Rescale Units Without Changing the Interaction Pattern

Why Rescaling Units Matters

A regression equation can describe the same relationship using different measurement units. Study time might be recorded in hours or minutes; distance might be recorded in meters or centimeters. Changing units changes the numerical values of the predictors, so the regression coefficients must also change. In an interaction model, this applies to the main-effect coefficients and to the coefficient multiplying the product of the predictors.

The important point is that a change in units does not, by itself, change the fitted outcomes or the underlying interaction pattern. It changes how the same relationship is written and the units used to describe its coefficients. This tutorial considers rescaling: replacing a predictor by a positive constant multiple of its original numerical value, without shifting its zero point.

Model: For two quantitative predictors $x_1$ and $x_2$, an interaction regression model is $$ Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3x_{1i}x_{2i}+\epsilon_i. $$ Here, $Y_i$ is the response for observation $i$, the $\beta$ values are coefficients, and $\epsilon_i$ is the error. The model assumes that the conditional mean of $Y_i$, given the predictors, has the displayed form and that the errors have conditional mean zero. Standard regression inference additionally relies on appropriate assumptions about the errors and observations; the unit-conversion algebra below does not require a normal-error assumption.

The fitted conditional mean is obtained by replacing each $\beta$ with its estimated coefficient. The main-effect coefficients describe the predictor slopes when the other predictor is zero, and the product coefficient describes how those slopes vary with the other predictor. These interpretations are tied to the predictors’ units.

The Coefficient Conversion Rule

Let $x_1^*$ and $x_2^*$ be the new numerical measurements of the predictors. Suppose the new units are defined by $x_1^*=c_1x_1$ and $x_2^*=c_2x_2$, where $c_1$ and $c_2$ are positive conversion constants. For example, if $x_1$ is measured in hours and $x_1^*$ in minutes, then $c_1=60$. To express the original model in the new units, substitute $x_1=x_1^*/c_1$ and $x_2=x_2^*/c_2$.

\begin{aligned} Y_i &=\beta_0+\beta_1\frac{x_{1i}^*}{c_1} +\beta_2\frac{x_{2i}^*}{c_2} +\beta_3\frac{x_{1i}^*}{c_1}\frac{x_{2i}^*}{c_2} +\epsilon_i\\ &=\beta_0+\frac{\beta_1}{c_1}x_{1i}^* +\frac{\beta_2}{c_2}x_{2i}^* +\frac{\beta_3}{c_1c_2}x_{1i}^*x_{2i}^* +\epsilon_i. \end{aligned}

Coefficient conversion rule: If $x_1^*=c_1x_1$ and $x_2^*=c_2x_2$, the coefficients in the new units are $$ \beta_0^*=\beta_0,\qquad \beta_1^*=\frac{\beta_1}{c_1},\qquad \beta_2^*=\frac{\beta_2}{c_2},\qquad \beta_3^*=\frac{\beta_3}{c_1c_2}. $$ The intercept is unchanged because rescaling leaves zero at zero. Each main-effect coefficient is divided by the scale factor for its predictor, and the product coefficient is divided by the product of both scale factors.

The rule follows directly from substitution: the equation in the new variables is the same equation written with the original predictor values replaced by their expressions in the new units. Because the scale factors are positive, zero and the direction of increasing values are preserved. The coefficient values may change substantially, but the model’s fitted value at any observation does not.

Worked Example: Converting Hours and Meters

Suppose a hypothetical fitted model predicts an outcome $Y$ from study time $x_1$, measured in hours, and travel distance $x_2$, measured in meters: $$ \hat{Y}=12+3x_1-2x_2+0.5x_1x_2. $$ Rewrite the equation using study time $x_1^*$ in minutes and distance $x_2^*$ in centimeters.

