Modelling Time to Events

Question

I am trying to see if there is a class of statistical regression models which can specifically be used to model the "time to event" (e.g. time at which a certain threshold is expected to be passed for the first time). So far I have seen models that can be used for modelling how different phenomena change with time (e.g. Hazard, Survival), but nothing which directly models the "first passage time to event".

Here is a review of the research I have done:

1) Joint Survival Models

I decided to write a Joint Survival Model based on AFT (Accelerated Failure Time) instead of Cox-PH since it is easier to determine the Survival and Hazard Function in AFT vs Cox-PH. In a joint model, the survival and longitudinal models are linked through shared random effects and/or correlated error terms. This allows the model to account for the correlation between the longitudinal and survival processes.

The common terms in both components are the covariates $X$ and the coefficients $\beta$. These represent the variables of interest and their effects on the response, respectively. In the survival model, they influence the survival time, while in the longitudinal model, they influence the evolution of the response over time.

Survival Model (AFT Model): The AFT model describes the survival time $T$ as a function of covariates $X$ and a random error term $\epsilon$. It can be written as:

$$\log T = -X'\beta + \epsilon$$

where $\beta$ is a vector of coefficients to be estimated, and $\epsilon$ follows some specified distribution, such as a Weibull or log-normal distribution.

In this case, let's assume that the Survival Times have a PDF of $f(t)$ and a CDF of $F(T)$. For $Z = -X'\beta$, then we have $\log T = Z + \epsilon$. This implies: $T = e^{Z + \epsilon} = e^Z e^\epsilon$

If we denote $Y = e^\epsilon$, then $T = e^Z Y$. The distribution of $T$ is specified, so we know the pdf and CDF of $T$, denoted as $f(t)$ and $F(t)$ respectively.

Given the transformation $T = e^Z Y$, we can derive the survival and hazard functions for $T$ as follows:

$$S_T(t) = P(T > t) = P(e^Z Y > t) = P(Y > t / e^Z) = 1 - F_Y(t / e^Z)$$

$$h_T(t) = f_T(t) / S_T(t) = f_Y(t / e^Z) / (1 - F_Y(t / e^Z))$$

where $f_Y$ and $F_Y$ are the pdf and CDF of $Y$.

Longitudinal Model: The longitudinal model describes the evolution of covariates over time. A common choice is the linear mixed effects model, which can be written as:

$$Y(t) = X(t)'\beta + Z(t)'\gamma + \epsilon(t)$$

where $Y(t)$ is the response at time $t$, $X(t)$ is a matrix of fixed effects covariates, $Z(t)$ is a matrix of random effects covariates, $\beta$ and $\gamma$ are vectors of fixed and random effects coefficients to be estimated, and $\epsilon(t)$ is a random error term.

2) ARIMA Model with Exogeneous Terms (i.e. ARIMAX):

Given a basic ARIMA model describing a Stochastic Process $Y_t$:

$$\Delta^d y_t = c + \phi_1 \Delta^d y_{t-1} + \ldots + \phi_p \Delta^d y_{t-p} + \theta_1 \varepsilon_{t-1} + \ldots + \theta_q \varepsilon_{t-q} + \varepsilon_t$$

We can modify this to turn it into a regression model:

$$\Delta^d y_t = c + \beta X_t + \phi_1 \Delta^d y_{t-1} + \ldots + \phi_p \Delta^d y_{t-p} + \theta_1 \varepsilon_{t-1} + \ldots + \theta_q \varepsilon_{t-q} + \varepsilon_t$$

where:

• $X_t$ is the exogenous variable.
• $\beta$ is the coefficient of the exogenous variable.

Problem: It seems to me that neither of these models (i.e. Joint Survival Models, ARIMAX) are explicitly modelling the "time to event". These models are modelling "what will happen at a certain time", but they are not directly modelling the probability the time at which the event will happen. I think these models can indirectly be used to model "time to event" via simulation and prediction, but this will not be an exact solution.

I tried to find some statistical models that address this specific problem and came across a concept known as "first passage time regression", e.g. https://www.jstatsoft.org/article/download/v066i08/879, https://www.wiley.com/en-br/First+Hitting+Time+Regression+Models:+Lifetime+Data+Analysis+Based+on+Underlying+Stochastic+Processes-p-9781848218895

If I understand correctly, we start by considering a stochastic process $X(t)$, where $t \geq 0$. The first hitting time $\tau_b$ of a level $b$ is defined as:

$$\tau_b = \inf\{t \geq 0: X(t) = b\}$$

This represents the first time $t$ that the process $X(t)$ reaches the level $b$.

