9597

• UCREFRP Catalog Number: 9597
• Author: U.S. Environmental Protection Agency.
• Title: Biological Field and Laboratory Methods for Measuring the Quality of Surface Waters and Effluents.
• USFW Year: 1973.
• USFW - Doc Type: Cincinnati, OH.
• Copyright Material: NO

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I

Source
So urce df MS F Total
Total n-l
Regression 1 MSR MSR/MSD Regression
Deviations from
regression n-2 MSD Deviations from
Lack of fit m-2 MSL MSL/MSE regression
Error nom MSE Lack of fit
Error

If the computed t exceeds the tabular t, then the
null hypothesis is rejected and the estimated
slope, b, is tentatively accepted. Other values of
/30 may be tested in the null hypothesis and in
the t-test statistic.
With data such as those in Table 12, another
hypothesis may be tested - that of lack of fit of
the model to the data, or bias. This idea must be
distinguished from random deviations from the
straight line. Lack of fit implies a nonlinear
trend as the true model, whereas random devia-
tions from the model imply that the model
adequately represents the trend. If more than
one Y observation is available for each X (3 in
the example Table 12), random fluctuations can
be separated from deviations from the model
,
Le., a random error may be computed at each
point so that deviations from regression may be
partitioned into random error and lack of fit.
The test is in the form of an analysis of vari-
ance and is illustrated in brief form symbolically
in Table 14. Here, the F ratio MSL/MSE tests
linearity, Le., whether a linear model is suffi-
cient; the ratio MSR/MSD tests whether the
slope is significantly different from zero.

TABLE 14. ILLUSTRA nON OF ANALYSIS
OF VARIANCE TESTING LINEARITY OF
REGRESSION AND SIGNIFICANCE OF
REGRESSION

To use this analysis, one set of computations
must be made in addition to those of Table 13.
The computation is the same as that for treat-
ment sums of squares in the analysis of variance
previously discussed; in this case, levels of X are
comparable to treatments. First compute the
sum of the V's, Ti> for each level of X. For
X = 1, T I = 224, etc. Then compute:

~ Tj2
ki

where kj = the number of observations for the

BIOMETRICS - LINEAR REGRESSION

level of X; in this case always 3. For the
example,

T.2
~2-=51341
k.
1

With this, the analysis of variance table (Table
15) may be constructed. In the first part of
Table 15, the sums of squares and degrees of
freedom are given symbolically to relate to the
computations of Table 13 and to the above
computations. The mean squares (MS) are always
obtained by dividing SS by df.
When the data for Table 12 are analyzed
(second part of Table 15), there is a very
unusual coincidence in the values of MS for
deviations from regression, lack of fit, and error.
Note that this is coincidence and they must
always be computed separately.
As already known from the graph, t-test, etc.,
the regression is highly significant. A negative
result from the test for nonlinearity (lack of fit)
was also suspected from the visually-satisfactory
fit of Figure 7. Therefore, for this range of data,
we can conclude that a linear (straight line) rela-

TABLE 15. ANALYSIS OF VARIANCE OF
THE DATA OF TABLE 12; TESTS FOR
LINEARITY AND SIGNIFICANCE OF
REGRESSION*

df
n-l

SS

~y2_cr
y
(~XY-CTxy)2
(~X2-CT x)

Total SS - Regression SS

Deviation SS - Error SS

~y2 _ ~i2
ki

*Symbols refer to quantities of Table 13 or to symbols de-
fined in the text immediately preceding this table.

n-2
m-2

nom

For- the data of Table 12:

Source df SS MS F
Total 17 10,587
Regression 1 10,139 10,139 362**
Deviations from
regression 16 448 28
Lack of fit 4 113 28 1 n.s.
Error 12 335 28
* * Significant at the 0.01 probability level.

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