FLOOD08610

• County: Statewide
• Basin: Statewide
• Title: Techniques of Water-Resources Investigations of the US Geological Survey Some Statistical Tools in Hydrology
• Date: 1/1/1969
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

34

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

Two general conditions for which an analysis
of covariance will produce conclusions different
from an analysis of variance are shown in
figure 23. In eR9h condition, Y is the variable
being analyzed and X is the independent varia-
ble. Plot A of figure 23 shows means of Y for
the two periods to be practically equal. For
this condition an analysis of varilLIlce would
show no significant difference between means.
But a major change in the relation of Y to X
occurred between periods 1 and 2, and it is this
change that the analysis of covariance can
identify.

PlatA

y

Xl Xt X2
X

X2 Xl
X

Figure 23.-Two conditions for which analysis of covariance
will produce conclusions different from those of analysis
of variance.

The analysis of covariance test is made on
deviations from regression rather than on
means. The test involves the sum of squares
of deviations from a regression defined by all
points plotted about their own period means
and the sum of squares of deviations from an
overall regression Jine (Dixon and Massey,
1957, p. 210). In effect the test indicates
whether the two periods are different when
adjusted to the same X value. As previously
stated, an analysis of variance of data of the
condition of plot A, figure 23, would indicate
no difference between periods because the
means Y, and Y, are nearly alike. But analysis
of covariance would show .a significant differ-
ence in Y values corresponding to the overall
mean X,.
Plot B of figure 23 shows two periods having
very different mean Y values but no real
difference in the regressions of Y on X for the
two periods. An analysis of variance would
show a significant difference between means,
but an analysis of covariance would show no
significant difference in regressions for the two

.

periods. The two results do not conflict. There
is a difference in means for the two periods,
but this difference is due to a difference in X
values for those periods.
Analysis of covariance requires that slopes
of the regression Jines for the individual periods
be virtually parallel. A test for parallelism has
been described by Dixon and Massey (1957,
p.218).

Table 5.-Annual precipitation index and annual runoff, for
example of analysis of covariance

Period 1

Period 2

Runoff
(Y)

Precipitation
index (X)
27___________
36___________
26_______n__
18___________
27_______0000
30__00_______
25______0000_
28__00_______
19____000000_
22_______00__
22___________
2900_________
2600_________
29____00_____
24_ __ _ 00 _ 00 __
388___00_____

Runoff Precipitation
(Y) index (X)
1~ 3 14__________
21.9 26__________
13.6 15__________
lQS 11__________
lR 7 19__________
20.7 2Loo_oooooo
16. 3 18__________
16.2 22__________
12.5 20__________
11.3 17._________
14.0 2900________
16.5 30000000____
15. 3 16__________
19.2 23__________
13.0 23__________
238. 3 304_________

6.4
15.2
9.7
4.4
9.9
11.9
11. 9
15.4
9.4
7.0
16.0
17.0
11.2
13. 2
11.5

.

T:=6D2j T~=408.4.

170.1

Details of an analysis-of-covariance compu-
tation are given below using (1) the same runoff
data as in the previous section for the analysis-
of-variance example and (2) some assumed
values of a precipitation index, all of which
are listed in table 5 and plotted on figure 24.
The plot indicates that there is no change in

20

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o Period 1

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ANNUAL PRECIPITATION INOEX

Figure 24.-Plot of data From table 5.

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