AFT Models

These industrial machines are all the same model, yet some fail after 300 hours while others keep running normally for more than 900 hours. Why? Does a higher operating temperature shorten the time to failure? How much can regular maintenance extend a machine's life?

In this way, the method that analyzes how the time until a particular event changes with various conditions is called the accelerated failure time model (AFT, Accelerated Failure Time Model).

If you analyze only whether a machine failed, the information about when it failed is lost. Where the Cox proportional hazards model describes the relative size of the instantaneous risk as a hazard ratio (HR), the AFT model describes how many times longer or shorter the time to the event itself becomes, using the time ratio (Time Ratio, TR). Records of machines that did not fail during the observation period can also be used as censored data.

It is widely used, from reliability analysis of machines and equipment to survival times in medicine and time until customer churn. In this article we use an example of industrial machine failures to look at how the AFT model works, how to interpret time ratios, how to choose a lifetime distribution, how it differs from the Cox model, and what to watch for when reporting real results.

The AFT model is a parametric survival model that explains the logarithm of the time to an event with explanatory variables. It is usually written in the form log(T) = β₀ + β₁X₁ + ··· + βₖXₖ + σϵ, with an assumption about the distribution of the error term.

For example, if under the same conditions machines that received maintenance take longer to fail than machines that did not, this shows up as a time ratio greater than 1. However, the effect must be interpreted in light of the distribution the model assumes and how the variables are coded.

Suppose machine A failed at 300 hours, while machine B kept working normally until the test ended at 800 hours. We do not know B's exact failure time, but we still have the useful information that it ran for at least 800 hours.

In an AFT model, holding other conditions equal, a one-unit increase in an explanatory variable multiplies the typical time to the event by exp(β). This is called the time ratio (TR) or acceleration factor. You can think of it as the time axis being stretched or shrunk relative to the reference condition.

This table is a conceptual example showing how the model's time multiplier is applied. It does not mean that an individual machine will fail at exactly that time.

Suppose we observed 240 industrial machines of the same model for up to 1,000 hours. For each machine we record the operating time until the end of observation and whether it failed, and use operating temperature, load level and maintenance interval as explanatory variables.

The time ratios and confidence intervals presented below are hypothetical values for explaining the results of the analysis. They are not estimates obtained by fitting real measurements from 240 machines.

If the time ratio for high load is 0.72, then with the other variables held equal, the time to failure is about 72% of that under the reference load. In other words, the model's time axis is shortened by about 28%.

Censored data tell us that the machine had not failed up to the time it was observed. Just because its failure time was not observed, you must not arbitrarily enter the machine as having failed at 1,000 hours. Also, if loss to follow-up is related to the machine's condition, you need to check whether the usual assumption of independent censoring is reasonable.

In the hypothetical results, higher temperature and high load are associated with shorter times to failure, and monthly maintenance is associated with relatively longer times to failure. You should check both whether the confidence interval includes 1 and whether the size of the estimate is practically meaningful.

These values are a consistent example set up for interpretation practice; they do not represent actual model estimates or goodness-of-fit test results.

AFT assumes a distribution for the time to the event. The Weibull, log-normal and log-logistic distributions are commonly used and are compared according to the failure mechanism and the shape of the data.

Rather than drawing mechanical conclusions from AIC or log-likelihood alone, it is better to check residuals, fitted survival curves, censoring patterns and field knowledge together.

When you want to describe directly how long machines last, the AFT time ratio can be intuitive. On the other hand, to compare the relative risk of failure at a particular point in time, the Cox model may be more suitable. AFT is not always more accurate, nor does it automatically solve violations of the Cox assumptions.

Distributional assumption: If the chosen Weibull, log-normal or log-logistic distribution does not match the actual pattern of failure times, the time ratios and predicted lifetimes can be distorted.

Censoring and start of observation: Check the reasons for test endings and stopped recordings, when the test started, and how equipment that was already operating before observation began was handled (whether there is left truncation).

Operating conditions that change over time: If temperature or load changes during follow-up, consider whether simple baseline variables alone can adequately capture that change.

Causal interpretation: Machines with shorter maintenance intervals may have been placed in riskier environments to begin with. In observational data, a time ratio is a conditional association; it does not automatically prove a causal effect of maintenance.

In particular, repeated failure data must not simply be treated like the first failure times of many independent machines. You need to reflect changes in risk after repair and the repeated-measures structure within each machine.

In the report, give the number of machines and the observation period, the definition of a failure event, the censoring rate, the chosen lifetime distribution, the units and reference categories of the explanatory variables, and the time ratios with 95% confidence intervals. In a real analysis, also describe the goodness-of-fit diagnostics for the distribution.

The numbers in the sentence above are hypothetical results for learning. In a real study, add them only after checking the name of the fitted distribution, the numbers of events and censored observations, and the model diagnostics.

Instead of looking only at whether an event occurred, the AFT model uses the time it took to occur to analyze differences in time across conditions. In the industrial machine example, we used time ratios (TR) to describe how temperature, load level and maintenance interval relate to failure time. The key point is that a TR below 1 means a shorter time to the event and a TR above 1 means a longer one.

However, a time ratio must not be reinterpreted as a failure probability or an instantaneous hazard ratio. You need to check how the censored data arose and whether the chosen lifetime distribution describes the data adequately. The strength of AFT is that it conveys lengthening or shortening of lifetimes intuitively, and its interpretation, too, must stay within the bounds of the model assumptions and the study design.