Time-dependent Predictive Accuracy In the Presence of Competing - PowerPoint PPT Presentation

Time-dependent Predictive Accuracy In the Presence of Competing Risks Paramita Saha psaha@u.washington.edu http://students.washington.edu/psaha/ ENAR Spring Meeting 2009

Background

1 ENAR, 2009 Background Diagnostic Accuracy for Retrospective Studies • Disease status - Case ( D ) / Control ( ¯ D ) • Marker - M (binary or continuous) • Higher marker is more indicative of disease Dept. of Biostatistics, University of Washington P. Saha

2 ENAR, 2009 Background Diagnostic Accuracy for Retrospective Studies - contd. • Sensitivity or True Positive (TP) - TP ( c ) = P ( M > c | D ) • Specificity or True Negative (1 - False Positive (FP)) - FP ( c ) = P ( M > c | ¯ D ) • ROC curve - plot (FP, TP) for all c ∈ ( −∞ , ∞ ) • AUC - Area under the curve Dept. of Biostatistics, University of Washington P. Saha

3 ENAR, 2009 Background Unique aspects of Time-to-event settings • Disease status can be defined in more than one way • Disease status can vary with time • Censored event time Dept. of Biostatistics, University of Washington P. Saha

4 ENAR, 2009 Background Notation Suppose, we are interested in the ability of the marker M to predict event-time T . • T i , M i - survival time and marker for i th subject ⊲ Marker can be time-dependent - M i ( t ) ⊲ Higher marker values ⇒ poor survival • X i - covariates • C i - (independent) censoring time • We observe: Z i = min ( T i , C i ) , δ i = 1 1 ( T i ≤ C i ) Dept. of Biostatistics, University of Washington P. Saha

5 ENAR, 2009 Background Cumulative/dynamic definition • Cumulative case - subject experienced event by t • Dynamic control - subject did not experience event by t • TP C t ( c ) = P ( M > c | T ≤ t ) • FP D t ( c ) = P ( M > c | T > t ) • At each time t , divide all the subjects as either a case or a control • Summary measures ⊲ ROC curve at time t ⊲ AUC at time t Dept. of Biostatistics, University of Washington P. Saha

Competing Risks ROC

6 ENAR, 2009 Competing Risks ROC Goal • Answer the following question ⊲ How well the marker distinguishes between subjects who experience event of type 1 (type 2) by time t and those who do not experience any type of event by time t? • Estimate time-dependent ROC and AUC • Account for competing risks events Dept. of Biostatistics, University of Washington P. Saha

7 ENAR, 2009 Competing Risks ROC • We observe time until first event and type of event ⊲ observed time until first event - Z i ⊲ δ i = 0 , 1 , 2 , . . . , C ⊲ δ i : censored = 0; different event types = 1 , 2 , . . . , C Dept. of Biostatistics, University of Washington P. Saha

8 ENAR, 2009 Competing Risks ROC What is different? • The risk of censored subjects are redistributed to the subjects present in the riskset • The subjects experiencing a competing risks event at t are not at risk of failure due to other causes after t ⇒ risk redistribution to the right is not meaningful in general Dept. of Biostatistics, University of Washington P. Saha

9 ENAR, 2009 Competing Risks ROC Review: Cumulative/dynamic definition • Cumulative case - subject experienced event by t • Dynamic control - subject did not experience event by t • TP C t ( c ) = P ( M > c | T ≤ t ) • FP D t ( c ) = P ( M > c | T > t ) • At each time t , divide all the subjects as either a case or a control Dept. of Biostatistics, University of Washington P. Saha

10 ENAR, 2009 Competing Risks ROC TP and FP for Cause-specific survival • Controls are free of all events FP D t ( c ) = P ( M > c | T > t ) • Cases are cause-specific - partition the cases into finer groups based on event-type TP C t ( c, d ) = P ( M > c | T ≤ t, δ = d ) • e.g. marker given AIDS by time t ( d = 1 ) • e.g. marker given death by time t ( d = 2 ) Dept. of Biostatistics, University of Washington P. Saha

11 ENAR, 2009 Competing Risks ROC Estimation P ( M > c, T ≤ t, δ = d ) P ( M > c | T ≤ t, δ = d ) = P ( T ≤ t, δ = d ) Dept. of Biostatistics, University of Washington P. Saha

11 ENAR, 2009 Competing Risks ROC Estimation P ( M > c, T ≤ t, δ = d ) P ( M > c | T ≤ t, δ = d ) = P ( T ≤ t, δ = d ) Numerator � ∞ P ( M > c, T ≤ t, δ = d ) = P ( T ≤ t, δ = d | M = m ) P ( M = m ) dm c � ∞ = P ( M = m ) dm c Dept. of Biostatistics, University of Washington P. Saha

