# It is possible to calculate the Hazard Ratio and 95% CI on different reference groups

I'm working on a meta-analysis and pooling hazard ratio (HR) using the metafor package in R. While most of the studies compare two categories (Biopsy vs Surgical resection) some studies report HR and 95% CI for 3 categories:

• Biopsy: 1 [Reference]
• Sub-total resection: 0.46(0.20-1.07)
• Gross-total resection: 0.85(0.42-1.70)

It is possible to combine these hazard ratios and the 95% confidence intervals for three categories in two categories ?

No. In this case, one cannot simply pool the log-hazard ratios, because they are correlated (the covariance is unlikely to be zero, since the relative treatment effects are based on the same reference group). The covariance cannot be calculated from the reported data.

What you can do is to obtain a conservative pooled estimate.

Steps

1. Perform a fixed-effect meta-analysis of the within-study log-hazard ratios
2. Multiply the standard error of the pooled effect by $$\sqrt{\frac{3}{2}}$$

Assumptions

a. The effect of the treatment/exposure is the same across the non-reference categories. You can also use a random-effects model - if this is more clinically appealing.

b.The category-specific standard errors are approximately equal.

This is an approximate adjustment, which results in the same pooled estimate, but a larger standard error. The study will contribute less to the meta-analysis.

Reference