Since one hour is 60 minutes and one meter is 100 centimeters, $x_1^*=60x_1$ and $x_2^*=100x_2$. Thus $c_1=60$, $c_2=100$, $x_1=x_1^*/60$, and $x_2=x_2^*/100$. Apply the conversion rule to each coefficient: $$ \begin{aligned} \beta_0^*&=12,\\ \beta_1^*&=\frac{3}{60}=0.05,\\ \beta_2^*&=\frac{-2}{100}=-0.02,\\ \beta_3^*&=\frac{0.5}{60(100)}=0.00008333. \end{aligned} $$

The equation in minutes and centimeters is therefore $$ \hat{Y}=12+0.05x_1^*-0.02x_2^*+0.00008333x_1^*x_2^*. $$

Check the fitted outcome for a hypothetical observation with $x_1=2$ hours and $x_2=3$ meters. In the original units, $$ \hat{Y}=12+3(2)-2(3)+0.5(2)(3)=15. $$ The same observation has $x_1^*=120$ minutes and $x_2^*=300$ centimeters. In the new units, $$ \hat{Y}=12+0.05(120)-0.02(300)+0.00008333(120)(300)\mathrel{\approx}15. $$ The small rounding in the displayed interaction coefficient accounts for the approximate equality; using the exact value $0.5/6000$ gives exactly 15. Both equations describe the same fitted outcome.

Why the Fitted Outcomes Stay the Same

The conversion rule is more than a way to calculate new coefficients. It shows that there is a one-to-one correspondence between coefficient sets in the old and new units. Given any original coefficients, the formulas above produce coefficients in the new units. Conversely, multiplying the new main-effect coefficients by their respective scale factors and the new product coefficient by $c_1c_2$ recovers the original coefficients.

For each observation, the new-unit predictors satisfy $x_{1i}^*=c_1x_{1i}$ and $x_{2i}^*=c_2x_{2i}$. Substituting those values and the converted coefficients into the new fitted equation gives $$ \begin{aligned} \hat{Y}_i^* &=\beta_0+\frac{\beta_1}{c_1}(c_1x_{1i}) +\frac{\beta_2}{c_2}(c_2x_{2i}) +\frac{\beta_3}{c_1c_2}(c_1x_{1i})(c_2x_{2i})\\ &=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3x_{1i}x_{2i}\\ &=\hat{Y}_i. \end{aligned} $$

The equality holds for every observation, not only for the one used in a worked check. If the coefficients are estimated by least squares, the same reasoning also explains why refitting in the new units gives the same fitted values. Every possible old coefficient set corresponds to a new one with identical residuals, and vice versa. Therefore the set of achievable residual sums of squares is unchanged, so minimizing that sum produces the same fitted outcomes.

Worked Example: Changing Only One Predictor’s Units

A hypothetical fitted model predicts a response from temperature $x_1$, measured in degrees on a chosen scale, and concentration $x_2$, measured in milligrams per liter: $$ \hat{Y}=8-1.2x_1+0.4x_2+0.3x_1x_2. $$ Suppose temperature is now recorded in tenths of the original unit, so $x_1^*=10x_1$, while concentration is left unchanged, $x_2^*=x_2$. Find the new equation and verify one prediction.

Here $c_1=10$ and $c_2=1$. The intercept remains 8. The $x_1$ coefficient is divided by 10; the $x_2$ coefficient is unchanged; and the product coefficient is divided by $10(1)$: $$ \beta_1^*=\frac{-1.2}{10}=-0.12,\qquad \beta_2^*=\frac{0.4}{1}=0.4,\qquad \beta_3^*=\frac{0.3}{10}=0.03. $$

Thus the equation becomes $$ \hat{Y}=8-0.12x_1^*+0.4x_2^*+0.03x_1^*x_2^*. $$ For a hypothetical observation with $x_1=2$ and $x_2=5$, the original fitted value is $$ 8-1.2(2)+0.4(5)+0.3(2)(5)=10.6. $$ The converted predictor value is $x_1^*=20$, so the new equation gives $$ 8-0.12(20)+0.4(5)+0.03(20)(5)=10.6. $$ Rescaling one predictor changes both the coefficient of that predictor and the product coefficient, even though it does not change the other predictor’s coefficient or the fitted outcome.

