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

2. 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. 3 Cox regression analysis: Covariate Treatment Sex Age Acetylcholinesterase Inflammation Baseline prothrombin Current prothrombin Model II -0.06 (0.14) 0.31 (0.15) 0.043 (0.008) -0.0015 (0.0006) -0.43 (0.15) -0.054 (0.004) Model I -0.28 (0.14) 0.27 (0.16) 0.041 (0.008) -0.0019 (0.0007) -0.47 (0.15) -0.014 (0.007) Model I gives an estimate of the total treatment effect Coding Placebo=0; Prednisone=1 Female=0; Male=1 Years (range 27-77) Range 42-556 Absent=0; Present=1 Percent of normal Model II shows the importance of prothrombin

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

5. 5 - 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 Outline: - Concluding comments

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

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

8. 8 Introduce M i (t) = N i (t) – L i (t). M i (t) is a martingale Note that observation signal noise For statistical modeling we focus on l i (t)

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

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

11. 11 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) Estimate by adding the estimated increments at all times s up to time t 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. 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. 13 Baseline Sex Thickness

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

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

16. 16 Treatment dN i (t) By a dynamic path analysis we may see how the two analyses fit together Current prothrombin total effectdirect effect indirect effect Due to additivity and least squares estimation: 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

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

18. 18 Direct effects on block II variables (ordinary least squares): Direct effects on current prothrombin (ordinary least squares): Age on acetylSex on inflammation TreatmentAcetylBaseline proth

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

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

21. 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. 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