More on Non Linear Relationships, Splines

Abstract

The general principle of regression analysis is that the best fit line/exponential-curve/curvilinear-curve etc is calculated, i.e., the one with the shortest distances to the data, and it is, subsequently, tested how far the data are from the curve. A significant correlation between the y (outcome data) and the x (exposure data) means that the data are closer to the model than will happen purely by chance. The level of significance is usually tested, simply, with t-tests or analysis of variance. The initial regression model is almost always a linear model.

Appendix

In this appendix the mathematical equations of the non linear models as reviewed are given. They are, particularly, helpful for those trying to understand the assumed relationships between the dependent and independent variables, but can be disregarded by those without affection to maths (ln  =  natural logarithm).

$\text{y}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	linear
$\text{y}=\text{a}+\text{bx}+{\text{cx}}^{2}+{\text{dx}}^{3}+\dots$	polynomial
$\text{y}=\text{a}+\text{sinus x}+\text{cosinus x}+\dots$	Fourier
$\text{Ln odds}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	logistic
$\text{Ln multinomial odds}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	multinomial logistic
$\text{Ln hazard}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	Cox
$\text{Ln rate}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	Poisson

Instead of ln odds (= logit) also probit (≈ πÖ3 x logit) is often used for transforming binomial data.

	probit
$\text{log y}=\text{a}+{\text{b}}_{1}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}$	logarithmic
or
$\text{y}=\text{a}+{\text{b}}_{1}\mathrm{log}{\text{x}}_{1}+{\text{b}}_{2}{\text{x}}_{2}+\dots \dots \dots.{\text{b}}_{10}{\text{x}}_{10}\text{etc}$	“trial and error”
transformation function of y  =  (yλ−1)/λ with λ as power parameter	Box-Cox
y  =  (transformation function−1) a  +  b1 log x1  +  b2 x 2  +  ..	ACE modeling
$\text{y}={\text{e}}^{{\text{x}}_{1}{\text{x}}_{2}\mathrm{sin}{\text{x}}_{3}}\text{etc}$	AVAS modeling
$\text{y}=\text{a}+{\text{e}}^{{\text{b}}_{1}}+{\text{e}}^{{\text{b}}_{\text{2}}}$	multi-exponential modeling

θ  =  magnitude of x-value (example)

$$ \begin{array}{l}{\theta }_{1}<\rm{x}<{\theta }_{2}\rm{y}={\rm{a}}_{1}+{\rm{b}}_{1}{\rm{x}}^{3}\rm{spline\,modeling}\\ {\theta }_{2}<\rm{x}<{\theta }_{3}\rm{y}={\rm{a}}_{2}+{\rm{b}}_{2}{\rm{x}}^{3}\\ {\theta }_{3}<\rm{x}<{\theta }_{4}\rm{y}={\rm{a}}_{3}+{\rm{b}}_{3}{\rm{x}}^{3}\end{array} $$