The hazard rate

In order to be able to understand what a hazard ratio is, you first need to know what a hazard rate is. The hazard rate (function) is formally defined as the conditional death (or event) rate calculated through time. What we mean by this is as follows. Suppose in a group of 1000 patients in month 1, 7 die the hazard rate for month 1 is 7/1000. Now suppose that 12 die in month 2 the hazard rate for month 2 is 12/993. If now 15 die in month 3 then the hazard rate for month 3 is 15/981 and so on. So the hazard rate is the death (event) rate for that time period amongst those patients still alive at the start of the period. 

There are several things to note about the hazard rate. Firstly, it is unlikely that the hazard rate will be constant over time. Secondly, even though we have introduced the concept of the hazard rate as taking values for monthly periods of time, we can think in terms of small time periods with the hazard rate being a continuous function through time. 

The hazard rate can be estimated from data by looking at the patterns of deaths (events) over time. This estimation process takes account of the censored values in ways similar to the way such observations were used in the Kaplan—Meier curves. 

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Estimate the Dow Fire and Explosion Index, and determine the hazard rating, for the processes listed below. 

ESD systems should be designed to be sufficiently reliable and fail safe that a (1) accidental initiation of the ESD is reduced to acceptable low levels or as low as reasonably practical, (2) availability is maximized as a function of the frequency of system testing and maintenance, and (3) the fractional MTBF of the system is sufficiently large to reduce the hazard rate to an acceptable level, consistent with the demand rate of the system. 

Even though the individual hazard rates seen in Figure 13.3 are not constant, it would be reasonable to assume, wherever we look in time, that the ratio of the hazard rates is approximately constant. In fact, these hazard rates have been specifically constructed to behave in this way. When this is the case, the ratio of the hazard rates will be a single value, which we call the hazard ratio. We will denote this ratio by X so that X = h /h. ... 

It is convention for the hazard rate for the test treatment group to appear as the numerator and the hazard rate for the control group to be the denominator. 

A hazard ratio of one corresponds to exactly equal treatments the hazard rate in the active group is exactly equal to the hazard rate in the placebo group. If we adopt the above convention and the event is death (or any other undesirable outcome) then a hazard ratio less than one is telling us that the active treatment is a better treatment. This is the situation we see in Figure 13.3. A hazard ratio greater than one is telling us that the active treatment is a poorer treatment. 

Even if the hazard ratio is not precisely a constant value as we move through time, the hazard ratio can still provide a valid summary provided the hazard rate for one of the treatment groups is always above the hazard rate for the other group. In this case the value we get for the hazard ratio from the data represents an average of that ratio over time. 

However, it is not always the case, by any means, that we see a constant or approximately constant hazard ratio. There will be situations, as seen in Figure 13.4, when the hazard rate for one group starts off lower than the hazard rate for a second group and then as we move through time they initially move closer together, but then a switch occurs. The hazard rate for the first group then overtakes that for the second group and they continue to move further apart from that point on. 

In an earlier section we saw two different patterns for two sets of survival curves. In Figure 13.2 a) the survival curves move further and further apart as time moves on. This pattern is consistent with one of the hazard rates (think in terms of death rates) being consistently above the other hazard rate. This in turn corresponds to a fairly constant hazard ratio, the situation we discussed in Section 13.4.1. So a constant hazard ratio manifests itself as a continuing separation in the two survival curves as in Figure 13.2 a). Note that the higher hazard rate (more deaths) gives the lower of the two survival curves. 

It is worth revisiting the calculation of the Kaplan—Meier curve following on from the discussion of the hazard rate, in order to see, firstly, how censoring is accounted for, and secondly, how the two are linked in terms of the calculation. 

This calculation in relation to both the hazard rate and the survival probabilities has been undertaken at intervals of one month. In practice we use intervals which correspond to the unit of measurement for the endpoint itself, usually days, in order to use the total amount of information available. 

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 ... 

As before, the coefficient c measures the effect of treatment on the hazard rate. If c < 0 then the log hazard rate, and therefore the hazard rate itself, in the active group is lower than the hazard rate in the control group. If c> 0 then the reverse... 

The following table provides a listing of chemicals along with their NFPA hazard ratings. Refer to Chapter 1 for a definition of the hazard ratings. The terms W and oxy refer to water reactive and oxidizer, respectively. 

It may be shown that the hazard rate H arising from a failed safety device is given by the formula ... 

It is very important to note that all this only applies to safety devices where the failure is hidden during normal plant operation. For, say, a normal control device where a failure would be immediately manifested by a malfunction of the plant, the hazard rate is simply the same as the failure rate. No amount of testing will help here. 

