# Can I validly expand the odds ratio analogously to the relationship between relative risk & the incidence rate ratio?

I am involved in a nested Case-Control study that involves cohorts of cases and controls entering a program, with the outcome of failure by six months. Cases and controls are individually matched on month / year of entry. I have a lot (>10,000) of both.

A primary risk factor of interest is receiving a medical diagnosis (psychiatric, musculoskeletal, respiratory) after entry and before 6 months. I have the person-time at risk for cases and controls to develop this risk factor, and can directly measure the incidence density of onset of risk factor in each cohort.

Now, for background:

The relative risk (RR) is defined as: $$\frac{\left(\frac{N_\text{Cases exposed}}{N_\text{population exposed}}\right)}{\left(\frac{N_\text{cases unexposed}}{N_\text{population unexposed}}\right)}.$$ The incidence rate ratio (IRR) is defined as: $$\frac{{\left(\frac{N_\text{Cases exposed}}{\text{Person-time exposed}}\right)}}{\left(\frac{N_\text{cases unexposed}}{\text{Person-time unexposed}}\right)}.$$ The odds ratio (OR) is defined as: $$\frac{N_\text{cases exposed}*N_\text{controls unexposed}}{N_\text{controls exposed}*N_\text{cases unexposed}}.$$ Under the rare disease assumption the OR approximates the RR.

My Questions:

Can I validly expand on the odds ratio using person-time: $$\frac{N_\text{cases exposed}*\text{Person-time unexposed}}{N_\text{controls exposed}*\text{Person-time exposed}}$$

• to approximate the IRR?
• If not, why not, and what alternative would you suggest?
• If I can do this, would logistic regression would be appropriate?
• What other analytic approaches would you suggest?

This link provides a decent discussion of this: https://www.ctspedia.org/do/view/CTSpedia/SampleIncidence

Your answer is yes, conditionally. If you are using incidence density sampling (that is that controls are sampled from the risk set each time a case is diagnosed) which effectively matches cases and controls for time at risk, you are indeed estimating the incidence rate ratio. This article might also be of interest.

Yes, if you can do this, logistic regression would be appropriate.

I'm not sure that your approach would give a valid $$IRR$$ approximation. First, you would be "looking into the future" to determine the person-time at risk through this approach, since you know cases/controls. Generally speaking, looking into the future like that causes all sorts of problems. Second, there is a stipulation to the rare-disease assumption, and that is that every strata is rare (generally defined as 10%). That means each strata of month/year (along with any other variables you stratify by) would need to satisfy the assumption. Third, if the rare-disease assumption is met, then the $$RR$$ will approximate the $$IRR$$ following from $$RR \le IRR \le OR$$ when $$RR \ge 1$$.

Instead of the above approach, I would recommend revising the control selection procedure since you have a nested case-control study. Depending on how controls are selected, the case-control $$OR$$ (abbreviated $$ccOR$$ hereafter) can approximate different cohort measures. There are three approaches I will briefly detail; case-cohort, density, and cumulative control sampling. (note: case-cohort and density sampling require careful determination of what the start time is)

For case-cohort sampling, controls are sampled from baseline of the cohort study irrespective of later case status. Under this scenario $$ccOR$$ is the cohort $$RR$$

For density sampling (this is the one you will want to use), controls are selected at every time point when a case occurs, irrespective of later case status. As an example, consider case-1 occurs at $$t=0.5$$. From all individuals who are not cases at $$t \le 5$$, you randomly select a control from that group. In this way, the control selection mimics person-time in the cohort study, and $$ccOR$$ approximates the $$IRR$$

Lastly, is cumulative sampling where controls are selected at the final follow-up time, where controls cannot be cases. In this scenario, the $$ccOR$$ approximates the cohort $$OR$$, which under the rare disease assumption the cohort $$OR$$ approximates the cohort $$RR$$

Finally, based on the matching procedure you would need to use conditional logistic regression. However, I would forgo matching since it will not necessarily improve efficiency and you will still need to include the matching criteria in your regression model despite matching. I would skip matching and directly include month/year of entry in the conditional logistic model

Sources / Further Reading

Case-control selection: Modern Epidemiology 3rd Edition pg 121-126

Matching in case-control: Pearce 2016, Mansournia 2018