Regression Slope Tests And Tests Of Zero Correlation

Why Slope and Correlation Tests Can Agree

In simple linear regression, a slope describes the direction and size of the mean response’s change per unit increase in the predictor. Correlation describes the direction and strength of a linear association on a unit-free scale. When the regression model and sampling assumptions are appropriate, a test of whether the population slope is zero is equivalent to a test of whether the population correlation is zero. The equivalence is useful because it lets us express the same evidence in regression language or correlation language.

The equivalence is not a license to treat every dataset with a zero sample correlation as having no relationship. It concerns a particular linear parameter, and exact t inference also depends on the sampling model. In particular, a fixed-predictor regression analysis and a test about a population correlation do not automatically have the same interpretation.

For a random predictor with finite, positive variance, the key population relationship is \(\operatorname{Cov}(X,Y)=\beta_1\operatorname{Var}(X)\), provided the simple linear conditional-mean model holds and the errors have conditional mean zero. Thus a zero slope gives zero covariance and hence zero correlation when both variables have positive variance. Conversely, zero correlation gives zero covariance and therefore a zero slope. The bivariate normal assumption supplies the familiar exact t-test result for correlation; a zero-covariance equivalence by itself does not guarantee that result.

The Sample Statistics and Their Equivalence

Write the centered sums of squares and cross-products as

When \(S_{xx}>0\), the least-squares slope is \(b_1=S_{xy}/S_{xx}\). When both \(S_{xx}\) and \(S_{yy}\) are positive, the sample correlation is \(r=S_{xy}/\sqrt{S_{xx}S_{yy}}\). The residual sum of squares is \(SSE=\sum_{i=1}^n(y_i-\hat{y}_i)^2\), and \(MSE=SSE/(n-2)\) estimates the error variance.

The condition \(|r|<1\) is essential: it ensures that the residual sum of squares is positive, so \(MSE>0\) and the displayed statistics are defined. If \(|r|=1\), all observed points lie exactly on a straight line; the usual residual-based standard error is zero, and this t statistic is not defined. The case \(n=2\) is also excluded because there are no residual degrees of freedom.

Why the Statistics Match

The equality follows from the definitions of \(b_1\), \(r\), and the regression residual sum of squares. In simple linear regression with an intercept, the fitted values use the least-squares slope, so the residual sum of squares is the total centered variation in \(y\) minus the variation explained by the fitted line:

Since \(MSE=SSE/(n-2)\), substituting \(b_1=S_{xy}/S_{xx}\) into the slope statistic gives

The first factor in the final expression is exactly the sample correlation, including its sign. Therefore the two statistics have the same sign and magnitude, and—under the conditions for exact t inference—the same degrees of freedom, p-value, and test decision. The algebraic identity uses the sample definitions; the claim that the statistic has an exact t distribution requires the model assumptions stated above.

Worked Example: Checking Both Forms of the Statistic

Using the Equivalence for a Test

For a two-sided test, the hypotheses \(H_0:\beta_1=0\) versus \(H_A:\beta_1\ne0\) correspond to \(H_0:\rho=0\) versus \(H_A:\rho\ne0\) under the random-predictor model described above. A positive-slope alternative corresponds to a positive-correlation alternative, and a negative-slope alternative corresponds to a negative-correlation alternative. Because the statistics are identical, the same t critical value or p-value is used. State the target parameter and the model before interpreting the result; a correlation test is not automatically justified just because a regression line was fitted.

What the Equivalence Does Not Say

A zero correlation does not prove independence, nor does it rule out a nonlinear relationship. In simple linear regression, the slope describes the linear component of the mean response. A curved pattern can have zero slope and zero correlation while still showing a strong dependence between the variables. Also, even where the population slope and correlation are both zero, an exact t test may fail if the normal-error or bivariate-normal assumptions do not hold.

The distinction between fixed and random predictors is another important exam point. With fixed \(x_i\), the exact regression t test concerns the slope conditional on those predictor values. It does not, by itself, test a population correlation, because the fixed design does not specify a population distribution for \(X\). For independently sampled random pairs under a bivariate normal model, the correlation is a population parameter and the two tests coincide.

Conditions and Exam Traps

• Check the degrees of freedom. The regression t statistic uses \(n-2\), so exact inference requires \(n>2\).
• Check that the statistic exists. Require \(S_{xx}>0\), \(S_{yy}>0\), and \(|r|<1\) for the displayed identity in its usual finite form. Perfect sample correlation makes the residual-based statistic undefined.
• Distinguish algebra from distribution. The two expressions are algebraically equal when defined. An exact t reference distribution requires the stated normal-error assumptions for fixed-design regression or the bivariate normal model for random pairs.
• Do not overstate a null result. Failure to reject zero slope or zero correlation is not proof of no association, and a zero correlation cannot exclude curvature.
• Keep the parameter clear. A fixed-predictor model supports inference about a conditional regression slope; a population correlation requires a random-pair population model.