Comparing Precision Across Regression Coefficients

Why Comparing Interval Widths Takes Care

A confidence interval gives a range of plausible values for a particular regression coefficient, on that coefficient’s own scale. It is tempting to look at several intervals and decide that the narrowest one is the “most precise.” That shortcut can fail: a slope per one milligram and a slope per one year are measured on different scales, and an intercept and a group difference describe different quantities even if both use the outcome’s units.

A useful comparison asks what uncertainty in each coefficient means for the outcome in a context that matters. For a slope, this often means considering a relevant change in its predictor. For any coefficient, it also helps to ask whether the interval is narrow enough for the practical decision or scientific question at hand.

Start With What Each Coefficient Means

For a fitted linear regression model, a slope coefficient describes the change in the model’s mean response associated with a one-unit increase in its predictor, holding other predictors fixed when they are in the model. Its units are outcome units per predictor unit. An indicator-variable coefficient describes the difference in the model’s mean response between the coded group and its reference group, holding other model terms fixed. An intercept describes the model’s mean response when the predictors are at their reference or zero values.

The conventional coefficient interval discussed here has the form \(\hat{\beta}_j \pm t^*SE(\hat{\beta}_j)\), where \(\hat{\beta}_j\) is the estimated coefficient, \(SE(\hat{\beta}_j)\) is its standard error, and \(t^*\) is the chosen critical value. Its usual interpretation relies on the regression conditions for conventional coefficient inference: the model’s mean structure is appropriate, errors are independent with mean zero and constant variance, and errors are normally distributed for exact small-sample \(t\)-based inference. In multiple regression, the model terms must also be estimable; in particular, there must not be perfect multicollinearity. Comparisons below assume the intervals were constructed appropriately and have the same confidence level unless stated otherwise.

The interval’s width is \(2t^*SE(\hat{\beta}_j)\). But it is in the units of \(\hat{\beta}_j\), not automatically in a common unit shared by every coefficient in a model. For a slope, the width might be outcome-units per hour; for another slope, outcome-units per kilogram. Directly ranking those numbers can have little meaning.

Translate Slope Intervals to Outcome Units

A slope interval can be translated to an interval for the model’s mean response change over a specified predictor increment. If the slope interval is \([L,U]\) and the predictor change is \(\Delta x\), then the change in the model’s mean response is \(\Delta x\beta\). When \(\Delta x\) is positive, multiplying every value in the original interval by \(\Delta x\) gives the translated interval. Its width is \(\Delta x(U-L)\). If \(\Delta x\) is negative, multiplication reverses the endpoint order, so the lower and upper endpoints must be rearranged.

This follows directly from scaling the set of possible coefficient values. For positive \(\Delta x\), the smallest product is \(\Delta xL\) and the largest is \(\Delta xU\), so:

For negative \(\Delta x\), the largest coefficient value produces the smallest product, giving \([\Delta xU,\Delta xL]\). This translation is an exact change of units for the coefficient interval; it does not alter the confidence level. The increment should be chosen because it is meaningful in context, not merely because it makes one interval look narrow.

Same Units Do Not Mean Same Question

Even when two intervals have the same units, their widths answer only a limited question: how large is each interval’s absolute span on that common scale? They do not tell us whether the coefficients are equally important or equally useful. An intercept, a slope translated over a chosen increment, and an indicator-variable contrast may all be expressed in outcome units while referring to different features of the model.

Practical comparison therefore requires a relevant reference scale. That scale might be a prespecified threshold for a meaningful outcome change, a tolerance for estimating a baseline, or a particular predictor difference of interest. A width can be judged against such a benchmark, but the benchmark must fit the estimand. There is no universal “percent precision” that is appropriate for every coefficient; dividing width by the estimate, for example, is especially unhelpful when an estimate is near zero.

A Practical Comparison Routine

When several regression intervals appear together, use a consistent sequence. First identify what each coefficient represents and write its units. Next check that the intervals have the same confidence level; different confidence levels can produce different widths for reasons unrelated to the coefficient estimates’ standard errors. For slopes, state a meaningful predictor increment and translate the interval into outcome units. Finally, interpret the range against a practical threshold or decision question, without treating a narrower interval as evidence of a larger effect.

Common Mistake: Ranking by the Smallest Number

A common error is to say, “The interval with width \(0.2\) is more precise than the interval with width \(5\), because \(0.2\) is smaller.” That conclusion ignores the units. A width of \(0.2\) outcome-points per hour and a width of \(5\) outcome-points per year are not directly comparable. Even when units match, equal or unequal widths alone do not settle which coefficient matters more for the research question.

The correction is to report each width with its units and explain the corresponding estimand. For slopes, translate intervals over meaningful predictor changes when that helps answer the question. Then discuss whether the resulting outcome ranges are sufficiently informative relative to a relevant practical benchmark. Precision concerns uncertainty in an estimate; practical importance concerns the size and consequences of the quantity being estimated. They are related considerations, but they are not the same.