IAI CS2 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Survival function S(t)

The probability that a lifetime exceeds t: S(t) = P(T > t) = 1 − F(t). It is non-increasing, equals 1 at t = 0 and tends to 0 as t grows.

Hazard function h(t)

The instantaneous failure rate given survival to t: h(t) = f(t)/S(t) = −d log S(t)/dt. It need not be monotone and is not a probability.

Cumulative hazard H(t)

The integral of the hazard from 0 to t: H(t) = ∫₀ᵗ h(u)du = −log S(t). Hence S(t) = exp(−H(t)) and H(t) is non-decreasing.

Exponential lifetime distribution

A distribution with constant hazard λ and survival S(t) = e^(−λt), mean 1/λ. It is memoryless, so it cannot represent increasing mortality with age.

Gompertz law of mortality

A hazard of the form h(t) = Bc^t with B > 0 and c > 1, giving exponentially increasing mortality. Makeham's extension adds a constant: h(t) = A + Bc^t.

Kaplan–Meier estimator

A nonparametric estimate of S(t): Ŝ(t) = Π(1 − d_i/n_i) over event times up to t, where d_i is the number of deaths and n_i the number at risk.

Greenwood's variance formula

An estimate of the variance of the Kaplan–Meier survival estimate: Var(Ŝ(t)) = Ŝ(t)² Σ d_i/[n_i(n_i − d_i)], summed over event times up to t.

Central death rate m_x

The ratio of deaths to central exposed to risk: m_x = d_x/E_x^c. Under a uniform distribution of deaths it approximates the force of mortality at age x + ½.

Initial death rate q_x

The probability a life aged x dies before x + 1, estimated by q_x = d_x/E_x^0 with E_x^0 = E_x^c + ½d_x. It satisfies q_x ≈ m_x/(1 + ½m_x).

Standardised mortality ratio

The ratio of observed to expected deaths: SMR = Σd_x / Σ E_x^c μ_x^std. A value above 1 indicates mortality higher than the standard population.

Graduation of mortality rates

The fitting of a smooth curve or formula to crude rates to remove sampling noise. Fit is judged by chi-squared tests, runs tests of deviations, and comparison of crude and graduated values.

Poisson model for deaths

Assumes D_x ~ Poisson(E_x^c μ_x), giving μ̂_x = d_x/E_x^c with Var(μ̂_x) = μ_x/E_x^c. It is appropriate when rates are small and populations large.

Cox proportional hazards model

A model h(t|x) = h_0(t)exp(βᵀx) in which covariates multiply an unspecified baseline hazard. Parameters are estimated by partial likelihood without specifying h_0(t).

Hazard ratio interpretation

In a Cox model, exp(β_j) is the ratio of hazards for a one-unit increase in x_j, holding other covariates fixed. It is constant over time under proportional hazards.

Partial likelihood

Cox's likelihood that, at each event time, conditions on which subject fails among those at risk. It eliminates the baseline hazard and yields consistent estimates of β.

Schoenfeld residuals

Residuals formed from covariate values of subjects who fail, compared with the risk-set average. Scaled residuals plotted against time test proportional hazards in cox.zph().

Accelerated failure time model

A model log T = μ + βᵀx + σε in which covariates rescale time: S_1(t) = S_0(t/exp(βᵀΔx)). Distributions include Weibull, lognormal and log-logistic.

Efron approximation for ties

A modification of the Cox partial likelihood for tied event times that adjusts the risk-set denominator after each tie. It is more accurate than Breslow's approximation.

GLM link function

The function g relating the linear predictor to the mean: g(μ) = η = Xβ. The canonical link for the binomial is the logit and for the Poisson the log.

Poisson log-linear model

A GLM with log μ = xᵀβ, variance function V(μ) = μ and canonical log link. Coefficients are log rate ratios and an offset handles exposure.

Logistic regression odds ratio

In logit(p) = xᵀβ, exp(β_j) is the multiplicative change in the odds of the outcome per unit increase in x_j, other covariates held constant.

Overdispersion in GLMs

Extra-binomial or extra-Poisson variation where the observed variance exceeds the model variance. Detected by Pearson χ²/df > 1 and handled by quasi-likelihood or negative binomial models.

Model deviance

Twice the difference in log-likelihood between the saturated and fitted models: D = 2[ℓ(sat) − ℓ(model)]. Differences in deviance test nested models against chi-squared.

Offset in a GLM

A predictor included with a fixed coefficient of 1. In Poisson models offset(log exposure) converts modelling of counts into modelling of rates.

Bayesian posterior distribution

The distribution of the parameter given the data: p(θ|y) ∝ p(y|θ)p(θ). It combines the likelihood with the prior and is normalised by the marginal likelihood.

Conjugate prior

A prior that yields a posterior in the same family. Examples are beta with binomial, gamma with Poisson, and normal with normal for known variance.

Credible interval

A range containing the parameter with a stated posterior probability, such as 0.95. An equal-tailed interval uses the 2.5% and 97.5% posterior quantiles.

Metropolis–Hastings algorithm

A Markov chain Monte Carlo method that proposes θ* and accepts it with probability min(1, ratio of posterior times proposal densities). Its stationary distribution is the posterior.

Gibbs sampler

An MCMC scheme that updates each parameter in turn from its full conditional distribution. Every proposal is accepted, so it is a special case of Metropolis–Hastings.

Gelman–Rubin statistic

The potential scale reduction factor R̂ comparing between-chain and within-chain variance. Values close to 1 indicate the chains have converged to the target distribution.

coxph function in R

The survival package function fitting Cox models: coxph(Surv(time, event) ~ x, data = d). It reports coefficients, hazard ratios, standard errors and likelihood tests.

survreg function in R

A survival package function fitting parametric accelerated failure time models by maximum likelihood, with dist set to weibull, lognormal, loglogistic or exponential.