1. Stata: Flexible Parametric Survival models 1h
Hein Stigum
Presentation, data and programs at:
courses
Feb-20
H.S.

2. Programs and Datasets Syntax File
0 Flexible Parametric Survival Course, Downloads
Programs
ssc install stpm2, replace
Datasets
Paul Dickman course files
Book files
Feb-20
H.S.

3. Credits: Patrick Royston Paul Lambert Paul Dickman Michael Crowther
Feb-20
H.S.

4. Introduction
Feb-20
H.S.

5. Effect measures Risk Rate Hazard probability, proportion, %
Km/h, cases/person-time
Example: Diabetes type 1
2 math quatities
prop: fraction, numerator (top) is part of denominator (bottom)
ex: colds in class=2/30=7% , no dimension, 0-1
rate: change in one quantity per change in another (time),
ex: speed, drive 100 km in 2 h then average speed is 50 km/h, dimension, no upper bound
Odds: disease per healthy person
Statistical concept:
risk: probability, no dimension,
ex: flip coin
ℎ 𝑡 = lim 𝛿𝑡→0 Pr⁡(𝑡≤𝑇<𝑡+𝛿𝑡|𝑇≥𝑡) 𝛿𝑡
Probability of event
in a small time interval,
given that you have survived
Feb-20
Feb-20
H.S.
H.S. 5 5 5

6. Cohorts Closed cohort Open cohort Count persons, risk
Count person-time, rate
Statistics
Open cohort, new time scale
Survival data
Line= follow up
Circle=event (disease)
Red line=exposed
Time to event data
Censored data
MF 9510 Logistic, Survival and Cox
Feb-20
H.S.

7. Timescale
Feb-20
H.S.

8. Timescale Possible timescales: Chose timescale: Stata commands, st:
Time since diagnosis
Age
Calendar time
Chose timescale:
the most important
and most non-linear
as analysis time
Stata commands, st:
In this study we follow subjects from diagnosis
to death, lost to follow-up or end of study
enter start of follow-up
exit end of follow-up
origin start of risktime
failure event (death) or not
Feb-20
H.S.

9. Survival time setup Time=years since diagnosis, based on a variable
stset ysdiag, failure(dead==1)
Time=years since diagnosis, based on dates
stset exit, failure(dead==1) origin(diag) scale(365.25)
Time=age, based on dates
stset exit, failure(dead==1) origin(birth) enter(diag) scale(365.25)
Feb-20
H.S.

10. Standard survival analysis
Kaplan-Meyer plot of survival by treatment
sts graph, by(trt) risktable
Log rank test of difference in survival
sts test trt
Cox proportional hazard model with one covariate
stcox trt
Predicted hazard curves from Cox model
stcurve, hazard at1(trt=0) at2(trt=1)
Feb-20
H.S.

11. Syntax “5,1 Flexible Parametric Survival, Basic“ “Standard Survival Analysis” line 24
Feb-20
H.S.

12. The hazard of hazard ratios
The hazard ratio is problematic under heterogeneity
RCT Example
Large variation in the risk of dying (heterogeneity, frailty)
Treatment halves the risk of dying (for all subjects)
Baseline: balanced, comparable groups
Over time treated: frail subjects will remain
Over time untreated: frail subjects will die out
HR may go from 0.5 (true) to 1.5 (misleading) over time
Use instead: survival, ?
Use HR with caution if large and varying risk of event
(Hernan 2010)
Feb-20
H.S.

13. Smoothers in regressions
Polynomials
x, x2, x3
Fractional polynomials (2 of 8)
x-2, x-1, x-0.5 log(x), x0.5 x, x2, x3
Splines (restricted cubic splines, rcs)
cubic
2 of 6 =28 models
2 of 8= c1 c2
Smoothers in survival:
Dose-response
Baseline hazards
Time Dependent effects
Polynomials: global
Splines: local
Feb-20
H.S.

14. A physical “spline”
Feb-20
H.S.

15. Reference risk, baseline hazard
Log-risk model
Twice the risk of what?
RR=2
Always report the reference (baseline) risk
Cox model
Twice the rate of what?
HR=2
Always report the baseline hazard, h(t)
Peak at 0.25 years
Still positive hazard
after 6 years
Hazards near 0 may give large HR
Feb-20
H.S.

