Supplement 1. R code for illustrating that standard linear regression is not valid for testing random walk null hypotheses for autoregressive time series.

File List deValpine_etal_invasions_supplement_randomWalkCode.R Description This code illustrates an aspect of time-series analysis that has been recognized for a long time but may be unfamiliar to some readers. The null hypothesis of no density-dependence in a population time-series, often called a random walk, can be represented as no relationship between log abundance at one time and the difference in log abundance from that time to the next time. However, the standard linear regression test of the hypothesis of a zero slope between two variables is not valid for testing the null hypothesis of a random walk in a time series. The code provided here uses simulations to illustrate that using standard linear regression to attempt to test a random walk null hypothesis leads to an incorrect probability of rejecting the null hypothesis even when it is true, i.e., Type I error rate. This code is provided for educational purposes. deValpine_etal_invasions_supplement_randomWalkCode.R contains annotated code comparing valid use of a linear regression hypothesis test to invalid use of such a test for a random walk null hypothesis for a time-series.