Testing a Zero Multiple Correlation

Why Test Multiple Correlation?

A response may be associated with a collection of predictors even when no single predictor has a strong marginal correlation with it. The multiple correlation summarizes the strength of the response’s linear association with the predictors taken together. The classical test asks whether that population association is zero, or equivalently whether the predictors jointly explain any linear variation in the response.

In a Gaussian linear model, this question has an exact finite-sample test. The test is the overall regression $F$ test: its numerator degrees of freedom count the predictors, and its denominator degrees of freedom count the residual variation left after fitting the intercept and slopes.

Population Multiple Correlation and the Regression Null

Definition: Let $X=(X_1,\ldots,X_p)^\top$ be a predictor vector with positive-definite covariance matrix $\Sigma_{XX}$, and let $Y$ have positive variance. The squared population multiple correlation is $$ \rho^2_{Y\cdot X} = \frac{\operatorname{Cov}(X,Y)^\top \Sigma_{XX}^{-1}\operatorname{Cov}(X,Y)} {\operatorname{Var}(Y)}. $$ It is the fraction of the variance of $Y$ accounted for by its best linear prediction from the predictors. The multiple correlation $\rho_{Y\cdot X}$ is the nonnegative square root of this quantity.

To connect this definition to a regression test, suppose the population model is $Y=\alpha+\beta^\top X+\varepsilon$, where $E[\varepsilon]=0$, $\operatorname{Cov}(X,\varepsilon)=0$, and $\operatorname{Var}(\varepsilon)=\sigma^2>0$. Then $\operatorname{Cov}(X,Y)=\Sigma_{XX}\beta$ and $\operatorname{Var}(Y)=\beta^\top\Sigma_{XX}\beta+\sigma^2$. Substitution gives $$ \rho^2_{Y\cdot X} = \frac{\beta^\top\Sigma_{XX}\beta} {\beta^\top\Sigma_{XX}\beta+\sigma^2}. $$

Result: Under this model, with $\Sigma_{XX}$ positive definite and $\sigma^2>0$, $\rho_{Y\cdot X}=0$ if and only if $\beta=0$. Indeed, positive definiteness gives $\beta^\top\Sigma_{XX}\beta=0$ exactly when $\beta=0$, and the displayed ratio is zero exactly when its numerator is zero. Thus testing zero population multiple correlation is a joint test of all $p$ slopes, not a test of the intercept.

The Gaussian Linear Model and the F Test

For $n$ observations, write the model as $y=\alpha\mathbf{1}+Z\beta+\varepsilon$, where $Z$ is the $n\times p$ predictor matrix and $\mathbf{1}$ is the vector of ones. For the exact test, assume that the design matrix $[\mathbf{1}\ Z]$ has full column rank $p+1$, so $n>p+1$, and that, conditional on the predictors, $\varepsilon\sim N_n(0,\sigma^2 I_n)$ with $\sigma^2>0$. The predictors may be treated as fixed; with random predictors, the same conditional result applies when these error assumptions hold. In a Gaussian population regression model, this tests $H_0:\beta=0$, equivalently $H_0:\rho_{Y\cdot X}=0$, against the alternative that at least one slope is nonzero.

Fit the intercept-only model and the model with all $p$ predictors. Let $\mathrm{SSE}_0$ be the residual sum of squares for the intercept-only model and $\mathrm{SSE}_1$ the residual sum of squares for the full model. Their difference, $\mathrm{SSR}=\mathrm{SSE}_0-\mathrm{SSE}_1$, is the extra sum of squares explained by adding the predictors. Define $R^2=\mathrm{SSR}/\mathrm{SSE}_0=1-\mathrm{SSE}_1/\mathrm{SSE}_0$, the sample coefficient of determination for the full model with intercept.

Theorem: Under $H_0:\beta=0$ and the stated Gaussian linear-model assumptions, the statistic $$ F = \frac{\mathrm{SSR}/p}{\mathrm{SSE}_1/(n-p-1)} = \frac{R^2/p}{(1-R^2)/(n-p-1)} $$ has an exact $F$ distribution with numerator degrees of freedom $p$ and denominator degrees of freedom $n-p-1$. The test rejects for large values of $F$; its exact $p$-value is $P(F_{p,n-p-1}\geq F_{\mathrm{obs}})$.

Proof. Let $P_0$ and $P_1$ be the orthogonal projection matrices onto the intercept-only and full-model column spaces, respectively. The full-model space contains the intercept-only space. Therefore $P_1-P_0$ is an orthogonal projection of rank $p$, and $I_n-P_1$ is an orthogonal projection of rank $n-p-1$. The sum of squares for adding predictors is $y^\top(P_1-P_0)y=\mathrm{SSR}$, while the full-model residual sum of squares is $y^\top(I_n-P_1)y=\mathrm{SSE}_1$.