Suppose now we decide to model this process using a Brownian Motion with drift $X_t = \mu(t) t + \sigma W_t$, where $W_t$ is a standard Brownian motion, $\mu(t)$ is the drift, and $\sigma$ is the standard deviation. We can model the drift term as a function of some covariates $Z(t)$:

$$\mu(t) = Z(t)'\beta$$

where $\beta$ is a vector of coefficients to be estimated. The first passage time $T_a$ for a fixed level $a > 0$ by $X_t$ is then distributed as an Inverse Gaussian (there is some mathematical property that says the first passage time of a Brownian Motion is naturally described by an Inverse Gaussian distribution, however I don't fully understand why to be honest) :

$$T_a \sim IG\left(\frac{a}{Z(t)'\beta}, \frac{a^2}{\sigma^2}\right)$$

The likelihood function for this model can be written as:

$$L(\beta | Z, T_a) = \sqrt{\frac{a^2}{2\pi \sigma^2 T_a^3}} \exp\left(-\frac{a^2(T_a - Z(t)'\beta)^2}{2\sigma^2 Z(t)'\beta^2 T_a}\right)$$

This likelihood function can be maximized to estimate the parameter $\beta$, giving us the maximum likelihood estimate (MLE) of $\beta$.

Thus, it appears that we can now exactly model the time at which a certain event might happen, by treating the underlying probability distribution of this event as a (covariate dependent) Stochastic Process and then analyzing this Stochastic Process using first passage times.

Is my understanding of this correct? Have I correctly described the shortcomings of the Joint Survival and ARIMAX models, and shown why an approach based on first passage times can remedy these shortcomings?

Answer 1

Thus it appears that we can now exactly model the time at which a certain event might happen, by treating the underlying probability distribution of this event as a (covariate dependent) Stochastic Process and then analyzing this Stochastic Process using first passage times. (Emphasis added.)

Your proposal doesn't give you any more of an "exact" event time than any other survival model. The "first passage times" for your "Stochastic Process" are themselves randomly distributed. As with any survival model, you end up with a distribution of event/first-passage times, often a very broad distribution. All you have done is to base your model on a particular set of underlying assumptions. See, for example, Figure 2 of a document you linked about "first passage time regression" based on a latent Wiener process. You still get a distribution of survival times with that approach, not "exact" times. For any particular application that set of assumptions might work better than another, but it isn't fundamentally different in what it produces: an estimate of a survival-time distribution, not any "exact" time.

In response to comments:

Although a Cox proportional hazards (PH) model only directly evaluates relative log-hazards as a function of covariates, you can use the results of the model to estimate the cumulative baseline hazard and thus the distribution of survival times for any set of covariate values. See this page, for example. In practice, the Cox model thus can give you something as equivalent to a "first hitting time" as an accelerated failure time (AFT) or other parametric model.

If the current behavior of the survival time density is influenced by previous behaviors of itself...

In the context of a survival model in which an individual can experience at most one event, I think what you're getting at is something that distinguishes PH models from other models. A PH model assumes that only the current values of covariates are associated with the current hazard of an event. The past history doesn't matter for fitting the model, even if covariate values have been changing over time. (That might affect the estimate of the baseline hazard, however.)

Beyond a PH model, a survival function can depend on the entire history of covariate values since time = 0. That's why "joint models" try to estimate both the true covariate history (typically from a restricted set of measurements that potentially include errors) and the association of covariate history with event risk. See, for example, the explanation of the R JM package by Dimitris Rizopoulos in Journal of Statistical Software 35: 1–33 (2010).

If an individual can experience multiple events, then intra-individual correlations might need to be taken into account even in the PH setting. A model of first passage time wouldn't, however, capture all that's of interest in that scenario.

Answer 2

The hazard $h(t)$ at time $t$ is the rate of change of the survival function $S(t)$

$$h(t) = -\frac{S'(t)}{S(t)}$$

For certain hitting time models, these hazards are not a constant ratio of time. This is for example the case with the inverse normal distribution.

Below is an example computation of the hazard and the relative hazard for two different hitting time distributions. The relative hazard is not constant in time.

library(statmod)
t = seq(0,10, 0.1)
h1 = -dinvgauss(t,1)/(1-pinvgauss(t,1))
h2 = -dinvgauss(t,2)/(1-pinvgauss(t,2))
plot(t,h1/h2, type = "l",
     main = "relative hazard for two inverse Gaussian distributed time to events")

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This dependency on time (if it is the case) makes a cox ph model less easily applicable.

On the other hand, an advantage of a cox ph model is that you do not need to explicitly model the absolute hazard and you only need to model the relative hazard. In such a case you only need to model the probability of the event to be of a certain class given that an event happened, and you do not care when the event happened. The advantage is that this requires less assumptions about the distribution of the time to events.

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Cox proportional hazards are well applicable when a model relates to the relative hazard. An accelerated failure time model is well applicable when a model relates to accelerated time until failure.

• An example of a relative hazard is a Poisson model where some failure happens when a first random hit occurs and the probability of a hit has a different rate.
• An example of an accelerated failure time model is when some failure happens due to an accumulation of little steps, like in a Brownian motion, and all steps occur with a faster rate or larger step sizes.