For a linear regression model:

Y = β_{0} + β_{1}X + β_{2}Z + **β _{3}XZ** + ε

If the coefficient of the interaction term β_{3} is statistically significant, then there is evidence of an interaction between X and Z. This means that the effect of X on the outcome Y is different for different sub-categories of Z, and vice-versa.

Information that should be reported:

- Why you included the interaction term in your regression model
- The regression output table: including the coefficients, standard errors, and p-values of the main effects and the interaction term
- The interpretation of the cause of the interaction

## 1. Reporting the reason for including the interaction term

The METHODS section of the research paper is where you justify testing for an interaction between 2 variables.

There are many reasons for adding an interaction term between 2 predictors in a regression model including:

- When they have large main effects.
- When the effect of one changes for various subgroups of the other.
- When the interaction has been proven in previous studies.
- When you want to explore new hypotheses.

(For a detailed discussion of these reasons, refer to: Why and When to Include Interactions)

Here are 2 examples of how to report such information:

**Example 1:** *“Since X and Z have large main effects on the outcome Y, we anticipate a possible interaction between the two. In order to evaluate such interaction, we will compare the fit (increase in R-squared and decrease in residual standard error) of a linear regression model with the interaction to one without the interaction”*.

**Example 2:** *“Based on a previous study that showed a statistically significant interaction between X and Z, we included an interaction term to model possible variation in the effect of X for different values of Z”*.

## 2. Reporting the model output

The following should be included in the RESULTS section of the research paper:

- The regression table: including the coefficients, standard errors, and p-values of the main effects and the interaction term.
- The interpretation of the coefficient of the interaction term.
- A comparison between a model with the interaction and a model without the interaction: For instance, comparing the models’ R-squared, AIC, or residual standard deviations.

**For example:**

*“Our model suggests that the presence of X and Z together has a larger effect on Y than the sum of each. Specifically, the presence of Z increases the effectiveness of X on Y by 0.189 (p < 0.001). This interaction explains 4.2% of the variation in the outcome Y (corresponding to an increase in R-squared of the model without interaction from 76.6% to 80.8% for the model with interaction).”*

## 3. Interpreting the cause of the interaction

In the DISCUSSION section of the research paper, you should explore alternative explanations of your results.

Getting a statistically significant interaction is not enough to conclude that the interaction is real. In fact, you should present enough evidence to eliminate alternative explanations such as:

- Bias: By eliminating the possibility of differential bias across strata.
- Confounding: By adjusting for confounding in the regression model used to assess the interaction.
- Multiple testing: By lowering the threshold of statistical significance when exploring multiple interactions.

## References

- Szklo M, Nieto FJ.
*Epidemiology: Beyond the Basics*. 4th edition. Jones & Bartlett Learning; 2018. - Celentano D, Szklo M.
*Gordis Epidemiology*. 6th edition. Elsevier; 2018.