A reviewer ranks predictor importance by raw coefficient size in a table with mixed units - how do you respond?

## Why the ranking is meaningless A regression coefficient is expressed in **outcome units per predictor unit**. The size of a coefficient therefore depends on two things: how strongly the predictor relates to the outcome, and how big the unit happens to be. The second has nothing to do with the data. Take a model with age in years, tenure in months and income in dollars. The income coefficient will be minuscule - a single dollar is a negligible difference - while age gets a large-looking number because a year is a big step. Refit the model with income in millions of dollars and, without a single value in the dataset changing meaning, income becomes the biggest coefficient in the table. A quantity that can be reordered by a choice of units is not measuring importance. This is the cleanest way to end the argument in a review: offer to make any predictor the "most important" one on request. Once the reviewer sees that the ranking is under your control, the conversation moves to what ranking would actually be defensible. ## Fix one: a common statistical yardstick Multiply each coefficient by the standard deviation of its predictor. That converts every effect into "outcome change per one standard deviation of this predictor", which is at least the same kind of step for each. Dividing additionally by the outcome's standard deviation gives fully standardised coefficients, which are dimensionless and directly comparable in size. This is a real improvement and it is what most people mean by comparable coefficients. It is also sample-dependent: standard deviations describe the spread in *this* dataset, so a predictor with a narrow observed range gets a small standardised effect even if the relationship is strong. ## Fix two: a decision-relevant step Usually better for a non-technical audience. Instead of one unit or one standard deviation, quote the effect of the change someone might actually see or make: a 10,000-dollar difference in income, one extra year of tenure, a move from the 25th to the 75th percentile of the predictor. The number stays in the outcome's own units, which stakeholders can reason about, and the step is concrete. "Moving a customer from the bottom quartile of tenure to the top is associated with 180 dollars more annual spend, with a 95% interval of 120 to 240" answers a business question. "The tenure coefficient is 0.42" does not. ## Fix three: how much the predictor explains A third angle, answering a different question: how much of the variation in the outcome does this predictor account for beyond the others? Compare the model's explained variation with and without the predictor - the incremental contribution to R-squared. This is not the same as effect size and should not be conflated with it: a predictor can move the outcome a lot per unit while varying so little in the sample that it explains almost nothing, and vice versa. Decide first which question you are answering - *how much does it move the outcome* or *how much of the spread does it account for* - because they have different answers and different uses. ## Always attach uncertainty A point estimate with no interval invites false precision, and rankings are especially vulnerable: two predictors whose intervals overlap heavily should not be ordered at all. Present intervals next to every effect and say explicitly when a ranking is not supported. ## What none of these fixes Be honest about the limits, because a good interviewer will push here. **Correlated predictors share credit.** When two predictors overlap, the fit splits the association between them in a way that depends on the specification. Neither "importance" number is a stable property of either variable alone. **Comparability is not causality.** All of these are descriptive comparisons within the fitted model. Whether any of them tells you what would happen if you changed the predictor is a separate question requiring separate assumptions. **Significance is not magnitude.** A p-value combines effect size, noise and sample size. With a large enough sample a trivially small effect is highly significant; with a small sample an important one may not be. Ranking by p-value is exactly as broken as ranking by raw coefficients, for different reasons. ## The response, in one breath "That table is ranking our unit choices. I can make income first or last by choosing dollars or millions. Let me re-present it two ways: effects per standard deviation so the magnitudes are comparable, and effects over a realistic step in the original units so they are quotable, both with intervals. Where the intervals overlap I will say the order is not determined by the data." That is the answer of someone who has been in the room when a table like this drove a decision.