Part IV Extensions: Competing Risks Endpoints and Non-Parametric AUC(t) Estimation

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1 Part IV Extensions: Competing Risks Endpoints and Non-Parametric AUC(t) Estimation Patrick J. Heagerty PhD Department of Biostatistics University of Washington 166 ISCB 2010

2 Session Four Outline Examples Breast Cancer: 70 gene prediction / validation HIV: markers of disease progression Competing Risks Data T P C and cause-specific endpoints / Estimation (non-parametric) T P I and cause-specific endpoints / Estimation (semi-parametric) Illustration / Software 167 ISCB 2010

3 Example: BC and 70-gene Signature among Node-negative Breast Cancer Prediction N = 307 women from (5) Euro Centers Endpoint(s): time-until-distant-metastases (next slide) disease-free-survival Predictive measurements: Clinicopathologic risk assessment 70-gene Signature Goal: validate (added) utility of signature Buyse et al. (2006) JNCI 168 ISCB 2010

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6 Example: Immune markers and disease progression Multicenter AIDS Cohort Study N = 447 men observed to seroconvert Endpoint(s): time-until-aids time-until-death Predictive measurements: CD4, CD8 at baseline CD4, CD8 measured every 6 months Goal: evaluate markers as predictors of disease-progression Saha and Heagerty (2010) 171 ISCB 2010

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8 Competing Risks Endpoints Observed time-until-event, and type of event. Death, cause = ( BC, other) Derived time-until-first-event, and type of event Time until progression or Death (first event, type) e.g. metastases, death (without metastases first) e.g. AIDS, death (without AIDS first) Representation (Ti, δ i) where δ i = 0, 1, 2,..., C δ i : censored = 0; types = 1, 2,... C 173 ISCB 2010

9 Sensitivity and Specificity for Survival (again!) Let T denote the survival time, and let N(t) denote the counting process for the uncensored outcome: N(t) = 1(T t) Possible definitions: CASE(t) : CONTROL(t) : Cumulative N(t) = 1 Incident dn(t) = 1 Static N(t ) = 0 Dynamic N(t) = 0 Where t is a fixed large time, t >> t. 174 ISCB 2010

10 Sensitivity and Specificity for Cause-specific Survival Define: sensitivity C (c, t; d) : P (M > c T t; δ = d) specificity D (c, t) : P (M c T > t) Cases are broken into finer groups based on the type of case. e.g. high marker given metastases by time t (d=1) e.g. high marker given death w/o metastases by time t (d=2) 175 ISCB 2010

11 Sensitivity and Specificity for Cause-specific Survival Example: d=1, 2 Case 1 : T i t, δ = 1 Case 2 : T i t, δ = 2 Control : T i > t, δ = [1, 2] T P C t (c, 1) = P (M > c T i t, δ = 1) T P C t (c, 2) = P (M > c T i t, δ = 2) F P D t (c) = P (M > c T i > t, δ = [1, 2]) 176 ISCB 2010

12 Estimation: Using local Cumulative Incidence Cause-specific Cumulative Incidence C d (t) = P (T t; δ = d) Percent of population with event of type d by time t. Non-parametric estimation (K&P 1980, p. 168) Ĉ d (t) = s t Ŝ(s ) λ d (s) 177 ISCB 2010

13 Estimation: Using local Cumulative Incidence Cumulative incidence estimator can handle censoring. Parallel the estimation of HLP(2000) using: P (M > c T t, δ = d) = P (M > c, T t, δ = d) C d (t) numerator = = c c P (T t, δ = d M = m) P (M = m) dm C d (t M = m) P (M = m) dm 178 ISCB 2010

14 Estimation: Using local Cumulative Incidence Use local cause-specific cumulative incidence to estimate C d (t M = m) and use empirical for P (M = m). Note: P (T > t M = m) = 1 d P (T t, δ = d M = m) Use above to estimate F P D t (c) such that joint distribution is proper. 179 ISCB 2010

15 Marker versus Disease status sensitivity ROC curves Case Type 1 Case Type specificity ISCB 2010

16 Software / Illustration Software: CRAN package for R called survivalroc we have extended this to implement the competing risks calculations. (P. Saha) MACS Data Baseline (e.g. seroconversion time) values of CD4 and CD8 Linear combination based on Cox regression Case Type 1 = AIDS Case Type 2 = death before AIDS Time for cumulative case status = 5 years 181 ISCB 2010

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18 Review: Sensitivity and Specificity for Survival Define: Heagerty and Zheng (2005) / Saha and Heagerty (2010) sensitivity I (c, t) : P [M(t) > c T =t] P [M(t) > c dn(t) = 1] specificity D (c, t) : P [M(t) c T > t] P [M(t) c N(t) = 0] T P I t (c) = P [M(t) > c dn(t)=1] F P D t (c) = P [M(t) > c N(t) = 0] 183 ISCB 2010

19 Sensitivity and Specificity for Cause-specific Survival Define: sensitivity I (c, t; d) : P (M > c T =t; δ = d) specificity D (c, t) : P (M c T > t) Cases are broken into finer groups based on the type of case. e.g. high marker given metastases at time t (d=1) e.g. high marker given death w/o metastases at time t (d=2) 184 ISCB 2010

20 Sensitivity and Specificity for Cause-specific Survival Example: d=1, 2 Case 1 : T i =t, δ = 1 Case 2 : T i =t, δ = 2 Control : T i > t, δ = [1, 2] T P I t (c, 1) = P (M > c T i =t, δ = 1) T P I t (c, 2) = P (M > c T i =t, δ = 2) F P D t (c) = P (M > c T i > t, δ = [1, 2]) 185 ISCB 2010

21 Estimation: Hazard as Bridge A general definition for the cause-specific hazard is Then using a little algebra yields λ (d) (t M i ) = P (T i = t, δ i = d M i ) P (T i t M i ) P (M i = m T i = t, δ i = d) λ (d) (t M i = m) }{{} P (M i = m T i t) }{{} Estimate = Smooth model + Empirical Note: direct (easy) generalization of the HZ(2005) methods. 186 ISCB 2010

22 Software / Illustration Software: CRAN package for R called risksetroc we have extended this to implement the competing risks calculations, and to handle time-dependent covariates. (P. Saha) MACS Data Longitudinal values of CD4 and CD8 Linear combination based on Cox regression Case Type 1 = AIDS (n=176) Case Type 2 = death before AIDS (n=34) ROC curve, and AUC versus time 187 ISCB 2010

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26 Summary Extension of time-dependent ROC methods to competing risks data. Cumulative Cases uses non-parametric methods based on local cumulative incidence calculations. Incident Cases uses semi-parametric methods that parallel those outlined in Heagerty and Zheng (2005). Time-dependent markers. 191 ISCB 2010

27 Thanks! 192 ISCB 2010

28 Nonparametric Estimation of AUC(t) Risk set ranks can be used to nonparametrically estimate AU C(t). AUC(t) = P [M j > M k dn j (t) = 1, N k (t) = 0] M (t) = M j for dn j (t) = 1 C(t) = {k : N k (t) = 0 } Saha & Heagerty (submitted) AUC(t) = E { 1[M (t) > M k k C(t)] } ÂUC(t) = 1 n t k C(t) 1[M (t) > M k ] ÂUC(t) = [risk set rank of M (t) ] 1 n t 193 ISCB 2010

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32 AUC Based on Risk Set Rank AUC(t) / Rank Time 197 ISCB 2010