csv(H: \BMTRY 722_Spring 2021\Tongue. Aalen, Fleming-Harrington), col=1: 3, lwd=2) Cumulative Hazard Interpreting S(t) and H(t) • General philosophy – Bad to extrapolate • In survival – Bad to put a lot of stock in estimates at late time points – Have less data at later times Observations? • Convergence to H(t) = lt with increasing N • Could apply to parametric smoothing to get estimate of h(t), just the slope of the line H(t) versus t • More divergence for the upper end, where denominator data (risk description is smaller • Textbook discusses bias in S(t) at tmax • Can estimate S(t) by 0 beyond tmax (negatively biased) • Can estimate S(t) = S(tmax) for t tmax (positively biased) • When there is no censoring, the product limit estimator reduces to the empirical survival function Point-wise Confidence Intervals • Constructed to ensure that the true value of S(t) at a particular t, falls in the interval with (1 – a)% confidence • Notation: • Recall that this is the sum in the Greenwood’s formula: “Linear” CIs • Most commonly estimated in stats package • It is a point-wise CI for t • For simplicity of notation, assume 95% confidence There are Other Better Options • Transformations have better properties • Two main approaches: – Log transformations: based on cumulative hazard approach – Arcsine square root Log Transformation • Define q: • Then, the 95% CI is Derivation of the Log Transformation Log-log transformation • Since the survival function estimates a probability, it is bounded by 0 and 1 • Taking the log results in bounds: • Taking the opposite results in bounds • Taking the double log results in bounds Complimentary log-log transformation Log-Log Transformation • Can estimate a confidence interval for the double log transformation – Estimate variance (delta method) – Use estimate to define CI according to: Log-Log Transformation • To get the CI for the survival function at time t – Must back transform from the double log Arcsin Squareroot • Very ugly: Cumulative Hazard CIs • Linear • Log • Arcsin square root • See KI Mo page 107 Which to Use When? • For N 25 and 50% censoring – Log and log-log are good – Arcsin square root good – Both given ~ nominal coverage for 95% CI – Exception: extreme right tail where there is little data • Linear approach requires much larger N for good coverage Which to Use When? • Arcsin square root – Slightly conservative – A little wider than necessary • Log – Slightly anti-conservative – A little too narrow • Linear – Overly anti-conservative – Too narrow • Large Samples: all about the same Remember… • Valid for point-wise intervals • Common incorrect interpretation: – Plot a set of point-wise 95% CIs – Interpret as confidence “band” – These “bands” are too narrow! Example: Tongue Cancer data R Code library(survival) tongue-read.
Statistical Inference of a Bivariate Proportional Hazard Model with Grouped DataBy
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Mark Yuying An
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Nonparametric Discrete Choice Methods for Measuring Economic ValuesBy
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