Comparative measures

In epidemiology, we frequently compare rates and probabilities between different groups or individuals. For example, individuals with a latent TB infection might experience a higher rate of transition to active TB disease if they have poor nutrition.

Rate ratio

The rate ratio compares rates between two groups:

\[\text{Rate ratio} = \frac{\lambda_1}{\lambda_2}\]

A rate ratio greater than 1.0 indicates that group 1 has a higher rate than group 2. Epidemiological models may have rate ratios as input parameters that modify baseline rates for different groups, like the TB + nutrition example above.

Hazard ratio

Another name for a rate ratio, but specifically in the context of survival analysis. It compares hazard rates between two groups at a specific time instant \(t\). It is defined as:

\[\text{Hazard ratio} = \frac{\lambda_1(t)}{\lambda_2(t)}\]

where \(\lambda_1(t)\) and \(\lambda_2(t)\) are the hazard rates for groups 1 and 2, respectively, at time instant \(t\). Again, the rates could be time-invariant or time-varying, as denoted above. A specific type of hazard ratio that you may encounter is a proportional hazards ratio, which assumes that two groups have hazard functions that are proportional over time.

Risk ratio

The risk ratio compares probabilities between two groups over the same time period:

\[\text{Risk ratio} = \frac{p_1}{p_2}\]

But care must be taken when interpreting risk ratios considering the dependence on the time period \(dt\) and the ease of applying a risk ratio that result in probabilities greater than 1.0.

Note

When probabilities are small (good timescale separation), risk ratio ≈ rate ratio. As probabilities increase, risk ratio becomes smaller than the rate ratio.

Odds ratio

You might also see odds ratios in epidemiological literature. The odds ratio compares the odds of an event occurring between two groups. The odds of an event is the probability of the event occurring divided by the probability of it not occurring. The odds ratio is defined as:

\[\text{Odds ratio} = \frac{p_1/(1-p_1)}{p_2/(1-p_2)}\]

Odds ratios are related to risk ratios. Let \(r\) represent the risk ratio between two groups or individuals, so that \(p_1 = r p_2\). Then the odds ratio can be written as:

\[ \begin{align*} \text{Odds ratio} &= \frac{r p_2/(1-r p_2)}{p_2/(1-p_2)} \\ &= r \frac{(1-p_2)}{(1-r p_2)} \end{align*} \]

This relationship shows two key properties of the odds ratio:

1. A risk ratio \(r\) can be determined from an odds ratio if the baseline probability \(p_2\) is known.
2. When the baseline probability, \(p_2\) in this example, is small, the odds ratio is approximately equal to the risk ratio, \(r\).