16. Proportional vs non-proportional
baseline
Hazards
how to model
baseline,
dose-response,
and TD
Non-
Hazard Ratios
time dependent, TD
Feb-20
H.S.

17. Baseline, dose-response and TD
t= time
E= exposure (cont.)
s(.)=spline
TD=Time Dependent
Generic models
l𝑛 ℎ 𝑡 =
𝑏 0 𝑡+ 𝑏 1 𝐸
Linear baseline, linear E
𝑠 0 (𝑡)+ 𝑏 1 𝐸
Flexible baseline
𝑠 0 (𝑡)+ 𝑠 1 (𝐸)
Flexible dose response
𝑠 0 𝑡 + 𝑏 1 𝐸+ b 2 t E
linear TD
𝑠 0 𝑡 + 𝑏 1 𝐸+ 𝑠 2 𝑡 E
Flexible TD
𝑠 0 𝑡 + 𝑠 1 𝐸 + 𝑠 2 𝑡 s 1 (E)
Flexible TD and dose response
Royston Parmar models (proportional hazard)
l𝑛 𝐻 𝑡 =𝑠( ln 𝑡)+ 𝑏 1 𝐸
Proportional hazard
l𝑛 𝐻 𝑡 =𝑠( ln 𝑡)+ 𝑏 1 𝐸+ 𝑠 2 ( ln 𝑡) 𝐸
Flexible TD
Feb-20
H.S.

18. Survival models Cox Parametric Flexible Baseline hazard: free Estimate
time
time
Baseline hazard:
free
Estimate
baseline hazard:
may be
unstable
easy, stable
easy, stable
Feb-20
H.S.

19. Royston-Parmar models
Feb-20
H.S.

20. Royston-Parmar models
3 flavors Generalized:
Proportional hazard Weibull
Proportional odds log logistic
Probit log normal
Stata syntax
stpm2 variables, scale(hazard) df(3)
Survival time
parametric model
version 2
Proportion hazard model
3 degrees of freedom
for baseline hazard =
3 cubic splines
df(1) = Weibull
R: flexsurv
Feb-20
H.S.

21. Syntax “5,1 Flexible Parametric Survival, Basic“ “Unstable hazards from Cox model” line 24
Feb-20
H.S.

22. Predict
Feb-20
H.S.

23. Powerful predict command
Survival
Hazard
Model
Proportional
hazards model
Survival difference
Hazard ratio
Hazard difference
Feb-20
H.S.

24. Predict Model Predict stpm2 year8594, scale(hazard) df(4) eform
predict s, survival
survival
predict sd, sdiff1(year8594 1) ci
survival difference
predict h, hazard per(1000)
hazard
predict hd, hdiff1(year8594 1) ci
hazard difference
predict hr, hrnumerator(year8594 1) ci
hazard ratio
+++
Feb-20
H.S.

25. Out of sample predictions
Predict on a new timescale to:
get nice plots in small data
speed up predictions in big data
predict beyond observed time
make a table of results
Model
stpm2 year8594 female, scale(hazard) df(4) eform
New time
range t
t2 from 0-20, 100 values
Predict
predict s0, survival zero timevar(t2)
OBS, all covariates must be specified when using a new timescale
Feb-20
H.S.

26. Syntax “5,1 Flexible Parametric Survival, Basic“ “Predict options” line 24
Feb-20
H.S.

27. Dose response modeling
Replace exposure variable with splines
no margins command
more difficult predictions
Four methods shown in the syntax
The last is perhaps the easiest
Feb-20
H.S.

28. Non-proportional hazards
Time dependent effects
Non-proportional hazards
Feb-20
H.S.

29. Proportional vs non-proportional
baseline
Hazards
how to model
baseline,
dose-response,
and TD
Non-
Hazard Ratios
time dependent, TD
Feb-20
H.S.

