Ørnulf Borgan Department of Mathematics University of Oslo

1. Using dynamic path analysis to estimate direct and indirect effects of treatment and otherfixed covariates in the presence of an internaltime-dependent covariate Ørnulf Borgan Department of Mathematics University of Oslo Based on joint work with Odd Aalen, Egil Ferkingstad and Johan Fosen

2. Motivating example: - Randomized trial of survival for patients with liver cirrhosis - Randomized to placebo or treatment with prednisone (a hormone) - Consider only the 386 patients without ascites (excess fluid in the abdomen). - Treatment: 191 patients, 94 deaths Placebo: 195 patients, 117 deaths - A number of covariates registered at entry - Prothrombin index was also registered at follow-up visits throughout the study

3. Cox regression analysis: Model I gives an estimate of the total treatment effect Model II shows the importance of prothrombin

4. Purpose: Get a better understanding of how treatment (and other fixed covariates) partly have a direct effect on survival and partly an indirecteffect operating via the internal time-dependent covariate (current prothrombin) This will be achieved by a combining classicalpath analysis with Aalen's additive regression model to obtain a dynamic path analysis for censored survival data

5. Outline: - Brief introduction to counting processes, intensity processes and martingales - Brief review of Aalen's additive regression model for censored survival data - Dynamic path analysis explained by means of the cirrhosis example - Concluding comments

6. Counting processes Have censored survival data (Ti, Di) status indicator (censored=0; failure=1) censored survival time Ni(t) counts the observed number of failures for indivdual i as a function of (study) time t Two possible outcomes for Ni(t): Ni(t) Ni(t) 1 1 0 0 t t TiDi= 0 TiDi= 1

7. Intensity processes and martingales Denote by Ft-all information available to the researcher "just before" time t (on failures, censorings, covariates, etc.) The intensity process li(t) of Ni(t) is given by where dNi(t) is the increment of Ni over [t,t+dt) Cumulative intensity process:

8. Introduce Mi(t) = Ni(t) – Li(t). Mi(t) is a martingale Note that signal observation noise For statistical modeling we focus on li(t)

9. Aalen's additive regression model Intensity process for individual i hazard at risk indicator (possibly) time dependent covariates for individual i (assumed predictable) Aalen's non-parametric additive model: excess risk at t per unit increase of xip(t) baseline hazard

10. It is difficult to estimate the bj(t) non-parametrically, so we focus on the cumulative regression functions: At each time s we have a linear model conditional on "the past" Fs- observation covariates parameters noise Estimate the increments by ordinary least squares at each time s when a failure occurs

11. Estimate by adding the estimated increments at all times s up to time t The vector of the is a multivariate "Nelson-Aalen type" estimator The statistical properties can be derived using results on counting processes, martingales, and stochastic integrals (e.g. Andersen et al. Springer, 1993) Software: "aareg" in Splus version 6.1 for Windows (not in R) "addreg" for Splus and R at www.med.uio.no/imb/stat/addreg/ "aalen" for Splus and R at www.biostat.ku.dk/~ts/timereg.html

12. Illustration of additive regression model: Survival after operation from malignant melanoma (cf. Andersen et al, 1993) 205 patients operated from malignant melanoma in Odense, Denmark, 1962-77 126 females, 28 deaths 79 males, 29 deaths Fit additive model with the fixed covariates: - Sex (0 = female, 1 = male) - Tumour thickness – 2.92

13. Baseline Thickness Sex

14. Cirrhosis example: only treatment and current prothrombin for a start Fit additive models: (i) with treatment as only covariate (marginal model) dNi(t) Treatment (ii) with treatment and current prothrombin (dynamic model) Treatment dNi(t) Current prothrombin

15. (ii) dynamic model: (i) marginal model: Treatment: Treatment: Current prothrombin: Total effect of treatment is underestimated in the dynamic model Current prothrombin has a strong effect on mortality

16. By a dynamic path analysis we may see how the two analyses fit together dNi(t) Treatment Current prothrombin Treatment on prothrombin: (least squares at each failure time) Treatment increases prothrombin, and high prothrombin reduces mortality Part of the treatment effect is mediated through prothrombin Due to additivity and least squares estimation: total effect direct effect indirect effect

17. Cirrhosis example: a dynamic path analysis with all covariates Block IV Block I Block II Block III

18. Direct effects on block II variables (ordinary least squares): Age on acetyl Sex on inflammation Direct effects on current prothrombin (ordinary least squares): Treatment Acetyl Baseline proth

19. Direct cumulative effects on death (additive regression): Age Inflammation Current prothrombin

20. Indirect cumulative effects on death: Treatment Baseline prothrombin Age Sex

21. Conclusions of the cirrhosis example: - The dynamic path analysis gives a detailed picture of how the effect of treatment operates via current prothrombin and how the effect is largest the first year - By not treating all fixed covariates on an equal footing we are able to distinguish between "basic" covariates (block I) and covariates that are a measure of progression of the disease (block II)

22. General conclusions: Aalen's additive regression model is a useful supplement to Cox's regression model Additivity and least squares estimation make dynamic path analysis feasible, including the concepts direct, indirect and total treatment effects Dynamic path analysis may be extended to recurrent event data with e.g. the previous number of events as an internal time-dependent covariate Much methodological work remains to be done on dynamic path analysis, e.g. on methods for model selection