This article from Longevity Bulletin provides an overview of mortality modeling under the survival analysis framework, machine learning methods, extensions to the GLMs, and examples of applying these models. With better predictions and understanding of mortality and longevity risks, (re)insurers and pension funds can leverage these advances to provide financial protection more effectively to more people.
The modeling and management of mortality and longevity risks are essential for insurers, reinsurers, pension funds, banks, and government agencies. More advanced models can yield a more accurate mortality estimation that can assist pricing, reserving, underwriting, solvency, and profitability.
In this article I set out an overview of mortality modeling using survival analysis techniques, a discussion of machine learning methods, extensions to the generalized linear models (GLM), and some examples of applying these models. The focus will be on the use of individual-level data in models that incorporate a multitude of risk factors, rather than group-level or population-level data.
Traditionally, actuaries and demographers made extensive use of mortality rates, i.e., the probability of a group of similar lives dying in one year. In more recent times, mortality modeling has advanced considerably, with model development employing a statistical framework capable of statistical tests and confidence intervals. At the same time, there has been a significant increase in the volume, veracity, velocity, and variety of data available for analysis, which encompasses policyholder data, demography, postcodes, electronic health records, lifestyle, and credit scores.
As insurers and pension schemes collect experience data on individual policyholders, this is essentially a form of longitudinal study because the individuals are observed for a period of time until a particular event of interest occurs (known as time-to-event or survival time).
The main challenge of time-to-event data is the presence of incomplete observations. This occurs due to censoring, i.e., when a death event happens outside of the observation period. For this reason, many classical statistical models and machine learning methods could not transpose directly to time-to-event data.
Modeling time-to-event data requires a specific approach called survival analysis. It is used to predict the target variable of survival time until mortality, while accounting for censoring, and the presence of explanatory variables that may affect survival time (Rodríguez, 2007).
Survival analysis is a very useful tool in evaluating risks such as mortality, longevity, morbidity, lapses, and demographic factors (e.g., marriage, migration, and fertility). As the name indicates, survival analysis has origins in the field of medical research to estimate the survival rate of patients after a medical treatment. It is also known as reliability analysis in engineering, duration analysis in economics, and event history analysis in sociology (Abbas et al., 2019).
Figure 1 shows a taxonomy of survival analysis models, which provides a holistic view of statistical and machine learning methods, categorized by continuous-time and discrete-time approaches (with separate charts for each).
Survival analysis theory focuses on two key concepts in continuous time:
a. the survival function S(t), i.e., the probability of being alive just before duration t
b. the hazard function h(t), i.e., the instantaneous death rate at time t, also known as the force of mortality by actuaries
There is a one-to-one relationship between the hazard function and the survival function. Whatever functional form is chosen for the hazard function, one could use it to derive the survival function. The integral of the survival function then gives the expectation of life, i.e., mean of survival time (Rodríguez, 2007).
The non-parametric methods are simple and require no assumptions on distributions. The Kaplan-Meier estimator, also known as the product limit estimator, provides an empirical estimate of the survival function. The Nelson- Aalen estimator approximates the cumulative hazard function. As the sample size gets very large, these two estimators are asymptotically equivalent (Jenkins, 2005). Kaplan-Meier and Nelson-Aalen are univariable methods and likely to be less predictive; therefore, considering multivariable methods is recommended if multiple explanatory variables are available.
In the semi-parametric category, the Cox proportional hazard model was proposed by Cox (1972) in perhaps the most often cited article on survival analysis. The hallmark of the Cox model is that it allows one to estimate the relationship between the hazard function and explanatory variables, without having to make any assumption on the baseline hazard function. Proportional hazards modeling assumes that the ratio of the hazards for any two individuals is constant over time. The fact that the hazards are proportional is helpful in making interpretations, such as when identifying the better treatment in medical trials or analyzing loadings of risk factors in underwriting. The Cox model can also be generalized to handle time-varying covariates and time-dependent effects (Rodríguez, 2007).
The Piecewise-Constant Exponential (PCE) model is another example of semi-parametric continuous-time model and can be seen as a special type of proportional hazards model (Jenkins, 2005). When the time axis is partitioned into a number of intervals in a PCE model, it assumes that the baseline hazard is constant within each interval. The advantage is that one does not have to impose the overall shape of the hazard function in advance. Another useful property of the PCE model is its equivalence to a certain Poisson GLM model; this will be discussed later.
