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Principles of Scientific Research

Linear and Nonlinear Relationships

by Cindy Ferraino

Fields of Study

Statistical Analysis; Relational Analysis

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Abstract

Linear and nonlinear relationships are elements that represent the statistical evaluation of a variable and a constant. A linear relationship produces a straight line on a graph, whereas a nonlinear relationship will not.

Principal Terms

  • direct relationship: a link between two variables in which if one variable increases, the other also increases, and if one decreases, the other also decreases.

  • inverse relationship: a link between two variables in which if one variable increases, the other decreases, and vice versa.

  • logarithmic: relating to the inverse of an exponent.

  • polynomial: a mathematical statement containing one or more variables and coefficients in which the only operations are addition, subtraction, multiplication, and non-negative exponentiation.

  • variable: in research or mathematics, an element that may take on different values under different conditions.

Straight or Curved Lines?

Scientists use the Cartesian coordinate system to evaluate the connection between data points from a set of variables that are plotted on a graph. René Descartes (1596–1650) and Pierre de Fermat (1601–65) figured out how to use ordered numbers with corresponding points to create algebraic equations. When graphed, the outcome resulted in a geometric shape that formed either a linear or nonlinear relationship.

In order for linear relationship to exist, three key features must be apparent: the equation can only contain two variables, the variables can only be taken to the first power, and the graph should display a straight line. The form of a linear relationship is often represented as

y = mx + b

In this equation, x and y represent the two variables, and b is a constant. (Sometimes c is used instead of b.) On a graph, this equation appears as a line on the xy plane with a slope of m and a y intercept (where y crosses the x axis) of b. The slope is the ratio between the change in x and the change in y. Slope can be either positive or negative. On a graph, a slope that increases from left to right is positive. A slope that decreases from left to right is negative.

When nonlinear relationships are plotted, they usually appear as curved lines. A linear approximation is a straight line that can be drawn tangent to part of the curve of interest. Linear approximations simplify complex functions such that a given area can be evaluated.

Determination of Linear and Nonlinear Relationships

Despite the different outcomes, similar procedures are followed to determine if a linear or nonlinear relationship exists. First, a plotting system, such as the Cartesian coordinate system, can be used to evaluate how the variables relate to data points on the graph. After the x and y variables are plotted on the graph, the set of data points will produce a geometric shape. If the shape is a straight line, then a linear relationship does exist. If the shape is curved, then a nonlinear relationship exists.

Another way to determine if a linear or nonlinear relationship exists is to model the relationship using equations and check their fit. Direct variation, or direct relationship, is a relationship where a change in one variable causes a proportional change in the other variable in the same direction. The equation for this relationship sets one variable as equal to the second variable multiplied by a constant value, as in

y = 3x

As x increases, y increases, and vice versa. Direct variation is linear.

Partial variation is a relationship that involves two variables with the addition of a fixed or constant value. One variable is equal to the value of the other variable multiplied by a constant value and changed with the addition of another constant value, which may be positive or negative. An example of a partial variation is

y = 3x + 2

Partial variation is linear as well.

Inverse variation, or inverse relationship, is a relationship where a change in one variable causes a proportional but opposite change in the other. One variable is equal to a constant divided by the second variable, as in

y = 30/x

Another way to describe this relationship is to say that one variable multiplied by the other variable is equal to a constant:

xy = 30

Thus, as x increases, y decreases, and vice versa. Inverse relationships are nonlinear.

Exponential, polynomial (including quadratic), logarithmic, and sinusoidal (wavelike trigonometric) equations all model nonlinear relationships as well.

Use in Statistical Analysis

Linear and nonlinear relationships can be used in statistical analysis of research data. In many studies, researchers are investigating whether a change in the value of an independent variable x causes a change in value in the dependent variable y. These values for x and y gathered from the study can be plotted, and a line of best fit can be drawn to try to account for the changes observed. How well that line accounts for all of the observed data points—in other words, how accurate the model is—can be determined through regression analysis. Regression calculates the R2 (pronounced “R squared”) value, or coefficient of determination, which expresses the ratio of the explained variation (that is, variation attributable to the independent variable) to the total variation. A higher R2 value usually indicates that the model fits the data well.

A coefficient of determination can be found if the presumed relationship between x and y is linear. For nonlinear relationships, linear approximations can be used to find the coefficient. However, an R2 value computed in this way is weaker than that of a linear relationship.

Applications

While linear relationships are simpler than nonlinear ones, this does not mean that they have less important applications in the real world. Telecommunications, for instance, often have a linear relationship between the input and output. There is a linear relationship between the bandwidth and data rate of a wireless communications channel, for example.

Equations describing nonlinear relationships are used to model some of the most complex phenomena found in nature, such as ocean currents or air flow around the wing of an aircraft. Fluid dynamics, a subfield of physics and engineering, studies many nonlinear relationships that are necessary for a wide range of applications, such as predicting weather or designing aircraft or oil and gas pipelines.

Linear and Nonlinear Relationships Sample Problem

At a local movie theater, the price of a box of candy is $2. The price of the candy is represented as the independent variable, x. The total price is represented as the dependent variable, y. The data is tabulated as follows:

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Using a linear model, the equation for the data would be

f(x) = 2x

with an R2 value 1. Using an exponential model, the equation for the data would be

f(x) = 1.6e0.4x

with an R2 value of 0.95. Based on the R2 values, which is the best model to use?

Answer:

Because the line with the best possible fit would have an R2 value of 1, the linear model is the best fit for this data. However, the R2 value of the exponential model is large enough that it could suffice if the linear model were not available.

Further Reading

1 

“Concepts: Linear and Nonlinear.” New England Complex Systems Institute, 2011, www.necsi.edu/guide/concepts/linearnonlinear.html. Accessed 13 Apr. 2017.

2 

Ellenberg, Jordan. How Not to Be Wrong: The Power of Mathematical Thinking. Penguin Books, 2014.

3 

Haslam, S. Alexander, and Craig McGarty. Research Methods and Statistics in Psychology. 2nd ed., Sage Publications, 2014.

4 

Lomax, Richard G., and Debbie L. Hahs-Vaughn. An Introduction to Statistical Concepts. 3rd ed., Routledge, 2012.

5 

Paulos, John Allen. Beyond Numeracy: Ruminations of a Numbers Man. Alfred A. Knopf, 1991.

6 

Rainville, Earl D., et al. Elementary Differential Equations. 8th ed., Prentice Hall, 1996.

Citation Types

Type
Format
MLA 9th
Ferraino, Cindy. "Linear And Nonlinear Relationships." Principles of Scientific Research, edited by Donald E. Franceschetti, Salem Press, 2017. Salem Online, online.salempress.com/articleDetails.do?articleName=POSR_0057.
APA 7th
Ferraino, C. (2017). Linear and Nonlinear Relationships. In D. E. Franceschetti (Ed.), Principles of Scientific Research. Salem Press. online.salempress.com.
CMOS 17th
Ferraino, Cindy. "Linear And Nonlinear Relationships." Edited by Donald E. Franceschetti. Principles of Scientific Research. Hackensack: Salem Press, 2017. Accessed December 14, 2025. online.salempress.com.