Nonextensive Statistical Mechanics in Survival Analysis
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The probability distribution which the lifetime data obey can be described approximately by the logistic system, and the method to analyze these data by its probability distribution is known as survival analysis$^1)$.
We take notice of the logistic system where its time dependence of population size $N(t)$ is described by $dNdt=-\mu(t)N$ with the force of mortality $\mu(t)$. Tsallis et al. have recently developed the nonextensive statistical mechanics$^2)$. According to this formulation, we introduce the $q-$exponential function $y=f(x)=e_q^-xłeft 1+(q-1)x\right^-1q-1$, and its inverse function, i.e., the $q-$logarithmic function $y=f^-1(x)=-łog_q x(1/x)^q-1-1q-1$.
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(A)\ In the case of $\mu(t)=łambda_qłeftN(t)\right^q-1$
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In this case, we define the modified time $t_a$ by
$$t_a\equivłambda_qłeftN(0)\right^(q-1)t,$$
so that the Hazard function $łambda(t)$ and the survival function $S(t)$ in survival analysis can be written respectively by
$$łambda(t)=łeft(\widetildet_at\right)łeft(e_q^-t_a\right)^q-1, S(t)=e_q^-t_a.$$
For this survival function $S(t)=e_q^-t_a$, we define its inverse function, that is, nonextensive entropy $S_q(t)$ as $S_q(t)-łog_q S(t)$ by using the $q-$logarithmic function, so that the nonextensive entropy can be expressed simply as
$$S_q(t)=t_a.$$
We can confirm that this nonextensive entropy satisfy a pseudo-additive property.
When we pay attention to the limit of $q1$, in particular, we have $łim_q1t_a=łambda_1 t$, and also have $łim_q1e_q^-t_a=e^-łambda_1 t$ in which $e^-łambda_1 t$ denotes the ordinary exponential function. As a result of these behaviors, we see that various quantities in survival analysis and the nonextensive entropy are independent of $N(0)$ which means the initial condition. That is to say, this implies the property referred to as the lack of memory property.
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(B)\ In the case of $\mu(t)=łambda_1+łambda_qłeftN(t)\right^q-1$
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In this case, we define the modified time $t_b$ by
$$t_błambda_qłeftN(0)\right^q-1łambda_11-e^-(q-1)łambda_1tq-1.$$
So the Hazard function $łambda(t)$ and the survival function $S(t)$ in this case can be written as
$$łambda(t)=łambda_1+łambda_qłeftN(0)\right^q-1e^-(q-1)łambda_1tłeft(e_q^-t_b\right)^q-1$$
$$S(t)=e^-łambda_1te_q^-t_b$$
respectively. This form of the survival function shows that it can be expressed as the product of the $q-$exponential function $S^(q)(t)e_q^-t_b$ and the ordinary exponential function $S^(1)e^-łambda_1t$, that is,
$$S(t)=S^(1)S^(q)(t).$$
For this survival function $S(t)=S^(1)S^(q)(t)$, the corresponding nonextensive entropy $S(t)-łog_qS(t)$ which is defined by using the $q-$logarithm has the following expression:
$$S(t)=łeft(1/S(t)\right)^q-1-1q-1=$$
$$=S_q^(1)+S_q^(q)(t)+(q-1)S_q^(1)S_q^(q)(t),$$
where $S_q^(1)(t)$ and $S_q^(q)$ are
$$ S_q^(1)(t)= -łog_qS^(1)(t) =1-e^(q-1)łambda_1tq-1$$
$$S_q^(q)(t)=-łog_qS^(q)=t_b,$$
respectively.
When we pay attention to the limit of $q1$, also, we can develop similar considerations as the case (A). Therefore, we have $łim_q1t_b=łambda_1 t$. As a result, we obtain in this limit that the Harzard function $łambda(t)$ is given by $łambda(t)=2łambda_1$, the survival function $S(t)$ is given by $S(t)=e^-2łambda_1 t$, and the nonextensive entropy $S(t)$ is given by $S(t)=2łambda_1 t$. In this limit of the case (B), we also show the lack of memory property which is independent of the initial condition $N(0)$.\\
J. F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd ed. (John Wiley & Sons, 2003)\\
C. Tsallis, in Nonextensive Entropy, edited by M. Gell-Mann and C. Tsallis (Oxford University Press, 2004)