Under the null, $y=\alpha\mathbf{1}+\varepsilon$. Both projections annihilate $\mathbf{1}$, so these two quadratic forms are the corresponding forms in $\varepsilon$. Orthogonal projection of the Gaussian error vector onto orthogonal subspaces gives independent Gaussian components. Consequently, $\mathrm{SSR}/\sigma^2\sim\chi^2_p$ and $\mathrm{SSE}_1/\sigma^2\sim\chi^2_{n-p-1}$, independently. Dividing each by its degrees of freedom and taking their ratio gives $F_{p,n-p-1}$. Finally, $\mathrm{SSE}_0=\mathrm{SSR}+\mathrm{SSE}_1$, so \[ \frac{\mathrm{SSR}/p}{\mathrm{SSE}_1/(n-p-1)} = \frac{R^2/p}{(1-R^2)/(n-p-1)}. \] This proves the stated distribution and formula. $\square$

The numerator degrees of freedom are $p$, because the model adds $p$ slopes. The denominator degrees of freedom are $n-p-1$: the full model estimates $p$ slopes and one intercept. The test is an upper-tail test because a larger $F$ means the added predictors explain more variation relative to the remaining residual variation.

Worked Calculations

Worked Example: Testing From Sums of Squares

A study has $n=18$ observations and $p=2$ predictors. The intercept-only residual sum of squares is $180$, and the full-model residual sum of squares is $108$. Test zero multiple correlation at the 5% level.

The predictors’ extra sum of squares is $\mathrm{SSR}=180-108=72$, and $R^2=72/180=0.40$. The degrees of freedom are $p=2$ and $n-p-1=18-2-1=15$. Thus $$ F=\frac{72/2}{108/15} =\frac{36}{7.2} =5.00. $$

Under the null this is compared with $F_{2,15}$. Its upper-tail probability is approximately $0.0217$, below $0.05$, so reject the null. There is evidence that the predictors jointly have a nonzero population linear association with the response. This conclusion concerns the set of predictors; it does not say that both slopes are individually nonzero.

Worked Example: Computing the Test From R-Squared

Suppose $n=24$, $p=2$, and the fitted model has $R^2=0.40$. The null degrees of freedom are $2$ and $24-2-1=21$. Compute $$ F=\frac{0.40/2}{(1-0.40)/21} =\frac{0.20}{0.60/21} =\frac{0.20}{0.028571\ldots} =7.00. $$

The exact null reference distribution is $F_{2,21}$. For an $F$ variable with 2 numerator degrees of freedom, the upper-tail probability at $f$ is $\left(\frac{21}{21+2f}\right)^{21/2}$. At $f=7$, this is $(21/35)^{10.5}=0.6^{10.5}\approx0.0047$. The result is significant at the 5% level, providing evidence against zero population multiple correlation.

Worked Example: A Larger R-Squared Need Not Mean a Significant Test

A smaller study has $n=14$, $p=5$, and $R^2=0.30$. The numerator degrees of freedom are $5$, while the denominator degrees of freedom are $14-5-1=8$. The statistic is $$ F=\frac{0.30/5}{0.70/8} =\frac{0.06}{0.0875} \approx0.686. $$

This value is not large for an $F_{5,8}$ distribution, so it does not reject the zero-correlation null at the 5% level. Although the fitted model accounts for 30% of the observed variation, the test must also account for the number of predictors and the limited residual degrees of freedom. A substantial sample $R^2$ alone does not determine statistical significance.

Assumptions, Scope, and a Common Pitfall

The exact $F$ reference distribution depends on the stated linear-model assumptions, especially Gaussian errors with a common variance and a full-rank design. If the design is rank deficient, the nominal predictor count does not equal the number of independently estimable slopes, and the formula’s degrees of freedom cannot be used as written. When the errors are not Gaussian, the same statistic may be useful in other settings, but the exact finite-sample null distribution established here is not guaranteed.

Common mistake: treating any positive sample multiple correlation as evidence against the null. Sample $R^2$ is nonnegative and will generally be positive even when the population slopes are all zero. The test assesses whether the observed $R^2$ is large relative to its null distribution, using both the predictor count and the residual degrees of freedom. Another frequent error is using $n-p$ for the denominator degrees of freedom despite fitting an intercept; the correct value is $n-p-1$.

Key takeaway: For a full-rank Gaussian linear model with an intercept and $p$ predictors, test zero population multiple correlation using the overall regression statistic $F=(R^2/p)/((1-R^2)/(n-p-1))$. Under the null its exact distribution is $F_{p,n-p-1}$.

Check Your Understanding

Use the definitions and test derivation to answer these questions.

In a model with an intercept and four predictors based on 30 observations, what are the numerator and denominator degrees of freedom?
Why does the zero-multiple-correlation test concern all slopes jointly rather than the intercept?
If $R^2=0.25$, $n=20$, and $p=3$, calculate the observed $F$ statistic and its degrees of freedom.
What Gaussian-model conditions make the stated finite-sample $F$ distribution exact?
Explain why a positive sample $R^2$ does not, by itself, establish a nonzero population multiple correlation.