To be or not to be valid in testing the significance of the slope in simple quantitative linear models with autocorrelated errors

In this article, the validity of procedures for testing the significance of the slope in quantitative linear models with one explanatory variable and first-order autoregressive [AR(1)] errors is analyzed in a Monte Carlo study conducted in the time domain. Two cases are considered for the regressor: fixed and trended versus random and AR(1). In addition to the classical t-test using the Ordinary Least Squares (OLS) estimator of the slope and its standard error, we consider seven t-tests with n - 2 df built on the Generalized Least Squares (GLS) estimator or an estimated GLS estimator, three variants of the classical t-test with different variances of the OLS estimator, two asymptotic tests built on the Maximum Likelihood (ML) estimator, the F-test for fixed effects based on the Restricted Maximum Likelihood (REML) estimator in the mixed-model approach, two t-tests with n - 2 df based on first differences (FD) and first-difference ratios (FDR), and four modified t-tests using various corrections of the number of degrees of freedom. The FDR t-test, the REML F-test and the modified t-test using Dutilleul's effective sample size are the most valid among the testing procedures that do not assume the complete knowledge of the covariance matrix of the errors. However, modified t-tests are not applicable and the FDR t-test suffers from a lack of power when the regressor is fixed and trended (i.e., FDR is the same as FD in this case when observations are equally spaced), whereas the REML algorithm fails to converge at small sample sizes. The classical t-test is valid when the regressor is fixed and trended and autocorrelation among errors is predominantly negative, and when the regressor is random and AR(1), like the errors, and autocorrelation is moderately negative or positive. We discuss the results graphically, in terms of the circularity condition defined in repeated measures ANOVA and of the effective sample size used in correlation analysis with autocorrelated sample data. An example with environmental data is presented.