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We compare two theoretically distinct approaches to generate artificial (or “surrogate”) data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single “best” model to the original data, and then to generate surrogate data sets that are “typical realizations” of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases; a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider test for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of …