Survival probabilities with time-dependent treatment indicator: quantities and non-parametric estimators.
The 'landmark' and 'Simon and Makuch' non-parametric estimators of the survival function are commonly used to contrast the survival experience of time-dependent treatment groups in applications such as stem cell transplant versus chemotherapy in leukemia. However, the theoretical survival functions corresponding to the second approach were not clearly defined in the literature, and the use of the 'Simon and Makuch' estimator was criticized in the biostatistical community. Here, we review the 'landmark' approach, showing that it focuses on the average survival of patients conditional on being failure free and on the treatment status assessed at the landmark time. We argue that the 'Simon and Makuch' approach represents counterfactual survival probabilities where treatment status is forced to be fixed: the patient is thought as under chemotherapy without possibility to switch treatment or as under transplant since the beginning of the follow-up. We argue that the 'Simon and Makuch' estimator leads to valid estimates only under the Markov assumption, which is however less likely to occur in practical applications. This motivates the development of a novel approach based on time rescaling, which leads to suitable estimates of the counterfactual probabilities in a semi-Markov process. The method is also extended to deal with a fixed landmark time of interest.
Center of Biostatistics for Clinical Epidemiology, Department of Health Sciences, University Milano-Bicocca, Monza, Italy.
Davide Paolo
DP
Bernasconi
Center of Biostatistics for Clinical Epidemiology, Department of Health Sciences, University Milano-Bicocca, Monza, Italy.
Paola
P
Rebora
Centro Interdipartimentale di Biostatistica a Bioinformatica, Università Tor Vergata, Rome, Italy.|Chronic Malignancies Working Party of the European Group for Blood and Marrow Transplantation, Leiden, The Netherlands.
Simona
S
Iacobelli
Center of Biostatistics for Clinical Epidemiology, Department of Health Sciences, University Milano-Bicocca, Monza, Italy.
Maria Grazia
MG
Valsecchi
Center of Biostatistics for Clinical Epidemiology, Department of Health Sciences, University Milano-Bicocca, Monza, Italy.
Laura
L
Antolini
D056808:Biostatistics; D003198:Computer Simulation; D006801:Humans; D053208:Kaplan-Meier Estimate; D008390:Markov Chains; D015233:Models, Statistical; D054198:Precursor Cell Lymphoblastic Leukemia-Lymphoma / Q000188:drug therapy / Q000401:mortality / Q000628:therapy; D011336:Probability; D018709:Statistics, Nonparametric; D033581:Stem Cell Transplantation; D016019:Survival Analysis; D013997:Time Factors
Kaplan-Meier; counterfactual survival; immortal time bias; landmark; potential quantities; time-dependent treatment