Why The Slope t Test Equals The Overall F Test

Why Two Regression Tests Give the Same Result

In simple linear regression, two familiar procedures can test whether the predictor is useful: a t test of the population slope and an overall F test of regression significance. Their statistics may look different, but when the model includes an intercept and there is just one predictor, the overall F statistic is exactly the square of the slope t statistic. This is more than a similarity in conclusions: it is an algebraic identity for the same fitted regression.

The identity is useful for checking calculations and understanding why the two-sided tests agree. It also has limits. The t statistic retains the estimated slope’s sign, while squaring it removes the direction. Thus, the direct equivalence is to the two-sided slope test, not to a one-sided test.

Model and assumptions: For observations \(i=1,\ldots,n\), use the simple linear regression model \(Y_i=\beta_0+\beta_1x_i+\varepsilon_i\), with an intercept and one predictor. Treat the predictor values \(x_i\) as fixed, require \(S_{xx}=\sum_{i=1}^n(x_i-\bar{x})^2>0\) and \(n>2\), and assume independent errors \(\varepsilon_i\sim N(0,\sigma^2)\) with common variance. The variance \(\sigma^2\) is unknown. These assumptions give the usual exact t and F reference distributions; the statistic identity itself follows from the ordinary least-squares formulas.

Write \(\hat{\beta}_0\) and \(\hat{\beta}_1\) for the least-squares estimates, and define the residual sum of squares by \(\mathrm{SSE}=\sum_{i=1}^n(Y_i-\hat{Y}_i)^2\). The estimated error variance is \(s^2=\mathrm{SSE}/(n-2)\). The test of \(H_0:\beta_1=0\) uses \(\hat{\beta}_1\) divided by its estimated standard error. The overall F test compares the regression sum of squares with the residual mean square. In simple regression, each test has one numerator degree of freedom, and each uses \(n-2\) residual degrees of freedom.

The Algebra Behind the Identity

With an intercept, the fitted values satisfy \(\hat{Y}_i=\bar{Y}+\hat{\beta}_1(x_i-\bar{x})\). Therefore, the regression sum of squares, \(\mathrm{SSR}=\sum_{i=1}^n(\hat{Y}_i-\bar{Y})^2\), can be written directly in terms of the estimated slope.

\mathrm{SSR} =\sum_{i=1}^n\left[\hat{\beta}_1(x_i-\bar{x})\right]^2 =\hat{\beta}_1^2 S_{xx}.

The usual estimated standard error of the slope is \(s/\sqrt{S_{xx}}\). Consequently, the slope t statistic is \(\hat{\beta}_1/(s/\sqrt{S_{xx}})\). The overall F statistic for one predictor is \(\mathrm{SSR}/(\mathrm{SSE}/(n-2))\). Substituting the expressions above shows the equality.

t^2 =\left(\frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}}\right)^2 =\frac{\hat{\beta}_1^2S_{xx}}{s^2} =\frac{\mathrm{SSR}}{\mathrm{SSE}/(n-2)} =F.

Result: In ordinary least-squares simple linear regression with an intercept and one predictor, \(F=t^2\), where \(t\) tests \(H_0:\beta_1=0\) and \(F\) tests the overall regression null \(H_0:\beta_1=0\). Under the stated normal-error assumptions, \(t\) has a \(t\) distribution with \(n-2\) degrees of freedom under the null, and \(F\) has an \(F\) distribution with \(1\) and \(n-2\) degrees of freedom under the null.

The squared-statistic relationship explains the matching two-sided results. For a chosen significance level, the two-sided t test rejects when \(|t|\) is sufficiently large. Since \(F=t^2\), this is exactly the same condition as the upper-tail F test rejecting for a sufficiently large \(F\). Equivalently, the two-sided t p-value equals the upper-tail F p-value for these corresponding distributions. A one-sided t test asks a different question: it distinguishes positive from negative slopes, while the F statistic cannot.