30. Baseline, dose-response and TD
t= time
E= exposure (cont.)
s(.)=spline
TD=Time Dependent
Generic models
l𝑛 ℎ 𝑡 =
𝑠 0 (𝑡)+ 𝑏 1 𝐸
Flexible baseline
𝑠 0 (𝑡)+ 𝑠 1 (𝐸)
Flexible dose response
𝑠 0 𝑡 + 𝑏 1 𝐸+ b 2 t E
linear TD
𝑠 0 𝑡 + 𝑏 1 𝐸+ 𝑠 2 𝑡 E
Flexible TD
Royston Parmar models (proportional hazard)
l𝑛 𝐻 𝑡 =𝑠( ln 𝑡)+ 𝑏 1 𝐸
Proportional hazard
l𝑛 𝐻 𝑡 =𝑠( ln 𝑡)+ 𝑏 1 𝐸+ 𝑠 2 ( ln 𝑡) 𝐸
Flexible TD
Feb-20
H.S.

31. Non-Proportional Cox vs Flexible par.
Non-Proportional Hazards = TD
= Time Varying Coefficients (tvc)
Cox model
time dependent effect
tvc(var) texp(_t)
tvc(var) texp(_t>5)
tvc(var) texp(ln(_t))
stsplit
Syntax:
TD effects in
Cox versus RP
Flexible parametric
tvc(var) dftvc(2)
two splines
Feb-20
H.S.

32. Time varying coefficient
Model commands
Proportional hazards
stpm2 sex, scale(hazard) df(4) eform
Non-proportional hazards = Time Dependent HR
stpm2 sex, scale(hazard) df(4) eform tvc(sex) dftvc(2)
Time varying coefficient
of sex, with 2 splines
Testing for non-proportional hazards
Standard method
Easier method
Fit at PH model
Plot Schoenfeld residuals
Add “tvc” terms as needed
Add “tvc” terms
(test PH vs TD with LR-test)
Predict and plot HR
Syntax: Test Proportional Hazards assumption
“tvc” for confounders?
Feb-20
H.S.

33. Syntax “5,2 Flexible Parametric Survival, Time Dependent“ “PH versus TD models using female as exposure” line 30
Feb-20
H.S.

34. Summary so far Effect of sex on melanoma survival Next
PH model  constant HR
Non-PH model  time dependent HR (TD)
Next
TD hazard ratios for age
Are the sex estimates the same?
Handle huge HR caused by hazards close to zero
Plot HR
Plot HD (hazard differences)
Feb-20
H.S.

35. Syntax “5,2 Flexible Parametric Survival, Time Dependent“ “PH versus TD models using age groups” line 85
Feb-20
H.S.

36. Summary so far TD hazard ratios for age
sex estimates the same (PH vs TD)
Plot HR
Table of HR
Huge HR caused by hazards close to zero
Plot HD (hazard differences) as better measure
Plot survival
Plot survival difference
Feb-20
H.S.

37. Time dependent HR
Huge HR
Use HD instead
Proportional HR
Feb-20
H.S.

38. Summing up Standard Cox Standard Royston-Parmar
Proportional Hazards  constant HR
Limited options for non-proportional hazards
Standard Royston-Parmar
Flexible non-proportional hazards (splines)
Predict and plot
Hazards
Hazard Ratios
Hazard Differences
Survival
Survival differences
Feb-20
H.S.

39. Other features of RP models
Relative survival. Competing risk
Joint modelling of longitudinal and survival data
Nested Case-Control and Case-Cohort
Multistate models (event history analysis)
Simulation of survival data
Feb-20
H.S.

