A simulation model for peak water flow from watersheds was tested by comparing measured peak flow (cfs) from 10 storms with predictions of peak flow obtained from the simulation model. Q o and Q p are...

A simulation model for peak water flow from watersheds was tested by comparing measured peak flow (cfs) from 10 storms with predictions of peak flow obtained from the simulation model. Q_o
and Q_p
are the observed and predicted peak flows, respectively. Four independent variables were recorded:

X₁
= area of watershed (mi₂),

X₂
= average slope of watershed (in percent), X₃
= surface absorbency index (0 = complete absorbency, 100 = no absorbency), and

X₄
= peak intensity of rainfall (in/hr) computed on half-hour time intervals.

(a) Use Y = ln(Q_o/Q_p) as the dependent variable. The dependent variable will have the value zero if the observed and predicted peak flows agree. Set up the regression problem to determine whether the discrepancy Y is related to any of the four independent variables. Use an intercept in the model. Estimate the regression equation.

(b) Further consideration of the problem suggested that the discrepancy between observed and predicted peak flow Y might go to zero as the values of the four independent variables approach zero. Redefine the regression problem to eliminate the intercept (force β₀
= 0), and estimate the regression equation.

(c) Rerun the regression (without the intercept) using only X₁
and X₄; that is, omit X₂
and X₃
from the model. Do the regression coefficients for X₁
and X₄
change? Explain why they do or do not change.

(d) Describe the change in the standard errors of the estimated regression coefficients as the intercept was dropped [Part (a) versus Part (b)] and as X₂
and X₃
were dropped from the model [Part (b) versus Part (c)].