Specify a Product-Term Model for Two Quantitative Predictors

Why Include a Product Term?

A model with two quantitative predictors can describe more than the separate contributions of each predictor. For example, a researcher might expect the relationship between training time and performance to depend on a participant’s starting level. A model containing only one term for training time and one for starting level represents an additive structure. Adding the product of the two predictors allows the model to represent a joint pattern that the additive equation cannot express.

The purpose here is to specify the equation and identify its parameters. Writing a product term does not, by itself, establish that an interaction is present in the data, explain the terms as causal effects, or calculate a fitted outcome. Those require further analysis and interpretation.

Definition: For two quantitative predictors $x_1$ and $x_2$, a linear regression model with a product term is $$ Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3(x_{1i}x_{2i})+\varepsilon_i,\qquad i=1,\ldots,n. $$ Here $Y_i$ is the outcome for observation $i$, $x_{1i}$ and $x_{2i}$ are that observation’s predictor values, $\beta_0,\beta_1,\beta_2,\beta_3$ are model parameters, and $\varepsilon_i$ is the error. The term $x_{1i}x_{2i}$ is the product term; including it makes this an interaction model.

The Model and Its Parameters

The equation is linear in its parameters: each $\beta$ appears as a coefficient multiplying a known term. The predictor values enter through four model terms: the constant $1$, $x_1$, $x_2$, and $x_1x_2$. In particular, the word “linear” in linear regression refers to the model’s form in the parameters; it does not require that the predictors appear only as separate, first-degree terms.

$\beta_0$ is the intercept parameter, attached to the constant term.
$\beta_1$ is the coefficient of the first predictor term $x_1$.
$\beta_2$ is the coefficient of the second predictor term $x_2$.
$\beta_3$ is the coefficient of the product term $x_1x_2$.

The coefficients have units determined by the outcome and predictor units. If $Y$ is measured in points, $x_1$ in hours, and $x_2$ in years, then $\beta_1$ has units of points per hour, $\beta_2$ has units of points per year, and $\beta_3$ has units of points per hour-year. The product coefficient’s units follow from the units of the product, not from either predictor alone.

A coefficient’s position in the equation identifies which term it multiplies; it should not be read as an independent, universal effect of that predictor. In an interaction model, the predictors appear together as well as separately. Detailed interpretation of the coefficients depends on the values and scale of both predictors.

Assumptions for the Regression Model

A model statement is an assumption about how the outcome relates to the predictors, not a fact guaranteed by writing an equation. For the usual linear regression setup, the conditional mean is specified by the equation’s non-error terms, and the error represents the difference between an observed outcome and that conditional mean.

Model assumptions: Given the observed predictor values, the conditional mean of $Y_i$ has the stated equation form, and $E(\varepsilon_i\mid x_{1i},x_{2i})=0$. For the usual variance-based regression procedures, observations are independent and the errors have a common variance $\sigma^2$. Normal errors are an additional assumption used for some small-sample inference procedures; they are not needed merely to write the product-term equation.

These assumptions concern the observations and the relationship being modeled. They do not require the predictors themselves to be normally distributed. In an applied analysis, the model form and error conditions need to be assessed using the study design and suitable diagnostics; a plausible equation alone does not verify them.

Interaction Model Versus Additive Model

The additive counterpart includes the intercept and both separate predictor terms but omits their product. It can be written as the same equation with $\beta_3=0$:

Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\varepsilon_i.

This comparison gives a direct way to distinguish the specifications. The interaction model contains four systematic terms, $1$, $x_1$, $x_2$, and $x_1x_2$; the additive model contains only the first three. The additive model is therefore a special case of the interaction model, obtained by setting the product-term parameter to zero.

This is a statement about model form, not a conclusion about which model should be used. Whether a product term is useful depends on the subject-matter question and evidence from the data. Nor does including it imply that the predictors cause the outcome: regression describes a conditional relationship under its assumptions.

Worked Example: Training Time and Starting Score

Suppose a hypothetical study records a trainee’s practice time in hours ($x_1$), a starting assessment score in points ($x_2$), and a later performance score in points ($Y$). For one trainee, the recorded values are $x_1=4$ hours, $x_2=60$ points, and $Y=72$ points. Specify the model with a product term and identify the term values for this observation.

The product is $x_1x_2=(4\text{ hours})(60\text{ points})=240\text{ hour-points}$. The model specification is

Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3(x_{1i}x_{2i})+\varepsilon_i.