Worked Calculations

Worked Example: Verifying the Equality From Regression Summaries

A hypothetical analysis relates a continuous response to one fixed predictor for \(n=8\) observations. The fitted model includes an intercept. Its summary gives \(\hat{\beta}_1=1.5\), \(S_{xx}=28\), and \(\mathrm{SSE}=12\). Assume independent normal errors with common unknown variance. Test \(H_0:\beta_1=0\) against a two-sided alternative at the 5% level, and verify the identity of the statistics.

Method. Calculate the residual mean square, then obtain the slope t statistic and the overall F statistic from their definitions. The common mistake to avoid is using the signed t statistic as if it were an F statistic.

There are \(n-2=8-2=6\) residual degrees of freedom. Thus,

s^2=\frac{\mathrm{SSE}}{n-2}=\frac{12}{6}=2, \qquad s=\sqrt{2}.

The slope standard error and t statistic are

\operatorname{SE}(\hat{\beta}_1) =\frac{s}{\sqrt{S_{xx}}} =\sqrt{\frac{2}{28}} =\sqrt{\frac{1}{14}} \approx 0.2673, \qquad t=\frac{1.5}{0.2673}\approx 5.612.

The regression sum of squares is \(\mathrm{SSR}=\hat{\beta}_1^2S_{xx}\), so

\mathrm{SSR}=(1.5)^2(28)=2.25(28)=63, \qquad F=\frac{63}{12/6}=\frac{63}{2}=31.5.

Checking directly, \(t^2\approx(5.612)^2\approx31.5\), agreeing with \(F\). For 6 degrees of freedom, the 5% two-sided t critical value is approximately \(2.447\), and the corresponding upper-tail F critical value is approximately \(5.987\). Since \(|5.612|>2.447\) and \(31.5>5.987\), both procedures reject \(H_0\). The conclusion is that this fitted sample provides evidence against a zero population slope. The identity is verified.

Worked Example: A Negative Slope and a Two-Sided Decision

In a hypothetical study of \(n=10\) units, a response is regressed on one fixed predictor using an intercept. The fitted slope is \(\hat{\beta}_1=-0.8\), \(S_{xx}=45\), and \(\mathrm{SSE}=28.8\). Assume independent normal errors with common unknown variance. Test for a nonzero slope at the 5% level and compare the t and F decisions.

Method. Use the sign of the slope for the t statistic and its square for F. The common mistake to avoid is treating a negative t statistic as a negative F statistic; an F statistic cannot be negative.

The residual degrees of freedom are \(10-2=8\), and

s^2=\frac{28.8}{8}=3.6, \qquad \operatorname{SE}(\hat{\beta}_1) =\sqrt{\frac{3.6}{45}} =\sqrt{0.08} \approx 0.2828.

Therefore,

t=\frac{-0.8}{0.2828}\approx -2.828, \qquad t^2\approx 8.00.

Using the overall F formula gives the same result:

\mathrm{SSR}=(-0.8)^2(45)=0.64(45)=28.8, \qquad F=\frac{28.8}{28.8/8}=\frac{28.8}{3.6}=8.00.

For 8 residual degrees of freedom, the 5% two-sided t critical value is approximately \(2.306\), and the 5% upper-tail F critical value with degrees of freedom \(1\) and \(8\) is approximately \(5.318\). Since \(|-2.828|>2.306\) and \(8.00>5.318\), both tests reject the zero-slope null. The negative sign identifies the fitted direction; the F test agrees on evidence against zero but does not report that direction. The statistics match.

Worked Example: Using the Correct Residual Degrees of Freedom

A hypothetical regression of a response on one fixed predictor includes an intercept and has \(n=12\), \(\hat{\beta}_1=2\), \(S_{xx}=30\), and \(\mathrm{SSE}=40\). Assume independent normal errors with common unknown variance. Calculate both test statistics for \(H_0:\beta_1=0\), and identify the error in a calculation that uses 11 residual degrees of freedom.

Method. Use \(n-2\) residual degrees of freedom because two parameters, the intercept and slope, have been estimated. The common mistake to avoid is using \(n-1\), as if only the intercept had been estimated.

The correct residual degrees of freedom are \(12-2=10\), so the residual mean square is \(40/10=4\). The regression sum of squares is

\mathrm{SSR}=\hat{\beta}_1^2S_{xx}=(2)^2(30)=120.

