Wilson D L
Department of Biology, University of Miami, Coral Gables, FL 33124.
Mech Ageing Dev. 1993 Jan;66(3):269-81. doi: 10.1016/0047-6374(93)90014-i.
The Gompertz mortality function, Rm = R0e alpha t, is frequently used to describe changes in mortality rate (Rm) with time (t). In this paper, four methods for determining the best fit values of the two parameters, R0 and alpha, are compared. Three of the four methods use the Gompertz mortality function with mortality rate estimates derived from survival data to determine the best fit values for the two parameters. All three confront problems. The fourth method uses the Gompertz survival function, which can be derived from the Gompertz mortality function and which allows one to use survival data directly. It thereby avoids the problems and generally gives the best estimates for the two parameters. The use of the mortality function, with mortality rate estimates, confronts four distinct problems. One of these is caused by time intervals when zero organisms die. A second is caused by errors produced in estimating mortality rates from survival data. If too high a proportion of a population die in a given time interval, the mortality rate estimates are too low. A third problem is the sensitivity of the mortality-equation-based analyses to values at the end of the survival curve, where scatter in mortality values tends to be greater. A final problem occurs when time intervals greater than one time unit (day, week, year, etc.) are used in the analysis. Such problems with the use of mortality rates to estimate parameter values are revealed when the calculated parameters are used to produce a survival curve, or when known values of R0 and alpha are used to generate survival data. This paper introduces a non-linear regression analysis, using a Simplex algorithm to fit parameters R0 and alpha in the Gompertz Survival function and concludes that it gives more reliable and consistent results with a variety of data than do three methods that use the mortality function.
冈珀茨死亡率函数(Rm = R0e^{\alpha t})经常用于描述死亡率（(Rm)）随时间（(t)）的变化。本文比较了确定两个参数(R0)和(\alpha)最佳拟合值的四种方法。四种方法中的三种使用冈珀茨死亡率函数，其死亡率估计值来自生存数据，以确定这两个参数的最佳拟合值。这三种方法都面临问题。第四种方法使用冈珀茨生存函数，它可以从冈珀茨死亡率函数推导得出，并且允许直接使用生存数据。因此，它避免了这些问题，并且通常能给出这两个参数的最佳估计值。使用带有死亡率估计值的死亡率函数面临四个不同的问题。其中一个是由零生物体死亡的时间间隔引起的。第二个是由从生存数据估计死亡率时产生的误差引起的。如果在给定的时间间隔内有过高比例的种群死亡，死亡率估计值就会过低。第三个问题是基于死亡率方程的分析对生存曲线末端的值很敏感，在那里死亡率值的离散度往往更大。当在分析中使用大于一个时间单位（天、周、年等）的时间间隔时，会出现最后一个问题。当使用计算出的参数来生成生存曲线，或者当使用(R0)和(\alpha)的已知值来生成生存数据时，使用死亡率来估计参数值的这些问题就会显现出来。本文介绍了一种非线性回归分析，使用单纯形算法来拟合冈珀茨生存函数中的参数(R0)和(\alpha)，并得出结论，与使用死亡率函数的三种方法相比，它对各种数据能给出更可靠和一致的结果。
