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Inference—Estimation
Published in Prabhanjan Narayanachar Tattar, H. J. Vaman, Survival Analysis, 2022
Prabhanjan Narayanachar Tattar, H. J. Vaman
Kaplan and Meier (1958) [61] makes a detailed presentation of the estimation of the survival function for right-censored data. We have stated in Section 2.6 that the Kaplan-Meier estimator is obtained by the formula S^(t)=∏j:T˜j≤tnj−djnj. Since the estimator has a product term, the Kaplan-Meier estimator is also called as the product-limit estimator. We will briefly discuss a heuritic justification of the form. Since the publication of the original paper, it has seen over sixty-thousand citations. It is no surprise that the paper is very relevant and it will seemingly continue to do so in the future too. The work is also considered as one of the twenty important contributions in the subject of the twentieth century.
Using Statistics in Clinical Practice: A Gap Between Training and Application
Published in Marilyn Sue Bogner, Human Error in Medicine, 2018
Roberta L. Klatzky, James Geiwitz, Susan C. Fischer
As has been noted, data constitute just one type of statistical knowledge; another concerns statistical concepts per se. One important concept, for example, is that of conditional probability, which underlies measures of risk. Concern with a risk factor focuses on the probability of incurring a disease on the condition that the factor is present. For example, the probability of a heart attack among people who smoke is a conditional probability. Comparative measures of risk are necessary to determine whether a potential risk factor actually has an effect. This is accomplished by comparing the risk within a population for which the factor is present to that of a control population, for which the factor is absent. When evaluating the comparative effect of some risk factor, physicians may encounter such terms as relative risk, odds ratio, or attributable risk. Risks may be combined (e.g., the smoker may also be obese), which leads to questions about the independence of multiple risk factors. It may be of interest to measure probabilities over time. For example, the survival function indicates the probability of being alive at each point in time after some life event has occurred, such as incurring cancer. Often, these functions have well-defined mathematical forms (e.g., exponential).
Introduction
Published in Brandon M. Greenwell, Tree-Based Methods for Statistical Learning in R, 2022
which describes the probability of surviving longer than time t. The Kaplan-Meier (or product limit) estimator is a nonparametric statistic used for estimating the survival function in the presence of censoring (if there isn't any censoring, then we could just use the ordinary empirical distribution function). The details are beyond the scope of this book, but the survfit function from package survival can do the heavy lifting for us.
Condition prediction and estimation of service life in the presence of data censoring and dependent competing risks
Published in International Journal of Pavement Engineering, 2019
Valentin Donev, Markus Hoffmann
Next, the computations are repeated with the interval-censored data using parametric models. With KM the overall survival function is overestimated in average by 2 years or half of the inspection interval, since the failures are recorded at the upper limit of the interval (Figure 12(a)). The same overestimation is observed for marginal distributions in the case of no correlation (Figure 12(b)). Surprisingly, the estimates with a parametric model are not identical to the true marginal survival functions. The reason for the remaining bias is interval censoring of the competing event. Censored sections due to competing risks are treated as right-censored with the integral of the survival function as contribution to the likelihood. The lower limit of this integral is assumed to be the time of the last inspection, but it should be prior to the inspection, because the competing (censoring) event has happened within the inspection interval. The size of this bias is less significant than the bias due to interval censoring of the event of interest because right-censored observations have a smaller contribution to the total likelihood as compared to interval censoring. In the case of rutting additional bias may be present due to the incorrect assumption of a Weibull distribution. In the case of positive correlation (Figure 12(c)) the bias due to interval censoring and bias due to dependent competing risks are in the same direction leading to increased total bias (summation). In contrast, in the case of negative correlation (Figure 12(d)) the bias due to interval censoring is in the opposite direction of the bias due to competing risks leading to a reduction of the total bias (cancellation).
Walking duration in daily travel: an analysis among males and females using a hazard-based model
Published in Transportmetrica A: Transport Science, 2021
Seyed Ahmad Reza Saeidi Hosseini, Yaser Hatamzadeh
This study develops a hazard-based duration model to analyze the time spent walking in each trip. These models are used to study the conditional probability of time duration closure at some time t, given that the duration has continued until time t (Van den Berg, Arentze, and Timmermans 2012). Hazard-based models can be described in terms of survival function S(t) or hazard function h(t) (Qi 2009). The hazard function is defined as the instantaneous probability that the walking duration will end in an infinitesimal time period t after time t, given that the walking duration has continued up to time t (Zhicai and Xianyu 2010). The hazard function is given by Equation (2). where T is the walking duration with a probability density function f(t) = P(T = t) and cumulative distribution function F(t) = P(T < t). The survival function S(t) is defined as the probability that the walking duration is greater or equal to time t (Equation 3) (Qi 2009). One noted method used for estimating the survival function is the Kaplan-Meier method. The Kaplan-Meier estimator SKM(tj) estimates the survival function by using Equation (4). where j is the number of all time periods, r(tk) is the number of individuals at risk of terminating their walking in time period k (number of individuals who have not finished their walking at the beginning of time period k), and d(tk) is the number of individuals terminating their walking in time period k (Qi 2009).