The causal risk ratio for X can be computed by standardization across strata Z as follows:

The causal odds ratio for X was considered non-collapsible because it could not be standardized across strata Z as follows which was the classical definition of non-collapsibility:

It seems that the causal odds ratio for X should be computed by standardization of probabilities alone across strata Z as follows since odds should emerge from risk standardization:

Thus for the data below

x | |||
---|---|---|---|

z | 0 | 1 | Total |

0 | Pr 0.2 | Pr 0.4 | 0.3 |

N=100 | N=100 | 200 | |

1 | Pr 0.6 | Pr 0.8 | 0.7 |

N=100 | N=100 | 200 | |

Total | 0.4 | 0.6 | 0.5 |

200 | 200 | 400 |

So does this not mean that odds ratios are collapsible given that the stratum specific ORs are 2.67 while the â€śrisk-standardizedâ€ť OR is 2.25 consistent with the marginal OR? The scientific question of interest here is whether the question raised about the collapsibility of odds (and thus ORs) is because instead of converting standardized risks to odds we directly attempted to standardize odds. Perhaps this demonstrates that we should always think in terms of risk regardless of the use of odds ratios - they are just a tool to enable us understand risks better.

Edit 1: Third equation edited

Edit 2: Second equation edited