If income is re-expressed in thousands of dollars, what happens to its regression coefficient and t-statistic?

## Linear rescaling is a change of units, not a change of model Suppose the fitted model is `y_hat = b0 + b1*income + b2*age`, with income in dollars. Now refit with income measured in thousands, which means every value of the predictor is divided by 1,000. Because the new predictor is 1,000 times smaller, its coefficient must be 1,000 times larger to produce the same fitted values: ``` b1 * income = (1000 * b1) * (income / 1000) ``` A coefficient of 0.004 dollars of outcome per dollar of income becomes 4 per thousand dollars of income. Nothing about the fit moved: the same predicted values, the same residuals, the same R-squared, the same coefficient on age. ## What happens to the standard error, t and p The standard error of the coefficient carries the same units as the coefficient, so it scales by exactly the same factor. The t-statistic is `t = b / SE(b)`, and multiplying numerator and denominator by 1,000 leaves it unchanged. The p-value, which is a function of t and the degrees of freedom, is unchanged too, and so is the confidence interval once you re-express its endpoints in the new units. This is the point worth internalising: **you cannot make a predictor significant by changing its units.** A candidate who says rescaling improved the p-value has misunderstood what a p-value measures. ## The general rule Replace a predictor `x` with `z = (x - c) / k`. Then: - the coefficient on `z` is `k` times the coefficient on `x` - stretching the axis by `k` stretches the slope's units by `k`; - the intercept changes to absorb the shift `c`, becoming `b0 + b*c` in the original coefficient's terms; - the coefficients on all the other predictors are unchanged; - fitted values, residuals, R-squared, F-statistics and every t-statistic are unchanged. Rescaling the **outcome** works the other way round: dividing `y` by 1,000 divides the intercept and every coefficient by 1,000, again leaving t-statistics and R-squared alone. ## The Celsius and Fahrenheit case Temperature makes both moves visible at once, because `F = 1.8*C + 32` is an affine transformation, not a pure scaling. If the fitted model in Fahrenheit is `y_hat = a + b*F`, substitute: ``` y_hat = a + b*(1.8*C + 32) = (a + 32*b) + (1.8*b)*C ``` So the Celsius slope is `1.8*b` - a one-degree Celsius change is a larger change than one degree Fahrenheit, so it buys 1.8 times as much outcome - and the intercept moves from `a` to `a + 32*b`, because zero Celsius is not zero Fahrenheit. The t-statistic on temperature is still identical, since the fitted model is the same model. The direction of the slope factor is where people slip. Ask which unit is bigger. A Celsius degree is bigger than a Fahrenheit degree, so its coefficient is bigger. A thousand dollars is bigger than a dollar, so income-in-thousands has the bigger coefficient. Reasoning from unit size beats memorising a rule. ## Why rescale at all Three honest reasons. **Readability.** A coefficient of 0.0000031 dollars of revenue per impression is unreadable; per million impressions it is 3.1, and a stakeholder can hold it in their head. Choose units so the coefficient is a number a human can say. **Meaningful step size.** The natural unit is rarely the interesting one. One extra dollar of income and one extra second of tenure are not changes anyone cares about. Rescaling to a decision-relevant step - per 10,000 dollars, per year, per interquartile range - makes the coefficient answer the question people are actually asking. **Numerical conditioning.** A design matrix mixing values in the millions with values near zero can be badly scaled. This matters much less than it used to, but it is real. There is one thing rescaling never buys you: evidence. Since the fit, the t-statistics and the p-values are all invariant, any argument that turns on a coefficient becoming "bigger" after rescaling is an argument about presentation, not about the data. ## The interview reflex When asked what happens to a coefficient under a change of units, answer in four parts: the coefficient scales by the size of the new unit relative to the old, the standard error scales identically, the t-statistic and p-value do not move, and the intercept shifts if the transformation includes an offset. Then name the units of the new coefficient out loud - that is what proves you understand rather than recall.