Relative Risk 1: Binary

Relative risk (RR) is the most common measurement type for the applications of MRTool. Here we take a chance to introduce the basic concepts regarding relative risk, and how we build different types of relative risk models in MRTool.

Relative risk is the probability ratio of a certain outcome between exposed and unexposed group. For more information please check the wiki page. Here we use smoking and lung cancer as a risk-outcome pair to explain the idea.

Imagine the experiment is conducted with two groups, smoking (e) and non-smoking (u) group. We record the probability of getting lung cancer among the two groups, \(P_e\), \(P_u\) and the relative risk can be expressed as,

\[RR = \frac{P_e}{P_u}.\]

To implement meta-analysis on the effect of smoking, we often convert the collected relative risks from different studies (longitudinal or not) to log space, for the convenience of removing the sign restriction,

\[\ln(RR) = \ln(P_e) - \ln(P_u).\]

To setup the binary model, we simply parametrize the log relative risk with an intercept,

\[\ln(RR) = \mathbf{1} (\beta + u),\]

where \(\beta\) is the fixed effect for intercept and \(u\) is the random effect. When \(\beta\) is significantly greater than zero, we say that it is harmful. For other risk outcome pair, there is possibility that \(\beta\) is significantly less than zero, in which case we will call it protective.

Very often instead of only considering smoking vs non-smoking (binary), we also want to study the effects under different exposure to smoking. The most common assumption is log linear, please check Relative Risk 2: Log Linear for the details.

Sample Code

To setup the problem, we will only need LinearCovModel.

from mrtool import MRData, LinearCovModel, MRBRT

data = MRData()
# `intercept` is automatically added to the data
# no need to pass it in `col_covs`
data.load_df(
    df=df,
    col_obs='ln_rr',
    col_obs_se='ln_rr_se',
    col_study_id='study_id'
)
cov_model = LinearCovModel('intercept', use_re=True)
model = MRBRT(data, cov_models=[cov_model])