Bassukas I D
Institute of Medical Radiation and Cell Research (MSZ), University of Würzburg, Germany.
Mech Ageing Dev. 1996 Aug 29;89(3):155-63. doi: 10.1016/0047-6374(96)01747-2.
The recursion formula of the Gompertz function is an established method for the analysis of growth processes. In the present study the recursion formula of the Gompertz survival function 1n S(t + s) = a + b x ln S(t) is introduced for the analysis of survival data, where S(t) is the survival fraction at age 1, s is the constant age increment between two consecutive measurements of the survival fraction and a and b are parameters. With the help of this method--and provided stroboscopial measurements of rates of survival are available--the Gompertz survival function, instead of the corresponding mortality function, can be determined directly using linear regression analysis. The application of the present algorithm is demonstrated by analysing two sets of data taken from the literature (survival of Drosophila imagoes and of female centenarians) using linear regression analysis to fit survival or mortality rates to the corresponding models. In both cases the quality of fit was superior by using the algorithm presently introduced. Moreover, survival functions calculated from the fits to the mortality law only poorly predict the survival data. On the contrary, the results of the present method not only fit to the measurements, but, for both sets of data the mortality parameters calculated by the present method are essentially identical to those obtained by a corresponding application of a non-linear Marquardt-Levenberg algorithm to fit the same sets of data to the explicit form of the Gompertz survival function. Taking into consideration the advantages of using a linear fit (goodness-of-fit test and efficient statistical comparison of survival patterns) the method of the recursion formula of the Gompertz survival function is the most preferable method to fit survival data to the Gompertz function.
冈珀茨函数的递归公式是分析生长过程的一种既定方法。在本研究中，引入冈珀茨生存函数的递归公式ln S(t + s) = a + b×ln S(t)来分析生存数据，其中S(t)是年龄为t时的生存分数，s是连续两次测量生存分数之间的恒定年龄增量，a和b是参数。借助这种方法——前提是有频闪观测的生存率测量数据——可以直接使用线性回归分析来确定冈珀茨生存函数，而不是相应的死亡率函数。通过对从文献中获取的两组数据（果蝇成虫和女性百岁老人的生存情况）进行线性回归分析，以将生存或死亡率拟合到相应模型，来证明本算法的应用。在这两种情况下，使用当前引入的算法拟合质量都更优。此外，根据死亡率定律拟合计算出的生存函数对生存数据的预测效果很差。相反，本方法的结果不仅与测量值拟合，而且对于这两组数据，本方法计算出的死亡率参数与通过将相同数据集拟合到冈珀茨生存函数的显式形式而相应应用非线性马夸特 - 列文伯格算法所获得的参数基本相同。考虑到使用线性拟合的优点（拟合优度检验和生存模式的有效统计比较），冈珀茨生存函数递归公式的方法是将生存数据拟合到冈珀茨函数的最优选方法。
