# Load libraries (assuming 'lmtest' and 'nlme' are installed) library(lmtest) library(nlme) # Read data (replace "path/to/tuna2.csv" with your actual data path) data <- read.csv("path/to/tuna2.csv") # Define the model formula model_formula <- log(St.Effort) ~ Tt # Linear regression with lm() lm_fit <- lm(model_formula, data = data) # Breusch-Pagan test for autocorrelation bp_test <- lmtest::bptest(lm_fit) # Check if p-value is significant for autocorrelation (usually < 0.05) if (bp_test$p.value < 0.05) { cat("Autocorrelation is an issue. ") } else { cat("Autocorrelation is not a significant issue. ") } # Robust SE approach with lmrob() lmrob_fit <- lmrob(model_formula, data = data) cat("Slope estimate (lmrob):", round(lmrob_fit$coefficients[["Tt"]], 3), " ") # t-statistic for slope (lmrob) t_lmrob <- lmrob_fit$coefficients[["Tt"]] / lmrob_fit$robust.se[["Tt"]] cat("t-statistic (lmrob):", round(t_lmrob, 1), " ") # gls() approach with AR(1) structure gls_fit <- gls(model_formula, correlation = corAR1(form = ~ 1), data = data) cat("Slope estimate (gls):", round(gls_fit$coefficients[["Tt"]], 3), " ") # t-statistic for slope (gls) t_gls <- gls_fit$coefficients[["Tt"]] / sqrt(gls_fit$varComp[1]) cat("t-statistic (gls):", round(t_gls, 1), " ") show how to do this simply in R