The hazard ratio

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. 

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. 

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. ... 

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. 

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...

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. 

The proportional hazards model, as the name suggests, assumes that the hazard ratio is a constant. As such it provides a direct extension of the logrank test, which is a simple two treatment group comparison. Indeed if the proportional hazards model is fitted to data without the inclusion of baseline factors then the p-value for the test Hg c = 0 will be essentially the same as the p-value arising out of the logrank test. 

As we have already seen, there will be settings where the pattern of differences between treatment groups does not conform to proportional hazards, where the hazard ratio is not a constant, single value. Such situations are best handled by using an alternative model to incorporate baseline factors. The accelerated failure time model is an analysis of variance technique which models the survival time itself, but on the log scale ... 

Each trial that is to be included in the meta-analysis will provide a measure of treatment effect (difference). For continuous data this could be the mean response on the active treatment minus the mean response in the placebo arm. Alternatively, for binary data the treatment effect could be captured by the difference in the cure rates, for example, or by the odds ratio. For survival data, the hazard ratio would often be the measure of treatment difference, but equally well it could be the difference in the two-year survival rates. 

If this confidence interval is on the log scale, for example with both the odds ratio and the hazard ratio, then both the lower and upper confidence limits should be converted by using the anti-log to give a confidence interval on the original odds ratio or hazard ratio scale. 

Kaplan EL and Meier P (1958) Non-parametric estimation from incomplete observations Journal of the American Statistical Association, 53, 457-M81 Kaul S and Diamond GA (2006) Good enough a primer on the analysis and interpretation of non-inferiority trials Annals of Internal Medicine, 145, 62-69 Kay R (1995) Some fundamental statistical concepts in clinical trials and their application in herpes zoster Antiviral Chemistry and Chemotherapy, 6, Supplement 1, 28-33 Kay R (2004) An explanation of the hazard ratio Pharmaceutical Statistics, 3, 295-297... 

We are not certain which comorbid risk factors cause mortality independent of sleep effects, and therefore, we cannot be certain whether we controlled too much or too little for comorbidities. For example, since short sleep or long sleep may cause a person to be sick at present or to get little exercise or to have heart disease (17), diabetes (18), etc., controlling for these possible mediating variables may have incorrectly minimized the hazards associated with sleep durations. This would be overcontrol. The hazard ratios for participants who were rather healthy at the time of the initial questionnaires were unlikely to be overcontrolled for initial illness. Since the 32-covariate models and the hazard ratios for initially healthy participants were similar, this similarity reduced concern that the 32-covariate models were overcontrolled. On the other hand, there may have been residual confounding processes that caused both short or long sleep and early death that we could not adequately control in the CPSII data set, either because available control variables did not adequately measure the confound or because the disease did not yet manifest itself. Depression, sleep apnea, and dysregulation of cytokines are plausible confounders that were not adequately controlled. It may be impossible to be confident that all conceivable confounds are adequately controlled in epidemiological studies of sleep. 

In contrast to Ato Z trial, the CRP levels fell from a median of 12,3 mg/L at baseline to 2,1 mg/L in the pravastatin group and 1,3 mg in the atorvastatin group in the PROVE-IT TIMI 22 trial (36), The primary end point of all caused death, myocardial infarction, and unstable angina requiring hospitalization was 26.3% in the pravastatin and 22.4% in the atorvastatin group. This represents a 16% reduction in the hazard ratio, favoring atorvastatin. 

Prediction of risk using models requires a computer, a pocket calculator with an exponential function or internet-access (the ECST model can be found at www.stroke. ox.ac.uk). As an alternative, a simplified risk score based on the hazard ratios derived from the relevant risk model can be derived. Table 27.3 shows a score for the five-year risk of stroke on medical treatment in patients with recently symptomatic carotid stenosis derived from the ECST model. As is shown in the example, the total risk score is the product of the scores for each risk factor. Fig. 27.7 shows a plot of the total risk score against the five-year predicted risk of ipsilateral carotid territory ischemic stroke derived from the full model and is used as a nomogram for the conversion of the score into a risk prediction. 

The risk of death in patients with supraventricular dysrhythmias taking dofetilide has been studied in a systematic review of randomized controlled trials (59). After adjusting for the effects of dysrhjdhmia diagnosis, age, sex, and structural heart disease, the hazard ratio was 1.1 (Cl = 0.3, 4.3). 

The association of nut intake and CHD mortality was exanfined in 399,633 subjects, including 1158 cases, enrolled in the EPIC study [59]. When comparing the highest (> 13 g/day) and the lowest (<1 g/day) nut intake categories, and after adjusting for coronary risk factors and dietary variables, the hazard ratio (HR) was 0.71 [95% confidence interval (Cl), 0.51-0.98]. An HR of 0.74 (0.57-0.96) was estimated for every 8 g/day of increased nut intake, and after adjustments, the HR was 0.89 (0.74-1.08). Nut intake was associated with a dcCTcased risk of CHD mortality. In approximately half of a population that rarely consumes nuts, an intake of only two servings of nuts per week (8 g/day) may reduce CHD mortality by 11%. 

The development plan aims to use phase 2 and 3 trials to evaluate if the upper bound of a 95% Cl of the hazard ratio (HR) of the experimental drug to the control group is below 1.3. A composite endpoint is considered time to CV deaths, stroke, or myocardial infarction (MI), whichever occurs first. Under this setting, the power is the probability that the upper 95% Cl of fhe HR < 1.3, when the true HR = 1. 

The same issue exists for time-to-event endpoints. While the most common metric for trials with these endpoints is the hazard ratio evaluated using survival analysis (usually a Cox proportional hazards model), absolute measures, such as the difference in event rates at a fixed follow-up time, are sometimes used. 

Patients with osteoarthritis or rheumatoid arthritis are randomized to one of three treatments, celecoxib, ibuprofen, or naproxen, and the primary endpoint is the occurrence of a cardiovascular endpoint a nonfatal myocardial infarction, a nonfatal stroke, or any cardiovascular death. Non-inferiority will be assessed for three different pairwise comparisons celecoxib versus ibuprofen, celecoxib versus naproxen, and ibuprofen versus naproxen. The definition of non-inferiority differs somewhat from the fixed margin approach describe earlier in that there are separate criteria for the confidence interval and the point estimate. The hazard ratio for each comparison will be calculated, and non-inferiority will be concluded if the upper end of the... 

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