## Analysis of variance – why it is more important than ever. (With discussions and rejoinder).(English)Zbl 1064.62082

Summary: 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 understapding a previously fit hierarchical model.

### MSC:

 62J10 Analysis of variance and covariance (ANOVA) 62F15 Bayesian inference 62J07 Ridge regression; shrinkage estimators (Lasso) 62J05 Linear regression; mixed models 62J12 Generalized linear models (logistic models)

### Software:

car; spatial; MLwiN; BayesDA; bootstrap; BUGS
Full Text:

### References:

 [1] Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679. · Zbl 0774.62031 · doi:10.2307/2290350 [2] Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581–598. · Zbl 0474.60044 · doi:10.1016/0047-259X(81)90099-3 [3] Bafumi, J., Gelman, A. and Park, D. K. (2002). State-level opinions from national polls. Technical report, Dept. Political Science, Columbia Univ. [4] 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. · Zbl 0951.62091 · doi:10.1111/1467-9868.00201 [5] Boscardin, W. J. (1996). Bayesian analysis for some hierarchical linear models. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley. [6] Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis . Addison–Wesley, Reading, MA. · Zbl 0271.62044 [7] Carlin, B. P. and Louis, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis . Chapman and Hall, London. · Zbl 0871.62012 [8] 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. [9] Cochran, W. G. and Cox, G. M. (1957). Experimental Designs , 2nd ed. Wiley, New York. · Zbl 0077.13205 [10] Cornfield, J. and Tukey, J. W. (1956). Average values of mean squares in factorials. Ann. Math. Statist. 27 907–949. · Zbl 0075.29404 · doi:10.1214/aoms/1177728067 [11] DeGroot, M. H. (1970). Optimal Statistical Decisions . McGraw-Hill, New York. · Zbl 0225.62006 [12] Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap . Chapman and Hall, New York. · Zbl 0835.62038 [13] Eisenhart, C. (1947). The assumptions underlying the analysis of variance. Biometrics 3 1–21. [14] Fox, J. (2002). An R and S-Plus Companion to Applied Regression . Sage, Thousand Oaks, CA. [15] 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. [16] Gelman, A. (1996). Discussion of “Hierarchical generalized linear models,” by Y. Lee and J. A. Nelder. J. Roy. Statist. Soc. Ser. B 58 668. [17] Gelman, A. (2000). Bayesiaanse variantieanalyse. Kwantitatieve Methoden 21 5–12. [18] Gelman, A. (2003). Bugs.R: Functions for running WinBugs from R. Available at www.stat. columbia.edu/ gelman/bugsR/. [19] Gelman, A. (2004). Parameterization and Bayesian modeling. J. Amer. Statist. Assoc. 99 537–545. · Zbl 1117.62343 · doi:10.1198/016214504000000458 [20] Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (1995). Bayesian Data Analysis . Chapman and Hall, London. · Zbl 1279.62004 [21] Gelman, A. and Little, T. C. (1997). Poststratification into many categories using hierarchical logistic regression. Survey Methodology 23 127–135. [22] Gelman, A., Pasarica, C. and Dodhia, R. M. (2002). Let’s practice what we preach: Turning tables into graphs. Amer. Statist. 56 121–130. · doi:10.1198/000313002317572790 [23] George, E. I. and McCulloch, R. E. (1993). Variable selection via Gibbs sampling. J. Amer. Statist. Assoc. 88 881–889. [24] Goldstein, H. (1995). Multilevel Statistical Models , 2nd ed. Arnold, London. · Zbl 1014.62126 [25] Green, B. F. and Tukey, J. W. (1960). Complex analyses of variance: General problems. Psychometrika 25 127–152. · Zbl 0094.33005 · doi:10.1007/BF02288577 [26] James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361–379. Univ. California Press, Berkeley. · Zbl 1281.62026 [27] 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. [28] Khuri, A. I., Mathew, T. and Sinha, B. K. (1998). Statistical Tests for Mixed Linear Models . Wiley, New York. · Zbl 0893.62009 [29] Kirk, R. E. (1995). Experimental Design : Procedures for the Behavioral Sciences , 3rd ed. Brooks/Cole, Belmont, MA. · Zbl 0943.62072 [30] Kreft, I. and de Leeuw, J. (1998). Introducing Multilevel Modeling . Sage, London. [31] 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. [32] Liu, C. (2002). Robit regression: A simple robust alternative to logistic and probit regression. Technical report, Bell Laboratories. [33] Liu, C., Rubin, D. B. and Wu, Y. N. (1998). Parameter expansion to accelerate EM—the PX-EM algorithm. Biometrika 85 755–770. · Zbl 0921.62071 · doi:10.1093/biomet/85.4.755 [34] Liu, J. and Wu, Y. N. (1999). Parameter expansion for data augmentation. J. Amer. Statist. Assoc. 