Hazard ratios: Risk over time

You've probably seen hazard ratios pop up in research papers and wondered what they actually tell you. They're everywhere in clinical trials and survival studies, but most explanations make them sound way more complicated than they need to be.

Here's the thing: hazard ratios are just a way to compare risks between groups over time. If you're running experiments or analyzing time-based outcomes, understanding these numbers can completely change how you interpret your results. Let's break this down in a way that actually makes sense.

Understanding hazard ratios and their role in assessing risk over time

Hazard ratios are fundamentally different from the risk metrics you might be used to. While relative risk and odds ratios give you a snapshot of cumulative risk, hazard ratios care about when things happen.

Think of it this way: if you're comparing two treatments, a hazard ratio tells you the instant-by-instant risk difference between groups. A hazard ratio of 0.5? That means at any given moment, your treatment group has half the risk of experiencing the event compared to your control group. Pretty straightforward when you strip away the jargon.

The real power comes from survival analysis, where timing matters as much as the outcome itself. You're not just asking "did this happen?" but "when did this happen?" - a crucial distinction when you're trying to understand treatment effects or user behavior patterns.

Most hazard ratios come from Cox proportional hazards models, which sound intimidating but are basically regression models that handle time-to-event data. The beauty is you can throw in covariates to adjust for confounding factors, making your comparisons more robust.

But here's where people get tripped up: interpreting hazard ratios requires understanding that they assume proportional hazards. Professor David Spiegelhalter puts it best when he emphasizes that we need to translate these statistics into plain language. Otherwise, we risk making decisions based on numbers we don't fully understand.

The importance of the proportional hazards assumption

The proportional hazards assumption is the backbone of hazard ratio interpretation, yet it's often glossed over. Simply put, it means the risk ratio between your groups stays constant over time. Violate this assumption, and your hazard ratio becomes misleading at best, completely wrong at worst.

Picture two groups with Kaplan-Meier curves that start parallel but then cross. That crossing point? It's screaming that your hazard ratio isn't telling the whole story. Maybe the treatment works great initially but loses effectiveness, or perhaps it takes time to kick in. A single hazard ratio can't capture that complexity.

So how do you check if this assumption holds? Visual inspection is your first line of defense:

• Plot those Kaplan-Meier curves

• Look for roughly parallel lines

• Watch for curves that cross or diverge dramatically

• Pay attention to how the gap between curves changes over time

For the statistically inclined, Schoenfeld residuals tests can give you a more formal assessment. A significant p-value here is bad news - it means your hazard ratio might be hiding important time-varying effects.

When the assumption fails (and it often does in real-world data), you've got options. Time-dependent covariates let the hazard ratio change over time, while stratified Cox models can handle different baseline hazards for subgroups. At Statsig, we've seen how these more nuanced approaches can reveal insights that a simple hazard ratio would miss entirely.

Interpreting and visualizing hazard ratios effectively

Let's get practical about what hazard ratios actually mean for your analysis. An HR of 2.0 doesn't mean "twice as likely" - it means "twice the instantaneous risk." That distinction trips up even experienced analysts.

The key to understanding hazard ratios lies in visualization. Kaplan-Meier curves aren't just pretty pictures; they're your window into what's really happening with your data. When you see those survival curves separating over time, that's your hazard ratio in action. The wider the gap, the stronger the effect.

But here's what most guides won't tell you: context is everything when interpreting hazard ratios. A hazard ratio of 0.8 might be:

• Clinically meaningful in a cancer trial (20% reduction in death risk)

• Practically irrelevant for user churn (barely moving the needle)

• Statistically significant but economically unimportant

The team at Stanford found that even medical professionals struggle with hazard ratio interpretation. Their solution? Always pair the number with a visual and a plain-English explanation. "Treatment A reduces the risk of progression by 30% at any given time" beats "HR = 0.7" every single time.

When presenting hazard ratios to stakeholders, focus on making them tangible. Use concrete timeframes, show the actual survival curves, and translate the statistics into business or clinical impact. Your audience doesn't need to understand Cox models - they need to know what the numbers mean for their decisions.

Applications and limitations of hazard ratios in research

Hazard ratios shine brightest in time-to-event analyses, particularly in medical research where they've become the gold standard for comparing treatments. Oncology trials practically live and breathe hazard ratios - they capture both whether patients survive and how long they survive.

But let's be honest about the limitations. The proportional hazards assumption is violated more often than researchers like to admit. When treatments have delayed effects or wear off over time, a single hazard ratio becomes an oversimplification. You're trying to summarize a complex, time-varying relationship with one number.

Here's where things get really tricky:

• Statistical vs. clinical significance: Your HR might have a tight confidence interval, but does a 5% risk reduction actually matter?

• Communication challenges: Try explaining hazard ratios to executives or patients without their eyes glazing over

• Misinterpretation risks: People instinctively think in terms of absolute risk, not instantaneous risk

The statistics community on Reddit regularly debates these issues, and for good reason. We've all seen papers where hazard ratios are presented without proper context or assumption checking. It's like showing only the headline without the story.

Despite these challenges, hazard ratios remain invaluable when used appropriately. They let us compare treatments while accounting for censoring, handle varying follow-up times elegantly, and provide insights that simpler metrics miss. At Statsig, we've found them particularly useful for understanding long-term user retention patterns where the timing of churn matters as much as whether it happens.

Closing thoughts

Hazard ratios don't have to be intimidating. Strip away the statistical complexity, and you're left with a powerful tool for comparing risks over time. The key is remembering that they tell you about instantaneous risk, not cumulative risk, and that the proportional hazards assumption can make or break your interpretation.

Next time you encounter a hazard ratio, ask yourself three questions: Does the proportional hazards assumption hold? What does this mean in practical terms? How can I communicate this clearly to my audience? Get those right, and you'll extract real insights from your time-to-event data.

Want to dive deeper? Check out the survival analysis course from Stanford, explore interactive visualizations of Kaplan-Meier curves, or experiment with your own data using Cox models. The more you work with hazard ratios, the more intuitive they become.

Hope you find this useful!