Parametric statistical methods assume that survival time follows a particular theoretical distribution (Wang et al., 2019). Commonly used distributions include exponential, Weibull, Normal, Gamma, log-logistic, log-normal, and Gompertz (Jenkins, 2005). If the survival time follows the assumed distribution, resulting outcomes are accurate, efficient, and easy to interpret, but if the assumption is violated parametric models can give sub-optimal results.
Another approach in the parametric category is the accelerated failure time (AFT) model. AFT assumes a linear relationship between the log of survival time and the explanatory variables. The effect of variables is to accelerate or decelerate the life course. The Weibull model is the only model that satisfies both proportional hazards and AFT assumptions (Rodríguez, 2007).
As discussed in Bayesian Survival Analysis (Ibrahim et al., 2001), Bayesian approaches can be applied to survival models, including parametric, proportional, and non-proportional models. Interpretability is a strength of Bayesian modeling. Other examples of survival models include machine learning methods, which will be discussed in Section 2, as well as competing-risk and multi-state models (Jenkins, 2005).
The survival analysis techniques discussed in the previous section assume continuous measurement of time. Although it is natural to consider time as a continuous variable, in practice observations are often on a discrete time scale, such as days, months or years (Jenkins, 2007). An advantage of discrete-time modeling is the embedding of the GLM framework.
Interestingly, the PCE model is equivalent to a GLM Poisson log-linear model for discretized pseudo-data, when the death indicator is the response, and the log of exposure times is the offset (Rodríguez, 2007). The likelihood function of PCE and independent Poisson observations happen to coincide and would therefore lead to the same estimates.
Generally, the choice of GLM for survival analysis depends on the nature of data (Rodríguez, 2007):
i. If data is continuous and if one is willing to assume hazard is constant in each interval, the Poisson GLM is appropriate as it allows use of partial exposures
ii. If data is truly discrete, logistic regression is recommended
iii. If data is continuous but only observed in grouped form, the complementary log-log link is preferable.
In recent years, machine learning models have achieved success in many areas. This is due to built-in strengths that include higher prediction accuracy, ability to model non-linear relationships, and less dependence on distribution assumptions. Nevertheless, some machine learning algorithms bring notable weaknesses as well, such as difficulty in interpretation, sensitivity to hyperparameters, and a tendency to overfit. Dealing with censored data presents the biggest challenge for using machine learning in survival analysis.
A deeper understanding of survival analysis, and how machine learning can overcome the challenge of censored data, is required in order to effectively adapt it to mortality modeling.
The discussion below starts with regularization – a versatile machine learning technique applicable to many approaches, including GLM and classical survival models. The total range of machine learning models is vast; therefore, I look just at continuous-time models and deliberately exclude some notable discrete-time approaches, for instance support vector machines and random forest. Discrete-time supervised machine learning models are discussed in Modeling Discrete Time to Event Data (Tutz and Schmid, 2018), while extensions of GLM are discussed in Section 3.
Regularization is a technique used to simplify a model and reduce overfitting by adding penalties or constraints to the model-fitting problem. The three main types of regularization are:
i. Ridge, also known as Tikhonov or L2 regularization, adds a penalty term based on the squared value of coefficients. It reduces the size of coefficients and deals with correlations between features simultaneously.
ii. Lasso (least absolute shrinkage and selection operator), also known as L1 regularization, adds a penalty term based on the absolute value of coefficients. In contrast to Ridge, Lasso can shrink coefficients to zero, which means it can perform automatic variable selection. Extensions of Lasso include Group Lasso, Fused Lasso, Adaptive Lasso, and Prior Lasso.
iii. Elastic net linearly combines the Ridge and Lasso penalty terms.
Introducing regularization into Cox proportional hazard models provides us with a form of machine learning – the resulting models include Ridge-Cox (Verweji and Van Houwelingen, 1994), Lasso-Cox (Tibshirani, 1997), and Elastic Net-Cox (Simon et al., 2011).
The Cox-Boost method (Binder and Schumacher, 2008) incorporates gradient boosting machines in Cox models. It is useful on high-dimensional data and considers some mandatory variables explicitly in the model.
Survival trees are classification and regression trees (CART) specifically designed to handle censored data (Gordon and Olshen, 1985). The data is recursively partitioned based on a splitting criterion and objects with similar survival times are grouped together. This approach is easier to interpret and does not rely on distribution assumptions.
In machine learning, ensemble learning is a method that takes a weighted vote from multiple models to obtain better predictive performance than could be obtained from any of the constituent models alone. Common types of ensembles include bagging, boosting, and stacking.
