Abstract
Comparison of population means is essential in quantitative research. For comparing means of three or more groups, analysis of variance (ANOVA) is the most frequently used statistical approach. Typically, ANOVA is used for continuous data, but discrete data are also common in practice. To compare means of binary or count data, the classical ANOVA and the corresponding power analysis are problematic, because the assumption of normality is violated. To address the issue, this study introduces an analogous ANOVA approach for binary or count data, as well as the corresponding methods for statistical power analysis. We first introduce an analogous ANOVA table and a likelihood ratio test statistic for comparing means with binary or count data. With the test statistic, we then define an effect size and propose a method to calculate statistical power. Finally, we develop and show software to conduct the proposed power analysis for both binary and count data.
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References
A. Agresti, A survey of exact inference for contingency tables. Stat. Sci. 7 (1), 131–153 (1992)
T. Baguley, Standardized or simple effect size: What should be reported? Br. J. Psychol. 100 (3), 603–617 (2009)
R. Bahadur, An optimal property of the likelihood ratio statistic, in Proceedings of Fifth Berkeley Symp. Math. Statist. Probab, vol. 1 (1967), pp. 13–26
M. Birch, Maximum likelihood in three-way contingency tables. J. R. Stat. Soc. Ser. B Methodol. 25 (1), 220–233 (1963)
Y.M. Bishop, S.E. Fienberg, P.W. Holland, Discrete Multivariate Analysis: Theory and Practice (MIT Press, Cambridge, 1975)
A.C. Cameron, F.A. Windmeijer, An r-squared measure of goodness of fit for some common nonlinear regression models. J. Econ. 77 (2), 329–342 (1997)
J. Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd edn. (Lawrence Erlbaum Associates, Hillsdale, 1988)
D. Collett, Modelling Binary Data (Chapman & Hall, New York, 1991)
D.R. Cox, E.J. Snell, Analysis of Binary Data, 2nd edn. (Chapman & Hall, New York, 1989)
B. Efron, Regression and anova with zero-one data: measures of residual variation. J. Am. Stat. Assoc. 73 (361), 113–121 (1978)
P.D. Ellis, The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results (Cambridge University Press, New York, 2010)
R.A. Fisher, On the “probable error” of a coefficient of correlation deduced from a small sample. Metron 1, 3–32 (1921)
R.A. Fisher, On the interpretation of χ 2 from contingency tables, and the calculation of p. J. R. Stat. Soc. 85 (1), 87–94 (1922)
J.L. Fleiss, Measures of effect size for categorical data, in The Handbook of Research Synthesis, ed. by H. Cooper, L.V. Hedges (Russell Sage Foundation, New York, 1994), pp. 245–260
M.F. Freeman, J.W. Tukey, Transformations related to the angular and the square root. Ann. Math. Stat. 21 (4), 607–611 (1950)
L.A. Goodman, The analysis of multidimensional contingency tables: stepwise procedures and direct estimation methods for building models for multiple classifications. Technometrics 13 (1), 33–61 (1971)
L.A. Goodman, W.H. Kruskal, Measures of association for cross classifications. J. Am. Stat. Assoc. 49 (285), 732–764 (1954)
D.M. Grove, Positive association in a two-way contingency table: likelihood ratio tests. Commun. Stat. Theory Methods 13 (8), 931–945 (1984)
W. Hoeffding, Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36 (2), 369–401 (1965)
D.C. Howell, Statistical Methods for Psychology (Duxbury/Thomson Learning, Pacific Grove, 2012)
K. Larntz, Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics. J. Am. Stat. Assoc. 73 (362), 253–263 (1978)
S.E. Maxwell, H.D. Delaney, Designing Experiments and Analyzing Data: A Model Comparison Perspective, 2nd edn. (Lawrence Erlbaum Associates, Mahwah, 2004)
C.R. Mehta, N.R. Patel, A.A. Tsiatis, Exact significance testing to establish treatment equivalence with ordered categorical data. Biometrics 40 (3), 819–825 (1984)
D.S. Moore, W. Notz, M.A. Fligner, Essential Statistics (W.H. Freeman and Company, New York, 2013)
E.S. Pearson, The choice of statistical tests illustrated on the interpretation of data classed in a 2× 2 table. Biometrika 34 (1/2), 139–167 (1947)
R. Rosenthal, Parametric measures of effect size, in The Handbook of Research Synthesis, ed. by H. Cooper, L.V. Hedges (Russell Sage Foundation, New York, 1994), pp. 231–244
J.S. Strickland, Predictive Analytics Using R (2014), Lulu.com
R.J. Turek, R.C. Suich, An exact test on the goodman-kruskal λ for prediction on a dichotomy: power considerations. Comput. Stat. Data Anal. 8 (2), 171–178 (1989)
S.S. Wilks, The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9 (1), 60–62 (1938)
D. Williams, Improved likelihood ratio tests for complete contingency tables. Biometrika 63 (1), 33–37 (1976)
D. Yates, D. Moore, G. McCabe, The Practice of Statistics (W.H. Freeman and Company, New York, 1999)
Acknowledgements
This research is supported by a grant from the Department of Education (R305D140037) awarded to Zhiyong Zhang and Ke-Hai Yuan. However, the contents of the paper do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.
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Mai, Y., Zhang, Z. (2017). Statistical Power Analysis for Comparing Means with Binary or Count Data Based on Analogous ANOVA. In: van der Ark, L.A., Wiberg, M., Culpepper, S.A., Douglas, J.A., Wang, WC. (eds) Quantitative Psychology. IMPS 2016. Springer Proceedings in Mathematics & Statistics, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-56294-0_33
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DOI: https://doi.org/10.1007/978-3-319-56294-0_33
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