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<br />onn240 <br /> <br />REGRESSION ANALYSIS <br /> <br />I. <br /> <br />Ordinary least-squares (OLS) and generalized <br />least-squares (GLS) regression analyses wcre used <br />in this study. OLS analysis was used for prcliminary <br />delineation of flood regions and selection of signifi- <br />cant explanatory characteristics. GLS regression <br />analysis was used to further define the explanatory <br />variables determined using OLS analysis and to <br />compute the final regression equations. GLS regres- <br />sion is a more appropriate method for developing <br />regional regression equations of streamflow character- <br />istics than OLS regression (Stedinger and Tasker, <br />1985) because using flood-frequency characteristics <br />at gaged sites as a response variable could violate two <br />assumptions of OLS regression. Those assumptions <br />are that the response variable at each site is indepen- <br />dent and has equal variance and that peak discharges <br />for nearby drainage basins may be correlated as a <br />result of similar climatic events. <br /> <br />, . <br /> <br />. <br />, <br /> <br />. <br />, <br /> <br />. <br /> <br />. <br />, <br /> <br />. <br /> <br />Multiple Regression and Drainage-Basin <br />Characteristics <br /> <br />, <br /> <br />, <br />, <br />, <br />, <br />, <br />., <br />., <br />., <br />., <br />., <br />., <br />, <br />, <br />, <br />, <br />., <br />., <br />., <br />, <br />, <br />, <br />, <br />, <br />, <br /> <br />Multiple-regression equations, expressing <br />flood magnitudes as a function of drainage-basin char- <br />acteristics, were developed for each hydrologic region. <br />Base 10 logarithmic transformations werc performed <br />on all strcamflow and drainage-basin-characteristic <br />data prior to the regression analyses. Thcsc data were <br />transformed to normalize the variables and residuals, <br />to obtain a constant variance about the rcgrcssion line, <br />and to obtain linear relations between dcpendent and <br />independent variables as required for regression anal- <br />yses (Stedinger and Tasker, 1985). The regression rela- <br />tions based on logarithmic transformation of the <br />variables were; <br /> <br />log Qt = log K +a log A + b log B+ .../llog N (1) <br /> <br />or, taking antilogs. <br /> <br />Q, <br /> <br />KAa Bb..N" <br /> <br />where <br />Q, (the response variable) is the estimated flood magni- <br />tude, in cubic feet per second, having a 'l~ycar recur- <br />rence interval; K is a regression constant; A. 8, . . , N <br /> <br />(the explanatory variables) are values of drainage- <br />basin characteristics, and a. b, . . . /l are regression <br />coefficients. <br />Based on the results of previous streamflow <br />regionalization studies (McCain and Jarrett. 1976; <br />Kircher and others. 1985) and on consideration of <br />physical characteristics that affect streamflow. a set of <br />drainage-basin and climatic cbaracteristics were evalu- <br />ated for inclusion as explanatory variables in the <br />regression equations, These characteristics included <br />drainage area. mean drainage-basin slope, mean <br />channel slope. gaging-station elevation. percentage <br />of drainage area covered by lakes and ponds, mean <br />annual precipitation, and percentage of drainage basin <br />covered by forest. Richter and others (1984) defined <br />these characteristics in greater detail and summarized <br />the characteristics for most of the drainage basins used <br />in this report. Drainage-basin-characteristic data used <br />in this study that have been determined since Richter <br />and others (1984) are unpublished. <br />Combinations of the explanatory variables <br />were evaluated using OLS multiple-regression <br />methods. Stepwise regression adds explanatory vari- <br />ables, one at a time, to the hasic regression equation <br />until all statistically significant variables have been <br />included, The statistical significance of certain vari- <br />ables already in the equation may change as other vari- <br />ables are added, Consequently. variables that may be <br />added at one step could be removed at a later step. The <br />purpose of using stepwise regression is to include all <br />the explanatory variables that have a great effect on the <br />response variable and to exclude the variables that <br />have little effect on the response variable. Drainage <br />area was the most statistically significant variable in <br />all of the regression equations. Other statistically <br />significant variables were mean annual precipitation <br />and mean drainage-basin slope. <br /> <br />Generalized Least-Squares Regression <br /> <br />(2) <br /> <br />After acceptable drainage-basin characteristics <br />were determined and the five hydrologic regions were <br />delineated. GLS regression was performed. Stedinger <br />and Tasker (1985) reported that the GLS method <br />provides more accurate estimates of the regression <br />coefficients, better estimates of the accuracy of the <br />regression equation, and almost unbiased estimates <br />of rhe model-error variance. The GLS analysis was <br />performed using ANNIEIWDM, a set of programs <br />designed for analyzing hydrologic data (Flynn and <br />others, 1995). <br /> <br />REGRESSION ANALYSIS <br /> <br />7 <br />