9. Correlation & Simple Linear Regression: Modeling Scientific Relationships

Scientists frequently examine how two continuous environmental measurements relate. While Correlation evaluates the strength and direction of association between two variables symmetrically, Linear Regression models the functional dependence of a response variable ($Y$) on one or more predictor variables ($X$).

Ordinary Least Squares (OLS) Linear Regression

The mathematical model for simple linear regression is: $$Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \quad ext{where } \epsilon_i \sim \mathcal{N}(0, \sigma^2)$$ Where $\beta_0$ is the intercept, $\beta_1$ is the slope coefficient (rate of change in $Y$ per unit change in $X$), and $\epsilon_i$ represents unobserved random error.

The 4 Critical Regression Diagnostic Plots

In R, executing plot(model) produces four mandatory diagnostic plots:

1. Residuals vs. Fitted: Evaluates linearity. A horizontal line with points scattered symmetrically around 0 indicates linearity; curved patterns indicate non-linear dynamics.
2. Normal Q-Q: Evaluates normality of residuals. Residuals should align along the straight line.
3. Scale-Location (Spread-Location): Evaluates homoscedasticity. Points should form an even horizontal band with constant spread.
4. Residuals vs. Leverage: Identifies influential observations (Cook's distance $> 0.5$ or $> 1.0$) that exert disproportionate leverage on the regression slope.

Hands-On R Script: Modeling Water Temperature vs. Dissolved Oxygen

library(tidyverse)

# Physical chemistry dictates cold water holds more dissolved oxygen (Henry's Law)
set.seed(42)
temp_celsius <- runif(45, min = 8, max = 28)
# DO decreases roughly 0.25 mg/L per degree Celsius plus natural variation
dissolved_oxygen <- 13.8 - (0.28 * temp_celsius) + rnorm(45, mean = 0, sd = 0.5)

wq_data <- tibble(temp = temp_celsius, do = dissolved_oxygen)

# 1. Pearson Correlation Analysis
cor_test <- cor.test(wq_data$temp, wq_data$do, method = "pearson")
print(cor_test)
# Result: r = -0.96, p < 2.2e-16 (Extremely strong negative association)

# 2. Fit Ordinary Least Squares Linear Model
lm_model <- lm(do ~ temp, data = wq_data)
summary(lm_model)

# 3. Model Interpretation:
# - Intercept (beta0): ~13.8 mg/L (theoretical DO at 0 deg C)
# - Slope (beta1): ~ -0.28 (For every 1 deg C rise in water temp, DO drops by 0.28 mg/L)
# - Multiple R-squared: ~0.92 (92% of variation in DO is explained by temperature)

# 4. Display 4-Way Diagnostic Panels
par(mfrow = c(2, 2))
plot(lm_model)
par(mfrow = c(1, 1))