The time between regular inspection of the relief valve is 1 year, and the failure rate (to danger) is 0.01 failures per year. So the hazard rate arising from this demand on the relief valve is given by Equation (1) above to yield... 

Many of PPG s high-pressure and high-temperature alarms are tested every six months. A significant number of the PPG Lake Charles Complex test frequencies have been developed using detailed reliability studies that consider the hazard rate (the acceptable probability of a process accident) and the demand rate (the number of times the critical alarm or shutdown function is required in service). [8]... 

The so-defined elements hij (t) of the transfer-intensity matrix are called the hazard rates, and define the conditional probability... 

These equations are linear differential equations with time-varying coefficients since the hazard rates are time-dependent and may be presented in matrix form as... 

Noteworthy is that only for the exponential distribution is the hazard rate h a) = f (a) /S (a) = k not a function of the age a, i.e., the molecule has no memory and this is the main characteristic of Markovian processes. In other words, the assumption of an exponential retention time is equivalent to the assumption of an age-independent hazard rate. One practical restriction of this model is that the transfer mechanism must not discriminate on the basis of the accrued age of a molecule in the compartment. In summary, it is clear that the formulations in the probabilistic transfer model and in the retention-time distribution model are equivalent. In the probabilistic transfer model we assume an age-independent hazard rate and derive the exponential distribution, whereas in the retention-time distribution model we assume an exponential distribution and derive an age-independent hazard rate. 

Figure 9.3 depicts state probability curves for the Erlang and the Weibull distributions. The hazard rates as functions of time are also illustrated. For v 1 and 1, we obtain the behavior corresponding to an exponential... 

The Weibull distribution allows noninteger shape parameter values, and the kinetic profile is similar to that obtained by the Erlang distribution for p, > 1. When 0 < p < 1, the kinetic profile presents a log-convex form and the hazard rate decreases monotonically. This may be the consequence of some saturated clearance mechanisms that have limited capacity to eliminate the molecules from the compartment. Whatever the value of p, all profiles have common ordinates, p(l/X) = exp(-l). 

In the extravascular case, the compartment is the absorption compartment and the hazard rate hev represents the absorption rate constant. If we assume that hev is not dependent on time, rearranging (9.17), taking the limit At —> 0, and solving the so obtained differential equation with initial condition pev (0) = 1, we obtain... 

In conclusion, the solutions E Qt (f)] for the expected values for such stochastic models are the same as the solutions qT (t) for the corresponding deterministic models, and the transfer-intensity matrix H is analogous to the fractional flow rates matrix K of the deterministic model. If the hazard rates are constant in time, we have the stochastic analogues of linear deterministic systems with constant coefficients. If the hazard rates depend on time, we have the stochastic analogues of linear deterministic systems with time-dependent coefficients. 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

We consider now a class of models that introduce particle heterogeneity through random rate coefficients. In this conceptualization, the particles are assumed different due to variability in such characteristics as age, size, molecular conformation, or chemical composition. The hazard rates h are now considered to be random variables that vary influenced by extraneous sources of fluctuation... 

Let also n0 be replicates of the above experiment where the hazard rate varies from experiment to experiment with probability density function / (h). From the previous relations, the unconditional expectation and variance are (cf. Appendix D)... 

Bedaux and Kooijman 1994 Kooijman 1996 Newman and McCloskey 1996, 2000 Zhao and Newman 2007). This is not just an academic discussion the 2 theories lead to different time courses of mortality at constant exposure (Kooijman 1996) (see Figure 2.10) and have very different consequences for sequential exposure (Newman and McCloskey 2000 Zhao and Newman 2007). In reality, both sensitivity difference and stochasticity are likely to play a role in mortality. Individuals also differ in sensitivity, especially in field populations, but there is clearly a substantial stochastic component involved in mortality that cannot be ignored. The method to deal with stochastic events in time is survival analysis or time-to-event analysis (see Bedaux and Kooijman 1994 Newman and McCloskey 1996). For industrial practices, this method has a long history as failure time analysis (see, e.g., Muenchow 1986). Bedaux and Kooijman (1994) link survival analysis to a TK model to describe survival as a function of time (i.e., the hazard rate is taken proportional to the concentration above a threshold value). Newman and McCloskey (1996) take an empirical relationship between external concentration and hazard rate. 

If the demand rate, 8, is the number of occasions per year that the protective device is actuated, then the hazard rate, or number of occasions per year when a hazardous condition exists, rj, is... 

The hazard rate can be reduced by using more reliable equipment (lower value of A), more frequent testing (lower t), or by making improvements that lead to steadier operation (lower 8). Alternatively, two protective systems in parallel can be installed, in which case the hazard rate becomes... 

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