The Difference between Linear Regression and Nonlinear Regression Models

Linear regression relates two variables with a straight line; nonlinear regression relates the variables using a curve. Minitab’s fitted line plot conveniently has the option to log-transform one or both sides of the model. So I’ve transformed just the predictor variable in the fitted line plot below. In the scatterplot below, I used the equations to plot fitted points for both models in the natural scale.

Transforming the Variables with Log Functions in Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables, meaning that the relationship can be represented by a straight line. On the other hand, nonlinear regression is a method used when the relationship between the variables is not linear and cannot be adequately represented by a straight line. Nonlinear regression models can take various forms, such as exponential, logarithmic, or polynomial, allowing for more flexibility in capturing complex relationships.

Linear regression shall better fit superficial relationships representing a straight line. On the other hand, data flexibility in nonlinear regression corresponds to higher nature data in handling. 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. Nonlinear regression can be a powerful alternative to linear regression because it provides the most flexible curve-fitting functionality. The trick is to find the nonlinear function that best fits the specific curve in your data.

Is a linear equation where X1, X2 are feature variables and W1, W2 are parameters. Nonlinear regression uses a different procedure than linear regression to minimize the sum of squares of the residual error (SSE). For a basic understanding of nonlinear regression, it is important to understand the similarities and differences between it and linear regression. From the above plot, we can see that there is a randomness in the residuals, explaining the variance and hence satisfying the assumption of the regression model.

Assumptions

In model III, the covariates were adjusted based on the principle that when added to the model, the matched odds ratio should change by at least 10%19. Account for non-linear correlation between DM duration and prevalence of DR, we also used generalized additive model (GAM) and a fitted smoothing curve (penalized spline method) to address nonlinearity. Besides, in cases where a nonlinear correlation was observed, a two-piecewise logistic regression model was utilized to calculate the saturation effect of DM duration on DR through a smoothing plot. The inflection point, at which the maximum likelihood model was applied, was automatically calculated using a recursive method when the ratio between DR and DM duration distinctly appeared in the smoothed curve20. Linear regression models the relationship between the independent and dependent variables with a straight line, while non-linear regression models more complex, non-linear relationships.

Define Sigmoid Function

For a model to be considered non-linear, Y hat must be a non-linear function of the parameters Theta, not necessarily the features X. When it comes to the non-linear equation, it can be the shape of exponential, logarithmic, logistic, or many other types. To overcome this problem, the model should incorporate time and distance as a factor.

1. This article delves into the key differences between these models, their applications, and their advantages and limitations.
2. While the flexibility to specify many different expectation functions is very powerful, it can also require great effort to determine the function that provides the optimal fit for your data.
3. For a linear regression model, the estimates of the parameters are unbiased, are normally distributed, and have the minimum possible variance among a class of estimators known as regular estimators.
4. The model is trained on a subset then makes predictions for withheld observations.

👉Residual Plot for Quadratic Model

Thus, the mean of \(y\) is a linear function of \(x\) although the variance of y does not depend on the value of \(x\). Furthermore, because the errors are uncorrelated, the response variables are also uncorrelated. Thus while powerful, non-linear models need expertise and care to yield their full potential while avoiding pitfalls. The tradeoff between flexibility and interpretability needs evaluation. In these examples, linear regression provides a computationally efficient way to difference between linear and nonlinear regression uncover trends and patterns that deliver actionable insights. The simplicity of the linear model also makes it easy to explain and trust the predictions.