The adjusted R2 can be negative, and its value will always be less than or equal to that of R2. Unlike R2, the adjusted R2 increases only when the increase in R2 (due to the inclusion of a new explanatory variable) is more than one would expect to see by chance. If a set of explanatory variables with a predetermined hierarchy of importance are introduced into a regression one at a time, with the adjusted R2 computed each time, the level at which adjusted R2 reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms. Adjusted R2 can be interpreted as a less biased estimator of the population R2, whereas the observed sample R2 is a positively biased estimate of the population value. Adjusted R2 is more appropriate when evaluating model fit (the variance in the dependent variable accounted for by the independent variables) and in comparing alternative models in the feature selection stage of model building. The principle behind the adjusted R2 statistic can be seen by rewriting the ordinary R2 as where and are the sample variances of the estimated residuals and the dependent variable respectively, which can be seen as biased estimates of the population variances of the errors and of the dependent variable. These estimates are replaced by statistically unbiased versions: and . Despite using unbiased estimators for the population variances of the error and the dependent variable, adjusted R2 is not an unbiased estimator of the population R2, which results by using the population variances of the errors and the dependent variable instead of estimating them. Ingram Olkin and John W. Pratt derived the Minimum-variance unbiased estimator for the population R2, which is known as Olkin-Pratt estimator. Comparisons of different approaches for adjusting R2 concluded that in most situations either an approximate version of the Olkin-Pratt estimator or the exact Olkin-Pratt estimator should be preferred over (Ezekiel) adjusted R2. Coefficient of partial determination The coefficient of partial determination can be defined as the proportion of variation that cannot be explained in a reduced model, but can be explained by the predictors specified in a full(er) model. This coefficient is used to provide insight into whether or not one or more additional predictors may be useful in a more fully specified regression model. The calculation for the partial R2 is relatively straightforward after estimating