# Proportional hazards

Survival analysis is a branch of statistics that allows us to model and analyze time-to-event data, such as the time it takes for a patient to experience a certain health outcome or the time it takes for a machine to fail. One important assumption that underlies most survival analysis methods is the proportional hazards assumption. In this Statistical Primer, we’ll discuss what this assumption is, why it’s important, and how to check it.

## What is the proportional hazards assumption?

The proportional hazards assumption states that the hazard ratio between any two groups is constant over time. The hazard ratio is the ratio of the hazard rate of one group (e.g., a treatment group) to the hazard rate of another group (e.g., a control group). Mathematically, the proportional hazards assumption can be expressed as:

$h(t) = h_0(t) * \exp(x \beta)$

where h(t) is the hazard function at time t, h0(t) is the baseline hazard function (i.e., the hazard function when all covariates are equal to 0), x is a vector of covariates, and $\beta$ is a vector of regression coefficients. The proportional hazards assumption holds if $\beta$ is constant over time.

## Why is the proportional hazards assumption important?

The proportional hazards assumption implies that the hazard ratio between two groups is the same at any point in time. If the proportional hazards assumption is violated, it means that the hazard ratio is not constant over time, which can lead to biased estimates of the treatment effect, incorrect confidence intervals, and misleading conclusions.

For example, if the hazard ratio between a treatment group and a control group changes over time, then the treatment effect may be overestimated in the short term and underestimated in the long term. This can have important implications for clinical practice, where decisions are often based on the long-term benefits and harms of treatments.

## How to check the proportional hazards assumption?

There are several ways to check the assumption. One common method is to plot the log minus log of the survival function for each group against log time and check if the curves are parallel. If the curves are parallel, it suggests that the proportional hazards assumption is met. If the curves cross, it suggests that the proportional hazards assumption is violated.

Another method is to use a statistical test, such as the Schoenfeld residual test or the scaled Schoenfeld residual test. These tests compare the observed residuals (i.e., the differences between the observed and expected values of the covariates) to their expected values under the proportional hazards assumption. If the test statistic is not significant, it suggests that the proportional hazards assumption is met. If the test statistic is significant, it suggests that the proportional hazards assumption is violated.

## Conclusion

The proportional hazards assumption is a key assumption in survival analysis that forces the hazard ratio between two groups to be constant over time. It is important to check this assumption to ensure that the estimates of the treatment effect are valid and reliable. There are several ways to check the proportional hazards assumption, including graphical methods and statistical tests. If the assumption is violated, it may be necessary to use alternative methods, such as modelling time-dependent effects or stratified analysis, to obtain valid estimates of the treatment effect.

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