Survival Models: Concepts and Applications | 生存模型的概念与应用

📚 Survival Models: Concepts and Applications | 生存模型的概念与应用

Survival analysis is a branch of statistics that focuses on the time until an event of interest occurs. Whether the event is death, machine failure, or patient recovery, survival models provide a powerful framework for analyzing time-to-event data that standard statistical methods cannot handle. In this article, we explore the core concepts of survival models, the key functions that define them, and their real-world applications in actuarial science, medicine, and engineering.

生存分析是统计学的一个分支，专门研究从起点到某一感兴趣事件发生所经历的时间。无论该事件是死亡、机器故障还是患者康复，生存模型都为分析”时间到事件”数据提供了强大的框架，而这些数据是标准统计方法无法处理的。在本文中，我们将深入探讨生存模型的核心概念、定义它们的关键函数，以及它们在精算科学、医学和工程学中的实际应用。

1. What Is a Survival Model? | 什么是生存模型

A survival model is a statistical model that describes the probability distribution of the time T until an event occurs. The time variable T is always non-negative, and the event of interest is often called a “failure” in engineering contexts or a “terminal event” in medical contexts. Unlike ordinary regression models, survival models must account for censoring — a situation where the event has not yet been observed for some individuals by the end of the study.

生存模型是一种描述事件发生时间 T 的概率分布的统计模型。时间变量 T 始终为非负数，感兴趣的事件在工程领域常被称为”失效”，在医学领域常被称为”终点事件”。与普通回归模型不同，生存模型必须处理删失——即某些研究对象在研究结束时尚未观察到事件发生的情况。

The fundamental quantity in any survival model is the random variable T, which has a probability density function f(t) and a cumulative distribution function F(t) = P(T ≤ t). From these basic building blocks, we derive two specialized functions that form the heart of survival analysis: the survival function and the hazard function.

任何生存模型的基本量都是随机变量 T，它具有概率密度函数 f(t) 和累积分布函数 F(t) = P(T ≤ t)。从这些基本构件出发，我们推导出构成生存分析核心的两个专门函数：生存函数和风险函数。

2. Survival Function S(t) | 生存函数 S(t)

The survival function, denoted S(t), gives the probability that an individual survives beyond time t. Formally, it is defined as the complement of the cumulative distribution function:

生存函数（记作 S(t)）表示个体存活超过时间 t 的概率。形式上，它被定义为累积分布函数的补函数：

S(t) = P(T > t) = 1 − F(t) = ∫ₜ^∞ f(u)du

The survival function has three key properties. First, S(0) = 1, meaning at the start of observation, all individuals are alive. Second, S(t) is a non-increasing function of t, reflecting that survival probability cannot increase over time. Third, as t → ∞, S(t) → 0, meaning eventually the event will occur for everyone.

生存函数具有三个关键性质。第一，S(0) = 1，意味着在观察开始时所有个体都存活。第二，S(t) 是 t 的非增函数，反映生存概率不会随时间而增加。第三，当 t → ∞ 时，S(t) → 0，意味着最终每个人都会发生事件。

In actuarial science, the survival function is intimately related to life tables. For a life aged x, the survival probability is often written as ₜpₓ = S(t), which denotes the probability that a person age x survives t more years. This quantity is the cornerstone of premium calculation and pension liability valuation.

在精算科学中，生存函数与生命表密切相关。对于年龄为 x 的个体，生存概率通常记为 ₜpₓ = S(t)，表示年龄为 x 的人再多存活 t 年的概率。这一量是保费计算和养老金负债评估的基石。

3. Hazard Function h(t) | 风险函数 h(t)

The hazard function, also called the hazard rate or force of mortality, measures the instantaneous rate at which the event occurs at time t, conditional on survival up to time t. It is defined as:

风险函数（也称为危险率或死亡力）衡量在存活到时间 t 的条件下，事件在 t 时刻发生的瞬时速率。其定义为：

h(t) = lim(Δt→0) P(t ≤ T < t + Δt | T ≥ t) / Δt

Equivalently, the hazard function can be expressed in terms of the density and survival functions:

等价地，风险函数可以用密度函数和生存函数来表达：

h(t) = f(t) / S(t)

Unlike the survival function, which decreases over time, the hazard function can take any shape. It may be constant (as in the exponential distribution), increasing (as in aging populations where mortality rises with age), decreasing (as in the “burn-in” phase of manufactured components), or bathtub-shaped (common in human mortality across the lifespan).

