# Nonlinear vs. Linear Regression: Key Differences Explained

![Nonlinear Regression: A form of regression analysis in which data is fit to a model and then expressed as a mathematical function.](https://www.investopedia.com/thmb/IO5ZSihG2AZpR68xq625vO4EuGU=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Nonlinear-regression_final-eb6cf9f18e794d63ab4ec9363f665f6b.png)

*— Investopedia / Michela Buttignol*

## Nonlinear vs. Linear Regression

Nonlinear regression is regression analysis that's used to model data that forms a curved relationship rather than a straight line. That is, simple linear [regression](https://www.investopedia.com/terms/r/regression.asp) links two variables, X and Y, with a straight line (y = mx + b), while nonlinear regression connects them in a curved line.

Nonlinear and linear regression both seek to graphically track a response from variables. Nonlinear models are more complex and harder to develop because they use iterative approximations, sometimes by trial and error. Established methods, such as the Gauss-Newton and the Levenberg-Marquardt, contribute to that complexity.

The model aims to minimize the [sum of the squares](https://www.investopedia.com/terms/s/sum-of-squares.asp), which measures how far Y observations differ from the nonlinear function predicting Y.

Often, regression models that appear nonlinear upon first glance are actually linear. The curve estimation procedure can be used to identify the nature of the functional relationships at play in your data, so you can choose the correct regression model, whether linear or nonlinear. Linear regression models, while they typically form a straight line, can also form curves, depending on the form of the linear regression equation. Likewise, it’s possible to use algebra to transform a nonlinear equation so that it mimics a linear equation—such a nonlinear equation is referred to as “intrinsically linear.”

Compute by finding the difference between the fitted nonlinear function and each Y data point, square those differences, then add them together. The smaller the sum of these squared figures, the better the function fits the data points in the set.

### Key Takeaways

*   Nonlinear regression involves fitting data to a model using mathematical functions beyond a straight line, allowing for the modeling of complex relationships between variables.
*   The primary goal of nonlinear regression is to minimize the sum of squared differences between observed data points and the model’s predictions, thereby improving model accuracy.
*   Nonlinear regression requires careful selection of the function to describe the relationship between variables, and precise starting values to ensure that the model converges to an accurate solution.
*   Consider using nonlinear regression for modeling phenomena with naturally curved data patterns, such as population growth over time, which cannot be effectively captured by linear regression.
*   The development and application of nonlinear regression models can be challenging, relying on iterative approximation methods like the Gauss-Newton or Levenberg-Marquardt methods.

> **Fast Fact**
>
> Linear regression relates two variables with a straight line; nonlinear regression relates the variables using a curve.

## Applications of Nonlinear Regression

One example of how nonlinear regression can be used is to predict population growth over time. A scatterplot of changing population data over time shows that there seems to be a relationship between time and population growth, but that it is a nonlinear relationship, requiring the use of a nonlinear regression model. A logistic population growth model can provide estimates of the population for periods that were not measured, and predictions of future population growth.

Independent and dependent variables used in nonlinear regression should be quantitative. Categorical variables, like region of residence or religion, should be coded as binary variables or other types of quantitative variables.

In order to obtain accurate results from the nonlinear regression model, you should make sure the function you specify describes the relationship between the independent and dependent variables accurately. Good starting values are also necessary. Poor starting values may result in a model that fails to converge, or a solution that is only optimal locally, rather than globally, even if you’ve specified the right functional form for the model.

## The Bottom Line

While linear regression uses a straight line to relate variables, nonlinear regression uses a curved relationship. Nonlinear regression can offer insights into complex relationships, such as predicting population growth over time. It's important to select the correct regression model for the goals established by understanding the nature of the data and the relationships between variables. Likewise, to ensure accurate representation and prediction, it's crucial that nonlinear regression models have good starting values.

> **Article Sources**
>
> 1.  IBM. "[Nonlinear Regression](https://www.ibm.com/docs/en/spss-statistics/25.0.0?topic=regression-nonlinear)."