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<br />Ilaving established that the historical regression mcthod is not appropriatc for the <br />evaluation of these operational sceding programs, the evaluation of all the targets in Table I was <br />conductcd using the method of ratio statistics (Gabriel, 1999). For single targets, as is the case <br />herc, J ratio statistics are relevant. The simplest ratio statistic is the Single Ratio (SR), the ratio of <br />the Average Target Streamflow during the operational period to the Averagc Targct Streamflow <br />during the historical period. \\'nen streamflow data for a control basin is available, more prccise <br />ratio statistics can be calculated. One can calculate the Double Ratio, the SR divided by the ratio of <br />A vcrage Control Streamflow during the opcrational pcriod to the Average Control Streamflow <br />during the historical period, which adjusts tor imbalances betwecn thc target operational period and <br />the target historical period. An cvcn more precise ratio statistic is thc Regression Ratio (RR), <br />especially when the target and control streamflow is highly correlalL'<i. In calculating the regression <br />ratio (RR), the secd/no seed ratio for the target station (SR) is divided by thc seed/no secd ratio <br />as predicted by the control station regression relationship (SRpRw) in order to adjust SR for <br />effccts due to natural difTerences in streamflow, i.e.. RR = SR I SRpRm. By laking advantage of <br />thc high correlation betwecn thc target and control station streamllows ovcr the entirc pcriod of <br />analysis (including both the historical and opcrational periods), the variance of the regression <br />ratio is reduced with respect to the variance of the single ratio lor the target station only, thereby <br />enabling the ddection of smaller effccts duc to seeding with greater probability. Gabriel (1999) <br />has shown that using ratios based on regression is always preferable to using single or double <br />ratios: therefore, the use of the Regression Ratio (RR) is cmphasized in this analysis. <br />These RR results werc then adjusted for biases that can occur when operational data are <br />compared to historical records in an a posteriori evaluation of non-randomized seeding <br />programs, Brier and Engcr (1952) have shown that thc variation bctwccn sequcnccs of years is <br />dillerent from the variation between random samples ofycars. Gabriel and PetTOndas (1983) have <br />shown that reliablc conclusions cannot be drawn from comparisons of operational data with <br />historical records, and have demonstratcd the problems encountered in trying to do so, The <br />calculated levels of significance (P-values) are likely to bc lowcr than the tme levcl of <br />significance and thc calculatcd confidence intervals are likely to bc narrower and more precise <br />than thcy rcally arc. Because of thcse problems, standard statistical methods are pronc to indicate <br />effects whcn thcrc might not be any. Gabriel and Petrondas (1983) suggest that the computcd P- <br />values from analyses of this type should be augmented by a factor whosc magnitude is proportional <br />to the number of years involved in thc evaluation in accordancc with a rough guideline that fit their <br />findings. An adjustment was, therefore, made to the RR results based on doubling the computed P- <br />value, this factor being somewhat smaller than that indicated by the rough guidcline of Gabriel and <br />Petrondas (1983). At various stages in thc cvaluation the results using the regression ratio that werc <br />adjusted for bias in this way \\'ere comparcd to those from re-randomization to verify their <br />accuracy for the sample sizes involved, Thc water ycar strcamllow was cvaluated by rc- <br />randomil'-3tion procedures using 30.000 pem1Utations (3 mns of 10,000 pern1Utations each to <br />establish the stability of the results). It was found that the results were indecd approximately equal; <br />there was no more than a J% differencc betwecn thc P-values from ratio statistics and those fTom <br />re-randomil'.3tion. Although randomization is the only surc way of safeguarding against bias and <br />its influence on the evaluation results, the factor of two adjustment is deemed adequatc to providc <br />suf1iciently precise results because (I) the historical and operational periods arc quite long so that <br />the potential cffl.'Ct on average strcamllo\\'s due to year-to.year variability and short-tcrm cycles is <br />mitigatcd. and (2) the RR takcs into account the effect of the long-ternl trend in natural streamflow <br />through the regression bctween the target and control. and (3) thc ratio statistics methodology is <br /> <br />39 <br />