# Collapsing hazard ratio comparisons from 3 groups to 2 groups before meta-analysis

#### __Oliver__

##### New Member
Hi there!

Having a stats problem; wondered if anyone could help

I am doing a meta analysis of various studies. I want to look for differences in hazards between treatment A, and controls B and C.
The problem is that all the studies (except my personal data) look at the analysis in a different way from this. They report hazards ratios for A versus B, and B versus C.

This is not what I want. I want to find a way of collapsing B and C into a single category and using a cox proportional hazards model comparing A against B AND C.
Could anyone advise me whether it is possible for me to take any of the associated statistics and back-calculate a single hazard ratio and standard errors for this hazard ratio? Note that I won't be able to access any raw data for the other studies.

I see that it's not possible from using hazard ratios alone, but perhaps with coefficients?
I have been trying to assess the literature, but haven't found anything of use so far.

Any help you may have would be greatly appreciated!
Thanks!
Oliver

#### hlsmith

##### Not a robit
So you are talking about A vs B+C? I haven't seen this before, may depend on how much info in the papers. May see if network meta analysis has some thing comparable. Lastly, one benefit of your predicament is if you can find a possible solution, you can test it out on your dataset first to see if it works.

#### j58

##### Active Member
@__Oliver__ - If I understand your problem correctly, for each study (other than your own) you have HR(A vs B) and HR(B vs C), but you want HR(A vs combined B and C). I think this problem is tractable. Here's a first shot at it:

Let X\Y denote the hazard ratio for X vs Y. Then for each study, you have A\B and B\C, but you want A\(B,C). It seems reasonable to define A\(B,C) as an average (appropriately weighted) of A\B and A\C: avg(A\B, A\C). Since you have A\B, and B\C, the point estimate of A\C is easy: A\C = A\B * B\C. Thus the point estimate of avg(A\B, A\C) is avg(A\B, A\C) = exp(w1*log(A\B) + w2*log(A\C)). Ideally, the weights, w1 and w2, would be chosen to be proportional to the inverses of the variances of log(A\B) and log(A\C). Note that log(A\B) for a study is the cox regression coefficient for that study, so you should have its variance from each paper. w2 might be difficult to estimate, since it requires knowing (or guessing) the covariance between log(A\B) and log(B\C). It might be reasonable to assume that w1=w2, which amounts to the weighted average reducing to a simple average.*

To properly use these quantities in a meta analysis you will also need to estimate their standard errors, which will require knowledge/estimation/guessing the covariance.*

*Edited to add: for your own data, you can compute the needed covariance. It may be reasonable to assume that this covariance applies to every paper. If so, problem solved.

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#### hlsmith

##### Not a robit
I am interested to see if the OP confirms the desired comparisons. I know in network meta-analyses getting at the SEs can be tricky. I just wanted to make one additional comment, it may be a moot point, but you will need to make sure follow-up times are comparable between studies as well.

#### j58

##### Active Member
@hlsmith - The follow-up times shouldn't be important, because Cox regression treats the hazard ratio as time-independent.

#### hlsmith

##### Not a robit
@j58 - It has been a few years since I have ran a PHREG model. Can you elucidate how getting HRs from one study with 6 month follow-up and another with 3 years of follow-up are collapsible?

#### j58

##### Active Member
@hlsmith - The hazard rate is an instantaneous risk per unit time. An assumption of the basic Cox model is that the hazard ratio is constant with respect to time; that is, it is the same at every time t under study. Under this assumption the length of follow-up is irrelevant, because you'll observe the same hazard rate at each point in time.

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#### hlsmith

##### Not a robit
Yes, this is ringing a little clearer. So if the variable was continuous it would hold (for unit increase) and if the proportionality assumption isn't broken then for categorical variables it would hold. I always get confused by the idea of the proportionality assumption and categorical variables since the lines need to cross and/or separate at some point for them to be different and statistical significant, but why doesn't that break the proportionality assumption.

#### j58

##### Active Member
I assume that the lines you are referring to are the survival functions. Yes, they have to separate for the variable defining the groups to be significant. But it's the hazard functions h(t), not the survival functions S(t) for the two groups that have to be proportional. The hazard function h(t) = -(dS(t)/dt) / S(t), making it awfully difficult to assess the proportional hazards assumption by looking at the survival functions. But if you take minus the log of h(t) twice, -ln(-ln(h(t)), for each group, the resulting curves will be (approximately) parallel if the hazards are proportional.