### Abstract

We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

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

Pages (from-to) | 212-226 |

Number of pages | 15 |

Journal | Journal of the American Statistical Association |

Volume | 95 |

Issue number | 449 |

DOIs | |

State | Published - Mar 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of the American Statistical Association*,

*95*(449), 212-226. https://doi.org/10.1080/01621459.2000.10473915

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*Journal of the American Statistical Association*, vol. 95, no. 449, pp. 212-226. https://doi.org/10.1080/01621459.2000.10473915

**Asymptotics for Analysis of Variance When the Number of Levels is Large.** / Akritas, Michael G.; Arnold, Steven.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Asymptotics for Analysis of Variance When the Number of Levels is Large

AU - Akritas, Michael G.

AU - Arnold, Steven

PY - 2000/3/1

Y1 - 2000/3/1

N2 - We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

AB - We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

UR - http://www.scopus.com/inward/record.url?scp=0442293927&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0442293927&partnerID=8YFLogxK

U2 - 10.1080/01621459.2000.10473915

DO - 10.1080/01621459.2000.10473915

M3 - Article

VL - 95

SP - 212

EP - 226

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 449

ER -