Describe the life table and the principles of survival analysis.

The Life Table and Survival Analysis

• Its columns, which should be known — x, the age; lx, the number surviving to exact age x; dx, the number dying between x and x+1; qx, the probability of dying in that interval; px, the probability of surviving it; Lx, the person-years lived in the interval; Tx, the total person-years remaining; and ex, the expectation of life at age x
• The two types — the current (period) life table, based on the age-specific mortality of a single period applied to a hypothetical cohort, which is what is normally published; and the cohort (generation) life table, which follows a real cohort from birth to death and therefore takes a lifetime to complete
• The crucial point about life expectancy AT birth, which is almost universally misunderstood: it is not a prediction of how long a baby born today will live. It is the average years that would be lived IF the current age-specific death rates persisted unchanged for that person’s whole life — which they certainly will not
• The second misunderstanding concerns historical comparison — a life expectancy at birth of about 30 in an earlier era did not mean people commonly died at 30
• It reflected very high infant and child mortality dragging the average down; those who survived childhood frequently lived into old age. This is why life expectancy at age 1 or at age 5 may exceed that at birth in a high infant-mortality population — a striking fact that makes the point immediately
• The uses of the life table — comparing the mortality of populations without the distortion of age structure; life insurance and actuarial work, for which it was invented; population projection; estimating survival after diagnosis or treatment; and calculating HALE and DALYs
• Survival analysis applies the same logic to clinical data, and exists because two features of follow-up data defeat ordinary methods:
• 1. The outcome is time-to-event, not merely whether the event occurred
• 2. Censoring — some patients are lost to follow-up, withdraw, or are still alive when the study ends, so their survival time is incomplete but not unknown — we know they survived AT least that long
• Discarding censored observations would waste that information and bias the result; survival analysis uses it
• The KAPLAN-MEIER method produces the familiar stepped survival curve, in which each step down represents an event and censored observations are marked as ticks
• The median survival is read where the curve crosses 50%, and is preferred to the mean because survival data are skewed and the mean cannot be calculated until everyone has had the event
• Two curves are compared by the log-rank test, and the COX proportional hazards model allows adjustment for several covariates and yields the hazard ratio
• Reading a survival curve critically — always look at the numbers AT risk printed beneath it, because the right-hand end of the curve is based on very few patients and is therefore unreliable, however dramatic the separation appears
• The assumption of the Cox model is proportional hazards — that the ratio of hazards stays constant over time — which should be checked, and fails visibly if the curves cross
• Life expectancy at birth is not a prediction, since it assumes current death rates persist for a whole lifetime
• A historical figure of thirty never meant dying at thirty; infant mortality was dragging the average down
• Life expectancy at age one may exceed that at birth where infant mortality is high, which makes the point immediately
• Check the numbers at risk under a survival curve, since its right-hand end rests on very few patients
• Censored observations carry real information; we know the patient survived at least that long
• Discarding censored cases wastes data and biases, which is precisely why survival analysis exists
• Median survival is read at fifty per cent, and is preferred to the mean because the data are skewed
• The mean cannot be calculated until everyone has died, which is a second practical reason for using the median
• The log-rank test compares two curves and Cox regression adjusts for several covariates at once
• Crossing curves violate proportional hazards, which invalidates the Cox model and is visible on inspection
• Current life tables use a hypothetical cohort, while cohort life tables follow real people and take a lifetime
• Life tables remove the distortion of age structure, which is why they permit fair comparison between populations
• Each step down marks an event and each tick a censored observation, which is how a Kaplan-Meier plot is read
• Kaplan-Meier suits any time-to-event outcome, not merely death, and is used for relapse, discharge and recurrence