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

Abstract

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 À 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 À 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.

Spatio‐temporal variability of ocean temperature in the Portugal Current System

“…The impact of serial correlation is so important that Santer et al [2000] and Yue et al [2002] have shown it to critically affect decisions on the significance of air temperature and river streamflow trends, respectively. The same studies, however, also showed that the various procedures available to account for temporal autocorrelation have differing impacts on trend significance [for another analysis, see Alpargu and Dutilleul , 2003]. When analyzing binned data, spatial correlations further reduce n eff , and the way how corrections are made to standard techniques becomes more complicated [see Yue and Wang , 2002], especially in the frequent cases where spatial and temporal features cannot be modeled separately.…”

Red herrings remain in geographical ecology: a reply to Hawkins et al. (2007)

“…These show clearly that the GLS estimates are much more precise than those from OLS. Alpargu and Dutilleul (2003) and others have used far more extensive simulations to explore different autocorrelation structures and relationships between variables, with the same general conclusions: for individual data sets, the OLS and GLS estimates may be very different, and that when residual autocorrelations are strong the OLS estimates are much less precise. Because of the low precision of the OLS estimates, it is also true that OLS estimates often have a greater absolute value than their GLS counterpart (Fig.…”

Small‐scale spatial autocorrelation and the interpretation of relationships between phenological parameters

“…This test requires confidence estimates of the magnitude of SA in the variables under study. Higher precision in the estimation of SA can be obtained by increasing the sample size (assuming the data are still partial realizations of the same spatial process), but also by replication, especially in the context of repeated measures designs (see Dutilleul & Pinel-Alloul 1996;Dutilleul et al 1998;Alpargu & Dutilleul 2003). With very large sample sizes (500 or more), correction for SA generally has no effect on the outcome of the t-test, modified or not (P. Dutilleul pers.…”