Data Transformations and Linearisation

Scope Label

Core 9758. Data transformations are explicitly part of the current correlation-and-regression scope.

Role in the Topic

This branch explains what to do when the original scatter diagram is not approximately linear but suggests a structured non-linear relationship.

Use it with:

• Correlation and Linear Regression
• Scatter, Correlation, and PMCC
• Regression Lines and Estimation

Why Transform Data?

Correlation and linear regression are designed for linear patterns.

If the scatter diagram is curved, a straight-line regression on the original variables may be misleading. However, some curved relationships can be rewritten as straight-line relationships by transforming one or both variables.

This is called linearisation.

Caption: Linearisation transforms a curved relationship into a new variable pair that can be modelled linearly.

The aim is not to force every dataset into a line. The aim is to test a plausible model form.

The Linearisation Workflow

A reliable workflow is:

1. Inspect the original scatter diagram.
2. Identify a plausible non-linear model.
3. Define the transformed variable or variables.
4. Draw or inspect the transformed scatter diagram.
5. Compute $r$ for the transformed variables.
6. Fit a regression line to the transformed variables.
7. Interpret the fitted model in the original context.

The model with $|r|$ closest to $1$ is often better supported, but it should still make contextual sense.

Common Transformation Forms

The goal is to rewrite the model in the form

$$Y=A+BX$$

for suitable transformed variables $X$ and $Y$.

Original model	Transformation	Linear form
$y=a+\frac{b}{x}$	let $X=\frac{1}{x}$	$y=a+bX$
$y=a+b{x}^2$	let $X=x^2$	$y=a+bX$
$y=a+b\ln x$	let $X=\ln x$	$y=a+bX$
$y=a{b}^x$	let $Y=\ln y$	$Y=\ln a+x\ln b$
$y=a{x}^b$	let $X=\ln x$, $Y=\ln y$	$Y=\ln a+bX$

Logarithmic transformations require positive values where the logarithm is taken. For example, $\ln y$ is only defined when $y>0$.

Reading Parameters After Linearisation

After transformation, the regression line may not directly use the original parameters.

For example, suppose

$$y=a{b}^x.$$

Taking logarithms gives

$$\ln y=\ln a+x\ln b.$$

If the transformed regression line is

$$\ln y=A+Bx,$$

then

$$A=\ln a,B=\ln b.$$

So

$$a=e^A,b=e^B.$$

The intercept and gradient of the transformed line must be translated back carefully.

Choosing Between Models

When several possible transformations are suggested, compare:

• the transformed scatter diagrams
• the values of $|r|$ for the transformed data
• the reasonableness of the model in context
• the domain restrictions introduced by the transformation

Do not choose a model from calculator output alone.

A good exam sentence is:

The plot of $\ln y$ against $x$ is approximately linear and has $|r|$ closest to $1$, so an exponential model is most suitable among the proposed models.

Core Example: Power Model

Suppose a relationship is suspected to have the form

$$y=a{x}^b,x>0,y>0.$$

Taking logarithms:

$$\ln y=\ln a+b\ln x.$$

So if we plot $\ln y$ against $\ln x$ and obtain an approximately straight-line pattern, the model is supported.

If the fitted transformed line is

$$\ln y=1.2+0.75\ln x,$$

then

$$\ln a=1.2,b=\mathrm{0.75.}$$

Hence

$$a=e^{1.2},$$

and the fitted original model is

$$y=e^{1.2}{x}^{0.75}.$$

Core Example: Reciprocal Model

Suppose the data decrease quickly at first and then level off. A possible model is

$$y=a+\frac{b}{x}.$$

Let

$$X=\frac{1}{x}.$$

Then

$$y=a+bX.$$

So the transformed scatter diagram should plot $y$ against $\frac{1}{x}$, not $y$ against $x$.

Common Pitfalls

• Applying linear regression to a clearly curved original scatter diagram without considering transformation.
• Choosing the largest $|r|$ without checking the transformed scatter diagram.
• Forgetting domain restrictions such as $x>0$ or $y>0$ before taking logarithms.
• Treating the intercept and gradient of the transformed line as if they were always the original parameters.
• Back-transforming incorrectly, especially in exponential and power models.
• Forgetting to interpret the final model in the original variables.

Revision Checklist

• Can you explain why linearisation is useful?
• Can you match common model forms to the correct transformed variables?
• Can you decide whether to plot $y$ against $1/x$, $y$ against $x^2$, $\ln y$ against $x$, or $\ln y$ against $\ln x$?
• Can you recover original parameters after a logarithmic transformation?
• Can you explain why a transformed model still needs contextual judgement?