The Annals of Statistics

Analysis of variance—why it is more important than ever

Andrew Gelman

Full-text: Open access


Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table.

We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects.

We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model.

Article information

Ann. Statist. Volume 33, Number 1 (2005), 1-53.

First available in Project Euclid: 8 April 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance 62J07: Ridge regression; shrinkage estimators 62F15: Bayesian inference 62J05: Linear regression 62J12: Generalized linear models

ANOVA Bayesian inference fixed effects hierarchical model linear regression multilevel model random effects variance components


Gelman, Andrew. Analysis of variance—why it is more important than ever. Ann. Statist. 33 (2005), no. 1, 1--53. doi:10.1214/009053604000001048.

Export citation


  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669--679.
  • Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581--598.
  • Bafumi, J., Gelman, A. and Park, D. K. (2002). State-level opinions from national polls. Technical report, Dept. Political Science, Columbia Univ.
  • Besag, J. and Higdon, D. (1999). Bayesian analysis of agricultural field experiments (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 61 691--746.
  • Boscardin, W. J. (1996). Bayesian analysis for some hierarchical linear models. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley.
  • Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison--Wesley, Reading, MA.
  • Carlin, B. P. and Louis, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall, London.
  • Chipman, H., George, E. I. and McCulloch, R. E. (2001). The practical implementation of Bayesian model selection. In Model Selection (P. Lahiri, ed.) 67--116. IMS, Beachwood, Ohio.
  • Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York.
  • Cornfield, J. and Tukey, J. W. (1956). Average values of mean squares in factorials. Ann. Math. Statist. 27 907--949.
  • DeGroot, M. H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York.
  • Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
  • Eisenhart, C. (1947). The assumptions underlying the analysis of variance. Biometrics 3 1--21.
  • Fox, J. (2002). An R and S-Plus Companion to Applied Regression. Sage, Thousand Oaks, CA.
  • Gelman, A. (1992). Discussion of ``Maximum entropy and the nearly black object,'' by D. Donoho et al. J. Roy. Statist. Soc. Ser. B 54 72--73.
  • Gelman, A. (1996). Discussion of ``Hierarchical generalized linear models,'' by Y. Lee and J. A. Nelder. J. Roy. Statist. Soc. Ser. B 58 668.
  • Gelman, A. (2000). Bayesiaanse variantieanalyse. Kwantitatieve Methoden 21 5--12.
  • Gelman, A. (2003). Bugs.R: Functions for running WinBugs from R. Available at www.stat.
  • Gelman, A. (2004). Parameterization and Bayesian modeling. J. Amer. Statist. Assoc. 99 537--545.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (1995). Bayesian Data Analysis. Chapman and Hall, London.
  • Gelman, A. and Little, T. C. (1997). Poststratification into many categories using hierarchical logistic regression. Survey Methodology 23 127--135.
  • Gelman, A., Pasarica, C. and Dodhia, R. M. (2002). Let's practice what we preach: Turning tables into graphs. Amer. Statist. 56 121--130.
  • George, E. I. and McCulloch, R. E. (1993). Variable selection via Gibbs sampling. J. Amer. Statist. Assoc. 88 881--889.
  • Goldstein, H. (1995). Multilevel Statistical Models, 2nd ed. Arnold, London.
  • Green, B. F. and Tukey, J. W. (1960). Complex analyses of variance: General problems. Psychometrika 25 127--152.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361--379. Univ. California Press, Berkeley.
  • Johnson, E. G. and Tukey, J. W. (1987). Graphical exploratory analysis of variance illustrated on a splitting of the Johnson and Tsao data. In Design, Data and Analysis by Some Friends of Cuthbert Daniel (C. Mallows, ed.) 171--244. Wiley, New York.
  • Khuri, A. I., Mathew, T. and Sinha, B. K. (1998). Statistical Tests for Mixed Linear Models. Wiley, New York.
  • Kirk, R. E. (1995). Experimental Design: Procedures for the Behavioral Sciences, 3rd ed. Brooks/Cole, Belmont, MA.
  • Kreft, I. and de Leeuw, J. (1998). Introducing Multilevel Modeling. Sage, London.
  • LaMotte, L. R. (1983). Fixed-, random-, and mixed-effects models. In Encyclopedia of Statistical Sciences (S. Kotz, N. L. Johnson and C. B. Read, eds.) 3 137--141. Wiley, New York.
  • Liu, C. (2002). Robit regression: A simple robust alternative to logistic and probit regression. Technical report, Bell Laboratories.
  • Liu, C., Rubin, D. B. and Wu, Y. N. (1998). Parameter expansion to accelerate EM---the PX-EM algorithm. Biometrika 85 755--770.
  • Liu, J. and Wu, Y. N. (1999). Parameter expansion for data augmentation. J. Amer. Statist. Assoc. 94 1264--1274.
  • Meng, X.-L. and van Dyk, D. (1997). The EM algorithm---an old folk-song sung to a fast new tune (with discussion). J. Roy. Statist. Soc. Ser. B 59 511--567.
  • Montgomery, D. C. (1986). Design and Analysis of Experiments, 2nd ed. Wiley, New York.
  • Nelder, J. A. (1965a). The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc. Roy. Soc. London Ser. A 283 147--162.