40. References
Bower H, Crowther MJ, Lambert PC Strcs: A command for fitting flexible parametric survival models on the log-hazard scale. Stata Journal 16:
Crowther MJ, Abrams KR, Lambert PC Flexible parametric joint modelling of longitudinal and survival data. Statistics in Medicine 31:
Crowther MJ, Lambert PC Framework and optimisation procedure for flexible parametric survival models reply. Statistics in Medicine 34:
Hinchliffe SR, Lambert PC. 2013a. Extending the flexible parametric survival model for competing risks. Stata Journal 13:
Hinchliffe SR, Lambert PC. 2013b. Flexible parametric modelling of cause-specific hazards to estimate cumulative incidence functions. BMC medical research methodology 13.
Lambert P, Royston P Flexible parametric alternatives to the cox model. In: Stata user group.
Lambert PC, Royston P Further development of flexible parametric models for survival analysis. Stata Journal 9:
Lambert PC, Wilkes SR, Crowther MJ Flexible parametric modelling of the cause-specific cumulative incidence function. Statistics in Medicine 36:
Mozumder SI, Rutherford MJ, Lambert PC A flexible parametric competing-risks model using a direct likelihood approach for the cause-specific cumulative incidence function. Stata Journal 17:
Nelson CP, Lambert PC, Squire IB, Jones DR Flexible parametric models for relative survival, with application in coronary heart disease. Statistics in Medicine 26:
Royston P, Parmar MKB Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine 21:
Wacholder S, Weinberg CR Flexible maximum-likelihood methods for assessing joint effects in case-control studies with complex sampling. Biometrics 50:
Feb-20
H.S.

41. Course
Feb-20
H.S.

42. Software
* From Statistical Software Components * Flexible Parametric Survival ssc install stpm2, replace * Restricted Cubic Splines ssc install rcsgen, replace ssc install strcs, replace * Competing risk ssc install stcompet, replace ssc install stcompadj, replace ssc install stpm2cm, replace ssc install stpm2cif, replace ssc install stpepemori, replace * Explained variance ssc install str2d, replace * Imputation ssc install stsurvimpute, replace * Symmetric nearest neighbor smoothing ssc install running, replace * Not from Statistical Software Components * Relative survival, manual install search strs
Feb-20
H.S.

43. Syntax and data files
* Download Paul Dickman course files mkdir c:\survival cd c:\survival net install all replace * Download Royston-Lambert book material net from net get fpsaus-dta net get fpsaus-do1 net get fpsaus-do2 * Syntax files for this course Opus>Metodelunsj>Delte dokumenter>Flexible Parametric Survival Course: 0 Flexible Parametric Survival Course, Downloads.do 1 Flexible Parametric Survival Course, Basic.do 2 Flexible Parametric Survival Course, TD.do 3 Flexible Parametric Survival Course, Spesial.do
Feb-20
H.S.

44. Extra slides
Feb-20
H.S.

45. Generalized Weibull model
Often used parametric model: Weibull
Log cumulative hazard is linear in log t:
ln 𝐻 𝑡 =𝑘 ln 𝑡− k ln 𝜆
Generalize to (any) smooth baseline hazard:
ln 𝐻 𝑡 =𝑠( ln 𝑡)− k ln 𝜆
s=spline
Stable estimates on the log cumulative hazard scale.
Splines generalize to (almost) any baseline hazard shape.
Proportional hazard model
Feb-20
H.S.

46. Purpose of regression Estimation Prediction
Estimate association between exposure and outcome adjusted for other covariates
Estimate the effect of smoking on lung cancer
Prediction
Use an estimated model to predict the outcome given covariates in a new dataset
Predict air pollution by distance from roads
DAGs, bias, precision
Predictive power, model fit, R2
Counfounding matter in the first
Fit of the models matters in the last
Feb-20
H.S.

47. Baseline, dose-response and TD
t= time
E= exposure (cont.)
s(.)=spline
tvc=time varying coeff
Generic model
l𝑛 ℎ 𝑡 =
𝑏 0 𝑡 𝑏 1 𝐸 b 2 t E
𝑠 0 𝑡
+ 𝑠 1 𝐸
+ 𝑠 2 𝑡 s 1 (E)
baseline
dose
response
Time
Dependent
Flexible parametric model
stpm2
agercs?,
df(3)
dftvc(2) tvc(agercs?)
Feb-20
H.S.

48. Cox versus Royston-Parmar-PH
Both are proportional hazards models Almost identical effect estimates But:
Feb-20
H.S.

49. Ex: unstable hazards from Cox model
FP=
Flexible
Parametric
Cox baseline
and treated
FP baseline
FP treated
Conclusion: unstable hazards from the Cox model (with artifacts)
Feb-20
H.S.

50. Relative survival Patient register Population Individual data
Tabular data
What is the survival of patients relative to the population?
SMR
Relative Survival
Feb-20
H.S.