Abstract
The two-way crossed model considered in Chapter 3 uses the simple additive model, which makes an important assumption that the value of the difference between the mean responses at two levels of A is the same at each level B. However, in many situations, this simple additive model may not be appropriate. When the differences between the mean response at different levels of A tend to vary over the different levels of B, it is said that the two factors interact. If an experimenter makes more than one observation per cell, it permits him to investigate not only the main effects of both factors but also their interaction. In this chapter, we consider a two-way crossed model with more than one observation per cell, which allows the investigation of interaction terms between the two factors.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
D. M. Andrews and A. Herzberg (1985), Data: A Collection of Problems from Many Fields for Students and Research Workers, Springer-Verlag, New York.
J. N. Arvesen (1976), Anote on the Tukey-Hooke variance components results, Ann. Inst. Statist. Math. (Japan), 28, 111–121.
N. J. Birch and R. K. Burdick (1989), Confidence intervals on the ratios of expected mean squares (θ1 + θ2 + θ3)/θ4, Statist. Probab. Lett, 7, 335–342.
G. E. P. Box and G. C. Tiao (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading, MA.
H. Bozivich, T. A. Bancroft, and H. O. Hartley (1956), Power of analysis of variance test procedures for certain incompletely specified model, Ann. Math. Statist., 27, 1017–1043.
L. D. Broemeling (1969a), Confidence intervals for measures of heritability, Biometrics, 25, 424–427.
L. D. Broemeling (1969b), Confidence intervals for variance ratios of random model, J. Amer. Statist. Assoc., 64, 660–664.
R. K. Burdick and F. A. Graybill (1992), Confidence Intervals on Variance Components, Marcel Dekker, New York.
R. K. Burdick and G. A. Larsen (1997), Confidence intervals on means of variability in R&R studies, J. Qual. Tech., 29, 261–273.
O. L. Davies and P. L. Goldsmith, eds. (1972), Statistical Methods in Research and Production, 4th ed., Oliver and Boyd, Edinburgh.
K. K. Dolezal, R. K. Burdick, and N. J. Birch (1998), Analysis of two-factor R&R study with fixed operators, J. Qual. Tech., 30, 163–170.
L. H. Gautschi (1959), Some remarks on Herbach’s paper, “Optimum nature of the F-test for Model II in the balanced case,” Ann. Math. Statist., 30, 960–963.
F. A. Graybill (1976), Theory and Application of the Linear Model, Duxbury, North Scituate, MA.
F. A. Graybill and C.-M. Wang (1980), Confidence intervals on nonnegative linear combinations of variances, J. Amer. Statist. Assoc., 75, 869–873.
R. Gui, F. A. Graybill, R. K. Burdick, and N. Ting (1995), Confidence intervals on ratios of linear combinations for non-disjoint sets of expected mean squares, J. Statist. Plann. Inference, 48, 215–227.
L. H. Herbach (1959), Properties of Model II type analysis of variance tests A: Optimum nature of the F-test for Model II in the balanced case, Ann. Math. Statist., 30, 939–959.
W. G. S. Hines (1996), Pragmatics of pooling in ANOVA tables, Amer. Statist., 50, 127–139.
R. R. Hocking, J. W. Green, and R. H. Bremer (1989), Variance component estimation with model-based diagnostics, Technometrics, 31, 227–240.
R. Hooke (1956), Some applications of biopolykays to the estimation of variance components and their moments, Ann. Math. Statist., 27, 80–98.
D. G. Janky (2000), Sometimes pooling for analysis of variance hypothesis tests: A review and study of a split-plot model, Amer. Statist., 54, 269–279.
H. Jeffreys (1961), Theory of Probability, 3rd ed., Clarendon Press, Oxford, UK; 1st ed., 1939; 2nd ed., 1948.
A. I. Khuri (1981), Simultaneous confidence intervals for functions of variance components in random models, J. Amer. Statist. Assoc., 76, 878–885.
