For a non-linear relationship, transform the variables to obtain a straight-line law Y=mX+c, then regress Y on X. Interpret the gradient and intercept in the original model. **Non-linear relationship** | **Linear form Y=mX+c** | **Y-axis** | **X-axis** | **Conditions** y=axn | ln y=nln x+ln a | ln y | ln x | x>0, y>0, a>0 y=sqrt(ax+b) | y2=ax+b | y2 | x | y 0, ax+b 0 y=abx | ln y=xln b+ln a | ln y | x | a>0, b>0, b 1 xy=bx+a | y=a(1x)+b | y | 1x | x 0 y= 1ax+b | 1y=ax+b | 1y | x | y 0, ax+b 0 y=ax+ bx | xy=ax2+b | xy | x2 | x 0 Plot and regress the transformed coordinates named in the axis columns. Do not report the transformed gradient or intercept as an original parameter until the linear form has been matched term by term.

For a non-linear relationship, transform the variables to obtain a straight-line law \(Y=mX+c\), then regress \(Y\) on \(X\). Interpret the gradient and intercept in the original model.

Plot and regress the transformed coordinates named in the axis columns. Do not report the transformed gradient or intercept as an original parameter until the linear form has been matched term by term.

Non-linear relationship	Linear form \(Y=mX+c\)	\(Y\)-axis	\(X\)-axis	Conditions
\(y=ax^n\)	\(\ln y=n\ln x+\ln a\)	\(\ln y\)	\(\ln x\)	\(x>0,\ y>0,\ a>0\)
\(y=\sqrt{ax+b}\)	\(y^2=ax+b\)	\(y^2\)	\(x\)	\(y\geq0,\ ax+b\geq0\)
\(y=ab^x\)	\(\ln y=x\ln b+\ln a\)	\(\ln y\)	\(x\)	\(a>0,\ b>0,\ b\ne1\)
\(xy=bx+a\)	\(y=a\left(\frac1x\right)+b\)	\(y\)	\(\frac1x\)	\(x\ne0\)
\(y=\frac1{ax+b}\)	\(\frac1y=ax+b\)	\(\frac1y\)	\(x\)	\(y\ne0,\ ax+b\ne0\)
\(y=ax+\frac bx\)	\(xy=ax^2+b\)	\(xy\)	\(x^2\)	\(x\ne0\)

Linearisation of Variables

Linearisation of Variables