FLOOD06431

• County: Statewide
• Basin: Statewide
• Title: Estimating Design-Flood Discharches for Streams in Iowa Using Drainage-Basin and Channel-Geometry Characteristics
• Date: 1/1/1993
• Prepared For: Iowa City
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

OLS regression analyses were performed
using the Statit! statistical procedures
ALLREG and SREGRES (Statware, Inc., 1990,
p. 6-10 - 6-27). Initial selections of significant
explanatory variables for the OLS regression
models were performed by using the ALLREG
procedure. The ALLREG procedure uses an
all-possible subsets regression to identify the
best-possible combinations of explanatory
variables on the basis of the Mallows' Cp
statistic (Mallows, 1973). The SREGRES
procedure, based on stepwise-regression
algorithms, then was used to perform an OLS
regression analysis on each best-possible
combination of explanatory variables. The final
selection of explanatory variables was based on
the following criteria as described by Koltun and
Roberts (1990, p. 11):

1. The selection of explanatory variables, as
well as the signs and magnitudes of their
respective regression coefficients, need to be
hydrologically valid in the context of flood
runoff. This criterion takes precedence over all
other criteria.

2. All explanatory
statistically significant
confidence level.

variables should be
at the 95-percent

3. The selection of explanatory variables,
within the constraints of criteria 1 and 2, should
minimize the prediction error sum of squares
[the PRESS statistic, an index of the prediction
error associated with the regression equation
(Allen, 1971; Montgomery and Peck, 1982)],
maximize the coefficient of determination (R2, a
measure of the proportion of the variation in the
response variable accounted for by the
regression equation), and mlIllmlze the
standard error of estimate. Correlation between
explanatory variables and the vClriance inflation
factor (VIF) (Marquardt, 1970; Montgomery and
Peck, 1982) was used to assess multicollinearity
in the regression models.

Weighted Least-Squares Regression

A weighted least-squares (WLS) regression
technique described by Tasker (1980 p. 1107-
1109) was used to develop the final

1 Use of brand names in this report is for identification
purposes only and dol's nut constitute endorsement by
the U.S. Geological Survey.

multiple-regression equations for both the
drainage-basin and channel-geometry flood-
estimation methods. WLS regression analyses
improved R2 values and standard errors of
estimate obtained in the majority of the OLS
regression analyses. Tasker (1980, p. 1107)
reports that OLS regression assumes that the
time-sampling variance in the response-
variable estimates (design-flood discharges) are
the same for each gaging station used in the
analysis (assumption of homoscedasticity). In
hydrologic regression, this assumption usually
is violated because the reliability of response-
variable estimates depends primarily on the
length of the observed annual-peak discharge
records. The error associated with
response-variable estimates is inversely
proportional to record length (Choquette, 1988,
p. 16-17). WLS regression adjusts for the
variation in the reliability of the response-
variable estimates by using a weighting
function to account for differences in the lengths
of observed annual-peak discharge records at
gaging stations. The weighting function
(Tasker, 1980, p. 1107) is based on theory and on
analysis of residuals from the initial OLS
regression equation. The weighting function
assumes that the response variable (the
design-flood discharge) is determined by fitting
the logarithms (base 10) of the observed
annual-peak discharges to a Pearson Type-III
distribution.

The variance of the response variable, QT, is
estimated to determine the appropriate weight
factors for the WLS regression analysis. To be
effective, the weight factors should be inversely
proportional to the variance of QT' The variance
of QT' for a selected l' -year recurrence interval,
can be partitioned into two componentsnthe
variance due to OLS regression-model error, co'
and the variance due to time-sampling error, ti,
which is related to the standard deviation and
weighted skew coefficient of the logarithms of
the observed annual-peak discharges and to the
number of years of effective record length
(Tasker, 1980, p, 1107-1109; Choquette, 1988, p.
16-171. The weight function has the form

It" :::
,

(4)

(Co + [I)

where Wi

'5 the weight factor for the ith
gaging station used in the

l
l

8 ESTIMATING DESIGN-FLOOD DISCHARGES FOR STREA..'\1S IN IOWA