94 1264–1274. · Zbl 1069.62514 · doi:10.2307/2669940 [35] 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. · Zbl 1090.62518 · doi:10.1111/1467-9868.00082 [36] Montgomery, D. C. (1986). Design and Analysis of Experiments , 2nd ed. Wiley, New York. · Zbl 0747.62072 [37] 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. · Zbl 0124.10703 · doi:10.1098/rspa.1965.0012 [38] 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. · Zbl 0124.10703 · doi:10.1098/rspa.1965.0012 [39] Nelder, J. A. (1977). A reformulation of linear models (with discussion). J. Roy. Statist. Soc. Ser. A 140 48–76. [40] Nelder, J. A. (1994). The statistics of linear models: Back to basics. Statist. Comput. 4 221–234. [41] Plackett, R. L. (1960). Models in the analysis of variance (with discussion). J. Roy. Statist. Soc. Ser. B 22 195–217. · Zbl 0109.37802 [42] R Project (2000). The R project for statistical computing. Available at www.r-project.org. [43] Ripley, B. D. (1981). Spatial Statistics . Wiley, New York. · Zbl 0583.62087 [44] Robinson, G. K. (1991). That BLUP is a good thing: The estimation of random effects (with discussion). Statist. Sci. 6 15–51. JSTOR: · Zbl 0955.62500 · doi:10.1214/ss/1177011926 [45] Robinson, G. K. (1998). Variance components. In Encyclopedia of Biostatistics (P. Armitage and T. Colton, eds.) 6 4713–4719. Wiley, Chichester. [46] Rubin, D. B. (1981). Estimation in parallel randomized experiments. J. Educational Statistics 6 377–401. [47] Sargent, D. J. and Hodges, J. S. (1997). Smoothed ANOVA with application to subgroup analysis. Technical report, Dept. Biostatistics, Univ. Minnesota. [48] Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components . Wiley, New York. · Zbl 0850.62007 [49] Snedecor, G. W. and Cochran, W. G. (1989). Statistical Methods , 8th ed. Iowa State Univ. Press, Ames, IA. · Zbl 0727.62003 [50] Snijders, T. A. B. and Bosker, R. J. (1999). Multilevel Analysis . Sage, London. · Zbl 0953.62127 [51] Speed, T. P. (1987). What is an analysis of variance? (with discussion). Ann. Statist. 15 885–941. JSTOR: · Zbl 0637.62070 · doi:10.1214/aos/1176350472 [52] 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 www.mrc-bsu.cam.ac.uk/bugs/. [53] 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. [54] 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. [55] Cochran, W. G. and Cox, G. M. (1957). Experimental Designs , 2nd ed. Wiley, New York. · Zbl 0077.13205 [56] Cox, D. R. (1984). Interaction (with discussion). Internat. Statist. Rev. 52 1–31. · Zbl 0562.62061 [57] Cox, D. R. and Snell, E. J. (1981). Applied Statistics : Principles and Examples . Chapman and Hall, London. · Zbl 0612.62002 [58] Joe, H. (1990). Extended use of paired comparison models with application to chess rankings. Appl. Statist. 39 85–93. · Zbl 0707.62149 · doi:10.2307/2347814 [59] McCullagh, P. (2000). Invariance and factorial models (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 62 209–256. · doi:10.1111/1467-9868.00229 [60] Stewart, J. Q. (1948). Demographic gravitation: Evidence and application. Sociometry 11 31–58. [61] Stigler, S. M. (1994). Citation patterns in the journals of statistics and probability. Statist. Sci. 9 94–108. [62] Tukey, J. W. (1974). Named and faceless values: An initial exploration in memory of Prasanta C. Mahalanobis. Sankhyā Ser. A 36 125–176. · Zbl 0401.62003 [63] 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. JSTOR: · Zbl 0596.65004 · doi:10.1214/aos/1176349743 [64] Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia. · Zbl 0813.62001 [65] Wu, C. F. J. and Hamada, M. (2000). Experiments : Planning , Analysis , and Parameter Design Optimization . Wiley, New York. · Zbl 0964.62065 [66] Berger, J. O. and Pericchi, L. (1996). The intrinsic Bayes factor for model selection and prediction. J. Amer. Statist. Assoc. 91 109–122. · Zbl 0870.62021 · doi:10.2307/2291387 [67] 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. [68] Lee, P. M. (1997). Bayesian Statistics : An Introduction . Arnold, London. · Zbl 0882.62017 [69] 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. [70] Meng, X.-L. (1994). Posterior predictive $$p$$-values. Ann. Statist. 22 1142–1160. JSTOR: · Zbl 0820.62027 · doi:10.1214/aos/1176325622 [71] Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 1151–1172. JSTOR: · Zbl 0555.62010 · doi:10.1214/aos/1176346785 [72] Wennberg, J. E. and Gittelsohn, A. (1982). Variations in medical care among small areas. Scientific American 246 (4) 120–134. [73] 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. [74] 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. · Zbl 0882.62103 · doi:10.2307/2291566 [75] Gelman, A. and Huang, Z. (2005). Estimating incumbency advantage and its variation, as an example of a before/after study. J. Amer. Statist. Assoc. · Zbl 1469.62397 [76] Louis, T. A. (1984). Estimating a population of parameter values using Bayes and empirical Bayes methods. J. Amer. Statist. Assoc. 79 393–398. [77] 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. [78] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.