Hazardous function

Failure Rate and Hazard Function. The failure rate is defined as the rate at which failures occur in a given time interval. Considering the time interval [/, the failure rate is given by... 

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

To determine which hazard paper is appropriate to use when plotting a set of multiply censored data, first rely on engineering experience. If that is not an option, try different papers until you find one that is suitable. To save time, it may be a good idea to try plotting the sample cumulative hazard function on exponential hazard paper, since it is just... 

The behavior of the failure rate as a function of time can be gaged from a hazard plot. If data are plotted on exponential hazard paper, the derivative of the cumulative hazard function at some time is the instantaneous failure rate at that time. Since time to failure is plotted as a function of the cumulative hazard, the instantaneous failure rate is actually the reciprocal of the slope of the plotted data, and the slope of the plotted data corresponds to the instantaneous mean time to failure. For the data that are plotted on one of the other hazard papers and that give a curved plot, one can determine from examining the changing slope of the plot whether the tme failure rate is increasing or decreasing relative to the failure rate of the theoretical distribution for the paper. Such information on the behavior of the failure rate cannot be obtained from probability plots. 

If estimated of distribution parameters are desired from data plotted on a hazard paper, then the straight line drawn through the data should be based primarily on a fit to the data points near the center of the distribution the sample is from and not be influenced overly by data points in the tails of the distribution. This is suggested because the smallest and largest times to failure in a sample tend to vary considerably from the true cumulative hazard function, and the middle times tend to lie close to it. Similar comments apply to the probability plotting. 

Hazard evaluation results, interpretation of, 14 219-220 HazardExpert, 6 19 Hazard function, 26 987-988 Hazard identification, 13 153 checklists, 13 155... 

Tarone s trend test is most powerful at detecting dose-related trends when tumor onset hazard functions are proportional to each other. For more power against other dose related group differences, weighted versions of the statistic are also available see Breslow (1984) or Crowley and Breslow (1984) for details. 

The method provides a model for the hazard function. As in Section 6.6, let z be an indicator variable for treatment taking the value one for patients in the active group and zero for patients in the control group and let Xj, X2, etc. denote the covariates. If we let t) denote the hazard rate as a function of t (time), the main effects model takes the form ... 

MOOLGAVKAR, S.H., DEWANJI, A. and VENZON, D.J. (1988). A stochastic two-stage model for cancer risk assessment The hazard function and the probability of tumor, Risk Anal. 8, 383-392. 

Figure 5.3 The spectral absorbance of macular pigment plotted with the blue light hazard function. (From Hammond, B.R. et al., Optom. Vis. Sci., 82(5), 387-404, 2005.) This function described the potential for photochemical damage to the retina resulting from exposure to light from about 400 to 500 nm as defined by the IESNA Photobiology Committee for ANSI (ANSI/ IESNA RP-27.1-05).

From the above relations, the hazard function h (a) is defined as... 

Then, the probabilistic hazard rate h is the particular hazard function value h (a) evaluated at a specified age a. For the retention-time distribution models, h (a) A a gives the conditional probability that a molecule that has remained in the compartment for age a leaves by a + Aa. In other words, the probabilistic hazard rate is the instantaneous speed of transfer. 

Figure 9.3 State probabilities and hazard functions with A = 0.5h 1, and u = 1,2, 3 and p = 0.5,1,1.5 for Erlang and Weibull distributions, respectively.

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain... 

Let us examine now the conditions for which a probabilistic transfer model is equivalent to a retention-time model, both using the same hazard functions. More precisely, for the irreversible multicompartment structures, the study can be reduced to the analysis of an irreversible two-compartment model, where the compartment n°l embodies all compartments before the compartment n°2. One has to compare two situations ... 

IF (ICALL.EQ.4) THEN FOR SIMULATION OF SURVIVAL TIMES GIVEN HAZARD FUNCTION... 

IN THIS CASE, THE HAZARD FUNCTION IS CONSTANT OVER TIME FOR EACH INDIVIDUAL... 

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