与随时间递减的生存函数不同，风险函数可以呈现各种形态。它可能是常数（如指数分布）、递增（如死亡率随年龄上升的老龄化人群）、递减（如制造部件的”磨合期”），或浴缸形（人类一生中的死亡率常见此形态）。

4. Relationship between Survival and Hazard | 生存函数与风险函数的关系

The survival and hazard functions are two sides of the same coin. From the definition h(t) = f(t)/S(t) and the fact that f(t) = −dS(t)/dt, we can derive a fundamental differential equation:

生存函数和风险函数是同一枚硬币的两面。根据 h(t) = f(t)/S(t) 的定义以及 f(t) = −dS(t)/dt 的事实，我们可以推导出一个基本的微分方程：

h(t) = −d[ln S(t)]/dt

Integrating both sides from 0 to t yields the crucial connection between cumulative hazard and survival:

两边从 0 到 t 积分，就得到累积风险与生存之间的关键联系：

S(t) = exp(−∫₀^t h(u)du) = exp(−H(t))

Here, H(t) = ∫₀^t h(u)du is called the cumulative hazard function. This elegant exponential relationship means that if we can estimate the hazard function, we automatically obtain the survival function, and vice versa. Furthermore, the density function can be recovered as f(t) = h(t) · S(t).

这里的 H(t) = ∫₀^t h(u)du 称为累积风险函数。这一优雅的指数关系意味着，如果我们能估计风险函数，就能自动获得生存函数，反之亦然。此外，密度函数也可以通过 f(t) = h(t) · S(t) 恢复。

For actuarial students, it is particularly useful to memorize the discrete analogue: for integer-age life tables, the force of mortality μₓ is approximately equal to −ln(pₓ), where pₓ is the one-year survival probability. This approximation is excellent when mortality rates are low.

对于精算专业的学生来说，记住离散形式的对应关系特别有用：对于整数年龄生命表，死亡力 μₓ 近似等于 −ln(pₓ)，其中 pₓ 是一年生存概率。当死亡率较低时，这种近似非常精确。

5. Censoring | 删失

Censoring is the defining feature that distinguishes survival analysis from ordinary statistical analysis. Right censoring occurs when a subject is lost to follow-up or survives beyond the end of the study — we know only that their event time exceeds a certain value. Left censoring occurs when the event has already happened before observation begins. Interval censoring arises when we only know the event occurred within a specific time interval.

删失是生存分析与普通统计分析相区别的标志性特征。右删失发生在受试者失访或在研究结束时仍存活的情况——我们只知道他们的事件时间超过了某个值。左删失发生在观察开始之前事件已经发生的情况。区间删失出现在我们只知道事件在某个特定时间区间内发生的情况。

Ignoring censoring leads to severe bias. If we simply discard censored observations, we underestimate survival because we ignore the information that these individuals survived for a substantial period. If we treat censored times as actual event times, we overestimate the hazard. Survival models are specifically designed to incorporate censored data without bias.

忽略删失会导致严重的偏差。如果我们简单地丢弃删失观测值，就会低估生存率，因为我们忽略了这些个体存活了相当长一段时间的信息。如果我们把删失时间当作实际事件时间，又会高估风险。生存模型正是专门为无偏地纳入删失数据而设计的。

A common assumption in survival analysis is non-informative censoring, meaning that the censoring mechanism is independent of the event time. Under this assumption, censored observations contribute their survival information up to the censoring time through the survival function S(cᵢ), where cᵢ is the censoring time for subject i.

生存分析中一个常见的假设是非信息性删失，即删失机制与事件时间独立。在此假设下，删失观测通过生存函数 S(cᵢ) 贡献其截至删失时间的生存信息，其中 cᵢ 是第 i 个受试者的删失时间。

6. Kaplan-Meier Estimator | Kaplan-Meier 估计量

The Kaplan-Meier (KM) estimator, also known as the product-limit estimator, is a non-parametric method for estimating the survival function from observed data. It handles right-censored data naturally and produces a step-function estimate of S(t). The idea is straightforward: at each distinct event time t(j), we multiply the current survival estimate by the fraction of individuals at risk who survive that instant.

