Glossary
The terms of survival analysis used in this tutorial, with the page where each is explained.
- Adjusted survival curve
- The survival curve each group would have if all groups had the same distribution of the other variables of a Cox model; the module computes it by direct standardisation. Adjusted curves
- At risk
- A patient is at risk at time \(t\) if they are alive (event-free) and still under observation just before \(t\). The number at risk \(n_j\) is the denominator of each Kaplan-Meier step. Kaplan-Meier
- Baseline hazard, \(h_0(t)\)
- The hazard of a patient with all variables at their reference level (or 0) in a Cox model. Its shape is not estimated as part of the hazard ratios. Cox
- C-index (concordance)
- The probability that, of two patients whose order of events is known, the one the model predicts to be at higher risk has the event first. 0.5 = chance, 1 = perfect. Cox
- Censoring (right censoring)
- Observation of a patient ends before the event: their survival time is known to be longer than the observed time. Administrative censoring happens when the study closes; informative censoring, when the reason for censoring is related to prognosis, biases the estimates. Censoring
- Competing risk
- An event that prevents the event of interest (death from another cause when the event is death from cancer). Treating it as censoring overestimates the cumulative incidence. Kaplan-Meier
- Confounding
- A distortion of the association between a variable and the outcome because of a third variable associated with both. In the tutorial, part of the crude effect of age was due to ECOG. Cox example
- Cox regression (proportional hazards model)
- \(h(t \mid x) = h_0(t)\exp(\beta_1 x_1 + \dots)\): a regression model for the hazard, which gives hazard ratios adjusted for the other variables. Cox
- Cumulative hazard, \(H(t)\)
- The hazard added up from 0 to \(t\); \(S(t) = e^{-H(t)}\). Time-to-event data
- Cumulative incidence
- The probability of having had the event by time \(t\), \(1 - S(t)\) when there are no competing risks.
- Direct standardisation
- Averaging the predicted survival curves of all patients after setting the variable of interest to the same level for everyone; used for adjusted survival curves.
- Effect modification (interaction)
- The effect of one variable depends on the value of another. Tested with an interaction term in a Cox model. Interactions
- Event
- What ends the survival time: death, relapse, progression. Its definition, and the time origin, must be stated.
- Events per variable
- The number of events divided by the number of coefficients of a model; roughly 10 or more is a common rule of thumb. Cox
- Greenwood’s formula
- The variance of the Kaplan-Meier estimate, used for its confidence interval. Kaplan-Meier
- Hazard, \(h(t)\)
- The rate of the event at time \(t\) among patients still event-free just before \(t\); a rate, with units (per month), not a probability. Time-to-event data
- Hazard ratio (HR)
- The ratio of the hazards of two groups (or per unit of a covariate). HR = 1: no difference; HR < 1: lower hazard. It is not a risk ratio. Cox
- Kaplan-Meier estimator
- The product of the conditional probabilities of surviving each event time, \(\prod (1 - d_j/n_j)\); the standard estimate of the survival curve. Kaplan-Meier
- Log-log plot
- \(\log(-\log \widehat S(t))\) against log time, by group; parallel curves support proportional hazards. Assumptions
- Log-rank test
- Compares survival curves by adding up observed minus expected events at each event time. Weighted versions: Gehan-Breslow, Tarone-Ware, Peto-Peto. Log-rank
- Median follow-up
- The median time patients were (or would have been) followed, estimated with the reverse Kaplan-Meier method: the event and censoring indicators are swapped. KM example
- Median survival
- The first time at which the Kaplan-Meier curve reaches 0.5 or less. “Not reached” if fewer than half of the patients had the event. Kaplan-Meier
- Partial likelihood
- The likelihood that the Cox model maximises: at each event time, the probability that the patient who died was the one to die among those at risk. The baseline hazard cancels out.
- Proportional hazards
- The assumption of the Cox model that each hazard ratio is constant over time. Assumptions
- Reference level
- The level of a factor that the others are compared with (HR = 1); in jamovi, the first level of the variable.
- Schoenfeld residuals
- For each death, the variable’s value in the patient who died minus its average in the risk set; plotted against time, they show whether a coefficient changes over time. Assumptions
- Stratified Cox model
- A Cox model with a separate baseline hazard in each stratum and common hazard ratios; the stratification variable gets no HR. Assumptions
- Survival function, \(S(t)\)
- The probability of being event-free beyond time \(t\), \(P(T > t)\).
- Trend test
- A 1-df test that survival (or the hazard) changes steadily along the levels of an ordered factor. Log-rank
- Univariable and multivariable
- A univariable (crude) model has one variable; a multivariable model estimates each variable’s effect adjusted for the others.
- Wald, likelihood ratio and score tests
- Three ways of testing coefficients of a Cox model; the global versions test that all hazard ratios are 1. With one factor, the score test is the log-rank test.