## Abstract

The veterinary research community has begun to use mixed effects logistic regression (MELR) for analyzing disease data obtained from groups of animals. In this article we discuss the issues of how to analyze these models and how to interpret MELR risk estimates and random effect variances (single and nested). We provide empirical evidence for their use and present equations for interpreting the results and comparing ordinary logistic regression (OLR) and MELR. These equations allow for a deeper interpretation of what random effects signify within the MELR model and help reveal the relationship between marginal (OLR) coefficients and conditional (MELR) coefficients. We used three veterinary data sets to illustrate our aims. The data sets contained data on vesicular stomatitis virus infection in cattle, Mycoplasma gallisepticum infection in chicken flocks, and three infectious conditions in puppies (respiratory, intestinal illness, and internal parasites). The chicken data had nested random effects such that 357 flocks were housed on 104 different farms operated by 45 different owners. Significant random effects were detected for all but intestinal illness in puppies and the nested farm random effect in the chicken data. The intra-group correlation coefficients on the logit scale, calculated from the random effect variances, were 0.47 and 0.55 for the cow and chicken data, respectively. This indicated that about 50% of the total variance on the logit scale for the probability of disease was attributable to unmeasured or unmeasurable group-level factors. Since the farm random effect was not significant once the owner random effect was controlled for in the chicken data, the unknown factor(s) inducing the intra-group correlation was operating at the owner level or higher. These data sets were also used to illustrate why predicted probabilities from the MELR model should not be presented as point estimates. For example, the predicted OLR probability of testing seropositive to vesicular stomatitis virus New Jersey serotype (VSV-NJ) for 5-year-old Bos taunts cattle living at an elevation of 0-500 m with a mean annual rainfall of 0-2 m was 74%. Given that significant herd random effects were present, however, the true probability of testing seropositive to VSV-NJ for such a cow should be formulated as a range of herd-specific probabilities: the probability varied from 48 to 97% for the central 65% of the herds and from 14 to 99% for the central 95% of the herds. We have also shown why marginal OLR coefficients are biased downward as estimates of conditional MELR coefficients owing to the intra-group correlation and non-linearity of the logistic regression model.

Original language | English (US) |
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Pages (from-to) | 187-201 |

Number of pages | 15 |

Journal | Preventive Veterinary Medicine |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - 1995 |

Externally published | Yes |

## Keywords

- Intra-group correlation
- Logistic regression
- Nested designs
- Random effects

## ASJC Scopus subject areas

- Animal Science and Zoology
- Food Animals