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Why must the R-squared value of a regression be less than 1? What does R-Squared value more than '1' indicate? Can a Regression Model with a Small R-squared Be Useful?

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In a general linear model, R-squared is $$R^2 = \frac{SS_{reg}}{SS_{tot}} $$ (Note: the above expression only holds if intercepts are included in the model -- see Whuber's comment below)

$SS_{reg}$ is the sum of squared differences between the regression and observations $$SS_{reg} = \sum_{i=0}^n(\bar{y}_i-\hat{y}_i)^2$$ and $SS_{tot}$ is the sum of squared differences between the mean and observations $$SS_{tot} = \sum_{i=0}^n(y_i-\bar{y}_i)^2$$

Why must the R-squared value of a regression be less than 1?

Under OLS regression, $0<SS_{reg}<SS_{tot}$, so $0<R^2<1$.

What does R-Squared value more than '1' indicate?

The important thing to note here is that $R^2$ is the ratio of explained variance ($SS_{reg}$) to total variance ($SS_{tot}$). There's not always a perfect analogue to this in other models, and other quantities are often reported as $R^2$ (see Pseudo-R2s in logistic regression), however, these values are not always guaranteed to be between 0 and 1.

Can a Regression Model with a Small R-squared Be Useful?

This depends what you want to use it for. If it is for prediction purposes, it is unlikely to be helpful, however, it is important to note that a large $R^2$ does not guarantee good prediction either. It is simply a measure of how well the model explains the variability in the observed data.

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    $\begingroup$ It is not true that $R^2$ must be positive: that result requires an intercept term be included. $\endgroup$
    – whuber
    Jun 1 '20 at 13:13
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    $\begingroup$ @whuber this is a fair point -- I've made a note in my answer addressing this. Seeing as the question only asks why $R^2 < 1$, and intercepts are generally included by default in OLS regression, I'm hesitant to go into more detail in the answer. I would have to start with $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$ and waffle on from there noting that $SS_{res}$ can be greater than $SS_{tot}$ .... $\endgroup$
    – Forrest
    Jun 1 '20 at 23:43
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R squared is a measure of how far variation in the dependent variable is explained by the independent variable.

Its value ranges from zero to one.

If the independent variable explains none of the variance the slope of the regression line is zero and the R squared is zero

If the slope of the regression line is positive or negative and all observed dependent variable values lie EXACTLY on the line then the model explains one hundred percent of the variance in the independent variable and then the R squared is one.

If the slope of the the line is greater than or less than zero, the independent variable has some explanatory value for the dependent variable. Depending on how closely the model (line) fits the observed data, R squared will have a value between zero and one. If the observed data tightly fits to the model (line), the sum of the squared errors will be small and the variance will be largely explained and the R squared value will be closer to one.

If the observed data is more dispersed around the line, the model explains less variance in the independent variable and the R squared will be closer to zero.

A model with a low R squared is not useful because it indicates the variance in the dependent variable is not well explained by the dependent variable.

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    $\begingroup$ You're assuming one predictor and a straight line fit but the idea of $R^2$ is much more general. $\endgroup$
    – Nick Cox
    Jun 1 '20 at 8:51

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