Flexible Survival Modeling in SASbeyond LIFETEST and PHREG

Presented to Nova Scotia Sas Users Group meeting, Feb 22, 2013

By Ron Dewar – Dalhousie University and Cancer Care Nova Scotia

Flexible survival modeling:today’s objectives

• Introduce time-to-event analysis• Survival analysis in sas• Survival modeling - some recent

developments• Availability of software (stata, R)• Progress in converting to sas macros:

stata stpm2, predict, rcsgen• The road ahead

Time-to-Event• Duration from start to event for each subject• Status at time (experienced event or still at risk)• Non-informative censoring: censoring does not

change risk of eventually experiencing event• Covariate patterns among subjects at risk (at the

time of an event)• Other aspects:

left truncation (delayed entry)attained age as time scalefixed or time-varying covariatesstrata informative censoring (competing risks)

Lifetest, Phreg• Lifetest

plotted output step/smoothed (survival, hazard)life table estimatessignificance testing between strata

• Phreg (cox regression)left truncationcovariates (numeric and categorical), strata,Wald tests, LR tests, AIC, BICplotted output(survival, hazard as step functions)regression diagnostics

Lifetest, Phreg (cont.)• Covariate time-dependency is common and

(maybe) more interesting• Hazard Ratio (HR) is difficult to interpret at an

individual level• Interest in other survival functions – hazard,

hazard differences, survival differences, relative survival, cumulative hazard, crude probability (of event)

• Parametric representation of survival functions and time-dependency relations (out of sample prediction)

Royston – Parmar survival model2002 Statistics in Medicine 21: 2175–2197

• Cumulative hazard scale – equiv. to hazard scale if no time-dependencies in covariates

• Restricted cubic splines for cum. hazard• Implemented in Stata 11 as Stpm2 (model fitting)

and Predict (post-estimation)• Similar (?) implementation in R• ‘cure’ models, left truncation, covariate time-

dependency, strata, relative survival (excess hazard), net survival, other scales

Development plan, resources

• Stata code and academic papers• Email support from module author• Replication of all features may not be feasible

(limited programming resources, obscure stata features)

• Basic functionality that replicates results• SAS code that can be used, understood,

modified, enhanced by collaborators

Proportional Hazards ModelBreast cancer survival with proportional hazards

Ods pdf file = “&loc.\doc\breast2004\cox regression.pdf”;proc phreg data = _events_;

class sstage (ref = first);model years*_death_(0) = sstage age1 age2 age3;title 'Breast cancer survival, 2004 – 2010, Nova Scotia’;title2 ‘followed to end of 2011’

title3 'Proportional Hazards model with stage, age';run;Ods pdf close;

…Number of Observations Read 5475Number of Observations Used 5475

Percent Total Event Censored Censored 5475 830 4645 84.84…

PHREG output

Parameter Standard Parameter DF Estimate Error Chi-Square Pr > ChiSq sstage II 1 0.77412 0.10221 57.3652 <.0001 sstage III 1 1.72777 0.10591 266.1493 <.0001sstage IV 1 3.23040 0.10948 870.6004 <.0001age1 1 0.18070 0.12381 2.1300 0.1444age2 1 0.84798 0.11689 52.6287 <.0001 age3 1 1.58877 0.11833 180.2785 <.0001

PHREG output

Outline of analysis stepsR-P model

• Create ‘standard’ dataset: dataset name, key variable names are fixed (%sas_stset)

• Describe and fit model (%sas_stpm2)• Estimate functions of fitted model parameters

(%predict) • plot predicted functions (eg, with Proc SGPlot)

Fit above model using stpm2()

• 3 stage and age binary variables, hazard scale, 5 df for baseline

• Stata command line Stpm2 st2 st3 st4 age1 age2 age3, scale(hazard) df(5)

• Sas macro call %Sas_stmp2( st2 st3 st4 age1 age2 age3, scale=hazard, df = 5);

Stpm2: baseline knots• Log cumulative hazard is parameterised with

restricted cubic spline functions:array z(*) rcs1 - rcsm < m spline variables> ;array k(*) < m knot values >; z(1) = y * log time to event; do j = 2 to m ; phi = (k(m) - k(j) )/(k(m) - k(1) ); z(j) = ( y > k(j) )*( y - k(j) )**3 - phi*( y > k(1) )*( y - k(1) )**3

- (1 - phi)*( y > k(m) )*( y - k(m) )**3; end;

How many knots to use?Where to put them?