The Interaction Pattern in New Units

The conditional slope for $x_1$ in the original model is $\beta_1+\beta_3x_2$. In the new units, the conditional slope with respect to $x_1^*$ is $\beta_1^*+\beta_3^*x_2^*$. Using the conversion rules and $x_2^*=c_2x_2$, this becomes $$ \beta_1^*+\beta_3^*x_2^* =\frac{\beta_1}{c_1} +\frac{\beta_3}{c_1c_2}(c_2x_2) =\frac{\beta_1+\beta_3x_2}{c_1}. $$

This is the original conditional slope per original unit of $x_1$, expressed per new unit of $x_1^*$. The corresponding result for $x_2$ is obtained in the same way: its new-unit slope is its original conditional slope divided by $c_2$. Thus the numerical slopes change to match the measurement scale, while predictions and the way slopes vary across predictor values are preserved.

Because the scale factors are positive, the product coefficient’s sign is unchanged, and a zero product coefficient remains zero. Its numerical size is not invariant: it is expressed per new unit of $x_1^*$ and per new unit of $x_2^*$. The same interaction can therefore have quite different-looking coefficient values across unit systems.

Worked Example: Interpreting a Rescaled Product Coefficient

Suppose the fitted model is $$ \hat{Y}=20+4x_1+2x_2+0.6x_1x_2, $$ where $x_1$ is exercise time in hours and $x_2$ is a score measured in points. Exercise time is converted to minutes, with $x_1^*=60x_1$, and the score remains in points.

The new coefficients are $4/60=0.06667$ for exercise time, $2/1=2$ for the score, and $0.6/60=0.01$ for the product. The new equation is $$ \hat{Y}=20+0.06667x_1^*+2x_2+0.01x_1^*x_2. $$

In the original units, the conditional slope of exercise time at $x_2=5$ points is $$ 4+0.6(5)=7 $$ response units per hour. In the new units it is $$ 0.06667+0.01(5)=0.11667 $$ response units per minute, approximately. Since 60 minutes make an hour, $60(0.11667)\mathrel{\approx}7$ response units per hour. The differing slope numbers express the same fitted relationship in different time units.

Common Mistakes and Practical Checks

Keeping the old coefficients after changing predictor values. If a predictor is multiplied by $c$, its main-effect coefficient must be divided by $c$. The product coefficient must account for the scale change in both predictors.
Dividing the product coefficient by only one conversion factor. If both predictors are rescaled, divide the product coefficient by $c_1c_2$. For example, changing hours to minutes and meters to centimeters requires division by $60(100)$.
Assuming a larger numerical coefficient means a stronger interaction. The coefficient’s numerical size depends on both predictor units. Before comparing coefficients, check that the predictors use the same units and scaling.
Treating any unit conversion as pure rescaling. The rules here cover conversions of the form $x^*=cx$, which preserve zero. A change that also shifts the zero point is not a pure rescaling and requires additional changes to the equation.
Thinking rescaling changes predictions. Check one observation by converting its predictor values and evaluating both equations. If the conversion and coefficient updates are correct, the fitted values agree, apart from rounding.

Key takeaway: Under $x_1^*=c_1x_1$ and $x_2^*=c_2x_2$, divide each main-effect coefficient by the conversion factor for its predictor, and divide the product coefficient by $c_1c_2$. The intercept and every fitted outcome stay the same; only the coefficient units and numerical values change.

Check Your Understanding

Use the coefficient conversion rule to answer each question.

A predictor changes from hours to minutes, so its new numerical values are 60 times the old values. By what factor does its main-effect coefficient change?
If both predictors are rescaled by factors of 10 and 100, respectively, by what factor is the product coefficient divided?
Explain why the intercept stays the same when both predictors are rescaled without shifting their zero points.
A fitted model is written in new predictor units. What equality should hold between its fitted value and the fitted value from the original equation for the same observation?
Why can the numerical size of a product coefficient not be compared across models without checking the units of both predictors?