Traditional statistical hypothesis tests are performed by comparing the value of a detection statistic with an estimate of its natural internal variability in the unperturbed climate. This estimate must be obtained from control climate simulations because detection statistics typically measure change on time-scales that are a substantial fraction of the length of the available instrumental record (see Appendix 12.4). Most "optimal" detection studies use two data sets from control climate simulations, one that is used to develop the optimal detection statistic and the other to independently estimate its natural variability. This is necessary to avoid underestimating natural variability. The p-value that is used in testing the no signal null hypothesis is often computed by assuming that both the observed and simulated projections on signal patterns are normally distributed. This is convenient, and is thought to be a reasonable assumption given the variables and the time and space scales used for detection and attribution. However, it leads to concern that very small p-values may be unreliable, because they correspond to events that have not been explored by the model in the available control integrations (Allen and Tett, 1999). They therefore recommend that p-values be limited to values that are consistent with the range visited in the available control integrations. A non-parametric approach is to estimate the p-value by comparing the value of the detection statistic with an empirical estimate of its distribution obtained from the second control simulation data set. If parametric methods are used to estimate the p-value, then very small values should be reported as being less than 1/np where np represents the equivalent number of independent real-isations of the detection statistic that are contained in the second control integration.
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