Sensitivity analyses for informative censoring in survival data: A trial example

ABSTRACT

In a controlled clinical trial comparing an experimental drug to a control using time to event analysis, the logrank test is normally used to test against the equality between two survival curves when the proportional hazard rate assumption is held, which of course requires non-informative censoring. The authors used an example from a randomized, double-blind, parallel group, low-dose active controlled study comparing the safety and efficacy of two doses (400 mg/day versus 50 mg/day) of study medication used as monotherapy for the treatment of newly diagnosed or recurrent epilepsy. This analysis imputes the event time of subjects considered to have problematic informative censoring to demonstrate the impact of violations in necessary assumptions, and assesses robustness of the p-value as calculated from imputed data as compared with un-imputed data. Assuming a parametric distribution for time to event, had these subjects resulted in an event in the trial after withdrawal, the expected additional time to event is formulated and calculated using methods developed in this article. Combining the imputed informative censoring subjects with the remainder of the original data, new p-values are obtained using the log-rank test and compared to the original p-value. KM plots are also compared.

KEYWORDS:

• Survival data
• informative censoring
• robustness
• sensitivity
• expected time to event

Acknowledgments

The author wishes to thank Dr. Bruce Turnbull, Dr. Pilar Lim, Dr. Isaac Nuamah and Dr. Akiko Okamoto for their consistent interest and support of research and thank the Janssen Topiramate team for allowing me to use their trial data. In addition, appreciation is extended two referees, whose constructive comments have substantially improved this article. This research was sponsored by Janssen Research & Development, LLC.

A.1. Appendix

Notations used are re-stated here to ensure completeness of this appendix and methodology is described using the control group as an example. To impute informative censoring subjects, let and , , represent random variables of time to event and time to censoring for subject treated with control ( or TPM 50) and treatment ( or TPM 400) medications, respectively. All calculations in treatment group will be defined similarly. For the subject in the control group TPM 50, and are the randomization date and the date of informative censoring (e.g., withdrawal due to adverse event or subject choice in this trial), respectively. Let be the time of administrative trial end date 26Feb2002, which is date that the last patient had end-of-study visit performed. As is the same for all subjects across two groups, we denote as in this article. Subscript however can’t be omitted in , and , as they are subject-level randomization dates and subject-level informative censoring date. It is known that the event time for subject will be at least due to early withdrawal at time . Assumed that this subject had resulted in an event between and , the first quantity to be calculated is the probability of having an event in , given that this subject is event-free at . Next, we return to our objective of calculating: Had this subject resulted in an event prior to , what would it be for the expected additional time of having this event after and prior to ? Before calculating the expected additional time to event for each imputed informative censoring subject, let’s calculate probability of having an event in , given that subject is event-free at , which is needed for calculation of expected additional time to event in Step 2) below.

Step 1): For these informative censoring subjects, probability of having an event in when there is an independent censoring process competes with event process is:

(1)

(2)

(3)

Equation 1 is based on independence of time to censoring (i.e., ) and event process (i.e., ). Equation 2 makes use of time to non-informative censoring, which is exponentially distributed with hazard rate . is the conditional exponential survival function for time to censoring, given that subject still in the risk set at time . is the probability of having an event in in the absence of censoring, given that the subject is still in the risk set at time . In order to calculate conditional probability of having an event in the presence of censoring, one component in the integral is taking derivative of conditional probability in the absence of censoring with respect to . That is and the second component is the conditional exponential survival function of the censoring variable (See in Equation 3).

Step 2): The expected time to event, had this informative censoring subject resulted in an event in is:

(4)

where the probability calculated in Equation 3 is now the denominator of the integrand in Equation 4 . To understand the above formulation, one way is to think of P(A|B)=P(AB)/P(B). P(B) is the conditional probability of have an event in for informative censoring subjects in the presence of censoring. For different parametric time to event distributions, density of event time (i.e., , row 1 in Table A.1.), is used to obtain conditional probability of having an event in , which is (row 2 of Table A.1). Subsequently, after taking derivative with respect to the random variable (row 3 in Table A.1.), conditional probability of having an event in , in the presence of censoring as in Equation 3 or row 4 of Table A.1. will be calculated for different parametric event distributions. Finally, the expected time to event in can be calculated, had this informative censoring subject resulted in an event before or on.

Table A.1. Ingredients for calculation of the expected additional time to event after withdrawal when parametric event distributions are exponential, Weibull, log normal and log logistic, respectively. Row 1, 2 and 3 display density of event distribution, conditional probability of having an event in in the absence of censoring and conditional density of having an event in in the absence of censoring, respectively. Row 3 is the first integrand component in calculating Row 4, which is the conditional probability of having an event in in the presence of exponential censoring. Row 5 is in the counterpart of Row 4 but with Weibull censoring.

In case of non-exponential censoring, other conditional survival density of time to censoring, which is the component of ( in Equation 1, will be plugged in Equations 2, 3 and 4 in order to calculate the expected time to event in for imputed subject . For example, in case time to censoring having Weibull distribution with parameters of and , time to censoring density function then becomes and survival function at time is , resulting in conditional survival density being .

Therefore Equations 2, 3, 4 will become Equations respectively as follows:

(2’)

(3’)

(4’)

And the rest for calculating expected additional time for imputed subjects remains the same as case of exponential time to censoring illustrated in Steps 1 and 2.

Calculation will be much simplified if there is no censoring process in competition with event process. Without considering censoring, the expected length time of being an event in for this informative censoring subject is then degenerated to:

(5)

The numerator of integrand is times quantity from row 3 in Table A.1. for respective parametric event distribution and the denominator is the conditional probability calculated in row 2 of Table A.1. Table A.1. contains necessary ingredients for computation, in which rows 3 is used in the numerator of integrand for both cases with or without considering censoring and row 2 and row 4 are used in the denominator part of the integrand for the case in the absence of censoring and the case in the presence of censoring, respectively.