### Abstract

Conditioning on a shared outcome of two variables can alter the association between these variables, possibly adding a bias component when estimating effects. In particular, if two causes are marginally independent, they might be dependent in strata of their common effect. Explanations of the phenomenon, however, do not explicitly state when dependence will be created and have been largely informal. We prove that two, marginally independent, causes will be dependent in a particular stratum of their shared outcome if and only if they modify each other's effects, on a probability ratio scale, on that value of the outcome variable. Using our result, we also qualify the claim that such causes will "almost certainly" be dependent in at least one stratum of the outcome: dependence must be created in one stratum of a binary outcome, and independence can be maintained in every stratum of a trinary outcome.

Language | English (US) |
---|---|

Article number | 20160055 |

Journal | International Journal of Biostatistics |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

### Fingerprint

### Keywords

- Causal diagram
- Colliding bias
- Effect modification
- Independence

### ASJC Scopus subject areas

- Statistics and Probability
- Medicine(all)
- Statistics, Probability and Uncertainty

### Cite this

*International Journal of Biostatistics*,

*13*(1), [20160055]. DOI: 10.1515/ijb-2016-0055

**A Theorem at the Core of Colliding Bias.** / Shahar, Doron J.; Shahar, Eyal.

Research output: Contribution to journal › Article

*International Journal of Biostatistics*, vol 13, no. 1, 20160055. DOI: 10.1515/ijb-2016-0055

}

TY - JOUR

T1 - A Theorem at the Core of Colliding Bias

AU - Shahar,Doron J.

AU - Shahar,Eyal

PY - 2017

Y1 - 2017

N2 - Conditioning on a shared outcome of two variables can alter the association between these variables, possibly adding a bias component when estimating effects. In particular, if two causes are marginally independent, they might be dependent in strata of their common effect. Explanations of the phenomenon, however, do not explicitly state when dependence will be created and have been largely informal. We prove that two, marginally independent, causes will be dependent in a particular stratum of their shared outcome if and only if they modify each other's effects, on a probability ratio scale, on that value of the outcome variable. Using our result, we also qualify the claim that such causes will "almost certainly" be dependent in at least one stratum of the outcome: dependence must be created in one stratum of a binary outcome, and independence can be maintained in every stratum of a trinary outcome.

AB - Conditioning on a shared outcome of two variables can alter the association between these variables, possibly adding a bias component when estimating effects. In particular, if two causes are marginally independent, they might be dependent in strata of their common effect. Explanations of the phenomenon, however, do not explicitly state when dependence will be created and have been largely informal. We prove that two, marginally independent, causes will be dependent in a particular stratum of their shared outcome if and only if they modify each other's effects, on a probability ratio scale, on that value of the outcome variable. Using our result, we also qualify the claim that such causes will "almost certainly" be dependent in at least one stratum of the outcome: dependence must be created in one stratum of a binary outcome, and independence can be maintained in every stratum of a trinary outcome.

KW - Causal diagram

KW - Colliding bias

KW - Effect modification

KW - Independence

UR - http://www.scopus.com/inward/record.url?scp=85020433697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020433697&partnerID=8YFLogxK

U2 - 10.1515/ijb-2016-0055

DO - 10.1515/ijb-2016-0055

M3 - Article

VL - 13

JO - International Journal of Biostatistics

T2 - International Journal of Biostatistics

JF - International Journal of Biostatistics

SN - 1557-4679

IS - 1

M1 - 20160055

ER -