{"id":11136,"date":"2024-05-04T10:33:38","date_gmt":"2024-05-04T14:33:38","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=11136"},"modified":"2024-05-04T14:47:32","modified_gmt":"2024-05-04T18:47:32","slug":"r-squared-is-bigger-better","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2024\/05\/04\/r-squared-is-bigger-better\/","title":{"rendered":"R-squared. Is Bigger Better?"},"content":{"rendered":"
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The coefficient of determination<\/i>, R-squared or R^2<\/tt>, is a popular statistic that describes how well a regression model fits data. It measures the proportion of variation in data that is predicted by a model. However, that is all<\/i> that R^2<\/tt> measures. It is not<\/i> appropriate for any other use. For example, it does not support extrapolation beyond the domain of the data. It does not suggest that one model is preferable to another.<\/p>\r\n

I recently watched high school students participate in the final round of a national mathematical modeling competition. The teams' presentations were excellent; they were well-prepared, mathematically sophisticated, and informative. Unfortunately, many of the presentations abused R^2<\/tt>. It was used to compare different fits, to justify extrapolation, and to recommend public policy.<\/p>\r\n

This was not the first time that I have seen abuses of R^2<\/tt>. As educators and authors of mathematical software, we must do more to expose its limitations. There are dozens of pages and videos on the web describing R^2<\/tt>, but few of them warn about possible misuse.<\/p>\r\n

\r\nR^2<\/tt> is easily computed. If y<\/tt> is a vector of observations, f<\/tt> is a fit to the data and ybar = mean(y)<\/tt>, then<\/p>\r\n

   R^2 = 1 - norm(y-f)^2\/norm(y-ybar)^2<\/pre>\r\n
If the data are centered, then ybar = 0<\/tt> and R^2<\/tt> is between zero and one.<\/p>\r\n\r\n
One of my favorite examples is the United States Census. Here is the population, in millions, every ten years since 1900.<\/p>\r\n
   t         p\r\n  ____    _______\r\n  1900     75.995\r\n  1910     91.972\r\n  1920    105.711\r\n  1930    123.203\r\n  1940    131.669\r\n  1950    150.697\r\n  1960    179.323\r\n  1970    203.212\r\n  1980    226.505\r\n  1990    249.633\r\n  2000    281.422\r\n  2010    308.746\r\n  2020    331.449<\/pre>\r\n
There are 13 observations. So, we can do a least-squares fit by a polynomial of any degree less than 12 and can interpolate by a polynomial of degree 12. Here are four such fits and the corresponding R^2<\/tt> values. As the degree increases, so does R^2<\/tt>. Interpolation fits the data exactly and earns a perfect core.<\/p>\r\n
Which fit would you choose to predict the population in 2030, or even to estimate the population between census years?<\/p>\r\n
R2_census\r\n<\/pre>\r\n\"\" 
Thanks to Peter Perkins and Tom Lane for help with this post.<\/p>\r\n