Stochastic Models of Tumor Latency and their Biostatistical Applications

Abstract

Printed in Singapore. PrefaceThe role of statistical methods in general, and the statistical analysis of survival data in particular, in cancer research cannot be emphasized enough. They have penetrated deeply into clinical and experimental oncology, especially into such important areas as cancer epidemiology and clinical trials. It would be difficult to find a paper in the current literature which does not contain a reference to the Kaplan-Meier esti mator or Cox's regression model. Despite the fact that a great variety of parametric survival models are available (Johnson and Kotz, 1970; Cohen and Whitten, 1988), nonparametric methods of estimation and hypotheses testing are used the most. This is because nonparametric estimators and tests are quite general and easy to compute. At the same time, they provide less information than would be desirable. Using twosample significance tests, one can give the answer to the question as to whether there is a statistically significant difference between the two groups of patients under study. What is perhaps the more fundamental question, though, still remains: how can the observed difference be explained in terms of biological mechanisms, the simpler the explanation the better. Yet another question is how to make extrapolations beyond the follow-up period. These are questions any investigator would be likely to raise; to answer them would require the parametric techiques and substantive models that underlie them. Many other practical problems call for parametric methods, the design of optimal surveillance strategies is among them. There are some statistical problems of survival analysis for which no satisfactory nonparametric solutions are currently available. Some examples of this sort are given in this book. Needless to say that parametric methods are more difficult to handle in practice. They create psychological problems, casting doubts upon the basic assumptions used for the derivation of a given para metric model, since their validation is an arduous task for each specific application.There is no conflict between the two methodologies, each of them having merits and demerits as is usual in real life. Moreover, they supplement and enrich each other when used in combination. There are examples illustrating this point in the book by Yakovlev and Tsodikov. The most challenging problem is to find a "good" mathematical model, as applied to the analysis of biomedical data. There are no perfectly objective criteria for distinguishing a "good" model from a "bad" one.

Cure fraction models using mixture and non-mixture models

“…An alternative non-mixture formulation has been suggested which defines an asymptote for the cumulative hazard and hence for the cure fraction, [4], [8], [10]. In this case, the survival function for non-mixture cure fraction model is…”

Regression models generated by gamma random variables with long-term survivors

“…. is surely a strong one, as remarked by Yakovlev and Tsodikov (1996). This option favors simplicity and analytical tractability at the expense of a more general formulation.…”

“…Figure 3 displays the estimated KaplanMeier survival function with a well-pronounced plateau. According to Yakovlev and Tsodikov (1996) this may be thought of as an indication of the presence of a proportion of patients for whom the gastric adenocarcinoma will never recur; therefore, the patients can be considered cured. The explanatory variable is related to the parameter θ according to the following structure.…”

“…In clinical studies, the event of interest can be the death of a patient (which can happen due to different competing causes) or a tumor recurrence (which can be attributed to metastasis-component tumor cells left active after an initial treatment). A metastasis-component tumor cell is a tumor cell with a potential to metastasize (Yakovlev and Tsodikov, 1996). The literature on the subject is significant and increasing.…”

Two-Sample Statistics for Testing the Equality of Survival Functions Against Improper Semi-parametric Accelerated Failure Time Alternatives: An Application to the Analysis of a Breast Cancer Clinical Trial

“…A different approach which overcomes these drawbacks relies on models that define the cumulative hazard as a bounded increasing positive function in a parametric (Aalen, 1992;Cantor and Shuster, 1992;Yakovlev and Tsodikov, 1996) or semi-parametric way (Tsodikov, 1998;Shen and Sinha, 2002;Ibrahim, 1999Ibrahim, , 2001Broët et al, 2001). Such semi-parametric modeling was used in our previous work to test for no short-term or no long-term effect against improper short-term proportional hazard alternatives (Broët et al, 2001).…”