Competing Risks: A Practical Perspective by Melania Pintilie

By Melania Pintilie

The necessity to comprehend, interpret and examine competing chance info is essential to many components of technology, fairly clinical study. there's a genuine want for a e-book that offers an summary of technique utilized in the translation and research of competing dangers, with a spotlight on sensible purposes to clinical difficulties, and incorporating sleek ideas. This publication fills that want via proposing the main up to date method, in a fashion that may be effortlessly understood, and utilized, through the practitioner.

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Although in this book the discrete situation is mostly of interest, this section gives the formulae for both cases. In addition to the usual functions given in statistics – the cumulative distribution function (cdf) and probability density function (pdf), those concepts specific to survival analysis – the survivor function and hazard function – are presented. Continuous distributions Let T denote the random variable for the time to event, considered as continuous. 1) If F t is the cdf of the time to event, then the survivor function can be expressed in terms of the cdf, and vice versa: S t = P T > t = 1−P T ≤ t = 1−F t Definitions and background formulae 25 The pdf, f t , can be found from either the survivor function or the cdf by differentiating: ft = dF t d 1−S t = dt dt =− dS t dt Alternatively, if the density function is known, both the cdf and the survivor function can be found by integrating the pdf.

Breslow (1974) and Efron (1977) suggested ways to handle tied observations. Easy-to-follow descriptions of these methods can be found in Collett (2003) and Therneau and Software for survival analysis 37 Grambsch (2000). The confidence interval for each coefficient ˆ k , k = 1, 2, , m, is constructed as usual, ˆ k ± z1− /2 ˆ k , where z is the quantile of the standard normal distribution. The Wald test of the hypothesis H0 k = 0 versus the alternative Ha k = 0 is the test statistic ˆ k / ˆ k which follows the standard normal distribution.

Suppose that where 0 ≤ t1 < t2 < and T can take on the values t1 t2 t3 its probability function is p tj = P T = tj . As in the continuous case, the survivor function is defined as S t = P T > t . 5) There is no uniform agreement on how best to define the cumulative hazard function for the discrete case. 1 Basic mathematical formulae. Continuous distributions Discrete distributions Function Definition Relationship to other functions Definition Relationship to other functions Cumulative distribution F t =P T ≤t Ft = 0 F t =P T ≤t Ft = Probability density ft = d St dt p tj = P T = tj p tj = S tj−1 − S tj Survivor S t =P T>t t d Ft dt f x dx f t =− S t = 1−F t = S t =P T>t ∗ f x dx h t = lim t→0 P t ≤ T < t+ t T > t t h t = f t /S t =− Cumulative hazard Ht = * Probability function t 0 h x dx d log S t dt H t = − log S t p tj S t = 1−F t = p tj tj >t t = exp −H2 t = exp −H t Hazard tj ≤t h tj = P T = tj T ≥ tj H1 t = tj ≤t H2 t = − h tj = p tj /S tj−1 h tj tj ≤t log 1 − h tj H2 t = − log S t 28 Survival – basic concepts appealing because it is intuitively easy to understand, especially in applied settings, is H1 t = h tj tj ≤t This definition, however, does not preserve the property S t = exp −H1 t that was seen in the continuous case.

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