Analysis of Mean Absolute Deviation for Randomized Block Design under Laplace Distribution
American Journal of Theoretical and Applied Statistics
Volume 4, Issue 3, May 2015, Pages: 138-149
Received: Apr. 2, 2015;
Accepted: Apr. 11, 2015;
Published: Apr. 21, 2015
Views 4182 Downloads 140
Elsayed A. H. Elamir, Department of Statistics and Mathematics, Benha University, Benha, Egypt & Management & Marketing Department, College of Business, University of Bahrain, Manama, Kingdom of Bahrain
Analysis of mean absolute deviation (ANOMAD) for randomized block design is derived where the total sum of absolute deviation (TSA) is partition into exact block sum of absolute deviation (BLSA), exact treatment sum of absolute deviation (TRSA) and within sum of absolute deviation (WSA). The exact partitions are derived by getting rid of the absolute function from MAD by using the idea of re-expressing the mean absolute deviation as a weighted average of data with sum of weights zero. ANOMAD has advantages: offers meaningful measure of dispersion, does not square data, and can be extended to other location measures such as median. Two ANOMAD graphs are proposed. However, the variance-gamma distribution is used to fit the sampling distributions for the mean of BLSA and the mean of TRSA. Consequently, two tests of equal means and medians are proposed under the assumption of Laplace distribution.
Elsayed A. H. Elamir,
Analysis of Mean Absolute Deviation for Randomized Block Design under Laplace Distribution, American Journal of Theoretical and Applied Statistics.
Vol. 4, No. 3,
2015, pp. 138-149.
Algina, J., Keselman, H.J., & Penfield, R.D. “An alternative to Cohen’s standardized mean difference effect size: A robust parameter and confidence interval in the two independent group’s case” Psychological Methods,10, 317-328 (2005).
Bakeman, R. “Recommended effect size statistics for repeated measures designs”, Behaiour Research Methods, 37, 379–384 (2005).
Cohen, J. “Statistical power analysis for the behavioral sciences”, (2nd ed.), Hillsdale, NJ: Erlbaum (1988).
Cohen, J. “The earth is round (p< .05)”, American Psychologist, 49, 997-1003 (1994).
Elamir, E.A.H. “On uses of mean absolute deviation: decomposition, skewness and correlation coefficients”, Metron: International Journal of Statistic, LXX, n.2-3, 145-164 (2012).
Gorard S. “Revisiting a 90-year-old debate: the advantages of the mean deviation”, British Journal of Educational Studies, 53, 417-430 (2005)
Gorard, S. “Introducing the mean absolute deviation 'effect' size”, International Journal of Research & Method in Education 38(2): 105-114 (2015).
Gradshteyn, I.S., and Ryzhik, I.M., “Table of Integrals, Series, and Products”, Academic Press (1980).
Granger, C.W.J. and Z. Ding, “Some properties of absolute return”, Annales D’economie et de Statistique, 40, 67–91 (1995).
Haas, M., Mittnik, S. and M.S. Paolella, “Modelling and predicting market risk with Laplace-Gaussian mixture distributions” Applied Financial Economics, 16, 1145–1162 (2006).
Habib, E.A.E “Correlation coefficients based on mean absolute deviation about median” International Journal of Statistics and Systems, 6, 413-428 (2011).
Kotz, S, Kozubowski, T. J., and Podgórski, K. “The Laplace Distribution and Generalizations” Birkhauser, Boston (2001).
Kozubowski,, T. and K. Podgorski, “Asymmetric Laplace laws and modelling financial data” Mathematical and Computer Modelling - special issue, Eds, Mitnik, S., Rachev, S.T.,: Stable non-Gaussian models in finance and econometrics, 34, 1003–1021 (2001).
Lakens, D. “Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs” Frontiers in Psychology, 4, 1-12 (2013).
Linden, M., “A model for stock return distribution” International Journal of Finance and Economics, 6, 159–169 (2001).
Mittnik, S., Paolella, M.S. and S.T. Rachev, “Unconditional and conditional distributional models for the Nikkei Index”, Asia Pacific Financial Markets, 5, 99–128 (1998).
Neter, J., Kutner, H., Nachtsheim, C. and Wasserman, W. “Applied linear statistical models” 4th ed., McGraw-Hill (1996).
Olejnik,S.,and Algina, “Measures of effect size for comparative studies: applications, interpretations, and limitations” Contemporary Educational Psychology, 25, 241–286 (2000).
Pham-Gia, T. and T. L. Hung “The mean and median absolute deviations” Mathematical and Computer Modeling 34, 921–936 (2001).
Puig, P. and Stephens, M. A. “Tests of fit for the Laplace distribution, with applications” Technometrics 42, 417-424 (2000).
Sabarinath, A. and A.K. Anilkumar. “Modeling of sunspot numbers by a modified binary mixture of Laplace distribution functions”, Solar Physics, 250, 183–197 (2008).
Seneta, E. “Fitting the variance-gamma model to financial data”, Journal of Applied Probability. 41A:177-187 (2004).
Srivastava, H.M., Nadarajah, S. and S. Kotz,. “Some generalizations of the Laplace distribution”, Applied Mathematics and Computation, 182, 223–231(2006).