A. W. Kimball (1951), On dependent tests of significance in the analysis of variance, Ann. Math. Statist, 22, 600–602.
R. A. Leiva and F. A. Graybill (1986), Confidence intervals for variance components in the balanced two-way model with interaction, Comm. Statist. B Simul. Comput., 15, 301–322.
R. Mead, T. A. Bancroft, and C. Hand (1975), Power of analysis of variance test procedures for incompletely specified fixed models, Ann. Statist., 3, 797–808.
G. A. Milliken and D. E. Johnson (1992), Analysis of Messy Data, Vol. 1, Chapman and Hall, London.
D. C. Montgomery and G. C. Runger (1994), Gauge capability and designed experiments, Part II: Experimental design model and variance component estimation, Qual. Engrg., 6, 289–305.
K. Paark and R. K. Burdick (1998), Confidence intervals for the mean in a balanced two-factor random effets model, Comm. Statist. A Theory Methods, 27, 2807–2825.
A. E. Pauli (1950), On a preliminary test for pooling mean squares in the analysis of variance, Ann. Math. Statist., 21, 539–556.
S. Portnoy (1971), Formal Bayes estimation with application to arandom effects model, Ann. Math. Statist., 42, 1379–1402.
C. V. Rao and K. P. Saxena (1979), A study of power of a test procedure based on two preliminary tests of significance, Estadística, 33, 201–214.
H. Sahai (1974), Simultaneous confidence intervals for variance components in some balanced random effects models, Sankhyā Ser. B, 36, 278–287.
H. Sahai and R. L. Anderson (1973), Confidence regions for variance ratios of random models for balanced data, J. Amer. Statist. Assoc., 68, 951–952.
H. Sahai and A. A. Ramírez-Martínez (1978), Estimadores formales de Bayes en el modelo aleatorio general de clasificación doble cruzado, Trab. Estadist., 29, 88–93.
W. A. Thompson, Jr. (1962), The problem of negative estimates of variance components, Ann. Math. Statist, 33, 273–289.
W. A. Thompson, Jr. and J. R. Moore (1963), Non-negative estimates of variance components, Technometrics, 5, 441–450.
N. Ting, R. K. Burdick, and F. A. Graybill (1991), Confidence intervals on ratios of positive linear combinations of variance components, Statist. Probab. Lett, 11, 523–528.
N. Ting, R. K. Burdick, F. A. Graybill, S. Jeyaratnam, and T.-F. C. Lu (1990), Confidence intervals on linear combinations of variance components that are unrestricted in sign, J. Statist. Comput. Simul, 35, 135–143.
N. Ting and F. A. Graybill (1991), Approximate confidence interval on ratio of two variances in a two-way crossed model, Biometrical J., 33, 547–558.
S. B. Vardeman and E. S. Van Valkenburg (1999), Two-way random-effects analyses and gauge in R&R studies, Technometrics, 41, 202–211.
C. M. Wang (1994), On estimating approximate degrees of freedom of chi-squared approximations, Comm. Statist. B Simul. Comput, 23, 769–788.
C. M. Wang and F. A. Graybill (1981), Confidence intervals on a ratio of variances in the two-factor nested components of variance model, Comm. Statist A Theory Methods, 10, 1357–1368.
J. S. Williams (1962), A confidence interval for variance components, Biometrika, 49, 278–281.
G. Wolde-Tsadik and A. A. Afifi (1980), A comparison of the “sometimes-pool,” “sometimes-switch,” and “never-pool” procedures in the two-way ANOVA random effects model, Technometrics, 22, 367–373.
S. Wolfram (1996), The Mathematica Book, 3rd ed., Cambridge University Press, Cambridge, UK.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Sahai, H., Ojeda, M.M. (2004). Two-Way Crossed Classification with Interaction. In: Analysis of Variance for Random Models. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8168-5_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8168-5_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6470-5
Online ISBN: 978-0-8176-8168-5
eBook Packages: Springer Book Archive