FLOOD09697

• County: Statewide
• Community: Nationwide
• Basin: Statewide
• Title: Mathematical Modeling of a Sociological and Hydrologic Decision System
• Date: 6/1/1978
• Prepared By: Institute for Social Science Research on Natural Resources, Utah State Univ.
• Floodplain - Doc Type: Educational/Technical/Reference Information

for future analyses of public evaluation of flood con-
trol proposals. Because of measurement difficulties
and real differences, the coefficienls can be expected
10 vary substantially from those specified in Equa.
tion 5.24. Much varialion can be eliminated by im-
proved measurement of the variables and by more
dala for calibration. If this was done, the predictive
ability of the equation should increase and the corre-
lation coefficient attained would be considered high
by traditional standards of nonexperimental research
in social science.

Equation V(b): Population evaluation
equation - categorized form

In cases where relationships are nonlinear, the
predictive capability of equations designed to reflect
these relations can be improved by incorporating the
nonlinear effects. Thus, the predictive capability of
Equation V(a) (5.24) can be improved by allowing
for nonlinear relationships. However, the inclusion
of nonlinear terms in an equation complicales its de-
velopment, calibration, and use. A way of accounting
for nonlinearities in an equation without having non-
linear terms is to stratify or categorize the independenl
variables into relatively homogeneous groups and to
formulate linear "sub-relationships" for each group.
The resulting equation produces a series of connecting
but discontinuous straight lines which in effect form
a quasi curvilinear function. The categorized form of
the unstandardized version of Equation V (5.24) is as
follows:

Unslandardized form:35

POPEVE = 3.30 + .1521/-1.1 + .2241#1.2 + .2271/-1.3
- .493 h4 - .110 1/.1.5 - .1341/-2.1
-.0841/-2.2 + .1361/-2.3 + .0821/-2.4
+ .3441/-3.1 + .0261#3.2 + .0221/-3.3
- .1733.4 - .1141#3.5 - .1051/-3.6 - .0464.1
- .0761/-4.2 - .0051/-4.3 + .1271/-4.4
- .1441/-5.1 - .2211/-5.2 - .013 1/-5.3
+ .1401/-5.4 + .2381/-5.5 - .6381/-6.1
- .5921/-6.2 - .0061/-6.3 + .3131/-6.4
+ .3681/-6.5 + .5551/-6.6 - .6511/-7.1
- .2431/-7.2 + .2411/-7.3 .2351#7.4
+ .4351/-7.5 - .5671/-8.1 - .1751/-8.2
+ .0671/-8.3 - .0021/-8.4 + .0061/-8.5
+ .6711/-8.6 - .2881/-9.1 - .3591/-9.2
+ .2831/-9.3 + .1701/-9.4 + .1941/-9.5

35The standardized form of Equation V(b) is not pre-
sented because it is not meaningful when the variables are
categorized.

- .0341/-10.1 - .3361/-10.2 - .1661/-10.3
+ .0931/-10.4 + .2811/-10.5 + .1621/-10.6
- .1121/-11.1 - .1341/-11.2 + .0341/-11.3
+.2151/-11.4-.0031/-11.5 . . (5.28)

The independent or I/- variables in this equation
are defined in Table 5.8. The first number of each I/-
subscripl refers to a term in Table 5.7. The second
number in each subscripl is the number of a category
encompassing a range of values for that item shown
in Table 5.8. Each range has been numbered in se-
quence from smaller to larger values.

The regression using this categorizalion of con.
tinuous variables increased r2 to 0.602. The increase
was not spurious. Since there were 275 0 bservations,
dividing them into five or six categories still leaves
many degrees of freedom. The cutting points divid.
ing categories were chosen to give sufficient observa-
tions in each category for stable statistics. Negative
and positive values were not permitted in the same
category, and each category was nearly equal in range.
The number of observations in each category is as
shown in Table 5.8.

The pattern of the values of the coefficients of
the I/- variables in Equation 5.28 suggests a direct reo
lationship with the dependent variable in cases where
the coefficients for a given term increase with the num.
ber of the category. The opposite trend suggests an
inverse relationship. Fluctuations in the pattern sug.
gest a nonmonotonic relationship.36 Most of the val-
ues of the coefficients for a given term in Equation
5.28 are consistent; indeed, they must be for the term
to be significant in a linear equation such as 5.24.

Another attempt to increase the predictive
abilily of Equation 5.27 used a multiplicative power
function with the same variables.37

Unstandardized form:
POPEVE = e - .31X - .049x - .053X .39x
1 2 3 5
- .16x12 .41(X4X7) .072(X9XIS)

.17(X10XI6) - .041

. (5.29)

36The coefficients of the category terms add to one be-
cause a dummy variable coding using O'sand l'swas used. If
there were equal numbers of cases in each category, each coeffi-
cientwould be equal to the differences between the mean of a
given group assigned 1 's and the group assigned O's tluoughout,
(Kerlinger and Pedhazur, 1973: 108). This is because the best
single predictor for a subject is a score equal to the man of
the group of values in which the subject's real value occurs.

37 All the variables in Equation 5.27 and listed in Table
5.8 were entered singly in development of the multiplicative
power function and, in addition, all the interaction terms
were tried.

78