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It is a first-blush indicator of a good model. Unlike R 2, the adjusted R 2 does not necessarily increase, if a predictor variable is added to a model. Therefore, the adjusted R 2 allows for an ‘apples-to-apples’ comparison between models with different numbers of variables and different sample sizes. Specifically, the adjusted R 2 adjusts the R 2 for the sample size and the number of variables in the regression model. The explanation of this statistic is the same as R 2, but it penalises the statistic when unnecessary variables are included in the model. Accordingly, an adjustment of R 2 was developed, appropriately called adjusted R 2. Modellers unwittingly may think that a ‘better’ model is being built, as s/he has a tendency to include more (unnecessary) predictor variables in the model. It can increase as the number of predictor variables in the model increases it does not decrease.
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It is the correlation coefficient between the observed and modelled (predicted) data values. The value of r 2, called the coefficient of determination, and denoted R 2 is typically interpreted as ‘the percent of variation in one variable explained by the other variable,’ or ‘the percent of variation shared between the two variables.’ Good things to know about R 2: Values between 0.7 and 1.0 (−0.7 and −1.0) indicate a strong positive (negative) linear relationship through a firm linear rule. Values between 0.3 and 0.7 (0.3 and −0.7) indicate a moderate positive (negative) linear relationship through a fuzzy-firm linear rule. Values between 0 and 0.3 (0 and −0.3) indicate a weak positive (negative) linear relationship through a shaky linear rule. −1 indicates a perfect negative linear relationship – as one variable increases in its values, the other variable decreases in its values through an exact linear rule. +1 indicates a perfect positive linear relationship – as one variable increases in its values, the other variable also increases in its values through an exact linear rule. The following points are the accepted guidelines for interpreting the correlation coefficient: The correlation coefficient can – by definition, that is, theoretically – assume any value in the interval between +1 and −1, including the end values +1 or −1. The well-known correlation coefficient is often misused, because its linearity assumption is not tested. The correlation coefficient, denoted by r, is a measure of the strength of the straight-line or linear relationship between two variables. The purpose of this article is (1) to introduce the effects the distributions of the two individual variables have on the correlation coefficient interval and (2) to provide a procedure for calculating an adjusted correlation coefficient, whose realised correlation coefficient interval is often shorter than the original one. Among the weaknesses, I have never seen the issue that the correlation coefficient interval is restricted by the individual distributions of the two variables being correlated.
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As a 15-year practiced consulting statistician, who also teaches statisticians continuing and professional studies for the Database Marketing/Data Mining Industry, I see too often that the weaknesses and warnings are not heeded. The correlation coefficient's weaknesses and warnings of misuse are well documented. It is one of the most used statistics today, second to the mean. Accordingly, this statistic is over a century old, and is still going strong. The ‘correlation coefficient’ was coined by Karl Pearson in 1896.