Recurrent Event Regression Analysis

In the same way that we want to understand the relationship between a single event outcome and a set of covariates, we may want to understand the relationship between a recurrent event process and a set of covariates. For example, we may want to understand the relationship between the number of times an item needs repairing based on the environment, the duty cycle, and the types of inputs it receives. For example, an electric motor in a hot and humid environment may need more repairs than one in a cool and dry environment. Additionally, a motor that is used more often may need more repairs than one that is used less often. Finally, a motor that is used in a factory that produces a lot of dust may need more repairs than one that is used in a cleaner environment. We can use recurrent event regression to understand the relationship between these covariates and the number of repairs that a motor needs.

We can start with the simplest version of recurrent event regression, a Homogeneous Poisson Process Regression. In this case we have a regular Homogeneous Poisson Process given by:

\[N(t) = \lambda t\]

But we extend this to include the impact that additional factors have on the cumulative count. So the model becomes:

\[N(t) = \phi\left( X \right) \lambda t\]

In this case, just as was the case with single event proportional hazard models, there is a factor that relates the covariates to the counting function. In doing so we can now model jointly the cumulative event process with factors that are likely to impact the rate at which the events occur. Again, repeating the lessons from single event survival analysis, a logical choice for the phi function would be the exponential function. This is because it ensures there is never a negative number and so will always provide a valid rate even during optimisation. That is, our model will be:

\[N(t) = e^{X \beta} \lambda t\]

This is the proportional intensity HPP model. The log-linear (exponential) link function \(e^{X\beta}\) is the standard choice because it guarantees a positive rate regardless of the sign of \(\beta\), mirrors the Cox model for single events, and gives regression coefficients a direct multiplicative interpretation on the rate.

A notable example within recurrent event regression is the Duane process, which is particularly relevant in reliability engineering. The Duane model is a form of non-homogeneous Poisson process (NHPP) that describes the improvement in reliability of a system or component over time, typically as a result of learning effects or reliability growth. The model posits that the failure rate of a system decreases as a function of cumulative operating time, reflecting the notion that systems become more reliable through usage and corrective actions. In the Duane process, the cumulative number of failures is modeled as a function of time, providing a way to quantify reliability growth and forecast future performance. The convetntional parameterisation for the Duane is:

\[N(t) = \alpha t^\beta\]

This can be interpreted as the number of events we can expect up to time t is given by the result of the equation.

In addition to specific models like the Duane process, recurrent event regression encompasses the broader class of proportional intensity models. These models, often used in the context of survival analysis, assume that the intensity function (or hazard function) for an individual’s time to the next event is proportional to a baseline intensity function, adjusted by the individual’s covariates. This assumption of proportionality allows for the straightforward interpretation of covariate effects on the hazard of an event occurring. This makes the comparison of risks between different groups or under different conditions easy and interpretable.

  1. The traditional Duane model’s cumulative number of failures as a function of time:

\[N(t) = \alpha t^\beta\]

  1. The failure intensity function derived from the Duane model:

\[\lambda(t) = \frac{dN(t)}{dt} = \alpha \beta t^{\beta - 1}\]

  1. The Proportional Intensity model incorporating a covariate \(x\):

\[\lambda(t \mid x) = \lambda_0(t) \exp(\gamma x)\]

  1. The adjusted failure intensity function with the covariate effect in the context of the Duane model:

\[\lambda(t \mid x) = \alpha \beta t^{\beta - 1} \exp(\gamma x)\]

These equations outline the framework for modelling the reliability growth of a system, incorporating the effects of covariates on the failure intensity.

These methods form a comprehensive toolkit for researchers and practitioners working with recurrent event data, enabling detailed analysis and prediction of event occurrences.