Time to dementia onset: competing risk analysis with Laplace regression

Transcription

1 Tme to dementa onset: competng rsk analyss wth Laplace regresson Gola Santon, Debora Rzzuto, Laura Fratglon 4 th Nordc and Baltc STATA Users group meetng, Stockholm, November 20 Agng Research Center (ARC), Department of Neurobology, Health Care Scences and Socety, Karolnska Insttutet, and Stockholm Gerontology Research Center, Stockholm, Sweden

2 Why competng rsks? Smulaton: 000 subjects 67 subjects (7%) (censored data) 423 subjects (42%) (.e. dementa onset) 40 subjects (4%) (.e. all cause of mortalty) Falure stcox True values for falure Tme (years) 2

3 Laplace regresson Laplace regresson estmates the condtonal quantles of a contnuous outcome varable gven a set of covarates n the presence of random censorng. We observe: Y The falure s defned as: ( T, C ),2 KN = mn = ( T C ) δ = I Assumptons: condtonally on the covarates, the outcome follows an asymmetrc Laplace dstrbuton. 3

4 Laplace regresson The coeffcents are found by maxmzng the lkelhood functon: Probablty of the event ( y x ) + ( δ ) log[ F( y x )] l = δ log f n Event Censored event Probablty of event at that moment Probablty of event after that moment Note:. The censorng can be dependent on the set of covarates ( C = f ( x ) ); 2. Laplace regresson s a robust regresson method; 3. Laplace regresson gves the smlar results as the Kaplan Meer when no covarates are ntroduced. 4

5 Competng rsks Laplace regresson Baselne Event δ = Event δ = 2 Event δ = k Censored Suppose δ=j s the event of nterest and δ j are the competng events. T s the tme to frst event or censorng. The falure varable s defned as: 0 censored Falure = M k event M event k 5

6 Competng rsks Laplace regresson For smplcty, consder two events only: δ = and δ = 2. Suppose δ = s the event of nterest and δ = 2 s the competng event. We analyze the new tme varable defned as: cause of falure: Falure * = 0 censored event and 2 T * ( δ 2) + T I ( = 2) = T I δ Elandt J. Scandnavan Actuaral Journal (976) Gray RJ. Annals of Statstcs (988) Start Tme T End Tme T * Advantages:. No assumpton on proportonalty of the regresson coeffcents; 2. Derve drectly the cumulatve ncdence functon (CIF) of the falure of nterest; 3. Intutve results; 4. Intutve concept of competng rsks. 6

7 Smulaton T T x x C 2 = e = e ~ 2x ~ U 0.5x + N ( 0,) 0.5 Bernoull ( 0,) x2 + N( 0,) ( 0.5) ( δ = X ) = 0.8I( x = 0) + 0.6I( x ) Pr Censored data 27% Event of nterest 52% Competng event 2% = // Generate new tme and falure varables sum tme gen tme_new = Tme*(falure!=2)+r(max)*2*(falure==2) gen falure_new = (falure!=0) // Ft Laplace model laplace tme_new x x2, fal(falure_new) q(2(2)80) 7

8 Smulaton T T x x C 2 = e = e ~ 2x ~ U 0.5x + N ( 0,) 0.5 Bernoull ( 0,) x2 + N( 0,) ( 0.5) ( δ = X ) = 0.8I( x = 0) + 0.6I( x ) Pr Censored data 27% Event of nterest 52% Competng event 2% = Falure Falure x = 0 x = Tme true Laplace stcrreg Tme 8

9 Tme to dementa onset N=307 Age 75+ years old Baselne 592 Event δ = : dementa Event δ = 2: death Censored Δt=0 years Data from Kungsholmen Project: longtudnal populaton based study on ageng and dementa Event Censored data 26% Dementa 29% Death 45% 9

10 Tme to dementa onset CIF Educaton: less than 8 years Educaton: 8 years or more laplace tme_new educaton age sex, fal(dementa) q( ) // Use post-estmaton commands to tabulate the results lncom [q0]_cons+[q0]educaton Age lncom [q0]_cons+[q0]educaton-[q5]_cons 0

11 Tme to dementa onset Educaton Percent Age at dementa CI <8years % 76.6 (76 4, 76.9) 0% 78.9 (78., 79.8) 5% 80.3 (78.8, 8.7) 20% 82.3 (8., 83.5) 8 years % 76.9 (76.6, 77.2) 0% 80.4 (79.7, 8.2) Δt=0.2 years p value=.825 CI(.2,.5) 5% 82.2 (8.2, 83.) 20% 83.9 (83.0, 84.8) Percent Δt (years) p value % 0.2 <0.09 0%.5 <0.00 5%.9 < %.7 <0.002

12 Questons? 2