Notation
EVENT
Suppose a clinical trial was conducted to evaluate the efficacy of a treatment on survival of patients in standard and experimental treatment groups. After randomization, each patient in the standard treatment could switch to the experimental treatment at time t (t > 0). If So that, Here, Also each patient may experience the event of interest, a competing event or is censored at the end of study without any event. The time interval between the entrance time point and the end of study is represented by Thus, the observed data for subject OrOn the other hand, some patients may be switched from their randomized assigned treatment to the other treatment, following the diagnosis and decision of the clinical physician. The switching usually occurs from the standard treatment to the experimental treatment groupThus, the observed survival time will be resulted as:The random variable And so, the distribution of survival time for switched patient in the experimental treatment will be as follows:So, if the treatment is beneficial, the survival time for a switched patient to the experimental treatment will be increased. Thus the likelihood for the On the other hand, the likelihood function for n patients in Accordingly, the contribution of the Aforementioned likelihood is the product of two terms, the first term is for patients that never switched and the second term is for the switched patients, where If Under some regularity conditions One approach for solving the likelihood equation in (