Hazard ratio

Adjuvant tamoxifen therapy generally is initiated shortly after surgery or as soon as pathology results are known and the decision to administer tamoxifen as adjuvant therapy is made. The administration of tamoxifen should be limited to administration after completion of chemotherapy based on results from a study that randomized patients to receive chemotherapy for six cycles with concurrent tamoxifen, followed by continued tamoxifen for a total of 5 years, or chemotherapy with sequential tamoxifen for 5 years.39 After a median follow-up of 8.5 years, the administration of sequential tamoxifen resulted in an estimated DFS advantage of 18% [hazard ratio (HR) = 1.18] compared with the concurrent use of tamoxifen with chemotherapy.39 It is believed the growth-inhibitory effect of... 

Note that hazard ratios can be plotted in the same way that odds ratios are plotted. 

Hazard ratios are created using the Cox proportional hazards model through PROC PHREG. 

Note that with PROC PHREG all covariates need to be numeric, so treatment and gender need to be numeric. The p-values and hazard ratios that are useful for your statistical tables can be found in the ProbChiSq and HazardRatio variables, respectively, in the pvalue data set. 

Table 3 describes the main parts of an environmental risk assessment (ERA) that are based on the two major elements characterisation of exposure and characterisation of effects [27, 51]. ERA uses a combination of exposure and effects data as a basis for assessing the likelihood and severity of adverse effects (risks) and feeds this into the decision-making process for managing risks. The process of assessing risk ranges from the simple calculation of hazard ratios to complex utilisation of probabilistic methods based on models and/or measured data sets. Setting of thresholds such as EQS and quality norms (QN) [27] relies primarily on... 

Fig. 10.8. Hazard ratio of presenting an endometrial cancer as the consequence of treatment with tamoxifen or raloxifene (MORE study). Reproduced with permission from Cuzick et al. (2003)...

Zhao et al. [105] have been calculated the hazard ratio (HR) of PFOS for fish consumption and the risks and potential effects of PFCs to health of coastal population in the Pearl River Delta. Due to the contamination levels of more consumed species (mandarin fish, bighead carp, grass carp and tUapia), the authors have concluded that the levels of PFCs in these fish species might pose an unacceptable risk to human health. 

In this study, CRT reduced the primary endpoint by 37% (hazard ratio 0.63 P < 0.001) and all-cause mortality by 36% (hazard ratio 0.64 P < 0.001). Although the relative risk reduction for mortality at first appears larger than that which trended in COMPANION (24% versus 36%), the mortality benefit was not evident in the early portion of the trial, but the benefit grew over time (see Fig. 4.3). Therefore, much, if not all, of the difference can be attributed to the longer follow-up (29.4 months in CARE-HF versus 14.8-16.5 months in COMPANION). 

Schmidt-Lucke et al. [46] followed 120 patients (43 control subjects, 44 patients with stable CAD, and 33 patients with acute coronary syndromes) for 10 months and recorded MACE events (Fig. 7.2). Patients with reduced EPCs had significantly higher rates of MACE. When the results were analyzed by multivariate analysis, reduced EPC levels were found to be an independent predictor of worse prognosis, even after adjustment for traditional cardiovascular risk factors and disease activity (hazard ratio, 3.9 P< 0.05). 

Note however that the hazard ratio in both high and low baseline LDL cholesterol subgroups is below 1.00 indicating a benefit of pravastatin (a quantitative interaction), although this benefit is marginal in those patients presenting with LDL cholesterol <4.0mmol/l. 

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. 

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. ... 

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. 

Confidence intervals for the hazard ratio are straightforward to calculate. Like the odds ratio (see Section 4.5.5), this confidence interval is firstly calculated on the log scale and then converted back to the hazard ratio scale by taking anti-logs of the ends of that confidence interval. 

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 this case it clearly makes no sense to assign a single value to the hazard ratio. The hazard ratio in this case will start off below one, say, increase towards one as... 

Figure 13.4 Hazard rates for two groups of patients where the hazard ratio is not constant...

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. 

The most popular method for analysis of covariance is the proportional hazards model. This model, originally developed by Cox (1972), is now used extensively in the analysis of survival data to incorporate and adjust for both centre and covariate effects. The model assumes that the hazard ratio for the treatment effect is constant. 

Figure 13.5 Hazard ratios (and 95 per cent confidence intervals) for death in subgroups defined by baseline characteristics (Packer M, Coats AJS, Fowler MB, et ai for the Carvedilol Prospective Randomised Cumulative Survival Study Group, Effect of carvedilol on survival in severe chronic heart failure. New England Journal of Medicine, 344, 1651-1658. Copyright (2001) Massachusetts Medical Society.

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