In a randomised clinical trial with a longitudinal outcome, analyses of the efficacy of the study treatments may be complicated by both non-trial interventions, which have not been administered by the researcher, and sparsely measured outcome values. The delay between the change in outcome and the starting of the non-trial intervention may be much shorter than the time intervals between the actual measurements. We propose a model that accounts for the possible dynamic interdependence between the longitudinal outcome and time-to-event data. The model is based on discretising time into short intervals. This results in a missing data problem, which we tackle using Bayesian inference and data augmentation. The method is based on the assumption that decisions to initiate non-trial interventions are not confounded by unobservable factors. The Helsinki Psychotherapy Study data are used as an illustration. Different psychotherapies were compared, and possible episodes of psychotropic medication were viewed as non-trial interventions. Simulation studies suggest that our method provides reasonable estimates of the effects of both the study treatment and the non-trial intervention also showing some robustness against possible latent background factors. An application of marginal structural modelling, however, appeared to underestimate the differences between the treatments.
WhiteIRCarpenterJPocockSJ. Adjusting treatment comparisons to account for non-randomized interventions: an example from an angina trial. Stat Med2003; 22(5): 781–793.
2.
EerolaMGasbarraDMäkeläPH. Joint modelling of recurrent infections and antibody response by bayesian data augmentation. Scand J Stat2003; 30(4): 677–698.
3.
ThomasAO'HaraBLiggesU. Making BUGS open. R News2006; 6(1): 12–17.
4.
RobinsJMHernanMABrumbackB. Marginal structural models and causal inference in epidemiology. Epidemiology2000; 11: 550–560.
5.
HernanMABrumbackBRobinsJM. Marginal structural models to estimate the causal effect of zidovudine on the survival of hiv-positive men. Epidemiology2000; 11: 561–570.
6.
HernanMALanoyECostagliolaD. Comparison of dynamic treatment regimes via inverse probability weighting. Basic Clin Pharmacol Toxicol2006; 98: 237–242.
7.
ImbensGWRubinDB. Bayesian inference for causal effects in randomized experiments with noncompliance. Ann Stat1997; 25(1): 305–327.
8.
FrangakisCERubinDB. Principal stratification in causal inference. Biometrics2002; 58(1): 21–29.
9.
KnektPLindforsOHärkänenT. Randomized trial on the effectiveness of long-and short-term psychodynamic psychotherapy and solution-focused therapy on psychiatric symptoms during a 3-year follow-up. Psychol Med2008; 38: 689–703.
10.
KnektPLindforsOA randomized trial of four forms of the psychotherapy on depressive and anxiety disordersNo. 77 in Studies in social security and health. The Social Insurance Institution, Finland, 2004.
11.
PerryJBondM. The sequence of recovery in long-term dynamic psychotherapy. J Nerv Ment Dis2009; 197(12): 930–937.
12.
BeckATWardCHMendelsonM. An inventory for measuring depression. Arch Gen Psychiatry1961; 4: 561–571.
13.
TsiatisAADavidianM. Joint modeling of longitudinal and time-to-event data: an overview. Stat Sinica2004; 14: 809–834.
14.
HoganJWDanielsMJ. A hierarchical modelling approach to analysing longitudinal data with drop-out and non-compliance, with application to an equivalence trial in paediatric acquired immune deficiency syndrome. Appl Stat2002; 51(1): 1–21.
15.
PearlJCausality: models, reasoning, and inferenceCambridge: Cambridge University Press, 2000.
16.
PearlJ. An introduction to causal inference. Int J Biostat2010; 6(2): Iss. 2, Article 7–Iss. 2, Article 7.
17.
R Development Core Team. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, 2010. ISBN 3-900051-07-0, http://www.R-project.org
18.
TannerMAWongWH. The calculation of posterior distributions by data augmentation. J Am Stat Assoc1987; 82: 528–550.
19.
van der WalWMGeskusRB. ipw: An R package for inverse probability weighting. J Stat Softw2011; 43(13): 1–23http://www.jstatsoft.org/v43/i13/
20.
YanJ. geepack: Yet another package for generalized estimating equations. R-News2006; 3(2): 12–14.
21.
KnektPLaaksonenMAHärkänenTLevyRAblonJSKächeleH. Helsinki Psychotherapy Study: effectiveness, sufficiency, and suitability of short- and long-term psychotherapy. Evidence-based practice and practice-based evidence seriesNew York: Humana Press, 2012.
22.
SalminenJKKarlssonHHietalaJ. Short-term psychodynamic psychotherapy and fluoxetine in major depressive disorder - a randomized comparative study. Psychother Psychosom2008; 77: 351–357.
23.
RobinsJMOrellanaLRotnitzkyA. Estimation and extrapolation of optimal treatment and testing strategies. Stat Medicine2008; 27(23): 4678–4721.
24.
PosternakMAMillerI. Untreated short-term course of major depression: a meta-analysis of outcomes from studies using wait-list control groups. J Affect Disord2001; 66: 139–146.
25.
WestMHarrisonJBayesian forecasting and dynamic modelsNew York: Springer, 1997.
26.
HarveyACForecasting, structural time series models and the Kalman filterCambridge: Cambridge University Press, 1989.
27.
CarterCKKohnR. On Gibbs sampling for state space models. Biometrics1994; 81: 541–553.
28.
Frühwirth-SchnatterS. Data augmentation and dynamic linear models. J Time Ser Analysis1994; 15: 183–202.
29.
DurbinJKoopmanSJ. A simple and efficient simulation smoother for state space time series analysis. Biometrics2002; 89: 603–616.
30.
BangHRobinsJM. Doubly robust estimation in missing data and causal inference models. Biometrics2005; 61(4): 962–972.
31.
RobinsJMLinDYHeagertyP. Optimal structural nested models for optimal sequential decisions. Proceedings of the second Seattle symposium on biostatisticsNew York: Springer, 2004.
32.
KangJDYSchaferJL. Demystifying double robustness: a comparison of alternative strategies for estimating a population mean from incomplete data. Stat Sci2007; 22(4): 523–539.
33.
NagelkerkeN. A note on a general definition of the coefficient of determination. Biometrics1991; 78(3): 691–692.
34.
GallopRSmallDSLinJY. Mediation analysis with principal stratification. Stat Med2009; 28(7): 1108–1130.
35.
LinJYHaveTRTElliottMR. Nested markov compliance class model in the presence of time-varying noncompliance. Biom2009; 65: 505–513.
36.
CommengesDGégout-PetitA. A general dynamical statistical model with causal interpretation. J Roy Stat Soc Ser B2009; 71(3): 719–736.
37.
GenelettiS. Identifying direct and indirect effects in a non-counterfactual framework. J Roy Stat Soc Ser B2007; 69(2): 199–215.
38.
ArjasE. Wiley, chap. Causal inference from observational data: a Bayesian predictive approach. Berzuini C, Dawid P and Bernardinelli L (eds) Causality: statistical perspectives and applicationsNew York: Wiley2012, 71–84.
