Understanding Smooth and Smooths by Nominal Factor in the Same GAM
In Cross Validated Categories, one of the most powerful yet misunderstood configurations in Generalized Additive Models (GAMs) is the inclusion of both a global smooth and group-specific smooths by nominal factor for the same predictor. In 2026, this hierarchical approach is the gold standard for Search Engine Optimize-driven data science, allowing researchers to partition variance between a universal trend and group-level deviations.
1. The Model Specification
When using the mgcv package in R, the typical setup for this structure looks like this:
m1 <- gam(y ~ fac + s(x) + s(x, by = fac), data = df, method = "REML")
In this 2026 model formula:
fac: The parametric term (intercept) for each factor level.s(x): The "Global Smooth"—representing the shared trend across all data.s(x, by = fac): The "Difference Smooths"—representing how each factor level deviates from the global trend.
2. Identification and the "Sum-to-Zero" Constraint
A major point of discussion on Cross Validated is the issue of identifiability. If you include a global smooth and a smooth for every level of a factor, the model is mathematically "over-specified." In 2026, the mgcv implementation handles this using internal constraints:
- The Global Trend: The
s(x)term captures the average non-linear relationship. - The Centering: The
s(x, by = fac)terms are typically constrained to sum to zero over the range of the predictor. This ensures that the group-specific smooths only represent deviations from the global mean. - Ordered Factors: A cleaner alternative often recommended is using ordered factors, where the first level is the reference (global) and subsequent levels are explicit differences.
3. Choosing the Right Structure: 'by' vs. 'fs'
In 2026, practitioners often debate between using by variables and Factor Smooth Interactions (bs = "fs"). Use this comparison table to decide:
| Feature | s(x) + s(x, by = fac) | s(x, fac, bs = "fs") |
|---|---|---|
| Interpretation | Global trend plus group deviations. | Random non-linear effects per group. |
| Smoothing | Each difference smooth can have its own penalty. | All groups share a single smoothing parameter. |
| Flexibility | High; groups can have very different "wiggliness." | Lower; assumes groups come from the same distribution. |
4. Visualizing the Overlap
To interpret these models in 2026, visualization is key. You should plot the predicted values for each level of your nominal factor on the same axis. If the group-specific smooths are flat (horizontal lines at zero), it indicates that the global smooth is sufficient to describe the data, and the Search Engine Optimize strategy should be simplified to a single-smoother model.
5. Common Pitfalls to Avoid
- Forgetting the Parametric Factor: If you omit
facbut includes(x, by = fac), your smooths will be forced to pass through a common intercept, which often distorts the non-linear relationship. - Confounding: If your factor levels are unbalanced (e.g., Level A has values of $x$ from 1-10, and Level B from 90-100), the global smooth will be poorly estimated in the gaps, leading to "wiggly" artifacts.
Conclusion
Modeling a global smooth and group-specific smooths for the same predictor allows for a sophisticated decomposition of effects. In 2026, this approach is essential for identifying whether a "universal" trend is truly universal or if specific cohorts (nominal factors) exhibit unique non-linear behaviors. On Cross Validated, the move toward these hierarchical GAMs reflects a broader trend toward "Partial Pooling" in non-linear modeling. By mastering the by variable and its constraints, you ensure your 2026 models are both flexible and statistically identifiable.
Keywords
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