The coefficient of determination |
James Dean Brown University of Hawai'i at Manoa |
Table 1. Sample data for tests A, B, and C | |||
Test A | Test B | Test C | |
9 | 8 | 1 | |
8 | 7 | 2 | |
7 | 6 | 3 | |
6 | 5 | 4 | |
5 | 4 | 5 | |
4 | 3 | 6 | |
3 | 2 | 7 | |
2 | 1 | 8 | |
1 | 0 | 9 | |
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Table 2. Sample correlation coefficients and corresponding coefficients of determination | ||
Correlation Coefficient (r_{xy}) | Coefficient of Determination (r_{xy}^{2}) | |
1.00 | 1.00 | |
0.90 | 0.81 | |
0.80 | 0.64 | |
0.70 | 0.49 | |
0.60 | 0.36 | |
0.50 | 0.25 | |
0.40 | 0.16 | |
0.30 | 0.09 | |
0.20 | 0.04 | |
0.10 | 0.01 | |
.90 squared equals .81 (i.e., 81%) or about four-fifths overlap,Notice how much more rapidly the coefficients of determination drop as you scan down Table 2 than do the correlation coefficients. This should help you recognize that correlation coefficients can be misleading. For example, I have seen a correlation coefficient of .60 called a "moderate" correlation. After all, .60 on a scale from .00 to 1.00 looks like it represents about three-fifths overlap. But when you square that value to find the coefficient of determination, you quickly realize that the proportion of relationship is .36, or 36%, which is only about one-third overlap. How can that be said to represent a moderate relationship? Even a "moderate" correlation of .70 is only .49 when squared, and thus represents less than one-half overlap between whatever two sets of numbers are involved. The bottom line is that the coefficient of determination transforms a correlation coefficient into a statistic that you can more readily interpret and compare to other coefficients.
.80 squared equals .64 (i.e., 64%) or about two-thirds overlap,
.70 squared equals .49 (i.e., 49%) or about one-half overlap,
.60 squared equals .36 (i.e., 36%) or about one-third overlap,
.50 squared equals .25 (i.e., 25%) or about one-quarter overlap,
.40 squared equals .16 (i.e., 16%) or about one-fifth overlap,
.30 squared equals .09 (i.e., 9%) or about one-tenth overlap,
.20 squared equals .04 (i.e., 4%) or about one-twenty-fifth overlap (almost nothing)
.10 squared equals .01, or less than one-hundredth overlap (definitely nothing)
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References
Brown, J. D. (1996). Testing in language programs. Upper Saddle River, NJ: Prentice Hall.
Brown, J. D. (translated into Japanese by M. Wada). (1999). Gengo tesuto no kisochishiki (Literally: Basic knowledge of language testing). Tokyo: Taishukan Shoten. Brown, J. D., Yamashiro, A. D., & Ogane, E. (2001). The emperor's new cloze: Strategies for revising cloze tests. In T. Hudson & J. D. Brown (Eds.), A focus on language test development: Expanding the language proficiency construct across a variety of tests. Honolulu, HI: University of Hawai'i Press. |
Where to Submit Questions: |
Please submit questions for this column to the following address: |
JD Brown Department of Second Language Studies University of Hawai'i at Manoa 1890 East-West Road Honolulu, HI 96822 USA |
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