For the stated trainee, the four non-error terms use values $1$, $4$, $60$, and $240$. Thus the corresponding model expression for the conditional mean is $\beta_0+\beta_1(4)+\beta_2(60)+\beta_3(240)$; no numerical mean can be calculated without parameter values. The units of $\beta_1$, $\beta_2$, and $\beta_3$ are respectively points per hour, points per point, and points per hour-point. The additive counterpart is obtained by omitting $\beta_3(x_{1i}x_{2i})$, equivalently setting $\beta_3=0$.

Worked Example: Temperature and Humidity

Imagine a hypothetical set of building observations where energy use $Y$ is recorded in kilowatt-hours, outdoor temperature $x_1$ in degrees Celsius, and relative humidity $x_2$ in percentage points. One observation has temperature $18^\circ\mathrm{C}$, humidity $50$ percentage points, and energy use $42$ kilowatt-hours. Write the product-term model and check the product’s units.

The product for this observation is $18\times50=900$ degree-Celsius percentage-points. The model is

Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3(x_{1i}x_{2i})+\varepsilon_i.

The product term for the observation is $\beta_3(900)$, with $900$ carrying units of degree-Celsius percentage-points. Since each term in the conditional mean must have the outcome’s units, $\beta_1$ is measured in kilowatt-hours per degree Celsius, $\beta_2$ in kilowatt-hours per percentage point, and $\beta_3$ in kilowatt-hours per degree-Celsius percentage-point. The additive specification for the same predictors retains the intercept, temperature, and humidity terms but excludes their product.

Here humidity is recorded on a 0-to-100 percentage-point scale. If it were instead recorded as a proportion from 0 to 1, the numerical product values and the corresponding product coefficient’s units would change. A model must therefore state the predictor scales clearly.

Worked Example: Building the Product Column

For a hypothetical set of observations, let $x_1$ be study hours, $x_2$ be attendance percentage points, and $Y$ be a course score in points. The observed predictor values are shown below. Form the product column and state the complete model specification.

Observation	Study hours $x_1$	Attendance points $x_2$	Product $x_1x_2$
1	2	80	160
2	5	90	450
3	3	70	210

For observation 1, $2\times80=160$; for observation 2, $5\times90=450$; and for observation 3, $3\times70=210$. The full model uses each row’s outcome with its study-hours value, attendance value, and calculated product:

Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3(x_{1i}x_{2i})+\varepsilon_i.

Here $x_1x_2$ is measured in hour-percentage-points, so $\beta_3$ has units of course-score points per hour-percentage-point. The additive version keeps the first three terms on the right-hand side and removes $\beta_3(x_{1i}x_{2i})$. The table supplies predictor values only; it does not supply parameter estimates or establish which specification fits better.

A Short Check on the Model Forms

The distinction between the two equations can be verified by substitution rather than by a special rule. Start with the interaction model and impose the restriction $\beta_3=0$:

\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3(x_{1i}x_{2i})+\varepsilon_i \quad\overset{\beta_3=0}{=}\quad \beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\varepsilon_i.

The resulting expression is exactly the additive model. Conversely, if $\beta_3$ is not constrained to zero, the specified mean includes an additional term involving both predictors. This confirms the basic structural distinction: the product-term model allows a joint component, while the additive form does not include one.

Common Specification Errors

Multiplying the outcome by a predictor. The interaction term is $x_1x_2$, not $Yx_1$, $Yx_2$, or $Yx_1x_2$. The outcome remains on the left side of the regression equation.
Forgetting a predictor’s separate term. A model containing $x_1x_2$ but omitting one or both of $x_1$ and $x_2$ is a different specification. When the intended model includes a product interaction, retain both lower-order predictor terms.
Calling the additive model an interaction model. Separate $x_1$ and $x_2$ terms alone do not create the product-term specification. The equation needs an explicit $x_1x_2$ term.
Ignoring scales and units. If $x_2$ is in percentage points rather than proportions, the numerical product and product coefficient’s units must reflect that choice.
Reading specification as proof of causation. Including a product term describes a model form; it does not show that either predictor causes changes in the outcome.

A reliable specification names the outcome and both quantitative predictors, states their measurement units, writes the product explicitly, and retains the separate predictor terms when an interaction model is intended. Keep the additive equation available as the clearly defined alternative, with the product coefficient fixed at zero.

Check Your Understanding

Use the equation structure and unit checks from this tutorial to answer the following questions.

Write the product-term model when $x_1$ is measured in months, $x_2$ in dollars, and $Y$ in units of output.
What restriction on the interaction model’s parameters gives the additive counterpart?
If $x_1=6$ hours and $x_2=75$ percentage points, calculate $x_1x_2$ and state its units.
In a model with $Y$ measured in points, $x_1$ in hours, and $x_2$ in years, what are the units of $\beta_3$?
Why is a model containing $x_1$ and $x_2$, but not $x_1x_2$, additive rather than an interaction model?