# Asymptotics for Analysis of Variance When the Number of Levels is Large

Michael Akritas, Steven Arnold

Research output: Contribution to journalArticle

35 Citations (Scopus)

### 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) 212-226 15 Journal of the American Statistical Association 95 449 https://doi.org/10.1080/01621459.2000.10473915 Published - Mar 1 2000

### Fingerprint

Analysis of variance
Fixed Effects Model
F-statistics
F Test
Numerator
Fixed Effects
Random Effects Model
Mixed Model
Denominator
Random Effects
Null
Degree of freedom
Factors
Alternatives
Model
Experiment
Observation

### All Science Journal Classification (ASJC) codes

• Statistics and Probability
• Statistics, Probability and Uncertainty

### Cite this

@article{74867aea35064bb3ae447e12446d4f7e,
title = "Asymptotics for Analysis of Variance When the Number of Levels is Large",
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.",
author = "Michael Akritas and Steven Arnold",
year = "2000",
month = "3",
day = "1",
doi = "10.1080/01621459.2000.10473915",
language = "English (US)",
volume = "95",
pages = "212--226",
journal = "Journal of the American Statistical Association",
issn = "0162-1459",
publisher = "Taylor and Francis Ltd.",
number = "449",

}

In: Journal of the American Statistical Association, Vol. 95, No. 449, 01.03.2000, p. 212-226.

Research output: Contribution to journalArticle

TY - JOUR

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

AU - Akritas, Michael

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

AN - SCOPUS:0442293927

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 -