Build on multiple regression by adding interaction terms (moderation effects) and quadratic terms (non-linear relationships). Understand how the effect of one predictor depends on another, or how relationships curve rather than staying linear.
This tool extends multiple linear regression to handle two powerful concepts:
With interaction term: $$ Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3 (X_{1i} \times X_{2i}) + \varepsilon_i $$
The interaction coefficient (\(\beta_3\)) tells you how much the effect of X₁ changes for each unit increase in X₂. Key insight: If \(\beta_3\) is significant, the relationship between X₁ and Y is not constant—it varies depending on X₂.
With quadratic term: $$ Y_i = \beta_0 + \beta_1 X_{i} + \beta_2 X_{i}^2 + \varepsilon_i $$
The quadratic coefficient (\(\beta_2\)) captures curvature. If \(\beta_2 < 0\), the relationship is an inverted U (increases then decreases). If \(\beta_2 > 0\), it's a U-shape (decreases then increases). Business relevance: Find optimal points (e.g., ideal price, optimal ad frequency).
Real-world effects rarely operate in isolation. Moderation (statistical interactions) recognizes that contexts matter:
Managerial implication: One-size-fits-all strategies miss opportunities. If an interaction is significant, you should tailor your approach based on the moderator.
When including interaction terms, mean-centering continuous predictors is recommended (and enabled by default in this tool). Here's why:
Example: If studying ad_spend × seasonality, centering ad_spend means the season coefficients represent effects "at average ad spend levels," not at zero spend (which never happens).
Advanced users can disable centering, but interpretations become more complex.
When an interaction is significant, the next step is simple slopes analysis: testing whether the focal predictor's effect is significant at different levels of the moderator.
For continuous moderators, we conventionally test at three levels:
For categorical moderators, we test the focal predictor effect within each category separately.
This tool visualizes these simple slopes in the interaction plots, making it easy to see where effects are strong vs. weak.
Quadratic effects capture non-monotonic relationships—where "more is better" only up to a point, then becomes "too much of a good thing."
The turning point (maximum or minimum) occurs at: $$ X^* = -\frac{\beta_1}{2 \beta_2} $$
Business applications:
Interpretation note: Check that the optimal point falls within your observed data range. Extrapolating beyond observed values is risky.
For educational clarity, this tool restricts you to one interaction or quadratic effect per model. This is not a limitation of regression in general—real models often include multiple interactions.
Why this restriction helps learning:
For professional analysis requiring multiple interactions or three-way interactions, use statistical software like R, Python, SPSS, or Stata.
Use presets to explore realistic scenarios demonstrating interactions and non-linear effects in marketing, pricing, and gaming contexts. Each scenario can be downloaded, edited in Excel, and re-uploaded.
Upload a CSV file with raw case-level data. Include one outcome variable and multiple predictors (numeric or categorical).
Drag & Drop raw data file (.csv, .tsv, .txt)
Include headers; at least one numeric outcome and 2+ predictors.
No file uploaded.
Set the significance level for hypothesis tests and confidence intervals.
Centering improves interpretability and reduces multicollinearity. Main effects then represent effects "at average levels" of other variables. Disable only if you have specific reasons.
Toggle visibility of shaded confidence intervals around predicted lines. Useful for assessing uncertainty in simple slopes.
Each point compares an observed outcome to its predicted value. Points near the 45-degree line indicate better fit. This plot helps diagnose whether the model (including interactions/quadratic) adequately captures the data structure.
| Variable | Mean | Median | Std. Dev. | Min | Max |
|---|---|---|---|---|---|
| Provide data to see summary statistics. | |||||
| Predictor | Level | Percent |
|---|---|---|
| Provide data to see level percentages. | ||
Provide data to see the fitted regression equation.
| Predictor | Term | Estimate | Std. Error | t | p-value | Lower CI | Upper CI |
|---|---|---|---|---|---|---|---|
| Provide data to see coefficient estimates. | |||||||
Run the analysis to see checks on multicollinearity, residual patterns, and model fit.