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<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.
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<br />Multiple Regression and Drainage-Basin
<br />Characteristics
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<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"
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<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
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<br />(2)
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<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
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