test_that("anytime-valid p-values are more conservative than standard p-values for linear models", {
  # Test with different precision parameters
  for (g_val in c(1, 2, 5)) {
    # Fit standard and anytime-valid linear models
    std_fit <- lm(mpg ~ wt + hp, data = mtcars)
    std_summary <- summary(std_fit)
    std_pvals <- std_summary$coefficients[, 4]
    
    av_fit <- av(std_fit, g = g_val)
    av_summary <- summary(av_fit)
    av_pvals <- av_summary$coefficients[, 4]
    
    # Check if all anytime-valid p-values are greater than or equal to standard p-values
    for (i in seq_along(std_pvals)) {
      expect_gte(av_pvals[i], std_pvals[i])
    }
  }
})

test_that("anytime-valid confidence intervals are supersets of standard confidence intervals for linear models", {
  # Test with different precision parameters and confidence levels
  for (g_val in c(1, 2, 5)) {
    for (level_val in c(0.90, 0.95, 0.99)) {
      # Fit standard and anytime-valid linear models
      std_fit <- lm(mpg ~ wt + hp, data = mtcars)
      std_ci <- confint(std_fit, level = level_val)
      
      av_fit <- av(std_fit, g = g_val)
      av_ci <- confint(av_fit, level = level_val)
      
      # Check if each anytime-valid CI encloses the standard CI
      for (i in 1:nrow(std_ci)) {
        coef_name <- rownames(std_ci)[i]
        expect_lte(av_ci[coef_name, 1], std_ci[coef_name, 1])
        expect_gte(av_ci[coef_name, 2], std_ci[coef_name, 2])
      }
    }
  }
})

test_that("sequential p-value computed via log_G_f is greater than the standard F-test p-value", {
  for (g_val in c(1, 2, 5)) {
    # Fit the ordinary linear model and compute its F-test p-value.
    std_fit <- lm(mpg ~ wt + hp, data = mtcars)
    std_f <- summary(std_fit)$fstatistic
    std_f_pvalue <- pf(std_f[1], std_f[2], std_f[3], lower.tail = FALSE)
    n <- length(std_fit$residuals)

    
    av_seq_pvalue <- p_G_f(log_G_f(std_f[1], std_f[2], std_f[3], n, g_val))
    
    # Assert that the standard F p-value is less than the anytime-valid (sequential) p-value.
    expect_lt(std_f_pvalue, av_seq_pvalue)
  }
})


test_that("anytime-valid methods work with increasingly complex linear models", {
  # Test using a more complex linear model with interactions
  complex_lm <- lm(mpg ~ wt * hp + factor(cyl), data = mtcars)
  av_complex_lm <- av(complex_lm)
  expect_no_error(summary(av_complex_lm))
  expect_no_error(confint(av_complex_lm))
})