Solved – Is $R^2$ value valid for insignificant OLS regression model

partitioningr-squaredregression

I am interested in stating that ___ % of the variance in Y is explained uniquely by $X_1$ and ___ % is explained uniquely by $X_2$.

  • Is there some way to obtain this from a multiple regression model, or do I need to obtain adjusted $R^2$ values from a series of residual regressions (sensu Legendre & Legendre et al.)?
  • That being asked, is $R^2$ value valid for insignificant OLS regression model?
  • Does the value of $R^2$ depend on a statistical test?

Best Answer

Yes, you're trying to calculate the Extra Sum of Squares. In short you are partitioning the regression sum of squares. Assume we have two $X$ variables, $X_1$ and $X_2$. The $SSTO$ (total sum of squares, made up of the SSR and SSE) is the same regardless of how many $X$ variables we have. Denote the $SSR$ and $SSE$ to indicate which $X$ variables are in the model: e.g.

$SSR(X_1,X_2) = 385$ and $SSE(X_1,X_2) = 110$

Now let's assume we did the regression just on $X_1$ e.g.

$SSR(X_1) = 352$ and $SSE(X_1) = 143$.

The (marginal) increase in the regression sum of squares in $X_2$ given that $X_1$ is already in the model is:

\begin{eqnarray} SSR(X_2|X_1)& = &SSR(X_1,X_2) - SSR(X_1)\\ & = & 385 - 352\\ & = & 33 \end{eqnarray}

or equivalently, the extra reduction in the error sum of squares associated with $X_2$ given that $X_1$ is already in the model is:

\begin{eqnarray} SSR(X_2|X_1) & = & SSE(X_1) - SSE(X_2,X_1)\\ &=& 143 - 110\\ &=& 33 \end{eqnarray}

In the same way we can find:

\begin{eqnarray} SSR(X_1|X_2) &=& SSE(X_2) - SSE(X_1,X_2)\\ &=& SSR(X_1,X_2) - SSR(X_2) \end{eqnarray}

Of course, this also works for more $X$ variables as well.

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