IAI CS2 key terms

The probability that a subject's lifetime exceeds a given time t, written S(t) = P(T > t). It starts at one and decreases, possibly to zero, as time increases.

The instantaneous rate of failure at time t given survival to that time. It describes the force of mortality at each instant and need not integrate to one over the lifetime.

The integral of the hazard function from zero to time t. It relates to the survival function through S(t) = exp(-H(t)), which links additive hazard models to multiplicative survival models.

Incomplete observation of the event time, usually because follow-up ended first (right censoring). Analysis must account for the information that the event occurred after the censoring time.

An assumption that the censoring time carries no information about the future event time beyond what is already observed. It is required for standard likelihood and partial likelihood methods to be valid.

Occurs when subjects only enter the risk set after a delayed entry time, so those who fail before entry are never observed. It requires adjusting risk sets, for example in the Nelson-Aalen or Cox partial likelihood.

A non-parametric product-limit estimate of the survival function that updates at each event time by multiplying by the proportion surviving that step. It handles right-censored data and yields a step function.

An approximate variance formula for the Kaplan-Meier estimator, summing the number of events divided by the product of risk set size and number at risk minus events at each event time. It underpins confidence intervals.

A transformation of the survival estimate, computing the log of minus the log of S(t), used to build confidence intervals for Kaplan-Meier curves that stay within zero and one and behave better in the tails.

A non-parametric estimator of the cumulative hazard function obtained by summing the reciprocal of the risk set size at each event time. It is the basis for the Breslow estimate of baseline hazard.

A score test comparing survival curves between groups under proportional hazards, weighting observed minus expected events equally over time. It is the standard unadjusted comparison of Kaplan-Meier curves.

The set of subjects still under observation and event-free just before each event time. Its size enters the partial likelihood, Kaplan-Meier and Nelson-Aalen calculations, and delayed entry changes its composition.

A survival estimate computed over fixed intervals, assuming events occur uniformly within each interval. It handles grouped data and is closely related to the Kaplan-Meier estimator as intervals shrink.

The process of fitting a smooth parametric curve to crude mortality or hazard estimates, reducing random noise while preserving the underlying pattern. It is central to constructing life tables and mortality tables.

The total time subjects spend under observation during a period, often approximated by the average population at risk times the length of the period. It is the denominator of mortality rates.

The number of observed events divided by the corresponding exposed to risk, without adjustment for age or other confounders. Crude rates are simple but can mislead when populations differ in structure.

The ratio of observed deaths to deaths expected from applying a reference set of age-specific rates to the study population. Values above one indicate higher mortality than the reference.

The assumption that covariate effects multiply the hazard by a constant factor over time, so hazard ratios do not change with follow-up time. It is checked with Schoenfeld residuals or time interaction tests.

A semi-parametric regression model for survival data that leaves the baseline hazard unspecified while estimating log hazard ratios for covariates. Estimation uses the partial likelihood, avoiding distributional assumptions about the baseline.

A likelihood for the ordering of events that conditions on the risk set at each event time, so the unspecified baseline hazard cancels. Cox's method yields consistent estimates of regression coefficients.

Occur when two or more subjects fail at the same recorded time, so the exact event ordering is unknown. Approximations such as Breslow or Efron, or exact methods, are used in the partial likelihood.

The ratio of hazards between two covariate values, obtained by exponentiating the corresponding regression coefficient. It is the standard effect measure from Cox and parametric proportional hazards models.

A non-parametric estimate of the cumulative baseline hazard in a Cox model, summing the reciprocal of the weighted risk set at each event time. It uses the fitted linear predictors as weights.

Residuals formed by comparing covariate values of subjects who fail with the weighted average over the risk set. Their pattern over time is used to test the proportional hazards assumption.

Residuals defined as the observed event indicator minus the estimated cumulative hazard for each subject. They are used to assess the functional form of covariates and to identify influential observations.

A parametric survival model in which covariates act multiplicatively on the time scale rather than the hazard, so a positive coefficient lengthens or shortens survival proportionally. It can be fitted with log-normal, Weibull or log-logistic errors.

A lifetime distribution with survival exp(-(lambda t)^k), where k is the shape and lambda the scale. Its hazard is monotone increasing, decreasing or constant, making it a flexible parametric proportional hazards choice.

A special case of the Weibull distribution with shape one, giving a constant hazard and memoryless survival. It is the simplest parametric lifetime model but rarely realistic for human mortality.

A lifetime distribution where the logarithm of the time is normally distributed. Its hazard rises then falls, making it useful for survival that initially worsens then improves, and it fits accelerated failure time models.

A regression framework specifying a distribution from the exponential family, a linear predictor and a link function connecting them. It unifies logistic, Poisson and normal regression under one estimation approach.

The function that maps the expected response to the linear predictor in a generalised linear model, such as the logit for binary data or the log for counts. The choice determines how effects are interpreted.

The link function that equates the linear predictor to the natural parameter of the exponential family distribution. Examples include the logit for binomial and the log for Poisson responses.

A generalised linear model for count data using a log link, so covariates act multiplicatively on the rate. With an offset for exposure it models mortality or incidence rates directly.

Extra variability in count or binomial data beyond that implied by the assumed distribution, leading to underestimated standard errors. It is addressed with quasi-likelihood, negative binomial models or robust variance estimates.

Twice the difference between the log-likelihood of a saturated model and that of the fitted model. It measures goodness of fit and is used in likelihood ratio tests and analysis of deviance.

An estimation approach that specifies only the mean and variance relationship rather than a full distribution, allowing overdispersion to be handled by scaling the variance. It gives consistent coefficient estimates and adjusted standard errors.

A rule updating prior beliefs with observed data, giving the posterior proportional to the prior times the likelihood. It is the foundation of Bayesian inference for survival and regression models.

The probability distribution expressing belief about a parameter before observing data. It may be informative, weakly informative or non-informative, and its influence diminishes as data accumulate.

The updated distribution of a parameter after combining the prior with the likelihood of the observed data. It summarises all inferential information, with summaries such as means, intervals and predictive quantities derived from it.

A prior that yields a posterior in the same distributional family, allowing closed-form updating. Examples include beta-binomial and normal-normal pairings, which simplify computation.

A Bayesian interval containing a parameter with a stated posterior probability, such as 95 percent. Unlike confidence intervals, it has a direct probability interpretation conditional on the data and prior.

A family of simulation algorithms that draw dependent samples from a target posterior distribution when direct sampling is impossible. Diagnostics assess convergence and mixing of the chains.

An MCMC algorithm that updates each parameter in turn by sampling from its full conditional posterior. It is efficient when these conditionals are standard distributions and requires no tuning.

An MCMC method that proposes new parameter values from a proposal distribution and accepts or rejects them based on a probability ratio. It works with unnormalised posteriors and requires tuning of the proposal.

R's basic tabular structure holding columns of possibly different types with equal length. Subsetting, merging and model formulas operate on data frames, making them central to data analysis in R.

R's syntax for specifying models, with the response on the left of a tilde and predictors on the right. Operators such as plus, colon and star define main effects and interactions.

An R data type for categorical variables that stores levels and is automatically expanded into indicator variables in model formulas. The first level is treated as the reference by default.

A structure in the survival package that combines follow-up time and event status, optionally with delayed entry. It is used as the response in Kaplan-Meier, Cox and parametric survival models.