A Note on the Noncollapsibility of Rate Differences and Rate Ratios

Area of Science:

• Epidemiology
• Biostatistics
• Statistical Modeling

Background:

• The odds ratio is known to be noncollapsible, meaning its value changes when conditioning on a covariate related to the outcome.
• Risk difference and risk ratio are collapsible, unlike the odds ratio.
• Rate difference and rate ratio are generally noncollapsible, which appears paradoxical given their relationship to risk.

Purpose of the Study:

• To explain why rate difference/ratio are noncollapsible and do not inherit collapsibility from risk difference/ratio.
• To clarify the collapsibility of the exposure coefficient in the Aalen additive hazards model.
• To resolve apparent contradictions regarding collapsibility in different statistical measures.

Main Methods:

• Graphical arguments to illustrate collapsibility concepts.
• Analysis of statistical properties of odds ratio, risk measures, and rate measures.
• Examination of the Aalen additive hazards model.

Main Results:

• Demonstration that rate difference/ratio do not inherit collapsibility from risk difference/ratio.
• Identification of conditions under which the exposure coefficient in the Aalen additive hazards model is collapsible.
• Explanation of the noncollapsibility of rate measures despite their relation to risk.

Conclusions:

• Rate measures (difference and ratio) are generally noncollapsible.
• The exposure coefficient in the Aalen additive hazards model exhibits collapsibility under specific conditions.
• Graphical methods provide clarity on complex collapsibility phenomena in statistical measures.