### Abstract

Clustered binary responses occur frequently in many fields of application. Examples include the development of tumors in one or more animals of a litter, the presence of retinitis in either or both eyes of an AIDS patient, and spontaneous abortion of one or more implanted fetuses. When a binary response is observed in multiple units from each subject, application of the usual Pearson chi-square statistic is invalid since such responses within the same subject are not independent. In estimating the common response probability in clustered binary data, two weighting systems have been most popular: equal weights to units, and equal weights to clusters. We also include an optimal weighting method that minimizes the variance of the response probability. The weighted chi-square statistics using the above three weighting systems are applied to the real data arising from a teratologic study and an ophthalmologic study. We perform the simulation study to evaluate the performance of the three weighted chi-square statistics in terms of empirical type I errors and empirical powers. The simulation study shows that the weighted chi-square statistic using an optimal weight yields higher empirical powers than the other weighted chi-square statistics and produces empirical type I errors close to a nominal value. The weighted chi-square statistics assigning equal weights to units (X^{2}_{D}) and optimal weights (X^{2}_{O}) are slightly anti-conservative when n_{1} = n_{2} = n = 10. We recommend using when n_{1} = n_{2} = n ≥ 20 since the differences in empirical type I errors are negligible among weighted chi-square statistics and X^{2}_{O} performs better than the other weighted chi-square test statistics in empirical powers.

Original language | English (US) |
---|---|

Pages (from-to) | 91-99 |

Number of pages | 9 |

Journal | Therapeutic Innovation & Regulatory Science |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 2003 |

### Fingerprint

### Keywords

- Chi-square statistic
- Intracluster correlation
- Optimal weight

### ASJC Scopus subject areas

- Pharmacology, Toxicology and Pharmaceutics (miscellaneous)
- Public Health, Environmental and Occupational Health
- Pharmacology (medical)

### Cite this

*Therapeutic Innovation & Regulatory Science*,

*37*(1), 91-99. https://doi.org/10.1177/009286150303700111

**An Evaluation of Weighted Chi-Square Statistics for Clustered Binary Data.** / Ahn, Chul; Jung, Sin HO; Kang, Seung ho.

Research output: Contribution to journal › Article

*Therapeutic Innovation & Regulatory Science*, vol. 37, no. 1, pp. 91-99. https://doi.org/10.1177/009286150303700111

}

TY - JOUR

T1 - An Evaluation of Weighted Chi-Square Statistics for Clustered Binary Data

AU - Ahn, Chul

AU - Jung, Sin HO

AU - Kang, Seung ho

PY - 2003

Y1 - 2003

N2 - Clustered binary responses occur frequently in many fields of application. Examples include the development of tumors in one or more animals of a litter, the presence of retinitis in either or both eyes of an AIDS patient, and spontaneous abortion of one or more implanted fetuses. When a binary response is observed in multiple units from each subject, application of the usual Pearson chi-square statistic is invalid since such responses within the same subject are not independent. In estimating the common response probability in clustered binary data, two weighting systems have been most popular: equal weights to units, and equal weights to clusters. We also include an optimal weighting method that minimizes the variance of the response probability. The weighted chi-square statistics using the above three weighting systems are applied to the real data arising from a teratologic study and an ophthalmologic study. We perform the simulation study to evaluate the performance of the three weighted chi-square statistics in terms of empirical type I errors and empirical powers. The simulation study shows that the weighted chi-square statistic using an optimal weight yields higher empirical powers than the other weighted chi-square statistics and produces empirical type I errors close to a nominal value. The weighted chi-square statistics assigning equal weights to units (X2D) and optimal weights (X2O) are slightly anti-conservative when n1 = n2 = n = 10. We recommend using when n1 = n2 = n ≥ 20 since the differences in empirical type I errors are negligible among weighted chi-square statistics and X2O performs better than the other weighted chi-square test statistics in empirical powers.

AB - Clustered binary responses occur frequently in many fields of application. Examples include the development of tumors in one or more animals of a litter, the presence of retinitis in either or both eyes of an AIDS patient, and spontaneous abortion of one or more implanted fetuses. When a binary response is observed in multiple units from each subject, application of the usual Pearson chi-square statistic is invalid since such responses within the same subject are not independent. In estimating the common response probability in clustered binary data, two weighting systems have been most popular: equal weights to units, and equal weights to clusters. We also include an optimal weighting method that minimizes the variance of the response probability. The weighted chi-square statistics using the above three weighting systems are applied to the real data arising from a teratologic study and an ophthalmologic study. We perform the simulation study to evaluate the performance of the three weighted chi-square statistics in terms of empirical type I errors and empirical powers. The simulation study shows that the weighted chi-square statistic using an optimal weight yields higher empirical powers than the other weighted chi-square statistics and produces empirical type I errors close to a nominal value. The weighted chi-square statistics assigning equal weights to units (X2D) and optimal weights (X2O) are slightly anti-conservative when n1 = n2 = n = 10. We recommend using when n1 = n2 = n ≥ 20 since the differences in empirical type I errors are negligible among weighted chi-square statistics and X2O performs better than the other weighted chi-square test statistics in empirical powers.

KW - Chi-square statistic

KW - Intracluster correlation

KW - Optimal weight

UR - http://www.scopus.com/inward/record.url?scp=84993778678&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84993778678&partnerID=8YFLogxK

U2 - 10.1177/009286150303700111

DO - 10.1177/009286150303700111

M3 - Article

AN - SCOPUS:84993778678

VL - 37

SP - 91

EP - 99

JO - Therapeutic Innovation and Regulatory Science

JF - Therapeutic Innovation and Regulatory Science

SN - 2168-4790

IS - 1

ER -