11 ENAR, 2009 Competing Risks ROC Estimation P ( M > c, T ≤ t, δ = d ) P ( M > c | T ≤ t, δ = d ) = P ( T ≤ t, δ = d ) Numerator � ∞ P ( M > c, T ≤ t, δ = d ) = P ( T ≤ t, δ = d | M = m ) P ( M = m ) dm c � ∞ = C d ( t | M = m ) � P ( M = m ) dm � �� c Dept. of Biostatistics, University of Washington P. Saha

12 ENAR, 2009 Competing Risks ROC Estimation - contd. � � S ǫ n ( u | M = m ) � � C d ( t | M = m ) = λ d ( u | M = m ) u ≤ t � S ǫ n ( u | M = m ) : use locally weighted KM estimator � λ d ( u | M = m ) : use local observed hazard Dept. of Biostatistics, University of Washington P. Saha

13 ENAR, 2009 Competing Risks ROC Estimation - contd. Dept. of Biostatistics, University of Washington P. Saha

13 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate TP � ∞ P ( M > c, T ≤ t, δ = d ) = P ( T ≤ t, δ = d | M = m ) P ( M = m ) dm c � ∞ = C d ( t | M = m ) � P ( M = m ) dm � �� c Dept. of Biostatistics, University of Washington P. Saha

13 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate TP � ∞ P ( M > c, T ≤ t, δ = d ) = P ( T ≤ t, δ = d | M = m ) P ( M = m ) dm c � ∞ = C d ( t | M = m ) � P ( M = m ) dm � �� c ⊲ Use local cause-specific cumulative incidence to estimate C d ( t | M = m ) ⊲ Use empirical estimator for P ( M = m ) Dept. of Biostatistics, University of Washington P. Saha

14 ENAR, 2009 Competing Risks ROC Estimation - contd. Dept. of Biostatistics, University of Washington P. Saha

14 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate FP - use previous expressions to estimate FP D t ( c ) Dept. of Biostatistics, University of Washington P. Saha

14 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate FP - use previous expressions to estimate FP D t ( c ) P ( M > c, T > t ) P ( M > c | T > t ) = P ( T > t ) � ∞ Numerator = P ( T > t | M = m ) P ( M = m ) dm c Dept. of Biostatistics, University of Washington P. Saha

14 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate FP - use previous expressions to estimate FP D t ( c ) P ( M > c, T > t ) P ( M > c | T > t ) = P ( T > t ) � ∞ Numerator = P ( T > t | M = m ) P ( M = m ) dm c � P ( T > t | M = m ) = 1 − P ( T ≤ t, δ = d | M = m ) d � = 1 − C d ( t | M = m ) d Dept. of Biostatistics, University of Washington P. Saha

15 ENAR, 2009 Competing Risks ROC Estimation - contd. Dept. of Biostatistics, University of Washington P. Saha

15 ENAR, 2009 Competing Risks ROC Estimation - contd. • Estimate TP and FP ⊲ Use cause-specific conditional cumulative incidence ⊲ Use empirical distribution for marker Dept. of Biostatistics, University of Washington P. Saha

16 ENAR, 2009 Event Type: 1 (C/D) Event Type: 2 (C/D) 1.0 1.0 0.8 0.8 0.6 0.6 TP 0.4 0.4 0.2 0.2 92 93 90 84 93 92 92 89 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 FP FP True ROC Estimated ROC Pointwise 90% Confidence band Figure 1: Bootstrapped ROC curves, confidence bands and coverage Dept. of Biostatistics, University of Washington P. Saha

17 ENAR, 2009 Event type: 1 � R i ( t ) Coverage a log(Time) AUC Mean SD 0.0 52.8 0.8446 0.8309 0.0344 89.20 Event type: 2 � R i ( t ) Coverage a log(Time) AUC Mean SD 0.0 52.8 0.6011 0.5899 0.0476 90.20 Table 1: Average of bootstrap mean, sd and coverage (percentile-based) of AUC a Nominal: 90.00 Dept. of Biostatistics, University of Washington P. Saha

18 ENAR, 2009 Competing Risks ROC Example Multicenter AIDS Cohort Study • N = 438 sero-convert patients • Endpoints - ⊲ time-to-AIDS ⊲ time-to-death • Predictive marker - ⊲ CD4 and CD8 measured at “baseline” • Goal: evaluate markers as predictors of disease-progression Dept. of Biostatistics, University of Washington P. Saha

19 ENAR, 2009 Time: 5 Years 1.0 AIDS 0.583 Death 0.506 All Cause 0.575 0.8 0.6 P T 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 F P Dept. of Biostatistics, University of Washington P. Saha

20 ENAR, 2009 Competing Risks ROC Summary • Two of the existing approaches can be modified to account for competing risks (Heagerty et al. (2000), Biometrics; Heagerty and Zheng (2005), Biometrics) • Answer the following question ⊲ How well the marker distinguishes between subjects who experience event of type 1 (type 2) by time t and those who do not experience any type of event by time t? (Saha and Heagerty, 2009) Dept. of Biostatistics, University of Washington P. Saha

21 ENAR, 2009 Thank you!! Dept. of Biostatistics, University of Washington P. Saha