Log hazard ratio

It would be quite wrong in my opinion to conclude that that what therefore needs to be done is to abandon the usual additive scales of analysis, such as log-odds ratios and log-hazard ratios, in favour of risk differences and differences in median survival. There is every reason to suppose that these will not transfer well from one trial to another and hence, of course, from clinical trial to clinical practice. 

A practical challenge of Bayesian meta-analysis for rare AE data is that noninformative priors may lead to convergence failure due to very sparse data. Weakly informative priors may be used to solve this issue. In the example of the previous Bayesian meta-analysis with piecewise exponential survival models, the following priors for log hazard ratio (HR) (see Table 14.1) were considered. Prior 1 assumes a nonzero treatment effect with a mean log(HR) of 0.7 and a standard deviation of 2. This roughly translates to that the 95% confidence interval (Cl) of HR is between 0.04 and 110, with an estimate of HR to be 2.0. Prior 2 assumes a 0 treatment effect, with a mean log(HR) of 0 and a standard deviation of 2. This roughly translates to the assumption that we are 95% sure that the HR for treatment effect is between 0.02 and 55, with an estimate of the mean hazard of 1.0. Prior 3 assumes a nonzero treatment effect that is more informative than that of Prior 1, with a mean log(HR) of 0.7 and a standard deviation of 0.7. This roughly translates to the assumption that we are 95% sure that the HR... 

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. 

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 ... 

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. 

Note that the confidence intervals in Figure 15.1 are not symmetric around the estimated hazard ratio. This is because confidence intervals for hazard ratios and odds ratio and indeed ratios in general are symmetric only on the log scale (see Section 4.5.5 for further details with regard to the odds ratio). Sometimes we see plots where the x-axis is on the log scale, although it will be calibrated in terms of the ratio itself, and in this case the confidence intervals appear symmetric. 

Approximate Number of Events Required for 80% Power with 5% Two-Sided Log-Rank Test for Comparing Randomized Arms of Design Shown in Fig. 1. Only Marker + Patients Are Randomized. Treatment Hazard Ratio for Marker + Patients Is Shown in First Column. Time-To-Event... 

Fig. 1. Hazard ratios, with 95% confidence intervals as floating absolute risks, as estimate of association between category of update mean HbAlc concentration and any end point or deaths related to diabetes and all cause mortality. Reference category (hazard ratio 1.0) is HbAlc <6% with log-linear scales, p-value reflects contribution of glycaemia to multivariate model. Data adjusted for age at diagnosis of diabetes, sex, ethnic group, smoking, presence of albuminuria, systolic blood pressure, high- and low-density lipoprotein cholesterol and triglycerides [2].

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