Noncollapsibility, confounding, and sparse-data bias. Part 1: The oddities of odds

To prevent statistical misinterpretations, it has long been advised to focus on estimation instead of statistical testing. This sound advice brings with it the need to choose the outcome and effect measures on which to focus. Measures based on odds or their logarithms have often been promoted due to their pleasing statistical properties, but have an undesirable property for risk summarization and communication: Noncollapsibility, defined as a failure of the measure when taken on a group to equal a simple average of the measure when taken on the group's members or subgroups. The present note illustrates this problem with a basic numeric example involving the odds, which is not collapsible when the odds vary across individuals and are not low in all subgroups. Its sequel will illustrate how this problem is amplified in odds ratios and logistic regression.