5.10 Evaluating a Transformed Linear Model

Assessing fit after transformation

After fitting a linear regression model to transformed data, check the fit the same way as any linear model: examine the residual plot for randomness (no pattern), check r² and s for strength and accuracy, and confirm the residuals show no leftover curvature. If residuals still show a pattern, the transformation didn't fully linearize the relationship.

Making predictions and transforming back

When the response variable was transformed (e.g., using ln(y)), predictions from the linear model are in the transformed scale. To get a prediction in the original units, you must back-transform using the inverse operation (e.g., exponentiate if you took a natural log). Forgetting to back-transform gives a nonsensical answer in the wrong units.

Comparing models

Use residual plots, r², and s to compare the original (untransformed) model against the transformed model. A good transformed model should have a more random residual plot, a higher r², and smaller s than the original nonlinear fit, indicating the transformation improved linearity and predictive accuracy.

Key terms

How do you evaluate a transformed linear model's fit?

Check the residual plot for random scatter (no pattern), and examine r² and s for strength and accuracy of the fit.

What must you do to predictions when the response variable y was log-transformed?

Back-transform (exponentiate) the predicted value to convert it from the transformed scale back to the original y units.

What residual plot pattern indicates a successful transformation?

A random scatter of points around zero with no curve or systematic pattern.

How do you compare an original nonlinear model to a transformed linear model?

Compare residual plots, r², and s — the better model typically has more random residuals, higher r², and lower s.

Why can't you rely on r² alone to judge a transformed model?

r² can be high even if curvature remains in the residuals, so you must also check the residual plot for randomness.

Try an AP-style question

A linear model's residual plot is curved; a log(y) model's residual plot is random. Which should you use?

1. Either one
2. Neither
3. The linear model
4. The log(y) model