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2008. Is spread of invasive species regulated? Using ecological theory to interpret statistical analysis. Appendix A. Technical note on testing random walk hypotheses, derivation of dynamics under "spread regulation" of new-site ratios, models used for simulated spread scenarios, and mathematical approximation showing approximate "spread regulation" expected slope of-1. A1. Statistical testing of random-walk null hypotheses. The main reasoning of our report accepts the statistical patterns shown by Arim et al. (2006) and focuses on their interpretation. However, we note that Arim et al. (2006) did not appear to test their regulation model against the null hypothesis of a random walk, for which their statistical significance calculations from simple linear regression are not valid. If S t of Arim et al. (2006) is simply taken as a time-series – regardless of all other interpretation issues – the null hypothesis of non-regulation would be for S t to be a random walk (cf. their Fig. 1), and random walks do not satisfy the ergodicity requirement for autoregression testing of time-series. Ecologists have a long history of grappling with this issue for testing density-dependence of process of our report the question was raised of whether linear regression does give a valid test of a random walk. We provide code (Supplement) for the S (R or S-PLUS) language that illustrates by simulation that linear regression does not give a valid test of a random walk. Nevertheless, many of the slopes in their Fig. 2 have a clear negative trend, so rather than worrying about their statistical significance, we focus on the bigger picture of what range of spread models are consistent with these slopes.