Master Linear Regression in R: From Random Data to Insightful Models

When studying two random variables with correlation, let Y be the dependent variable and X the independent variable. The error term ε follows a normal distribution. The regression coefficients β0 (intercept) and β1 (slope) are estimated by ordinary least squares, minimizing the sum of squared differences between observed values and predicted values. Assumptions: independence, homoscedasticity, normality.

In R, the lm() function implements linear models and can perform simple regression, multiple regression, and ANOVA. Its basic syntax is:

lm(formula, data, subset, weights, na.action, method = "qr", model = TRUE, x = FALSE, y = FALSE, qr = TRUE, singular.ok = TRUE, contrasts = NULL, offset, ...)

Example: a simple linear regression using a randomly generated dataset.

x <- rnorm(20, 4, 1)
y <- 2*x + rnorm(20)
lm.xy <- lm(formula = y ~ x)
summary(lm.xy)

The output shows the estimated intercept (≈2.05) and slope (≈1.42), their standard errors, t‑values, and p‑values, as well as residual standard error, R‑squared (≈0.72), adjusted R‑squared, and the F‑statistic, indicating a highly significant linear relationship.

Using summary(lm.xy) provides coefficient estimates, hypothesis tests, and goodness‑of‑fit measures that correspond to the results of a correlation test ( cor.test()). Note that regression results can be sensitive to sample selection and outliers, so diagnostics are important.

Source: Liu Hongde, Sun Xiao, Xie Jianming, Bio Data Analysis and Practice