A Multivariate Test for Three-Factor Interaction in 3-Way Contingency Table under the Multiplicative Model
Author(s)
Njoku O. Ama
ABSTRACT
Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables with sufficiently large sample size, have an asymptotic chi-square distribution with degrees of freedom (df) equal to the number of free parameters in the saturated model. For example under the hypothesis of independence of the row and column conditioned on the layer in an I × J × K contingency table, the df is K(I –1)(J– 1). These test statistics, in large sized tables, will have less power since they have large degrees of freedom. This paper proposes a product effect model, which combines the advantages of the multiplicative models over the additive, for analysing the interaction between the row and column of the 3-way table conditioned on the layer. The derived statistics is shown to be asymptotically chi-square with a small degree of freedom, K – 1, for the I × J × K contingency table. The performance of the developed statistic is compared with the Pearson’s chi-square statistic and the likelihood ratio statistic test using an illustrative example. The results show that the product effect test can detect interaction even when some of the main effects are not significant and can perform better than the other competitors having smaller degree of freedom in large sized tables.
Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables with sufficiently large sample size, have an asymptotic chi-square distribution with degrees of freedom (df) equal to the number of free parameters in the saturated model. For example under the hypothesis of independence of the row and column conditioned on the layer in an I × J × K contingency table, the df is K(I –1)(J– 1). These test statistics, in large sized tables, will have less power since they have large degrees of freedom. This paper proposes a product effect model, which combines the advantages of the multiplicative models over the additive, for analysing the interaction between the row and column of the 3-way table conditioned on the layer. The derived statistics is shown to be asymptotically chi-square with a small degree of freedom, K – 1, for the I × J × K contingency table. The performance of the developed statistic is compared with the Pearson’s chi-square statistic and the likelihood ratio statistic test using an illustrative example. The results show that the product effect test can detect interaction even when some of the main effects are not significant and can perform better than the other competitors having smaller degree of freedom in large sized tables.
Cite this paper
Ama, N. (2014) A Multivariate Test for Three-Factor Interaction in 3-Way Contingency Table under the Multiplicative Model. Open Journal of Statistics, 4, 586-596. doi: 10.4236/ojs.2014.48055.
Ama, N. (2014) A Multivariate Test for Three-Factor Interaction in 3-Way Contingency Table under the Multiplicative Model. Open Journal of Statistics, 4, 586-596. doi: 10.4236/ojs.2014.48055.
References
[1] Cheng, P.E., Liou, M. and Aston, J.A.D. (2009) Likelihood Ratio Tests with Three-Way Tables. Institute of Statistical Science, Academia Sinica 2CRiSM, Department of Statistics, The University of Warwick, UK.
http://www3.stat.sinica.edu.tw/library/c_tec_rep/2007-3-20090206.pdf [2] Agresti, A. (2002) Categorical Data Analysis. 2nd Edition, John Wiley and Sons Inc., Hoboken, 132.
http://dx.doi.org/10.1002/0471249688 [3] Christensen, R. (1997) Loglinear Models and Logistic Regression. 2nd Edition, Springer-Verlag, New York Inc., New York. [4] Grizzle, J.E., Starmer, F. and Koch, G.G. (1969) Analysis of Categorical Data by Linear Models. Biometrics, 25, 489-504.
http://dx.doi.org/10.2307/2528901 [5] Darroch, J.N. (1962) Interactions in Multi-Factor Contingency Tables. Journal of the Royal Statistical Society, B24, 251-263. [6] Johnson, D.E. and Graybill, F.A. (1972) An Analysis of a Two-Way Model with Interaction and No Replication. Journal of the American Statistical Association, 67, 862-868.
http://dx.doi.org/10.1080/01621459.1972.10481307 [7] Tukey, T.W. (1949) One Degree of Freedom for Non-Additivity. Biometrics, 5, 232-242.
http://dx.doi.org/10.2307/3001938 [8] Cheng, P.E., Liou, M. and Aston, J.A.D. (2010) Likelihood Ratio Tests with Three-Way Tables. Journal of the American Statistical Association, 105, 740-749.
