Abstract
There are many situations in law that require estimation of the time to failure, e.g., how long someone will live or something will last. The event of failure may be anything—death, filing a claim after exposure to a carcinogen, peeling of defective paint on house sidings, or termination of an at-will relationship. If there are comparable populations in which all or almost all members have already failed, the probabilities of survivorship and death at various ages can be estimated directly from data and presented in the form of a life table. The classic life table tracks an initial cohort of people year by year from birth, showing in each year how many people die and how many survive, until the entire cohort has died. If there are no comparable populations with complete survival data, the problem of estimation is more difficult. The average life of those who have already failed would seriously underestimate the average life of those still surviving. An estimate of the likelihood of early failure is still possible; but at later times, to take account of the fact that many in the population have not yet failed, one must assume that the force of mortality has a certain mathematical form or model as a function of time and estimate the model’s parameters from the data. The future rate of mortality is then extrapolated from the model.
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Notes
- 1.
The standard error is given by
$$ s.e.\left\{\widehat{S}(t)\right\}=\widehat{S}(t){\left\{{\displaystyle \sum_{j:{t}_j<t}\frac{d_j}{n_j\left({n}_j-{d}_j\right)}}\right\}}^{1/2}=\widehat{S}(t){\left\{{\displaystyle \sum_{j:{t}_j<t}\frac{d_j/{n}_j}{n_j\left(1-{d}_j/{n}_j\right)}}\right\}}^{1/2}, $$where the sum is taken over all death times prior to time t.
- 2.
See Devanand et al., Relapse Risk after Discontinuation of Risperidone in Alzheimer’s Disease, 367 New Engl. J. Med. 1497 (2012) for the primary findings of this randomized clinical trial.
- 3.
Multiplying these conditional probabilities together, even though they are not strictly independent, can be justified by partial likelihood theory, the properties of which are similar to standard likelihood theory. See Section 5.6.
- 4.
Score tests are based on the statistic that results from differentiating the logarithm of the likelihood function. In this instance, the score test is the derivative of the log partial likelihood, evaluated at β = 0.
- 5.
The study also identified elevated levels of birth defects and other health problems, but these are not covered here.
- 6.
See Armitage & Doll, Stochastic models for carcinogenesis, Proceedings of the Fourth Berkeley Symposium of Mathematical Statistics and Probability 19 (1961).
- 7.
An equivalent continuous dose that would yield the same total exposure (the actual numbers differ slightly from those that would be produced by a simple pro-rating).
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Finkelstein, M.O., Levin, B. (2015). Survival Analysis. In: Statistics for Lawyers. Statistics for Social and Behavioral Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5985-0_11
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DOI: https://doi.org/10.1007/978-1-4419-5985-0_11
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