Bagging survival trees involves taking a number of bootstrap samples from the survival data, building a survival tree for each sample, and then averaging the tree nodes’ predictions (Hothorn et al., 2004).
Random survival forest is similar to bagging, but random forest uses only a random subset of the features for selection at each tree node. This helps reduce the correlation between trees and improves predictions. Random survival forest does not depend on distribution assumptions and can be used to avoid the proportional hazards constraint of a Cox model (Ishwaran et al., 2008).
Boosting combines a set of simple models into a weighted sum and is iteratively fitted to the residuals based on the gradient descent algorithm. Hothorn et al. (2006) proposed gradient boosting to account for censored data.
Stacking combines the output of multiple survival models and runs it through another model. Wey et al. (2015) proposed a framework of stacked survival models that combines parametric, semi-parametric and non-parametric survival models. This approach has performed well by adaptively balancing the strengths and weaknesses of individual survival models.
Artificial neural networks (ANN) consist of layers of neurons interconnected as a network to solve optimization problems. The adjective ‘deep’ in deep learning refers to the use of multiple layers in the network. Neural networks and survival forests are examples of non-linear survival methods.
The initial adaptation of survival analysis to neural networks sought to generalize Cox with only one single hidden layer (Farragi and Simon, 1995). Katzman et al. (2018) later proposed DeepSurv, a deep feed-forward neural network generalizing the Cox proportional hazards model. It has the advantage of not requiring a priori selection of covariates, by learning them adaptively.
DeepHit is a deep neural network that learns the distribution of survival times directly (Lee et al., 2018). Unlike parametric approaches, it makes no assumption of the underlying stochastic processes and allows for the relationship between covariates and risk to change over time. DeepHit can be used for survival datasets with a single mortality risk as well as multiple competing risks.
GLM is a popular tool in survival analysis due to its versatility, interpretability, predictive power, and availability in many software packages. Section 1 demonstrated that GLM Poisson, Logistic and C-Log-Log models can perform survival analysis. However, GLMs elicit two common negative views: they are restricted by distribution assumption, and they do not account for non-linear relationships, which reduces predictive performance.
The first view is refutable because, as discussed in Section 1, a GLM is merely a device to derive the underlying survival model, so the model is not restricted by distribution assumptions of GLM.
The second issue can be mitigated using approaches such as these to extend or enhance GLM. Note that these are not restricted to survival analysis and can be applied to GLMs in general. Some practitioners would view these as ways to combine the advantages of GLMs (for instance, interpretability) with the power of machine learning:
Tedesco et al. (2021) constructed machine learning models to predict all-cause mortality in a two- to seven-year time frame in a cohort of healthy older adults. The models were built on features including anthropometric variables, physical and lab examinations, questionnaires, and lifestyle factors, as well as wearable data. Random forest showed the best performance, followed by logistic regression, AdaBoost, and decision tree. Additional insights could be extracted to gain understanding on healthy aging and long-term care.
Using the MIMIC-III dataset on long-term mortality after cardiac surgery and the AUC metric, the researchers observed the order of model performance, from highest to lowest, to be AdaBoost, logistic regression, neural network, random forest, Naïve Bayes, XGBoost, bagged trees and gradient-boosting machine (Yu et al., 2022).
The OpenSAFELY paper (Williamson, 2020) applied the multivariable Cox model to analyze data from 17 million patients in England and subsequently identified a range of risk factors for Covid-19 mortality This was instrumental in helping to identify high-risk population subgroups, as Dan Ryan describes elsewhere in this Bulletin. Later that year, RGA (Ng et al., 2020) published a paper that cross-compared an all-cause mortality model with OpenSAFELY’s Covid-19 model in a parallel and multivariable way. This revealed insights on excess mortality risk from certain factors, which were useful to actuaries and underwriters. Six months later, the OpenSAFELY team published another paper (Bhaskaran et al., 2021) analyzing Covid-19 and non-Covid-19 mortality odds ratios, by using logistic regression. The team produced results that were very consistent with RGA’s.
The goal of mortality modeling is to predict and understand mortality and longevity. This article provides a survey and taxonomy of mortality modeling under the survival analysis framework, structured by continuous-time and discrete-time, as well as statistical methods and machine learning. The choice of model depends on the nature of the data and the purpose – whether it is solely about predictive accuracy or if interpretability is important.
Due to the increasing availability of data, technology and development in survival analysis and machine learning, financial services providers, such as insurers and pension funds, can leverage advances in these areas to provide financial protection more effectively to more people.
Reprinted with permission of The Institute and Faculty of Actuaries.
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