The F statistic is

F=\frac{120}{40/10}=\frac{120}{4}=30.

The slope standard error and t statistic are

\operatorname{SE}(\hat{\beta}_1) =\sqrt{\frac{4}{30}} =\sqrt{\frac{2}{15}} \approx0.3651, \qquad t=\frac{2}{0.3651}\approx5.477.

Squaring gives \(t^2\approx30\), as required. The incorrect degrees-of-freedom calculation would give \(\mathrm{SSE}/11=40/11\approx3.6364\), and hence \(120/(40/11)=33\). This is wrong because the fitted slope also uses a degree of freedom. Using \(n-2\) restores the correct F statistic, and the identity checks.

Scope and Common Traps

The equality is specific to the ordinary overall F test in simple linear regression with an intercept. The F test in a multiple-predictor regression tests whether all slopes are zero together; its numerator degrees of freedom equal the number of predictors, so it is not generally the square of any one slope t statistic. A test of several restrictions is also a different comparison. Those procedures require their own test statistics.

The intercept matters as well. The identity above uses centered predictor values and the relation \(\mathrm{SSR}=\hat{\beta}_1^2S_{xx}\) for the fitted model with an intercept. Do not transfer the formula automatically to a regression forced through the origin. Also distinguish the algebra from its reference distributions: \(F=t^2\) for these ordinary least-squares statistics, but the stated exact t and F p-values rely on the model assumptions. If errors are not independent, have unequal variances, or are not normal, the usual exact inference may not apply. Alternative standard errors or test procedures need not preserve this particular t-squared/F identity.

Key takeaway: In simple linear regression with an intercept and one predictor, \(\mathrm{SSR}=\hat{\beta}_1^2S_{xx}\) and \(\operatorname{SE}(\hat{\beta}_1)=s/\sqrt{S_{xx}}\), which together give \(F=t^2\). This makes the overall F test and the two-sided zero-slope t test equivalent under the usual model assumptions; it does not make the F test directional.

Check Your Understanding

For each problem, use the ordinary simple-regression formulas and state the relevant degrees of freedom. Assume an intercept unless the problem says otherwise.

A fitted regression has \(n=9\), \(S_{xx}=16\), \(\hat{\beta}_1=-0.75\), and \(\mathrm{SSE}=28\). Find the slope t statistic and overall F statistic. (Answer: \(t=-1.500\) with 7 degrees of freedom; \(F=2.250\) with degrees of freedom \(1,7\).)
A regression has \(n=14\), \(S_{xx}=50\), \(\hat{\beta}_1=1.2\), and \(\mathrm{SSE}=60\). Calculate \(F\), then obtain \(|t|\) from the identity. (Answer: \(F=14.400\) with degrees of freedom \(1,12\); \(|t|=\sqrt{14.4}\approx3.795\) with 12 degrees of freedom.)
A simple regression has \(n=11\), \(\mathrm{SSR}=36\), and \(\mathrm{SSE}=24\). Find the overall F statistic. If the slope estimate is \(2\), verify the identity by finding \(S_{xx}\) and \(t^2\). (Answer: \(F=13.500\); \(S_{xx}=9\), and \(t^2=13.500\), with 9 residual degrees of freedom.)
A regression has \(n=10\), \(S_{xx}=20\), \(\hat{\beta}_1=-1\), and \(\mathrm{SSE}=32\). Find \(t\) and \(F\). At the 5% level, compare the two-sided slope test with a one-sided test of a negative slope. (Answer: \(t\approx-2.236\) and \(F=5.000\); the two-sided test does not reject using \(t_{0.975,8}\approx2.306\) or \(F_{0.95;1,8}\approx5.318\), while the one-sided negative-slope test rejects using \(t_{0.95,8}\approx1.860\).)
Explain why the overall F statistic in a simple regression with an intercept is not, in general, the square of the t statistic for the intercept. (Answer: the overall F test tests the slope null \(\beta_1=0\), while the intercept t test tests \(\beta_0=0\); they concern different hypotheses.)

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