We are 2 psychology students writing a systematic review. We need to convert the effect sizes from the different studies to Cohen's d (or another standardized effect size).
Can anyone help converting Hazard Rate Ratio (HHR) and Relative Risk (RR) to standardized effect size (eg Cohen's d)? Or how to convert HHR and RR to Odds Ratio, which we know how to convert to Cohen's d?
A risk ratio (RR) can be converted to an odds ratio (OR) with the following equation, where $p_2$ is the proportion of events for the control group: $$ \textrm{OR} = \frac{RR p_2 (1-p_2)}{p_2(1-RR p_2)} ~. $$ Unfortunately, this is not very useful. In a situation where you have $p_2$ you are likely to also have $p_1$ (the proportion of events for the treatment group), in which case you can compute the odds ratio directly and then convert to Cohen's $d$ if that is what you are trying to accomplish.
While the hazard rate and risk rate are not the same (i.e., former takes into account the timing of the events whereas the latter only reflects the occurrence of the event by the end of the study) the hazard rate is scaled as a risk ratio. Thus, you can use the above to estimate an odds ratio using a sensible value for $p_2$, such as the proportion surviving to the end of the study in the control group, often determined from a graph. It is important to recognize that this will not give you the same odds ratio that you would get from a simple 2 by 2 of group by event status at the end of the study. You will need to decide if this estimate is sensible in your context. As an alternative, you might be able to determine the proportion of events per group at the end of the observation period from a survival graph and use that directly to compute the odds ratio.
Another issue is the standard error of Cohen's d. In these cases you should not use the typical equation that assumes you are working with means and standard deviations. As I'm sure you are aware, there is an equation for converting the standard error (well, variance) of the odds ratio into the standard error for $d$. However, you can't compute this (and shouldn't) when using the above as you don't have the 2 by 2 frequencies. A sensible approach is to rescale the standard error from the hazard model to maintain the same coefficient/standard error ratio (i.e., maintain the same z-value). The equation is:
$$ se_d = d/z $$
where $z$ is the $z$ value associated with the hazard rate.
