Uncertainty for a Rescaled Predictor Contrast

Why Rescale Both the Estimate and Its Uncertainty?

A regression slope describes an expected outcome change for a one-unit predictor change. That unit may not be the change that matters in an application. If a slope is reported in score points per study hour, for example, a reader may want the fitted difference associated with five additional hours. Multiplying the slope estimate by five gives the estimated difference, but the uncertainty must be expressed on that same scale too.

This is a transformation of an estimate and its confidence interval, not a new fit of the regression model. The key is to apply the specified contrast consistently, and to reverse the order of the interval endpoints when the contrast is negative.

Model, Slope, and Predictor Contrast

For a simple linear regression with quantitative response \(Y\) and predictor \(x\), write the model for observation \(i\) as

Here \(\beta_0\) is the intercept, \(\beta_1\) is the population slope, and \(\varepsilon_i\) is the error for observation \(i\). For a specified predictor contrast \(c\), the corresponding slope-based quantity is \(c\beta_1\). When the predictor changes from \(x\) to \(x+c\), the model's expected response changes by \(c\beta_1\), because

This interpretation holds for the stated straight-line model. It describes a difference in fitted conditional means, not necessarily a causal effect. The contrast \(c\) has predictor units; \(c\beta_1\) has outcome units.

Assumptions Behind the Confidence Interval

In the usual simple linear regression confidence interval for \(\beta_1\), the errors are independent, have mean zero, have a common variance \(\sigma^2\), and are normally distributed. The regression mean is linear in \(x\); the predictor values are treated as fixed, or inference is understood conditional on the observed predictor values. Under these conditions, the usual two-sided \(100(1-\alpha)\%\) interval is

The \(t\) critical value has \(n-2\) degrees of freedom. The standard error accounts for sampling variability in the slope estimate under the model. If the assumptions are not reasonable, the usual interval may not have its stated coverage, so multiplying its endpoints cannot repair that problem.

For the transformed interval, \(c\) must be a fixed, known contrast for the stated coverage argument. If the contrast is chosen after examining the estimated slope or its interval, the ordinary interval calculation does not automatically account for that data-dependent choice. Also, interpreting \(c\beta_1\) as the expected change from \(x\) to \(x+c\) relies on the linear mean model applying over that range.

Why the Endpoint Rule Works

Let the original confidence interval for \(\beta_1\) be \([L,U]\), where \(L\leq U\). If \(c\) is positive, multiplying preserves order: \(cL\leq cU\). If \(c\) is negative, multiplication reverses order: \(cU\leq cL\). Thus, for any nonzero \(c\), the correctly ordered transformed interval is

This transformation also explains the standard error rule: scaling a random estimate by \(c\) scales its standard deviation by \(|c|\). For \(c\ne0\), the transformed interval contains \(c\beta_1\) exactly when the original interval contains \(\beta_1\). Therefore, if the original interval has its stated confidence level under the model assumptions, the correctly ordered transformed interval has that same level for \(c\beta_1\).

There is an important exception. When \(c=0\), the target is the known value \(0\), and both transformed endpoints are zero. The resulting interval \([0,0]\) covers that target with probability 1, or 100%, not generally with the original nominal confidence level. This case is not an uncertain estimate of a nonzero contrast.

What the Rescaling Does—and Does Not—Change

For a nonzero fixed contrast, rescaling changes the units and magnitude of the estimate and its uncertainty together. The point estimate, standard error, and interval width all scale by \(|c|\) in magnitude. The confidence level does not increase merely because the contrast is larger; it remains the same when the original interval is valid and the contrast is nonzero and fixed. Nor does rescaling make the underlying slope estimate more precise.

The choice of contrast should make sense for the data and question. A contrast far outside the observed predictor range may require extrapolation: the arithmetic still works, but the linear model may not describe the mean response there. For a contrast within the observed range, the result remains an estimate based on the model, not a guarantee that every pair of observations separated by that amount differs by the estimated quantity.

Common Mistakes

• Scaling the estimate but not its interval. A five-unit contrast needs both endpoints, and the standard error, scaled by five as well. Otherwise, the estimate and uncertainty are on different scales.
• Leaving negative endpoints in the wrong order. Multiplication by a negative contrast reverses order. Report the smaller product first.
• Multiplying a standard error by a negative number. Standard errors cannot be negative; the transformed standard error is \(|c|SE(\hat{\beta}_1)\).
• Claiming the same nominal coverage when \(c=0\). The interval then collapses to \([0,0]\) and covers the known target with probability 100%.
• Interpreting the result causally without justification. A rescaled regression slope describes a model-based association in expected outcomes. Causal language needs a study design and assumptions that support it.
• Ignoring the predictor range or model form. A multiplication is algebraically valid, but its practical interpretation depends on the linear model being appropriate for the contrast.