Information Technology Reference
In-Depth Information
To calculate the percentage of the total variance, a particular poly-
nomial component (e.g., linear, quadratic) accounts for
sum of squares for the polynomial component
sum of squares for total variance
eta squared
=
.
(7.8)
If we were to perform these computations on the results of our trend
analysis (which we will do in a moment), we can then speculate that the
linear trend would show up as being quite strong whereas the quadratic
trend is likely to be quite weak (assuming that it was statistically significant
in the first place).
7.22 PERFORMING A TREND ANALYSIS BY HAND
The statistically significant omnibus
F
test, which we computed previously,
on the SAT data suggested that study preparation time affected subsequent
SAT scores. As with any omnibus ANOVA employing an independent
variable with more than two treatment conditions, we cannot discern the
true nature of the relationship between study time and SAT scores, only
that there is a statistically significant difference among the means. The
plot of the SAT study time means suggests that a straight line fits these
data fairly well, indicating the presence of a
linear trend
. The flattening
out of this function for the six month and eight month treatment groups
may be indicative of a quadratic component to this trend.
Our analysis of linear trend will assess the following statistical hypothe-
ses:
H
0
: linear trend is absent (slope of line
=
zero).
H
1
: linear trend is present (slope of line
=
zero).
7.22.1 LINEAR SUM OF SQUARES CALCULATION
To assess for the presence of linear trend we need to calculate a special
F
ratio. One important constituent of this
F
ratio is the linear sum of
squares (
SS
A
linear
). We will now demonstrate the calculation of
SS
A
linear
and
its subsequent
F
ratio.
The formula for
SS
A
linear
is as follows:
n
(
ˆ
ψ
linear
)
2
SS
A
linear
=
,
(7.9)
c
2
where
n
=
sample size,
ˆ
ψ
linear
=
(read “psi hat linear”) the sum of the treatment means
weighted (multiplied) by a set of linear coefficients,
c
=
linear coefficients.
