Abelson's paradox

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Abelson's paradox is an applied statistics paradox identified by Robert P. Abelson. [1] [2] [3] The paradox pertains to a possible paradoxical relationship between the magnitude of the r2 (i.e., coefficient of determination) effect size and its practical meaning.

Paradox statement that apparently contradicts itself and yet might be true

A paradox is a statement that, despite apparently valid reasoning from true premises, leads to an apparently-self-contradictory or logically unacceptable conclusion. A paradox involves contradictory-yet-interrelated elements that exist simultaneously and persist over time.

Coefficient of determination indicator for how well data points fit a line or curve

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

In statistics, an effect size is a quantitative measure of the magnitude of a phenomenon. Examples of effect sizes are the correlation between two variables, the regression coefficient in a regression, the mean difference, or even the risk with which something happens, such as how many people survive after a heart attack for every one person that does not survive. For most types of effect size, a larger absolute value always indicates a stronger effect, with the main exception being if the effect size is an odds ratio. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. They are the first item (magnitude) in the MAGIC criteria for evaluating the strength of a statistical claim. Especially in meta-analysis, where the purpose is to combine multiple effect sizes, the standard error (S.E.) of the effect size is of critical importance. The S.E. of the effect size is used to weigh effect sizes when combining studies, so that large studies are considered more important than small studies in the analysis. The S.E. of the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (N), or the number of observations in each group.

Abelson's example was obtained from the analysis of the r2 of batting average in baseball and skill level. Although batting average is considered among the most significant characteristics necessary for success, the effect size was only a tiny [4] [5] [6] [7] [8] [9] 0.003.

Batting average (baseball)

In baseball, the batting average (BA) is defined by the number of hits divided by at bats. It is usually reported to three decimal places and read without the decimal: A player with a batting average of .300 is "batting three-hundred." If necessary to break ties, batting averages could be taken beyond the .001 measurement. In this context, a .001 is considered a "point," such that a .235 batter is 5 points higher than a .230 batter.

Baseball Sport

Baseball is a bat-and-ball game played between two opposing teams who take turns batting and fielding. The game proceeds when a player on the fielding team, called the pitcher, throws a ball which a player on the batting team tries to hit with a bat. The objectives of the offensive team are to hit the ball into the field of play, and to run the bases—having its runners advance counter-clockwise around four bases to score what are called "runs". The objective of the defensive team is to prevent batters from becoming runners, and to prevent runners' advance around the bases. A run is scored when a runner legally advances around the bases in order and touches home plate. The team that scores the most runs by the end of the game is the winner.

See also

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References

  1. ↑ Abelson, R. P. (1985). "A variance explanation paradox: When a little is a lot." Psychological Bulletin, 97, 129–133. The phrase "Abelson's paradox," stated explicitly in citations below, is derived from the title of Abelson's article.
  2. ↑ Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, New York: Erlbaum, p. 535.
  3. ↑ Sawilowsky, S., Sawilowsky, J., & Grissom, R. J. (2010). Effect size. In M. Lovric, (Ed.), International Encyclopedia of Statistical Science. NY: Springer.
  4. ↑ Ellis, P. D. (2010). The essential guide to effect sizes: Statistical power, meta-analysis, and the interpretation of research results. Cambridge: Cambridge University Press, p. 44.
  5. ↑ Pratkanis, A. R., & Greenwald, A. G. (1989). "A sociocognitive model of attitude structure and function." In Advances in Experimental Social Psychology, Leonard Berkowitz (ed.), Academic Press, 22, 245-285. ISSN 0065-2601, ISBN   978-0-12-015222-3.
  6. ↑ Borenstein, M. (1998). "The shift from significance testing to effect size estimation." In Comprehensive Clinical Psychology, eds. A. S. Bellack and M. Hersen, Pergamon, Oxford, 313-349. ISBN   978-0-08-042707-2
  7. ↑ Marzano, R. J. (2003). What works in schools, Alexandria, Virginia: Association for Supervision and Curriculum Development, p. 190.
  8. ↑ Sawilowsky, S. (2005). "Abelson’s paradox and the Michelson-Morley experiment." Journal of Modern Applied Statistical Methods, 4, 352.
  9. ↑ Roseman, I. J., & Read, S. J. (2007). "Psychologist at play: Abelson's life and contributions to psychological science." Perspectives on Psychological Science, 2(1), p. 91. doi : 10.1111/j.1745-6916.2007.00031.x


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