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Recommend a transformed regression model

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Two transformed models have residual plots and r² values. What should be checked first?

Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, Calculator allowedBuild a worksheet

Comparing transformed models

Use residual evidence first. Make the response scale and comparison rule explicit, then qualify a prediction using the original input range and the purpose of the model

\[r^{2}=\left(r\right)^{2},\quad\text{explained percentage}=100r^{2}\]
  1. Use shape to find candidates A scatterplot can suggest transformations to try, but it does not establish a final model \(\displaystyle \text{shape}\Rightarrow\text{candidate set}\)
  2. Inspect residual structure Reject a candidate whose residual plot has a clear curve, trend or changing spread around zero \(\displaystyle \text{clear residual pattern}\Rightarrow\text{reject model}\)
  3. State the response scale For the same observations and unchanged response, residual sum of squares and r² compare variation on a common scale. If the response transformation changes, the percentages refer to different response coordinates. Do not call a larger percentage proof of a better original-response prediction \(\displaystyle r^2=1-\frac{\mathrm{SSE}}{\mathrm{SST}}\)
  4. Qualify the recommendation Use the selected model for prediction and judge interpolation or extrapolation from the original explanatory range \(\displaystyle x\notin[10,40]\Rightarrow\text{extrapolation}\)

Worked example

Two candidate transformed models are fitted to the same paired data. Model A has a curved residual plot and r = 0.92. Model B has no clear residual pattern and r = 0.78. Recommend a model using the residual evidence and r²

\[A:\ r=0.92\text{ with a curved residual plot},\quad B:\ r=0.78\text{ with no clear residual pattern}\]
  1. \(\displaystyle A:\text{curved residuals}\Rightarrow\text{reject A}\)The clear curve is evidence against the linearity assumption for model A.
  2. \(\displaystyle B:\text{no clear residual pattern}\Rightarrow\text{retain B}\)Model B passes the residual-structure check
  3. \(\displaystyle r_B^{2}=\left(0.78\right)^{2}=0.6084=60.84\%\)Square the acceptable model's correlation to find its explained-variation percentage
  4. \(\displaystyle \text{recommend model B}\)Only model B has an acceptable residual plot, so it is selected even though model A has a larger r²

Common mistakes

Comparing r² before residuals
✕\(\displaystyle A:\ r^{2}=0.81\text{ and patterned},\quad B:\ r^{2}=0.64\text{ and unpatterned}\Rightarrow\text{recommend A}\)✓\(\displaystyle A:\ r^{2}=0.81\text{ and patterned}\Rightarrow\text{reject A};\quad B\Rightarrow\text{recommend B}\)
A model with patterned residuals is rejected before its r² is compared with an acceptable model
Treating a candidate as guaranteed
✕\(\displaystyle \text{curved raw scatter}\Rightarrow\text{the listed transformation is accepted}\)✓\(\displaystyle \text{curved raw scatter}\Rightarrow\text{try the transformation and inspect residuals}\)
A scatterplot shape only suggests transformations for trial; it does not prove that a model is linear
Using r instead of r²
✕\(\displaystyle r=0.8\Rightarrow\text{explained variation}=80\%\)✓\(\displaystyle r=0.8\Rightarrow r^{2}=0.64\Rightarrow\text{explained variation}=64\%\)
The coefficient of determination is the square of r before it is converted to a percentage

Difficulty levels

introCompare two residual plots and select the model with no clear residual pattern
standardCompare r² values after both candidate residual plots support linearity
challengeComplete the candidate, residual, r², prediction and reliability chain
examChoose a recommendation tied to the correct order of residual and explained-variation evidence

Questions about this topic

Why are residual plots checked before r²?+

A model can have a large r² while its residuals still show curvature, a trend or changing spread. The residual plot checks whether a linear model is appropriate

What does r² measure for a transformed model?+

It measures the proportion of variation in the fitted response coordinate explained by variation in the fitted explanatory coordinate

What if two acceptable models have the same reported percentage?+

Equal rounded percentages need not mean equal exact r². Use an explicitly stated task preference if one is supplied. Without one, report a tie at the shown precision and consider purpose, domain, interpretation and further validation rather than inventing a unique winner

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 4: Data transformation

In this chapter

Apply a transformation to one variableSelect plausible transformations from scatterplot shapeFit and write a transformed least-squares equationPredict from a transformed regression modelUse a residual plot to assess a transformed modelRecommend a transformed regression model