# Computing event rates given RR + CI and total sample size in each treatment group

I am looking at some data for the risk of mortality in patients undergoing treatment A vs treatment B and I am given the total number of patients in each treatment arm and the relative risk + confidence intervals of mortality. How would I go about finding the actual number of deaths in each treatment arm?

Here are the hard numbers:

Study 1
Adjusted total # of patients for treatment A: 8366
Adjusted total # of patients for treatment B: 10251
Adjusted RR for mortality: 0.72 (0.66-0.78)

Study 1
Adjusted total # of patients for treatment A: 23113
Adjusted total # of patients for treatment B: 26819
Adjusted RR for mortality: 0.78 (0.72-0.83)


So I would like to compute the actual number of patients who died in each treatment arm for both studies 1 and 2.

Not sure if this helps but the paper states that:

adjusted survival curves were estimated with the use of the inverse-probability-weighting approach of Cole and Hernan. For each treatment group, the survival curves adjusted with the use of inverse probability weighting represent the expected rate of survival if the treatment of interest were applied to all study patients. Using estimated rates of survival among patients undergoing A and among those undergoing B, we calculated risk ratios at specific time points and used bootstrap methods to obtain 95% confidence intervals.

I had a look at the Wikipedia article for RR (http://en.wikipedia.org/wiki/Relative_risk) but its not the most helpful.

Thanks!

• I am not sure how the weighting will affect the results, but if you're just looking for the raw counts, you can use the formula of RR and the formula of its standard error to derive two equations. Two equation, two unknown, and you would be able to find out what is the number of deaths in treatments A and b. Reference formulas here. According to my calculation, the #s of deaths in group A in your first example is about 842; group B, 1433. – Penguin_Knight Mar 23 '13 at 6:08

Email the authors.

Without some serious guesswork, there's no way I can think of to pull the unweighted data back out of a weighted data set like this without knowing the weights. For reference, here's what happened in that study:

• Each patient has an exposure X, and outcome Y, and a set of confounding covariates Z
• A model is built to estimate the probability of X given Z for each patient, usually using something like logistic regression.
• 1/the probability in the last step is that individuals "weight" in the model - this gives you a dataset with the same overall number of patients as the original dataset, but the influence of any given patient is modified by their weight. There's some adjustments you can do from here to handle really absurd weights.
• You run your analysis on the weighted data set, and theoretically get an estimate of the marginal effect of X on Y, having controlled for Z. This does assume you haven't missed anything, your weighting model isn't misspecified, etc.

This technique provides several nice properties, which are mentioned in your snippet: You get adjusted, marginal effect estimates, and when you plot survival curves for the weighted data, those survival curves are adjusted for covariates.

But unless you know the underlying weights, I don't think it will be trivial to extract the unweighted data. As I said in the beginning, it's likely much faster and less error prone to drop the authors an email.

• Ok thank you for your comments! Your explanation of how to the weighting works was great! I have emailed the author regarding getting the raw counts. Also do you have any idea what this 'bootstraping' is that they mention in my snippet? – BYS2 Jun 10 '13 at 8:14
• @BYS2 'Bootstrapping' is a resampling technique to help you obtain an estimate of variance in situations where conventionally calculated variance estimates are difficult/impossible to get. – Fomite Jun 10 '13 at 14:53

It seems surprising that the paper would not report the number of events, but if it reports the crude survival curves, either in a table or figure, then you can easily get the risk and thus the number of events, at any time t: risk(t) = 1-survival(t). Or if you have the overall survival probability: risk = 1-survival. Then the number of events is N*risk.

If you don't have the survival, there are several issues that may make this infeasible:

(1) You provide the adjusted RR and adjusted sample sizes. What are these adjusted for, and do you know what the frequency of the adjustment variables is in the sample? If not, without the crude risk ratio (or crude survival), any attempt to reconstruct the event counts would likely be incorrect.

(2) You don't say what type of study this is, and the study design can affect whether it is valid to estimate the risk. I assume from the information you provided (risk ratio and survival curves) that this is a cohort study or RCT. In which case, you could validly use an approach such as that suggested by Penguin_Knight, or calculate the risk from the survival.

(3) IP weighting provides you with an estimate of the risk ratio in the pseudo-population created by the weights. This validly estimates the causal effect of the exposure on the outcome (assuming no unmeasured or residual confounding), but does not necessarily validly estimate the association of the exposure with the outcome - and may be quite biased for the association, since you are removing the effects of all measured confounders. I suspect this means you cannot get a valid estimate of the event counts by working backwards from the IPW risk ratio, without knowing the weights. This is a similar issue to (1); the event counts you want are crude data, but the measures you are given are all adjusted. Without more information on that adjustment, you may not be able to reconstruct the correct event counts.

• Thank you for your comments! The study is actually an observational registry study. Yes unfortunately the paper does not give the raw values, only the adjusted values :( – BYS2 Jun 10 '13 at 8:05