New Challenges for Longitudinal Data Analysis Joint modelling of Longitudinal and Competing risks data

Transcription

1 New Challenges for Longiudinal Daa Analysis Join modelling of Longiudinal and Compeing risks daa Ruwanhi Kolamunnage-Dona Universiy of Liverpool Acknowledgmen Paula Williamson, Pee Philipson, Tony Marson, Rob Henderson Royal Saisical Sociey, 12 Errol Sree, London. 25 Feb

2 Ouline Inroducion ani-epilepic drug (AED rials Compeing risks Quesions of ineres Join modelling and Exension o compeing risks Resuls from Simulaion sudy Applicaion Relaive effecs of AEDs on reamen failure 2

3 Epilepsy drug rials Epilepsy a common neurological condiion characerised by seizures Newly diagnosed usually prescribed single ani-epilepic drug (AED reamen AED considered successful if he person aking i becomes seizure free wih lile in he way of side effecs SANAD (Sandard and new aniepilepic drugs rial recruied 2500 Sandard AED - Carbamazepine (CBZ New AEDs - Lamorigine (LTG, Gabapenin, ec Primary endpoin of AED rials - Time o reamen failure Defined as he ime o wihdrawal of a randomised drug or addiion of anoher/swich o an alernaive AED 3

4 Compeing risks even ime daa Paiens decided o swich o an alernaive AED because of inadequae seizure conrol (ISC or o wihdraw from a reamen because of unaccepable adverse effecs (UAE. When here are several reasons why he even can occur, or some informaive censoring occurs, i is known as compeing risks. Compeing risks - considers reasons for reamen failure: ISC and UAE 4

5 Compeing risks of reamen failure Analysis of overall reamen failure failure for any reason Fail o examine differenial effecs of AEDs on he reason for wihdrawal Assumes reasons of failure are of equal imporance (which may no be he case Differen consequences: loss of driving license due o coninued seizures vs common side effecs such as nausea, dizziness or rash AEDs are considered equivalen as a resul of similar overall reamen failure when in fac he drugs have very differen effecs on wihdrawal due o adverse effecs and poor seizure conrol Ignoring his aspec of an oucome by analysing evens overall can resul in misleading conclusions 5

6 Quesions of ineres CBZ (sandard Vs LTG (new includes 605 paiens. Reasons for reamen failure compeing risks Inadequae seizure conrol (ISC Unaccepable adverse effecs (UAE 94 (15% paiens wihdrew from he randomised AED due o UAE while 120 (20% wihdrew due o ISC. Is LTG superior o CBZ in erms of seizure conrol? olerabiliy? 6

7 Marson e al, Lance; 2007 LTG is significanly more olerable han CBZ LTG is similar o CBZ in erms of seizure conrol 7

8 Biased in favour of new drug? No blinded Differen iraion raes may have been o he disadvanage of sandard drug CBZ AED iraed more quickly brings benefis in erms of seizure conrol bu be more likely o cause adverse effecs Criicism a previous AED rials 8

9 Wha is he effec of drug iraion? Invesigae he effec of drug iraion on he relaive effecs of LTG and CBZ on reasons of reamen failure A large variaion across paiens in he iniial iraed dose was observed Average dose increased over ime bu he rae of increase was observed o vary across individuals Afer adjusing for drug iraion is LTG sill superior o CBZ in erms of UAE/ISC? 9

10 Model Does rae of drug iraion affec risk of drug wihdrawal? Treamen CBZ LTG UAE Drug iraion over ime Laen variables Longiudinal rajecory ISC Compeing risks of reamen failure 10

11 Join model In many sudies, inference abou a longiudinal oucome is of primary ineres problem of non-ignorable missing daa can be addressed hrough join modelling of he ime o dropou Join models can also address quesions concerning he associaion beween a longiudinal oucome and clinically defined ime o even oucome. e.g. relaionship beween prohrombin index and survival in cirrhosis paiens prognosic value of CD4 cell coun in relaion o he ime of AIDS onse 11

12 Sandard join model Longiudinal oucome W Time o even oucome Longiudinal sub-model Gaussian linear model where W 1 is a laen process. Even imes are associaed wih he longiudinal response hrough a second laen process W 2 Condiional on W 2, proporional hazards model is assumed for ime o even oucome Allow for a single ime o even oucome. Y X W ( 1( 1 1 ( X, W2 0( exp{ X 2( 2 W2 ( 2 } 12

13 Compeing risks join model Analyse daa arising from compeing evens and longiudinal processes simulaneously exploiing dependencies beween he componens Associaion beween longiudinal and compeing risks daa via laen processes Compeing Risks S 1 Y Longiudinal process Laen variables W (l S 2 S K 13

14 Compeing risks join model Le η be he cause of failure and compeing even ype indicaor is defined by l {I(T C, } 0 if l, l 1,..., K non - informaive censoring where T = min (T 1,, T K and C is non-informaive censoring imes Allow for compeing risks hrough a cause-specific hazards submodel wih a separae laen associaion beween longiudinal process and each cause of failure Longiudinal sub-model Gaussian linear model Y X W ( 1( 1 1 Sub-model for each compeing cause of failure follow a cause-specific proporional hazard; and for cause l ( l ( X 2, W 2 ( l 0 ( exp{ X 2 ( ( l 2 W ( l 2 ( } 14

15 Laen associaions In he model fied here, W 1 and W 2 (l are assumed o be proporional: ( l ( l W2 ( W1 (, l 1,..., K γ (l indicaes he level of associaion beween he l h compeing even and longiudinal process Longiudinal responses and compeing risks even ime are assumed o be condiionally independen given W 1 and W 2 (l 15

