The odds ratio always confounds: while it may be constant across different groups or clusters, the risk ratios or risk differences across those groups may vary quite substantially. This makes it really hard to interpret an effect. And then there is inconsistency between marginal and conditional odds ratios, a topic I seem to be visiting frequently, most recently last month.
My aim here is to generate a few figures that might highlight some of these issues.
Assume that there is some exposure (indicated by the use of a \(1\) or \(0\) subscript) applied across a number of different groups or clusters of people (think different regions, hospitals, schools, etc.) - indicated by some number or letter \(i\). Furthermore, assume that the total number exposed at each location is the same as the number unexposed: \(N_{i0} = N_{i1} = N = 100\).
The number of folks with exposure at a particular location \(i\) who have a poor outcome is \(n_{i1}\) and the number with a good outcome is \(N-n_{i1}\). Likewise, the corresponding measures for folks not exposed are \(n_{i0}\) and \(N-n_{i0}\). The probabilities of a poor outcome for exposed and non-exposed are \(n_{i1}/N\) and \(n_{i0}/N\). The relative risk of a poor outcome for those exposed compared to those non exposed is
\[\text{RR}_i = \frac{n_{i1}/N}{n_{i0}/N} = \frac{n_{i1}}{n_{i0}},\]
the risk difference between exposed and unexposed groups is
\[ \text{RD}_i = \frac{n_{i1}}{N}-\frac{n_{i0}}{N} = \frac{n_{i1} - n_{i0}}{N},\]
and the odds ratio is
\[ \text{OR}_i = \frac{[n_{i1}/N] / [(N - n_{i1})/N]}{[n_{i0}/N] / [(N - n_{i0})/N]} \]
\[= \frac{n_{i1}(N-n_{i0})}{n_{i0}(N-n_{i1})}.\]
The simple conditional logistic regression model that includes a group-level random effect \(b_i\) assumes a constant odds ratio between exposed and unexposed individuals across the different clusters:
\[\text{logit} (Y_{ij}) = \beta_0 + \beta_1 E_{ij} + b_i,\]
where \(E_{ij}\) is an exposure indicator for person \(j\) in group \(i\). The parameter \(\text{exp}(\beta_1)\) is an estimate of the odds ratio defined above.
The point of all of this is to illustrate that although the odds-ratio is the same across all groups/clusters (i.e., there is no \(i\) subscript in \(\beta_1\) and \(\text{OR}_i = \text{OR}\)), the risk ratios and risk differences can vary greatly across groups, particularly if the \(b\)’s vary considerably.
Constant odds ratio, different risk ratios and differences
If the odds ratio is constant and we know \(n_{i1}\), we can perform a little algebraic maneuvering on the \(\text{OR}\) formula above to find \(n_{i0}\):
\[ n_{i0} = \frac{N \times n_{i1}}{\text{OR} \times (N - n_{i1}) + n_{i1}}\]
If we assume that the \(n_{i1}\)’s can range from 2 to 98 (out of 100), we can see how the risk ratios and risk differences vary considerably even though we fix the odds ratio fixed at a value of 3 (don’t pay too close attention to the fact the \(n_0\) is not an integer - this is just an illustration that makes a few violations - if I had used \(N=1000\), we could have called this rounding error):
N