Why Test Against a Nonzero Benchmark?
A regression coefficient is often compared with zero, but zero is not always the most useful reference value. A researcher may want to know whether an estimated slope differs from a target rate, or whether a group difference differs from a prespecified value that would matter in practice. The same t-test framework applies: the benchmark changes, while the test still measures how far the estimate lies from the value specified by the null hypothesis.
For instance, a slope of 2.4 points per study hour may be statistically distinguishable from zero but not from a target of 2 points per hour. A test against that target directly addresses the question of interest. It does not establish that the target is practically important; that judgment comes from the context and the reason for choosing the benchmark.
Hypotheses and the Test Statistic
Let \(\beta_j\) be the population value of one regression coefficient, and let \(b_j\) be its estimate from the fitted regression. Write \(c\) for the specified benchmark, measured in the same units as \(\beta_j\). For the two-sided test in this tutorial, the hypotheses are:
The benchmark \(c\) may be nonzero. The standard error \(SE(b_j)\) describes the estimated sampling variability of the coefficient estimate. Under the null hypothesis, the t statistic is the distance between the estimate and the benchmark, measured in standard-error units:
For a fitted model with \(n\) observations and \(q\) estimated regression coefficients, including the intercept when there is one, the usual residual degrees of freedom are \(df=n-q\), provided the model matrix has full rank. The relevant degrees of freedom are those for the fitted regression, not a separate number chosen for the coefficient being tested.
The test requires the conditions behind conventional regression coefficient inference. The regression mean structure must be appropriate for the question, the errors must be independent with mean zero and constant variance, and the model must be estimable without exact collinearity. For an exact t distribution in a finite sample, the errors are also assumed to be normally distributed. In practice, the design and residual diagnostics inform whether these conditions are reasonable; a calculated statistic cannot repair a serious violation.
Why the Formula Uses the Benchmark
Under \(H_0\), the coefficient’s hypothesized value is \(c\), so the relevant discrepancy is \(b_j-c\), not simply \(b_j\). Dividing that discrepancy by its standard error expresses it in units of the estimate’s expected sampling variability. A value of \(t=2\), for example, means the estimate is two standard errors above the benchmark; \(t=-2\) means it is two standard errors below it. The two-sided test treats those distances symmetrically.
This also shows why a test against a nonzero value is not a different kind of calculation. The null distribution is centered at zero after subtracting the hypothesized benchmark. If \(c=0\), the numerator reduces to \(b_j\); if \(c\ne0\), it remains the difference between the estimate and \(c\).
Worked Example: A Plant-Growth Slope Versus a Target Rate
Hypothetical data. A simple linear regression relates daily plant growth, in millimeters per day, to daily light exposure, in hours. Suppose the fitted slope is \(b_1=2.4\) millimeters per day per additional hour of light, with \(SE(b_1)=0.5\) in the same units. There are \(n=20\) plants. Test whether the population slope differs from the nonzero target \(c=1.8\) millimeters per day per additional hour, using a two-sided test.
Conditions and hypotheses. Suppose the plants were independently observed, the linear mean relationship and constant error variance are reasonable, and the residuals are consistent with normal errors. These assumptions make the conventional small-sample t test appropriate. The hypotheses are \(H_0:\beta_1=1.8\) versus \(H_a:\beta_1\ne1.8\), with slope units of millimeters per day per hour of light.
The model estimates an intercept and a slope, so \(q=2\) and \(df=n-q=20-2=18\). The observed statistic is
For a t distribution with 18 degrees of freedom, the two-sided p-value is approximately \(0.2457\). At significance level \(\alpha=0.05\), this p-value is larger than \(0.05\), so we fail to reject \(H_0\). These data do not provide strong evidence that the population slope differs from 1.8 millimeters per day per additional hour of light. This conclusion does not establish that the slope equals 1.8.
Worked Example: A Quiz-Score Slope Versus a Target
Hypothetical data. In a simple linear regression of quiz score on study time, the estimated slope is \(b_1=2.84\) score points per additional study hour, and its standard error is \(0.30\) points per hour. The sample size is \(n=20\). Test whether the population slope differs from the target \(c=2.00\) points per hour.