Kaplan-Meier（KM）估计量，也称为乘积极限估计量，是一种从观测数据估计生存函数的非参数方法。它自然地处理右删失数据，并产生 S(t) 的阶梯函数估计。其思想非常直接：在每个不同的事件时间 t(j) 处，将当前生存估计乘以在风险集中能存活过那一时刻的个体比例。

Ŝ(t) = ∏_{t(j) ≤ t} [1 − d(j)/n(j)]

where d(j) is the number of events at time t(j) and n(j) is the number of individuals at risk just before t(j). Censored observations are removed from the risk set at their censoring time but do not otherwise contribute a multiplicative factor.

其中 d(j) 是在时间 t(j) 发生的事件数，n(j) 是恰好在 t(j) 之前处于风险集中的个体数。删失观测在其删失时间从风险集中移除，但不另外贡献乘性因子。

The KM estimator is widely used to compare survival between groups — for example, comparing the survival of patients receiving two different treatments. A log-rank test can then be used to test whether the survival curves differ significantly. In actuarial work, KM methods are applied to analyse policy lapse rates and mortality improvement trends from portfolio experience data.

KM 估计量被广泛用于比较不同组间的生存——例如，比较接受两种不同治疗的患者生存率。随后可以使用对数秩检验来检验生存曲线是否存在显著差异。在精算工作中，KM 方法用于分析保单失效率和来自投资组合经验数据的死亡率改善趋势。

7. Parametric Models: Exponential and Weibull | 参数模型：指数分布与威布尔分布

While non-parametric methods are flexible, parametric models offer smoother estimates and allow extrapolation beyond the observation period. The simplest parametric model is the exponential distribution, which assumes a constant hazard:

虽然非参数方法灵活，但参数模型能够提供更平滑的估计，并允许超出观察期进行外推。最简单的参数模型是指数分布，它假设风险为常数：

h(t) = λ, S(t) = e^(−λt), 其中 t ≥ 0, λ > 0

The exponential model has the memoryless property: the probability of surviving an additional interval does not depend on how long one has already survived. This is unrealistic for human lifetimes but adequate for certain electronic components where failure is purely random.

指数模型具有无记忆性：再存活一个额外区间的概率不取决于已经存活了多久。这对于人类寿命来说是不现实的，但对于某些故障纯属随机的电子元件来说是合适的。

The Weibull distribution generalizes the exponential model by allowing the hazard to be monotonic. Its hazard function is:

威布尔分布通过允许风险为单调函数来推广指数模型。其风险函数为：

h(t) = λκ(λt)^(κ−1), κ > 0

When the shape parameter κ = 1, the Weibull reduces to the exponential distribution. When κ > 1, the hazard increases over time (appropriate for aging populations); when κ < 1, the hazard decreases (appropriate for infant mortality or burn-in failures). The Weibull survival function is S(t) = exp(−(λt)^κ).

当形状参数 κ = 1 时，威布尔分布退化为指数分布。当 κ > 1 时，风险随时间递增（适用于老龄化人群）；当 κ < 1 时，风险递减（适用于婴儿死亡率或磨合期故障）。威布尔生存函数为 S(t) = exp(−(λt)^κ)。

8. Cox Proportional Hazards Model | Cox 比例风险模型

The Cox proportional hazards model is the most widely used regression model in survival analysis. It relates the hazard function to a set of covariates without assuming a particular form for the baseline hazard:

Cox 比例风险模型是生存分析中应用最广泛的回归模型。它将风险函数与一组协变量联系起来，而无需假设基线风险的具体形式：

h(t | X) = h₀(t) · exp(β₁X₁ + β₂X₂ + … + β_pX_p)

Here, h₀(t) is an unspecified baseline hazard, and the covariates X₁, …, X_p enter through a multiplicative exponential term. The key assumption is proportionality: the hazard ratio between two individuals with different covariates is constant over time, since the ratio exp(βX₁)/exp(βX₂) does not depend on t.

其中 h₀(t) 是未指定的基线风险，协变量 X₁, …, X_p 通过乘性指数项进入模型。关键假设是比例性：具有不同协变量的两个个体之间的风险比随时间保持恒定，因为比值 exp(βX₁)/exp(βX₂) 不依赖于 t。

The coefficients β are estimated by maximizing the partial likelihood — a clever construction that cancels out the baseline hazard h₀(t). Interpretation of coefficients is in terms of log hazard ratios. For example, a coefficient β = 0.5 for a binary treatment indicator means that treatment multiplies the hazard by e^0.5 ≈ 1.65, so the treatment group has 65% higher instantaneous risk.

