Perform Polynomial Evaluation of Linearity

Evaluates whether the relationship between two vectors is linear or nonlinear. Performs a test to determine how well a linear model fits the data compared to higher order polynomial models. Jhang et al. (2004) .

R package for polynomial evaluation of linearity.

About

lin.eval is a R package for performing polynomial evaluation of linearity.

Installation

lin.eval can be installed via Github:

if (!require(devtools)) {  
    install.packages('devtools')  
}  
devtools::install_github('vishesh-shrivastav/lin.eval')

How it works

Polynomial evaluation of linearity is a technique of assessing if the best way to describe the relationship between two vectors.

1. Fit three models - linear, second-order polynomial and third-order polynomial

2. Find out best-fitting model among the three by comparing their p-values. Model with the lowest p-value out of the three is the best-fitting one.

3. If the best-fitting model is linear, linearity is established and no further steps need to be carried out. This is called Linear 1 type.

4. Else, best-fitting model is either second or third order polynomoal model. In this case, calculate average deviation from linearity (adl). This is given by:

   

   where, l is the vector of predictions from linear model and p is the vector of predictions from best-fitting polynomial model.

5. If adl is greater than or equal to the threshold value for deviation from linearity, conclude that the relationship is non-linear.

6. Else if adl is less than the threshold value for deviation from linearity, conclude that although the best-fitting model is not linear, deviation from linearity is not significant and hence, it is still a linear relationship. This is called a Linear 2 type.

Usage

Call the poly_eval() function with the following parameters:
y: vector of response values
x: vector of predictor values
threshold: threshold value for average deviation from linearity as percentage. Defaults to 5.

> foo <- c(165.3929, 165.3929, 1119.5714, 1119.5714, 2073.7500, 2073.7500, 3027.9286, 3027.9286, 3982.1071, 3982.1071, 4936.2857, 4936.2857, 5890.4643, 5890.4643)
> bar <- c(386.2143,  386.2143, 840.6548, 840.6548, 1829.6905, 1829.6905, 3074.4048, 3074.4048, 4295.8810, 4295.8810, 5215.2024, 5215.2024, 5553.4524, 5553.4524)
> derp <- poly_eval(bar, foo, 30)
Best fitting model is third-order polynomial.
Computing average deviation from linearity:
Average Deviation from Linearity: 27.28 %
Although the best fitting model is nonlinear, since average deviation from linearity is 27.28; which is less than or equal to 30; linearity is established. We call this linearity type as Linear 2

You can check the values stored in the result variable:

> derp$p1
[1] 8.851095e-12
> derp$p2
[1] 2.514044e-10
> derp$p3
[1] 1.930392e-78
> derp$adl
[1] 27.28302

More examples

Usage without passing in optional argument for adl:

> xx <- c(0, 1, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30)
> yy <- c(126.6, 101.8, 71.6, 101.6, 68.1, 62.9, 45.5, 41.9, 46.3, 34.1, 38.2, 41.7, 24.7, 41.5, 36.6, 19.6, 22.8, 29.6, 23.5, 15.3, 13.4, 26.8, 9.8, 18.8, 25.9, 19.3)
> poly_eval(yy, xx)
Best fitting model is second-order polynomial.
Computing average deviation from linearity...
Average Deviation from Linearity: 70.42 %
Since, average deviation from linearity is greater than 5, nonlinearity is established.
The relationship between the two input vectors is best described by a second order polynomial