Conditions and hypotheses. Assume the observations are independent, a linear mean relationship and constant error variance are reasonable, and normal errors are a suitable approximation. The model has an intercept and one slope, giving \(df=20-2=18\). The hypotheses are \(H_0:\beta_1=2.00\) versus \(H_a:\beta_1\ne2.00\).
Substituting the estimate, benchmark, and standard error gives
For \(df=18\), the two-sided p-value is approximately \(0.01184\). Because \(0.01184<0.05\), we reject \(H_0\) at the five-percent significance level. The data provide evidence that the population slope differs from 2.00 score points per additional study hour. The estimated slope is above the target, but this is a two-sided test: its p-value measures evidence of a difference in either direction, not just evidence that the slope is greater than the target.
Worked Example: An Estimate Exactly at the Benchmark
Hypothetical data. A regression uses an indicator variable to represent membership in one of two groups. Suppose its coefficient estimates an adjusted difference in outcome means, in outcome points, of \(b_j=4.5\), with \(SE(b_j)=0.8\). Suppose there are 15 residual degrees of freedom. Test whether the population difference differs from the nonzero benchmark \(c=4.5\) points.
Conditions and hypotheses. Suppose the regression specification is appropriate, observations are independent, error variance is constant, and normal errors are reasonable for this sample. The hypotheses are \(H_0:\beta_j=4.5\) versus \(H_a:\beta_j\ne4.5\), with both coefficient and benchmark measured in outcome points.
The observed statistic is
A t statistic of zero is exactly at the center of the null distribution, so the two-sided p-value is \(1\). At \(\alpha=0.05\), we fail to reject \(H_0\). The estimate is exactly the benchmark in this sample, but the test does not prove that the population difference equals 4.5 points. Sampling uncertainty remains, and a different sample could produce a different estimate.
A Reliable Testing Sequence
The calculation is short, but a sound conclusion depends on stating the target and keeping the units and direction of the test clear. Use this sequence for an individual coefficient test.
Identify which population coefficient \(\beta_j\) is of interest and express the specified value \(c\) in the same units.
For a two-sided question, write \(H_0:\beta_j=c\) and \(H_a:\beta_j\ne c\). Do not silently replace a meaningful nonzero target with zero.
Consider whether the model form, independence, constant error variance, and—when relying on an exact small-sample t distribution—normality are reasonable.
Compute \(t=(b_j-c)/SE(b_j)\), use the residual degrees of freedom, and obtain the two-sided tail probability \(P(|T|\ge|t|)\).
Compare the p-value with \(\alpha\), then state whether the data provide evidence that the coefficient differs from the benchmark, including the relevant units.
Common Mistakes and Why the Test Matters
A common mistake is to use \(b_j/SE(b_j)\) even when the null value is nonzero. That statistic measures distance from zero, not distance from the stated target. For a test of \(H_0:\beta_j=c\), the numerator must be \(b_j-c\). Another mistake is to say that a large p-value shows the coefficient equals the benchmark. A large p-value means the data do not provide sufficient evidence against that null value; it does not demonstrate equality.
The p-value also is not the probability that \(H_0\) is true. It is calculated under the assumption that \(H_0\) is true and describes how unusual a statistic at least this far from zero would be. Finally, statistical evidence of a difference does not by itself show that the difference is practically important. The benchmark, the estimated difference, its units, and the consequences in the application all matter.
Check Your Understanding
Answer each question using the coefficient’s context and units.
- A fitted slope is \(3.1\) outcome units per predictor unit, its standard error is \(0.4\), and the proposed benchmark is \(2.5\). Write the two-sided hypotheses and calculate the t statistic.
- Why is \(b_j/SE(b_j)\) not the correct statistic for testing \(H_0:\beta_j=1.2\)?
- For a test with p-value \(0.08\) and \(\alpha=0.05\), state the decision and explain why it does not show that the coefficient equals the benchmark.
- In a two-sided test, what does the sign of the observed t statistic say, and what does it not change about the p-value calculation?
- Give one reason a coefficient might be tested against a nonzero benchmark rather than zero.