By Matthias Kaeding
Matthias Kaeding discusses Bayesian equipment for studying discrete and non-stop failure occasions the place the impact of time and/or covariates is modeled through P-splines and extra uncomplicated functionality expansions, permitting the substitute of linear results by way of extra common services. The MCMC method for those types is gifted in a unified framework and utilized on info units. between others, latest algorithms for the grouped Cox and the piecewise exponential version below period censoring are mixed with a knowledge augmentation step for the functions. the writer exhibits that the ensuing Gibbs sampler works good for the grouped Cox and is simply enough for the piecewise exponential model.
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Additional resources for Bayesian Analysis of Failure Time Data Using P-Splines
The criterion can be used to compare arbitrary models under right and interval censoring, as long as draws from the posterior predictive distribution can be obtained. Several methods for obtaining random deviates from failure time random variables using hazard rates are given by Devroye (1986, pp. 260–285). The simplest method is based on inverting the cumulative hazard. If H(t) can be inverted, a draw from f (t) = h(t) exp(−H(t)) can be obtained by H −1 (e), where e is a draw from a standard exponential distribution with cdf 1 − exp(−t); P(H −1 (e) ≤ t) = P(e ≤ H(t)) = 1 − exp(−H(t)) = F(t).
Z2 − km−1 )l I[z2 > km−1 ]⎥ ⎥ ⎥. .. ⎥ . ⎦ xn2 . . xnl (zn − k2 )l I[zn > k2 ] . . (zn − km−1 )l I[zn > km−1 ] In general, the design matrix B consists of evaluations of the basis functions, so that B i j gives the value of the jth basis function, evaluated at the covariate value xi . B B )−1B y , Given an estimate for ζ , for example the least squares estimate (B the function value f (·) at covariate value xi is estimated by f (xi ) = xi0 ζ1 + ... + xil ζl+1 + term 1 (xi − k2 ) I[xi > k2 ]ζl+2 + ...
In general obtaining draws from the predictive distribution is not always easy or feasible, especially for some nonparametric models considered here where the log-baseline is modeled via P-splines. Apart from this difﬁculty the criterion suffers from the disadvantage of using the data twice for estimation and model evaluation; every data point inﬂuences its own prediction positively, which can lead to an overestimation of prediction accuracy (Hastie et al. 2009, p. 228). Crossvalidation based criterions, measuring out-of-sample prediction accuracy avoid this.