16 Choice of laen processes Linear combinaions of Gaussian random effecs where U 0 and U 1 are random inercep and random slope, follow zero-mean bivariae Gaussian process or or ( l 1 W ( U 0 or W1 ( U 0 U1 and W2 ( W1 ( ( W l 2 ( 1U 0 2U1 3( U0 U1 ( W l 2 1U 0 2U1 3( U0 U1 ( U More complex models including a coninuous-ime sochasic processes 3 16

17 Likelihood funcion where K l Y l Y L Y L Y L 1 (, (,, ( }, ( { ( 2 ( ( 2 ( 2 l W Y W Y l W l L E L l l 17 Condiional on laen processes, he compeing risks are independen of hemselves and of he measuremens Y Facorise he likelihood for observed daa as he produc of he marginal disribuion of Y and he condiional disribuions of compeing evens η ε (1,, K given he observed values of Y θ denoe he combined vecor of unknown parameers, ( Y L Y is he sandard likelihood corresponding o he marginal mulivariae normal disribuion of Y

18 Esimaion Esimae he parameers of ineres by maximising he likelihood of he observed daa Deploy he EM algorihm E-sep: Expecaion of he complee daa log-likelihood is evaluaed. Evaluae expecaion of he form E{ g( W Y, S1,,, ˆ} Compuaionally burdensome and less mahemaically ransparen. M-sep: Parameer esimaes are compued via maximisaion of he expeced log-likelihood Plausible saring values esimaes from separae analyses of he longiudinal and compeing risk componens For more deails of he esimaion process, see Williamson e al S K 18

19 Simulaion sudy Parameer Posiively associaed risks True value Negaively associaed risks Esimae (SE True value Esimae (SE Longiudinal Inercep β ( (0.06 Coninuous covariae β ( (0.09 Binary covariae β ( (0.10 Time β ( (0.09 Survival Coninuous covariae β ( ( (0.15 Binary covariae β ( ( (0.15 γ ( ( (0.21 Coninuous covariae β ( ( (0.18 Binary covariae β ( ( (0.19 γ ( ( (0.28 Variances σ U ( (0.12 σ U ( (0.08 σ Z ( (

20 Applicaion SANAD rial Sandard AED - CBZ vs New AED - LTG Compeing reasons for reamen failure (1 Inadequae seizure conrol (ISC (2 Unaccepable adverse effecs (UAE Afer adjusing for drug iraion is LTG sill superior o CBZ in erms of UAE/ISC? 20

21 Longiudinal oucome - Calibraed dose Sandardise he dose of each drug relaive o he midpoin of he mainenance dose range for ha paricular drug. Change-poin a = 500 days Spline (piecewise mixed-effec model Define a new ime scale d = 500, and se an indicaor for he change poin 1 if if

22 Longiudinal sub-model Piecewise mixed-effec model i i i i U d d U U W (1 ( , ~ N U U U i i i 22 Longiudinal sub-model is defined by where X 1 is he reamen group. Laen process W 1 is defined by W X X d d Y ( (1 1 (

23 Compeing risks join model esimaes (longiudinal sub-model Parameer coef (95% CI coef (95% CI Inercep (2.129, Before 500 days Afer 500 days Time (Slope (0.0006, ( , Drug LTG vs CBZ (-0.002, (0.023, Drug x Time ineracion (0.0005, ( ,

24 Compeing risks join model esimaes (cause-specific hazards sub-model Parameers UAE: LTG vs CBZ Compeing risks join analysis Sandard compeing risks analysis* HR (95% CI (0.358, (0.359, 0.826* γ (1 (95% CI (-0.890, ISC: LTG vs CBZ HR (95% CI (0.709, (0.709, 1.450* γ (2 (95% CI (0.167, * Fiing cause-specific model o each compeing even alone (as in Marson e al, Lance;

25 Conclusion Is LTG superior o CBZ afer adjusing for iraion in erms of UAE? ISC? If LTG is iraed a he same rae as CBZ, he beneficial effec of LTG on UAE would sill be eviden LTG and CBZ sill appear o provide similar seizure conrol 25

26 Model diagnosic The marginal assumpion of normaliy of random effecs in he measuremen sub-model and proporionaliy assumpion of causespecific hazards sub-model hold reasonably well. Deviaions from assumpion of Gaussian random effecs in longiudinal sub-model has lile impac on he model esimaes. Fied an acceleraed failure ime sub-model in a single even ime seing where he proporionaliy assumpion was no saisfied. Furher work is needed o develop/exend diagnosic mehods for join models which include a compeing risks survival sub-model. Sofware is developed in R language (joiner library in CRAN; proposed EM algorihm converged in < 4 minues. 26

27 References Williamson PR, Kolamnunnage-Dona R, Philipson P, Marson AG. Join modelling of longiudinal and compeing risks daa. Saisics in Medicine 2008; Saisics in Medicine 27(30: Williamson PR, Kolamnunnage-Dona R, Tudur Smih C. The influence of compeing risks seing on he choice of hypohesis es for reamen effec. Biosaisics 2006; 8: Marson AG e al, on behalf of he SANAD Sudy group. Carbamazepine, gabapenin, lamorigine, oxcarbazepine or opiramae for parial epilepsy: resuls from he SANAD rial. Lance 2007; 369: Dobson A and Henderson R. Diagnosics for join longiudinal and dropou ime modelling. Biomerics 2003; 59, Tseng, Y-K, Hsieh, F, Wang, J-L. Join modelling of acceleraed failure ime and longiudinal daa. Biomerika 2005; 92, Wulfsohn MS, Tsiais AA. A join model for survival and longiudinal daa measured wih error. Biomerics 1997; 53,