FLOOD01900

• County: Statewide
• Community: Nationwide
• Title: Techniques of Weather-Resources Investigations of the USGS Book 4, Chapter A1
• Date: 1/1/1968
• Floodplain - Doc Type: Educational/Technical/Reference Information

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SOME STATISTICAL TOOLS IN HYDROLOGY

35

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relation between periods and that the difference
in runoff means between periods was due to
differences in precipitation. For these data it
would not be necessary to make a covariance
analysis. However, if separate regressions were
indicated by the plotted points, it might be
desirable to make a covariance analysis in order
to test whether the two regressions were signifi-
cantly different statistically.
The following computation illustrates the
procedure:
Total sum of products = ::8X"Y,,- TzT,/nk,
where Tz and T, are grand totals of X and Y,
n is the number of items in each period, and k
is the number of periods.

Between-means sum of products

=::8 Tz,T,Jn- TzT,7nk,

where Tz' and Tv< are column (period) totals.
For this example,

the total sum of products = (27) (17.3)

+ (36) (21.9)....+ (23) (11.5)

- (692) (408.4)/(15) (2)

= 10,054. 7 -9,420.5=634.3

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Between-means sum of products

= (388) (328.3) /15+ (304) (170.1) /15

- (692) (408.4)/30=9,611.4

-9,420.4= 191.0

Total sum of squares on X=::8X'<a

- Tx'/N=16,898-15,962=936

Between-periods sum of squares on X

=~T;Jn- Tx'/N=16,197 -15,962=235

Sums of squares on Yare taken from the
analysis-of-variance example.
Deviations from regression are computed by
the formula

::8y'- (::8xy)'/::8x',

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which, for totals, =507.6- (634.3)'/936=507.6
-429.8=77.8.
For within periods, the deviations from regres-
sion are
352.6- (443.3)'/701 =352.6-280.3=72.3,
and the between-means deviation from regres-

sion is obtained by subtraction. The data are
shown in the following covariance table.

Data

DevIations
Degrees Mean
of !:(Y- YP square
freedom

Source

Degrees
of Xx2 J;xy Xyl
freedom

Between
means. n 1 225 101 155.0 5.5 5.5
Within
periods__ 28 701 443.3 352.6 Zi 72.3 27
TotaL "" 936 634.3 507.6 28 77.8

Sums of squares fOf within periQds (In the first part of the table) are
obtained by subtraction. Degrees of freedom for deviations from regreg-
sion,l:(Y-Y)2, for within periods and total are one less than for means.

The test of significance compares F (the ratio
of mean squares from the covariance table), to
values of the distribution of F at the 5-percent
and 10-percent levels. For this example they are

F=5.5/2.7=2.0,

F1,27.0.05=4.2J

and

F1,27,O.1O=2.9.

Because 2.0 is less than 2.9, the difference in
periods is not significant at the lO-percent level
when runoffs are adjusted for precipitation.
See the article by Wilm (1943), which includes
a discussion by Davenport, for an application of
covariance analysis to a hydrologic problem.

Multivariate analysis

Multiple regression on independent variables
which are related among themselves sometimes
produces inconsistent results from diflerent sets
of data. For example, the regression coefficient
of an independent variable may range from
positive to negative in diflerent regressions and
yet test statistically significant in each. Under
these conditions, the conclusions regarding the
effect of that variable on the dependent variable
might be wrong if only one set of data was
analyzed. The use of multivariate analysis has
been proposed as a way out of this dilemma.
Multivariate analysis is concerned with the
relationship of sets of dependent variates and
includes several different procedures, each in-
tended to accomplish a different objective.
Snyder (1962) investigated the use of multi-
variate analysis in hydrology where the struc-
ture of the solution was of primary interest.
Kendall (1957) described the theory.