2016 Stata London Users Group Meeting stpm2cr : A Stata module for direct likelihood inference on the cause-specific cumulative incidence function within the flexible parametric modelling framework Sarwar Islam Mozumder 1 , Mark J Rutherford 1 & Paul C Lambert 1, 2 1 Department of Health Sciences, University of Leicester, Leicester, UK 2 Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Multi-state Model Death from Cancer, k = 1 Death from Alive at Diagnosis Cause k = 2 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 2/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Multi-state Model Death from Cancer, k = 1 Death from Alive at Diagnosis Cause k = 2 Death from Cause k = K Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 2/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Cumulative Incidence Function (CIF) Cause-specific CIF, F k ( t ) F k ( t ) = P ( T ≤ t , D = k ) Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 3/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Cumulative Incidence Function (CIF) Cause-specific CIF, F k ( t ) The probability that a patient will die from cause D = k by time t whilst also being at risk from dying of other causes We obtain this by either, Estimating using all cause-specific hazard functions, or 1 Transforming using a direct relationship with the 2 subdistribution hazard functions Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 3/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Cumulative Incidence Function (CIF) Cause-specific CIF, F k ( t ) The probability that a patient will die from cause D = k by time t whilst also being at risk from dying of other causes We obtain this by either, Estimating using all cause-specific hazard functions, or 1 Transforming using a direct relationship with the 2 subdistribution hazard functions Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 3/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (1) Cause-specific Hazards, h cs k ( t ) P ( t < T ≤ t + ∆ t , D = k | T > t ) h cs k ( t ) = lim ∆ t ∆ t → 0 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 4/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (1) Cause-specific Hazards, h cs k ( t ) Instantaneous conditional rate of mortality from cause D = k given that the patient is still alive at time t Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 4/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (1) Cause-specific Hazards, h cs k ( t ) Instantaneous conditional rate of mortality from cause D = k given that the patient is still alive at time t Estimating Cause-specific CIF using CSH � t S ( u ) h cs F k ( t ) = k ( u ) du 0 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 4/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (1) Cause-specific Hazards, h cs k ( t ) Instantaneous conditional rate of mortality from cause D = k given that the patient is still alive at time t Estimating Cause-specific CIF using CSH � t � s K � h cs h cs F k ( t ) = exp − j ( u ) du k ( s ) ds 0 0 j =1 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 4/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (2) Subdistribution Hazards, h sd k ( t ) P ( t < T ≤ t + ∆ t , D = k | T > t ∪ ( T ≤ t ∩ cause � = k ) h sd k ( t ) = lim ∆ t → 0 ∆ t Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 5/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (2) Subdistribution Hazards, h sd k ( t ) The instantaneous rate of failure at time t from cause D = k amongst those who have not died, or have died from any of the other causes, where D � = k Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 5/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Approach (2) Subdistribution Hazards, h sd k ( t ) The instantaneous rate of failure at time t from cause D = k amongst those who have not died, or have died from any of the other causes, where D � = k Direct Transformation of the Cause-specific CIF � t � � h sd F k ( t ) = 1 − exp − k ( u ) du 0 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 5/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Regression Modelling SDH Regression Model � � h sd k ( t | x ) = h sd β sd 0 , k ( t ) exp β x k β k � β sd � Subdistribution hazard ratio = exp β β k Association on the effect of a covariate on risk Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 6/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Why Flexible Parametric Survival Models? [Royston and Lambert, 2011] Models baseline (log-cumulative) SDH function using restricted cubic splines Log-Cumulative SDH Flexible Parametric Model ln( H sd k ( t | x ik )) = s k (ln( t ) | γ k , m 0 k ) + x ik β β β k Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 7/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions Why Flexible Parametric Survival Models? [Royston and Lambert, 2011] Models baseline (log-cumulative) SDH function using restricted cubic splines Log-Cumulative SDH Flexible Parametric Model ln( H sd k ( t | x ik )) = s k (ln( t ) | γ k , m 0 k ) + x ik β β β k Easy to include time-dependent effects Relaxing Assumption of Proportionality E ln( H sd � k ( t )) = s k (ln( t ); γ γ k , m 0 k ) + x k β γ β β k + s k (ln( t ); α α α lk , m lk ) x lk l =1 Can predict time-dependent HRs, absolute differences and more... Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 7/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Likelihood Function [Jeong and Fine, 2006] Direct Parametrisation (competing risks) n K [ S ( t | x )] 1 − � K � � � � j =1 δ ij ( f s j ( t i | x j )) δ ij L = i =1 j =1 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 8/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Likelihood Function [Jeong and Fine, 2006] Direct Parametrisation (competing risks) n K [ S ( t | x )] 1 − � K � � � � j =1 δ ij ( f s j ( t i | x j )) δ ij L = i =1 j =1 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 8/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Likelihood Function [Jeong and Fine, 2006] Direct Parametrisation (competing risks) n K [ S ( t | x )] 1 − � K � � � � j =1 δ ij ( f s j ( t i | x j )) δ ij L = i =1 j =1 CSH Approach n K � � [ S ( t | x )] 1 − � K � � j =1 δ ij ( S ( t | x ) h cs j ( t i | x j )) δ ij L = i =1 j =1 Estimates covariate effects on the cause-specific CIF rather than the CSH rate Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 8/22
Introduction Relationship with the Cause-specific CIF Flexible Parametric Models for the Cause-Specific CIF Conclusions The Likelihood Function [Jeong and Fine, 2006] Direct Parametrisation (competing risks) 1 − � K j =1 δ ij n K K � � � � ( f s j ( t i | x j )) δ ij � L = 1 − F j ( t | x j ) i =1 j =1 j =1 Sarwar I Mozumder Direct Likelihood FPM Approach for the Cause-specific CIF 9 Sept 2016 8/22
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