# Stretched Exponential Survival Analysis for South Korean Females
## 3 Results and Discussion  
### Survival Analysis  
The survival curves for women were obtained at the complete life tables taken from the Korean Statistical Information Service ([[http://kosis.kr/eng](http://kosis.kr/eng)]([http://kosis.kr/eng](http://kosis.kr/eng))). Figure 1 shows how we evaluate the survival curves for Korean female in the most recent decades, 1987-2016. The summarized survival curves in Figure 1a show the historic trends in the survival rates for Korean female over three recent decades. The survival curves for Korean female show that \(s(x)\) gradually decreases with age and historically shifts rightwards and towards a rectangular shape over three recent decades, as consistent with Swedish female [5; 14].  
The characteristic life, \(\alpha\), indicating the scale effect of the individual survival curve, was measured for each survival curve through the graphical analysis at \(s(\alpha)=\exp(-1)\approx 0.367879\). The recent historic trends in the \(\alpha\) values were summarized in Figure 1b. The characteristic life for Korean female was depicted as a function of the calendar year (1987-2016) in Figure 1b. This result shows that the \(\alpha\) value linearly increases at a constant rate of \(\sim\)2.88 years per decade (as marked by the solid line in Figure 1b). This steady increase rate for South Korean female is much higher than the rate of \(\sim\)1.20 years per decade for Swedish female in the previous analyses (data from the Human Mortality Database, [[http://www.mortality.org](http://www.mortality.org)]([http://www.mortality.org](http://www.mortality.org))) [5, 14]. The historic trends in the characteristic life clearly demonstrate that South Korean women have lived longer in the past three decades.  
The \(\beta(x)\) value, indicating the shape effect of the individual survival curve, was obtained from \(\beta(x)=-x\ln(x/\alpha)\frac{dB(x)}{dx}\) as shown in Figure 1c. Particularly as demonstrated in Figure 1d, old-age patterns of \(\beta(x)\) at ages over 60 years could be empirically fitted with the quadratic formula as \(\beta(x)=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}\) as suggested previously [5, 12, 13]. The modified stretched exponential function is successfully applicable to the survival curves and to the age-dependent stretched exponents for all ages (Figure 1c) and for old ages over 60 years (Figure 1d). Most remarkably, the \(\beta(x)\) curve smoothly varies with age at old ages over 60 years and old-age \(\beta(x)\) patterns over 90 years can be depicted with  
Figure 1: Survival analysis. (**a**) The survival curves for Korean female in recent decades, 1987–2016, were obtained at the complete life tables taken from the Korean Statistical Information Service ([[http://kosis.kr/eng](http://kosis.kr/eng)]([http://kosis.kr/eng](http://kosis.kr/eng))). (**b**) The characteristic life of the given survival curve. (**c**) The age-dependent stretched exponent. (**d**) The old-age patterns of the stretched exponent at ages over 60 years could be fitted with the quadratic formula of age. The phase transition in the stretched exponent with age appears for 1995–2016 but not for 1987–1994.  
the quadratic function of age. As found previously, the \(\beta(x)\) patterns for old-age populations undergo the phase transition with age: \(\beta(x)\) increases with age at \(x<\alpha\) and decreases with age at \(x>\alpha\)[12]. The phase transition appears in South Korean female datasets after the calendar year of 1995: there are phase transitions for 1995-2016 and no transitions for 1987-1994, as separately marked in Figure 1d. Considering old-age \(\beta(x)\) patterns, the female lifespan of South Korea would experience the critical phase change through the calendar year of 1995. The historic trends in age dependence in \(\beta(x)\) clearly show how the female lifespan in South Korea has been growing older during 1987-2016.