When is R squared negative?

Question

My understanding is that R squared cannot be negative as it is the square of R. However I ran a simple linear regression in SPSS with a single independent variable and a dependent variable. My SPSS output give me a negative value for R-squared. If I was to calculate this by hand from R then R squared would be positive. What has SPSS done to calculate this as negative?

R=-.395
R squared =-.156
B (un-standardized)=-1261.611

Code I've used:

DATASET ACTIVATE DataSet1. 
REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA 
           /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN 
           /DEPENDENT valueP /METHOD=ENTER ageP

I get a negative value. Can anyone explain what this means?

Thanks.

Answer 1

R2 compares the fit of the chosen model with that of a horizontal straight line (the null hypothesis). If the chosen model fits worse than a horizontal line, then R2 is negative. Note that R2 is not always the square of anything, so it can have a negative value without violating any rules of math. R2 is negative only when the chosen model does not follow the trend of the data, so fits worse than a horizontal line.

Here is an example. We fit data to a linear regression model constrained so that the Y intercept must equal 1500.

The model makes no sense at all given these data. It is clearly the wrong model,perhaps chosen by accident.

The fit of the model (a straight line constrained to go through the point 0,1500) is worse than the fit of a horizontal line. Thus the sum-of-squares from the model (SSreg) is larger than the sum-of-squares from the horizontal line (SStot). R2 is computed as 1 - SSreg/SStot. When SSreg is greater than SStot, that equation computes a negative value for R2.

With linear regression with no constraints, R2 must be positive (or zero) and equals the square of the correlation coefficient, r. A negative R2 is only possible with linear regression when either the intercept or the slope are constrained so that the "best-fit" line (given the constraint) fits worse than a horizontal line. With nonlinear regression, the R2 can be negative whenever the best-fit model (given the chosen equation, and its constraints, if any) fits the data worse than a horizontal line.

Bottom line: A negative R2 is not a mathematical impossibility or the sign of a computer bug. It simply means that the chosen model (with its constraints) fits the data really poorly.

Answer 2

Have you forgotten to include an intercept in your regression? I'm not familiar with SPSS code, but on page 21 of Hayashi's Econometrics:

If the regressors do not include a constant but (as some regression software packages do) you nevertheless calculate $R^2$ by the formula

$R^2=1-\frac{\sum_{i=1}^{n}e_i^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}$

then the $R^2$ can be negative. This is because, without the benefit of an intercept, the regression could do worse than the sample mean in terms of tracking the dependent variable (i.e., the numerator could be greater than the denominator).

I'd check and make sure that SPSS is including an intercept in your regression.

Answer 3

This can happen if you have a time series that is N.i.i.d. and you construct an inappropriate ARIMA model of the form(0,1,0) which is a first difference random walk model with no drift then the variance (sum of squares - SSE ) of the residuals will be larger than the variance (sum of squares SSO) of the original series. Thus the equation 1-SSE/SSO will yield a negative number as SSE execeedS SSO . We have seen this when users simply fit an assumed model or use inadequate procedures to identify/form an appropriate ARIMA structure. The larger message IS that a model can distort (much like a pair of bad glasses ) your vision. Without having access to your data I would otherwise have a problem in explaining your faulty results. Have you brought this to the attention of IBM ?

The idea of an assumed model being counter-productive has been echoed by Harvey Motulsky. Great post Harvey !