Introduction
Regression analysis is one of the most widely used statistical methods in data science, research, and applied mathematics. At its core, regression helps us understand the relationship between dependent and independent variables. Linear regression is often the first technique analysts turn to because it assumes a straight-line relationship between predictors (independent variables) and the outcome (dependent variable). However, real-world data often doesn’t follow a neat straight line.
Many natural and social phenomena are nonlinear in nature. For example, human decision-making is rarely linear—when choosing where to eat dinner, our choices are influenced by multiple nonlinear factors such as mood, taste preference, price, weather, and prior experiences. Similarly, processes in biology, economics, and engineering often display exponential, quadratic, or logistic behaviors. To model such complexities, nonlinear regression provides a far more accurate and flexible solution.
Nonlinear regression works similarly to linear regression—it attempts to fit a curve to observed data while minimizing errors. The key difference is that instead of a straight line, nonlinear regression fits a curve that can take different mathematical forms depending on the underlying relationship. In R, this process is implemented using the nls() function, which stands for nonlinear least squares.
Linear vs Nonlinear Regression: A Quick Illustration
In R, linear regression can be performed using the lm() function, while nonlinear regression requires the nls() function. To illustrate the difference, let’s consider a dataset that follows an exponential curve.
set.seed(23)
x <- seq(0, 100, 1)
y <- runif(1, 0, 20) * exp(runif(1, 0.005, 0.075) * x) + runif(101, 0, 5)
plot(x, y)
The plot shows a clear nonlinear pattern. If we apply linear regression:
lin_mod <- lm(y ~ x)
plot(x, y)
abline(lin_mod)
The fitted line fails to capture the data’s true shape, leaving large gaps between predictions and actual values.
Now let’s try nonlinear regression with an exponential model:
nonlin_mod <- nls(y ~ a * exp(b * x), start = list(a = 13, b = 0.1))
plot(x, y)
lines(x, predict(nonlin_mod), col = "red")
This curve fits the data much better, accurately passing through most points.
Error comparison shows the improvement:
error <- lin_mod$residuals
lm_error <- sqrt(mean(error^2)) # ~5.96
error2 <- y - predict(nonlin_mod)
nlm_error <- sqrt(mean(error2^2)) # ~1.52
The nonlinear model has less than one-third of the error compared to the linear model, proving that nonlinear regression is more effective for curved relationships.
Breaking Down the nls() Function
The nls() function is central to nonlinear regression in R. It requires two key inputs:
Formula – Defines the mathematical model, e.g., y ~ a * exp(b * x).
Start Values – Provides initial guesses for parameters.
For instance, in our exponential model:
a was chosen close to the minimum value of y (~13).
b was a small increment for the exponential growth rate.
R then iterates to estimate the best-fitting values:
summary(nonlin_mod)
Output:
a = 13.60391
b = 0.01911
These estimates are very close to our starting guesses, showing convergence. However, choosing poor starting values can cause the model to fail. For example, setting a = 1 and b = 1 might prevent convergence entirely.
The Challenge of Choosing Start Values
One major challenge with nonlinear regression is estimating appropriate starting values. Beginners often struggle because the choice of starting values determines whether the model converges to a solution.
Let’s consider the Puromycin dataset in R, which records enzyme reaction rates. The Michaelis-Menten equation, commonly used in enzyme kinetics, can model this data:
mm <- function(conc, vmax, k) vmax * conc / (k + conc)
mm1 <- nls(rate ~ mm(conc, vmax, k), data = Puromycin,
start = c(vmax = 50, k = 0.05), subset = state == "treated")
mm2 <- nls(rate ~ mm(conc, vmax, k), data = Puromycin,
start = c(vmax = 50, k = 0.05), subset = state == "untreated")
Even though our starting values (50, 0.05) differ from the actual data’s behavior (closer to 200, 0.01), the models still converged—though more slowly.
Self-Starting Functions in R
To solve the problem of picking starting values, R provides self-starting functions such as SSmicmen() for Michaelis-Menten equations. These functions automatically determine initial estimates, reducing manual effort.
mm3 <- nls(rate ~ SSmicmen(conc, vmax, k), data = Puromycin, subset = state == "treated")
mm4 <- nls(rate ~ SSmicmen(conc, vmax, k), data = Puromycin, subset = state == "untreated")
Comparisons show that both manual (mm1, mm2) and self-starting (mm3, mm4) models produce nearly identical coefficients, but the self-starting approach converges faster.
Available self-starting functions in R include:
SSasymp – asymptotic regression models
SSbiexp – biexponential models
SSfpl – four-parameter logistic models
SSgompertz – Gompertz growth models
SSlogis – logistic models
SSmicmen – Michaelis-Menten models
SSweibull – Weibull growth curve models
Goodness of Fit
After fitting a nonlinear model, it’s important to check how well it predicts actual data. The correlation between predicted and observed values provides a simple measure:
cor(y, predict(nonlin_mod)) # ~0.998
Values close to 1 indicate excellent fit, as seen with our examples. Both the Puromycin and exponential datasets show high correlations, confirming that nonlinear regression captured the underlying patterns effectively.
Summary
Nonlinear regression extends the power of regression analysis to situations where data doesn’t follow a straight-line relationship. In R, the nls() function makes it possible to estimate parameters of nonlinear models, provided we choose good starting values or use self-starting functions.
Key takeaways:
Linear regression struggles with curved data, but nonlinear regression handles it well.
Proper starting values are critical, but self-starting functions simplify the process.
R provides built-in datasets (like Puromycin) and functions (nls(), SSmicmen()) for practice.
Goodness-of-fit checks ensure reliable models.
As real-world data often exhibits nonlinear relationships, mastering nonlinear regression in R equips analysts with a powerful tool for accurate modeling in fields like biology, economics, and social sciences.
This article was originally published on Perceptive Analytics.
In United States, our mission is simple — to enable businesses to unlock value in data. For over 20 years, we’ve partnered with more than 100 clients — from Fortune 500 companies to mid-sized firms — helping them solve complex data analytics challenges. As a leading Microsoft excel expert, we turn raw data into strategic insights that drive better decisions.
Top comments (0)