  • Nelder, J. A. (1965b). The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proc. Roy. Soc. London Ser. A 283 163--178.
  • Nelder, J. A. (1977). A reformulation of linear models (with discussion). J. Roy. Statist. Soc. Ser. A 140 48--76.
  • Nelder, J. A. (1994). The statistics of linear models: Back to basics. Statist. Comput. 4 221--234.
  • Plackett, R. L. (1960). Models in the analysis of variance (with discussion). J. Roy. Statist. Soc. Ser. B 22 195--217.
  • R Project (2000). The R project for statistical computing. Available at
  • Ripley, B. D. (1981). Spatial Statistics. Wiley, New York.
  • Robinson, G. K. (1991). That BLUP is a good thing: The estimation of random effects (with discussion). Statist. Sci. 6 15--51.
  • Robinson, G. K. (1998). Variance components. In Encyclopedia of Biostatistics (P. Armitage and T. Colton, eds.) 6 4713--4719. Wiley, Chichester.
  • Rubin, D. B. (1981). Estimation in parallel randomized experiments. J. Educational Statistics 6 377--401.
  • Sargent, D. J. and Hodges, J. S. (1997). Smoothed ANOVA with application to subgroup analysis. Technical report, Dept. Biostatistics, Univ. Minnesota.
  • Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.
  • Snedecor, G. W. and Cochran, W. G. (1989). Statistical Methods, 8th ed. Iowa State Univ. Press, Ames, IA.
  • Snijders, T. A. B. and Bosker, R. J. (1999). Multilevel Analysis. Sage, London.
  • Speed, T. P. (1987). What is an analysis of variance? (with discussion). Ann. Statist. 15 885--941.
  • Spiegelhalter, D., Thomas, A., Best, N. and Lunn, D. (2002). BUGS: Bayesian inference using Gibbs sampling, version 1.4. MRC Biostatistics Unit, Cambridge, England. Available at
  • Voss, D. S., Gelman, A. and King, G. (1995). Pre-election survey methodology: Details from eight polling organizations, 1988 and 1992. Public Opinion Quarterly 59 98--132.
  • Yates, F. (1967). A fresh look at the basic principles of the design and analysis of experiments. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 4 777--790. Univ. California Press, Berkeley.
  • Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York.
  • Cox, D. R. (1984). Interaction (with discussion). Internat. Statist. Rev. 52 1--31.
  • Cox, D. R. and Snell, E. J. (1981). Applied Statistics: Principles and Examples. Chapman and Hall, London.
  • Joe, H. (1990). Extended use of paired comparison models with application to chess rankings. Appl. Statist. 39 85--93.
  • McCullagh, P. (2000). Invariance and factorial models (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 62 209--256.
  • Stewart, J. Q. (1948). Demographic gravitation: Evidence and application. Sociometry 11 31--58.
  • Stigler, S. M. (1994). Citation patterns in the journals of statistics and probability. Statist. Sci. 9 94--108.
  • Tukey, J. W. (1974). Named and faceless values: An initial exploration in memory of Prasanta C. Mahalanobis. Sankhyā Ser. A 36 125--176.
  • Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Statist. 13 1378--1402.
  • Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
  • Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.
  • Berger, J. O. and Pericchi, L. (1996). The intrinsic Bayes factor for model selection and prediction. J. Amer. Statist. Assoc. 91 109--122.
  • Goldstein, H., Rasbash, J., Plewis, I., Draper, D., Browne, W., Yang, M., Woodhouse, G. and Healy, M. (1998). A User's Guide to MLwiN. Institute of Education, Univ. London.
  • Lee, P. M. (1997). Bayesian Statistics: An Introduction. Arnold, London.
  • Ayanian, J. Z., Zaslavsky, A. M., Fuchs, C. S., Guadagnoli, E., Creech, C. M., Cress, R. D., O'Connor, L. C., West, D. W., Allen, M. E., Wolf, R. E. and Wright, W. E. (2003). Use of adjuvant chemotherapy and radiation therapy for colorectal cancer in a population-based cohort. J. Clinical Oncology 21 1293--1300.
  • Meng, X.-L. (1994). Posterior predictive $p$-values. Ann. Statist. 22 1142--1160.
  • Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 1151--1172.
  • Wennberg, J. E. and Gittelsohn, A. (1982). Variations in medical care among small areas. Scientific American 246(4) 120--134.
  • Zaslavsky, A. M., Zaborski, L. B. and Cleary, P. D. (2004). Plan, geographical, and temporal variation of consumer assessments of ambulatory health care. Health Services Res. 39 1467--1485.
  • Gelman, A., Bois, F. Y. and Jiang, J. (1996). Physiological pharmacokinetic analysis using population modeling and informative prior distributions. J. Amer. Statist. Assoc. 91 1400--1412.
  • Gelman, A. and Huang, Z. (2005). Estimating incumbency advantage and its variation, as an example of a before/after study. J. Amer. Statist. Assoc. To appear.
  • Louis, T. A. (1984). Estimating a population of parameter values using Bayes and empirical Bayes methods. J. Amer. Statist. Assoc. 79 393--398.
  • Meulders, M., De Boeck, P., Van Mechelen, I., Gelman, A. and Maris, E. (2001). Bayesian inference with probability matrix decomposition models. J. Educational and Behavioral Statistics 26 153--179.
  • Park, D. K., Gelman, A. and Bafumi, J. (2004). Bayesian multilevel estimation with poststratification: State-level estimates from national polls. Political Analysis 12 375--385.