系数 β 通过最大化偏似然来估计——这是一种巧妙的构造，能够消去基线风险 h₀(t)。系数的解释以对数风险比的形式进行。例如，对于二元治疗指标，系数 β = 0.5 意味着治疗使风险乘以 e^0.5 ≈ 1.65，即治疗组的瞬时风险高出 65%。

9. Applications in Actuarial Science and Medicine | 在精算科学与医学中的应用

Survival models are indispensable in actuarial science. Life insurance pricing relies on survival functions from life tables to calculate net premiums. Pension actuaries use survival models to project the duration of benefit payments and to value defined-benefit pension plan liabilities. In health insurance, survival models help forecast the incidence of chronic diseases and the cost trajectory of long-term care policies.

生存模型在精算科学中不可或缺。人寿保险定价依赖生命表中的生存函数来计算净保费。养老金精算师使用生存模型来预测福利支付的持续时间，并对固定收益养老金计划的负债进行评估。在健康保险中，生存模型帮助预测慢性病的发病率和长期护理保单的成本轨迹。

In medical statistics, survival models are used in clinical trials to compare the efficacy of new drugs versus placebos, using endpoints such as overall survival or disease-free survival. The Kaplan-Meier curve, often accompanied by a log-rank test, is the standard output of any oncology trial. Cox regression adjusts for confounding factors like age, sex, and disease stage, enabling researchers to isolate the true treatment effect.

在医学统计中，生存模型在临床试验中用于比较新药与安慰剂的疗效，终点指标包括总生存期或无病生存期。Kaplan-Meier 曲线通常伴随对数秩检验，是任何肿瘤学试验的标准输出。Cox 回归调整年龄、性别和疾病分期等混杂因素，使研究人员能够分离出真正的治疗效果。

Beyond these fields, survival models power reliability engineering — predicting the lifetime of industrial components and scheduling preventive maintenance. In economics, they model the duration of unemployment spells; in sociology, the time to marriage or divorce; in customer analytics, the time until subscription cancellation. Anywhere “time until an event” matters, survival analysis provides the rigorous statistical toolkit.

除了这些领域，生存模型还驱动着可靠性工程——预测工业部件的寿命并安排预防性维护。在经济学中，它们用于建模失业持续期；在社会学中，用于分析结婚或离婚的时间；在客户分析中，用于研究订阅取消的时间。凡是”到事件发生的时间”重要的地方，生存分析都能提供严谨的统计学工具。

10. Conclusion | 结语

Survival models offer a coherent framework for analysing time-to-event data that ordinary regression cannot accommodate. The survival function S(t) describes the probability of surviving beyond time t; the hazard function h(t) captures the instantaneous risk; and the relationship S(t) = exp(−H(t)) unifies them. Censoring is handled naturally through likelihood-based inference, and tools such as the Kaplan-Meier estimator, Weibull models, and the Cox proportional hazards model provide a spectrum from fully non-parametric to fully parametric approaches.

生存模型为分析”时间到事件”数据提供了一个连贯的框架，而普通回归无法处理这类数据。生存函数 S(t) 描述存活超过时间 t 的概率；风险函数 h(t) 捕捉瞬时风险；关系式 S(t) = exp(−H(t)) 将二者统一起来。删失通过基于似然的推断被自然处理，Kaplan-Meier 估计量、威布尔模型和 Cox 比例风险模型等工具提供了从完全非参数到完全参数方法的光谱。

For students preparing for mathematics and actuarial examinations, mastering the definitions, interrelationships, and estimation techniques of survival models is not merely an academic exercise — it is a directly applicable skill in insurance, healthcare research, and data science. We encourage learners to practice deriving the key formulas from first principles and to apply them to real datasets. Understanding survival models will serve you well in any career where timing and risk intersect.

对于准备数学和精算考试的学生而言，掌握生存模型的定义、相互关系和估计技术不仅仅是学术练习——它在保险、医疗研究和数据科学中都是直接可用的技能。我们鼓励学习者从基本原理出发练习推导关键公式，并将其应用于真实数据集。理解生存模型将有助于你从事任何时间与风险交汇的职业。

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Survival Models: Concepts and Applications | 生存模型的概念与应用