http://dx.doi.org/10.1198/jasa.2010.tm09061 [9] Ama, N.O. (1991) On Multiplicative Models for Interaction in Contingency Tables. Unpublished Ph.D. Thesis, University of Nigeria, Nsukka. [10] Hélie, S. (2007) Understanding Statistical Power Using Noncentral Probability Distributions: Chi-Squared, G-Squared, and ANOVA. Tutorials in Quantitative Methods for Psychology, 3, 63-69. [11] Lancaster, H.O. (1969) The Chi-Squared Distribution. Wiley & Sons, Inc., New York. [12] Darroch, J.N. (1974) Multiplicative and Additive Interaction in Contingency Tables. Biometrika, 61, 207-214.
http://dx.doi.org/10.1093/biomet/61.1.207 [13] Graybill, F.A. (1961) An Introduction to Linear Statistical Models. McGraw-Hill, New York. [14] Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate Analysis. Academic Press, London. [15] Kshirsagar, A.M. (1972) Multivariate Analysis. Marcel Dekker, Inc., New York. [16] Bartlett, M.S. (1938) Further Aspects of the Theory of Multiple Regression. Mathematical Proceedings of the Cambridge Philosophical Society, 34, 33-40.
http://dx.doi.org/10.1017/S0305004100019897 [17] Onukogu, I.B. (1985) An Analysis of Variance of Nominal Data. Statistica, anno, XLIV, No. 1, 87-96.
[1] Cheng, P.E., Liou, M. and Aston, J.A.D. (2009) Likelihood Ratio Tests with Three-Way Tables. Institute of Statistical Science, Academia Sinica 2CRiSM, Department of Statistics, The University of Warwick, UK.
http://www3.stat.sinica.edu.tw/library/c_tec_rep/2007-3-20090206.pdf [2] Agresti, A. (2002) Categorical Data Analysis. 2nd Edition, John Wiley and Sons Inc., Hoboken, 132.
http://dx.doi.org/10.1002/0471249688 [3] Christensen, R. (1997) Loglinear Models and Logistic Regression. 2nd Edition, Springer-Verlag, New York Inc., New York. [4] Grizzle, J.E., Starmer, F. and Koch, G.G. (1969) Analysis of Categorical Data by Linear Models. Biometrics, 25, 489-504.
http://dx.doi.org/10.2307/2528901 [5] Darroch, J.N. (1962) Interactions in Multi-Factor Contingency Tables. Journal of the Royal Statistical Society, B24, 251-263. [6] Johnson, D.E. and Graybill, F.A. (1972) An Analysis of a Two-Way Model with Interaction and No Replication. Journal of the American Statistical Association, 67, 862-868.
http://dx.doi.org/10.1080/01621459.1972.10481307 [7] Tukey, T.W. (1949) One Degree of Freedom for Non-Additivity. Biometrics, 5, 232-242.
http://dx.doi.org/10.2307/3001938 [8] Cheng, P.E., Liou, M. and Aston, J.A.D. (2010) Likelihood Ratio Tests with Three-Way Tables. Journal of the American Statistical Association, 105, 740-749.
http://dx.doi.org/10.1198/jasa.2010.tm09061 [9] Ama, N.O. (1991) On Multiplicative Models for Interaction in Contingency Tables. Unpublished Ph.D. Thesis, University of Nigeria, Nsukka. [10] Hélie, S. (2007) Understanding Statistical Power Using Noncentral Probability Distributions: Chi-Squared, G-Squared, and ANOVA. Tutorials in Quantitative Methods for Psychology, 3, 63-69. [11] Lancaster, H.O. (1969) The Chi-Squared Distribution. Wiley & Sons, Inc., New York. [12] Darroch, J.N. (1974) Multiplicative and Additive Interaction in Contingency Tables. Biometrika, 61, 207-214.
http://dx.doi.org/10.1093/biomet/61.1.207 [13] Graybill, F.A. (1961) An Introduction to Linear Statistical Models. McGraw-Hill, New York. [14] Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate Analysis. Academic Press, London. [15] Kshirsagar, A.M. (1972) Multivariate Analysis. Marcel Dekker, Inc., New York. [16] Bartlett, M.S. (1938) Further Aspects of the Theory of Multiple Regression. Mathematical Proceedings of the Cambridge Philosophical Society, 34, 33-40.
http://dx.doi.org/10.1017/S0305004100019897 [17] Onukogu, I.B. (1985) An Analysis of Variance of Nominal Data. Statistica, anno, XLIV, No. 1, 87-96.