• Too few: unrealistic representation of hazard• Too many: over-parameterisation. Unrealistic lumps

and bumps• AIC and BIC may be helpful. LR tests are not. models

are not nested• Some subject matter knowledge can be helpful• Choice of position probably doesn’t matter too much• ‘standard’ positions : centile points of cumulative

distribution of times of non-censored events

Effect of choice df

AIC BIC stage II stage III stage Iv1 5171.4 5224.2 2.2 5.6 24.52 5136.8 5214.2 2.2 5.6 25.33 5136.5 5222.6 2.2 5.6 25.44 5127.9 5222.5 2.2 5.6 25.45 5124.9 5228.2 2.2 5.6 25.46 5119.2 5231.1 2.2 5.6 25.47 5106.4 5226.9 2.2 5.6 25.48 5104.1 5233.2 2.2 5.6 25.4

PHREG 2.2 5.6 25.3

Hazard Ratio estimates

interior knots for baseline hazard

Programming in %sas_stpm2()

• Describe model in macro call• Internal macro strings drive subsequent

processing:• compute spline functions • 1st derivatives• orthogonalisation• Define linear predictor, log likelihood• derive initial values for optimisation• fit model (maximum likelihood with proc nlmixed)• save results for later processing

Key macro strings• _null_ data step to build macro strings• call symput(‘macro_var’, string)• Linear predictor:

independent variables parameters to be estimated

• Log likelihood:function to be maximisedlinear predictor1st derivative of linear predictorcensor indicator

Linear predictor, Likelihood %Sas_stmp2( st2 st3 st4 age1 age2 age3,

scale=hazard, df = 5);Linear predictor: &xb.cons*_cons + st2*_st2 + st3*_st3 + st4*_st4 + age1*_age1 + age2*_age2 + age3*_age3 + rcs1*_rcs1 + rcs2*_rcs2 + rcs3*_rcs3 + rcs4*_rcs4 + rcs5*_rcs5

1st derivative: &dxb.rcs1*_drcs1 + rcs2*_drcs2 + rcs3*_drcs3 + rcs4*_drcs4 + rcs5*_drcs5Log likelihood:_death_*((&dxb.) + &xb.) – exp(&xb.)

Linear predictorCovariates:xb = ifc (&int., 'cons*_cons + ', ' ');do _i_ = 1 to &n_cov.;

var = scan("&covar.",_i_);xb = trim(xb)||' '||trim(var)||'*_'||trim(var)||' + ' ;

end;Splines:do _i_ = 1 to &df.;

xb = trim(xb) ||' rcs'||put(_i_,1.)||'*'||'_rcs'||put(_i_,1.)|| ifc(_i_< &df.," + ", " ");

end;

Example: colon cancer* data must be sorted by unique ID;proc sort data = example; by pid;run;* set up a standardised survival dataset;%sas_stset(example, censor(0), years , pid ) ;* fit model of interest;%sas_stpm2(st1 st2 st3 nsex, scale=hazard, df=3 );* predicted hazard, survival functions for IIb cases;%predict(haz, hazard, at = st2:1 zero);%predict(surv, survival, at = st2:1 zero);

Example: time-varying covariate%sas_stpm2(st1 st2 st3 nsex, scale=hazard, df=3,

tvc= st2 st3 ,dftvc= st2:2 1 );

* hazard prediction;%predict(haz1, hazard, at = st2:1 zero);

* Hazard ratio prediction;%predict(hr1, hratio,

hrnum = st2:1 zero, hrdenom= st2:0 zero);

Example 2: time to initiation of chronic opiod use in new cancer patients

• t0: date of new cancer diagnosis• t1: date of initiation of chronic opiod pain medication• Censor: death, end of study period (or 365 days)• Tier: cancer type grouped by 5-year survival probability• Age: 10-year age groups• Other covariates: year of diagnosis, urban/rural, sex

%sas_stpm2(t2 t3 a1 a2 a3 a4 a5 a6 nsex nurb y, scale=hazard, df=3, tvc= a2 a6, dftvc = 2);

Example 2%macro int(row =, col =, sel = );

…%predict(surv, survival, at = &sel. nsex:1 nurb:1 y:3 zero);

…%mend;

And then, each for of the 21 combinations of 3 tiers X 7 age groups:…%int( row = t2, col = ag2, sel = a1:1 t2:1);%int( row = t3, col = ag2, sel = a1:1 t3:1);

…

The road ahead• Documentation (!no, really?)• Consistency checking• Confidence intervals for cumulative functions

(survival, cumulative hazard)• Out of sample estimation is inefficient• Other survival scales (cumulative log odds,

probit…)• Cure models• Stratified analysis• Competing risks framework

The future• Use of an optimsation routine that permits

analytic 1st and 2nd derivatives (gradient & hessian)

more efficient predictionout of sample prediction

• Re-code string modification in %predict()