Game 14 · Testing and estimation
Who lives longer, and how do you show it?
Kaplan-Meier curves, censoring, median survival, log-rank test and hazard ratio.
Read with: Risk and survival, Cox regression26 DNB questions
Play
Build it, read it, compare two of them
Build a Kaplan-Meier curve one death at a time, read median survival and survival at a time point by clicking the plot, estimate the hazard ratio between two arms, and spot when one hazard ratio is the wrong summary. You have three lives.
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The Kaplan-Meier method
Multiply the fractions that survive each death
Patients join a study on different dates and are followed for different lengths of time. Some are still alive when the study closes or are lost to follow-up. Their time is censored: we know they survived at least that long. The Kaplan-Meier method uses every patient's time up to death or censoring.
S(t) = S(previous death time) × (n − d) ÷ n
- S(t)
- probability of surviving beyond time t
- n
- patients still at risk just before that time: not dead and not yet censored
- d
- deaths at that time
A censored patient leaves the number at risk but does not cause a drop. The curve is drawn as steps, with a short tick for each censored patient. Removing censored patients would underestimate survival, and counting them as alive for the whole period would overestimate it. The method assumes censored patients have the same prospects as those still followed.
Reading the curve
Median survival, survival at a time, and numbers at risk
Median survival is the time at which the curve first reaches 50% or below. Go across from 50% to the curve and down. If the curve never falls to 50%, the median is reported as “not reached”. Survival at a time, such as 1-year survival, is read by going up from that time to the curve and across. The numbers at risk under the axis show how few patients support the right-hand end, where each death makes a large step and the estimate is least certain.
Comparing curves
Log-rank test and hazard ratio
Expected deaths in group A at a death time = all deaths × (A at risk ÷ total at risk)
χ² = Σ (O − E)² ÷ E, 1 degree of freedom for two groups
- O
- observed deaths in a group, added over all death times
- E
- expected deaths in that group if both groups had the same survival
The log-rank test compares whole curves and gives a P value only. The size of the difference is the hazard ratio, usually from Cox regression. Hazard is the event rate at a moment among patients still at risk. HR 0.6 means that at any time the treated group has the event at about 60% of the control rate. For a harmful event, HR below 1 favours the new treatment.
| Feature | Hazards proportional | Curves cross |
|---|---|---|
| Hazard ratio over time | Roughly constant | Changes, for example early harm and late benefit |
| One HR from Cox | A fair summary | An average that can hide both effects |
| What to report | HR with 95% CI, log-rank P, medians | HR for early and late periods, or restricted mean survival time |
| How to check | Log-minus-log lines parallel | Schoenfeld residuals show a trend over time |
Worked tables are in the risk and survival chapter, and the proportional hazards checks in the Cox regression chapter.
Writing the answer
How to write this in a 10-mark DNB answer
- Define survival analysis as the analysis of time to an event, and define censoring with its three types (right, left, interval).
- Write the Kaplan-Meier formula and work a table of 8 to 10 made-up patients: time, at risk, deaths, (n − d) ÷ n, S(t).
- Draw the step curve with censoring ticks, mark the median survival, and add numbers at risk under the axis.
- Explain the log-rank test (observed against expected deaths) and say it gives a P value only.
- Define the hazard ratio, interpret one example below 1, and name Cox regression and its proportional hazards assumption.
- End with the uses (cancer, cardiology, neonatology) and the limits: few patients at the right-hand end, crossing curves.
The one thing to remember
The curve drops only at deaths, censored patients leave quietly, and the steps get larger as fewer patients remain at risk.
Past papers
DNB questions this game helps you answer
Questions from NBE theory papers for DNB, DrNB and Diploma courses, most often asked first. The tag after each question is the specialty and the year it was set.
- ×2Survival analysis
- ×1Kaplan-Meier curves. [2.5]
- ×1Kaplan-Meier survival curve. [4]
- ×1Kaplan-Meier Curve and its significance.
- ×1Kaplan-Meier Survival Curves.
- ×1Write notes on Sensitivity, Specificity, Positive Predictive Value and Hazard ratio. Give Examples.
- ×1Describe Kaplan-Meier curve for survival analysis. Write a brief account on Cox proportional hazard analysis model.
- ×1Briefly discuss Receiver operating charecteristics and Kaplan-Meier analysis.
- ×1Confidence Interval (CI) and Hazard Ratio (HR) in biostatistics.
- ×1Discuss Survival Analysis.
- ×1c) What is odds ratio, relative risk and hazard ratio?
- ×1Kaplan Meier analysis
- ×1Kaplan meir survival curve.
- ×1Write briefly about Cox Regression model.
- ×1Kaplan Meier survival analysis.
- ×1Log Rank Test.
- ×1Kaplan Meier Survival Curve.
- ×1Kaplan-Meier Analysis.
- ×1Survival Analysis.
- ×1Cox regression analysis.
- ×1What do the terms - odds ratio, hazard ratio, relative risk, absolute risk and confidence interval mean and where are these used?
- ×1Hazard Ratio.
- ×1What is vital statistics? Where can we use life table and survival analysis in medical science? Explain with example.
- ×1What is 'crude' and 'actuarial survival'? How is a Kaplan-Meier curve computed?
- ×1Kaplan Meier graphs
- ×1Mention the various statistical tools used for ‘survival analysis’. Plan a statistical analysis to determine prognostic factors for patients receiving chemotherapy for lung cancer.
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