⚙️ R – ANCOVA & Nonlinear Least Squares Regression Explained with Examples

---

🧲 Introduction – Advanced Statistical Modeling in R

When your data goes beyond simple linear assumptions, R provides two powerful tools:

• ANCOVA (Analysis of Covariance): Combines regression and ANOVA
• Nonlinear Least Squares (NLS): Fits curved models to data

These advanced models allow you to:

• Adjust for continuous covariates while comparing group means (ANCOVA)
• Fit nonlinear trends like exponential or logistic curves (NLS)

🎯 In this guide, you’ll learn:

• How to implement ANCOVA using aov() and lm()
• Use nls() for fitting nonlinear models
• Interpret coefficients, test interaction, and plot fitted curves

---

🔍 1. ANCOVA in R – Combine Regression and ANOVA

ANCOVA tests whether group means differ after adjusting for a continuous variable.

✅ Simulated Example

set.seed(123)
group <- factor(rep(c("A", "B"), each = 10))
x <- rnorm(20, mean = 50, sd = 5)
y <- 3 + 0.5 * x + ifelse(group == "B", 5, 0) + rnorm(20, 0, 2)

df <- data.frame(group, x, y)

✅ ANCOVA Model

model_ancova <- aov(y ~ x + group, data = df)
summary(model_ancova)

🔍 Interpretation:

• x: Continuous covariate
• group: Factor (A/B)
• Tests if group means differ after controlling for x

---

✅ Test for Interaction

model_interact <- aov(y ~ x * group, data = df)
summary(model_interact)

🧠 If interaction is not significant, use additive model (no x:group)

---

✅ Visualize ANCOVA

library(ggplot2)
ggplot(df, aes(x = x, y = y, color = group)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(title = "ANCOVA: Adjusted Means by Group")

---

🔁 2. Nonlinear Least Squares (NLS) in R

When the relationship between variables is curved, use nls() to fit nonlinear models.

✅ Simulated Data (Exponential Growth)

set.seed(123)
x <- 1:20
y <- 5 * exp(0.2 * x) + rnorm(20, 0, 5)
df_nls <- data.frame(x, y)

---

✅ Fit NLS Model

model_nls <- nls(y ~ a * exp(b * x), data = df_nls, start = list(a = 1, b = 0.1))
summary(model_nls)

🔍 Explanation:

• y ~ a * exp(b * x): Exponential model
• start: Required to provide initial guesses for parameters

---

✅ Extract Coefficients

coef(model_nls)

---

✅ Plot Curve Fit

plot(df_nls$x, df_nls$y, main = "NLS Curve Fit", xlab = "x", ylab = "y")
lines(df_nls$x, predict(model_nls), col = "blue", lwd = 2)

---

📐 When to Use ANCOVA vs NLS

Feature	ANCOVA	NLS (Nonlinear Least Squares)
Model Type	Linear + Categorical	Nonlinear (e.g., exponential)
Covariate Adjustment	Yes	Not required
Interactions Allowed	Yes (x * group)	Not typical
Data Pattern	Linear trends	Curved/nonlinear trends

---

📌 Summary – Recap & Next Steps

Both ANCOVA and NLS help model complex relationships in data:

• ANCOVA adjusts for continuous variables while comparing groups
• NLS fits non-linear curves when linear models fall short

🔍 Key Takeaways:

• Use aov(y ~ x + group) for ANCOVA
• Use nls() with a formula and start values for NLS
• Always plot your results to assess model fit visually
• Check for interaction terms in ANCOVA

⚙️ Real-World Relevance:
Used in experimental design, pharmacokinetics, growth modeling, economics, and behavioral studies.

---

❓ FAQs – ANCOVA & Nonlinear Regression in R

❓ When should I use ANCOVA?
✅ When you want to compare group means while adjusting for a continuous variable.

❓ What is the difference between aov() and lm()?
✅ Both fit linear models; aov() is used for ANOVA/ANCOVA-style summaries, lm() is more general-purpose.

❓ What is required for using nls() in R?
✅ A nonlinear formula and starting values for each parameter.

❓ How do I choose good start values for NLS?
✅ Plot your data, test rough values manually, or fit a linearized version for hints.

❓ Can NLS handle multiple predictors?
✅ Yes, but the model must still be nonlinear in parameters, not in variables alone.

---

Share Now :