntax: stcox , nohr
![Page 41: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/41.jpg)
41
The Stata Output
Output with “nohr” option
![Page 42: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/42.jpg)
42 Continued
Accounting for Sampling Variability
The coefficient and hazard ratio estimates are based on an imperfect sample of 312 subjects from a large population/process
To complete the story, we need to incorporate sampling error into these estimates via a con#dence interval and a p-value
![Page 43: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/43.jpg)
43 Continued
Accounting for Sampling Variability
Luckily, Stata does this for us!
![Page 44: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/44.jpg)
44 Continued
Accounting for Sampling Variability
The con#dence interval gives a range of plausible values for the true hazard ratio of death for females compared to males in the population of PBC patients
The p-values are testing . . .
: hazard ratio =1
: hazard ratio ≠ 1
![Page 45: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/45.jpg)
45 Continued
Accounting for Sampling Variability
Describing the results
In a sample of 312 PBC patients, females had a lower hazard (risk) of death than males. The estimated hazard ratio was .62 indicating that females had 38% lower hazard (risk) of death than males.
Accounting for sampling variability, the decrease in risk for females could be as large as 62% or as small as 3% (95% CI for the hazard ratio 0.38–0.97).
![Page 46: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/46.jpg)
46 Continued
Accounting for Sampling Variability
Where do the CIs and p-value come from?
It turns out all inference is done on the coefficient scale
The 95% con#dence interval for is
The endpoints of this 95% CI can be exponentiated to get the 95% CI for the hazard (risk) ratio
![Page 47: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/47.jpg)
47
Accounting for Sampling Variability
Testing
: = 0 is equivalent to testing
: (hazard ratio) = 1
This is done by computing a test statistic :
And comparing to standard normal curve to get p-value
So its business as usual!
![Page 48: This work is licensed under a Creative Commons Attribution ... · Using Stata for Survival Analysis Let’s look at Kaplan-Meier curves by sex Command: sts graph, by(sex) Continued](https://reader035.fdocuments.in/reader035/viewer/2022070816/5f105bd37e708231d448b7bb/html5/thumbnails/48.jpg)
48
The Stata Output
Try it out using results below !