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Class Notes
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Canada
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University of Alberta
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Statistics
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STAT368
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Douglas Wiens
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Lecture

Description

133
18. 2 factorials
All of this generalizes to the 2ctorial, in which
factors are investigated, each at two levels. To
easily write down the estimates of the e ects, and
the contrasts, we start with a table of ± signs,
done here for = 3. Label the rows (1), then
the product of with (1). Then all products
of with the terms which are already there: ×
(1) = × = . Then all products of
with the terms which are already there. (This
is the standard order.) Now put in the signs.
Start with 2 = 8 +s under the I, then alternate
s and +s, then in groups of 2, nally (under
C) in groups of 4 (= 2 1). Then write in the
products under the interaction terms. 134
E ect
I A B C AB AC BC ABC
(1) + + + +
+ + +
+ + +
+ + + +
+ + +
+ + +
+ + +
+ + + + +
Interpretation: Assign the appropriate signs to the
combinations (1). E ect estimates are
[(1) + + + ]
=
4
+[ + + + ]
4
[ + + + ]
[(1) + + + ]
=
4
all with 2in the denominator. 135
³ ´
These are all of the form 2 1 for a con-
trast in t³e s´ms (1) ; the corresponding
is 2 2 . For example
( )2
[ + + + ]
[(1) + + + ]
=
8
The sums of squares are all on 1 d.f. (including
, which uses the 1 d.f.usually subtracted
from = 2 for the estimation of the overall
mean ), so that , obtained by subtraction,
is on 2 = 2 ( 1) d.f. Then, e.g., the
F-ratio to test the e ect of factor A is
0=
where = and = ( ).
The p-value for the hypothesis of no e ect is
µ ¶
1( 1) 0 .
2 136
Suppose = 1, so that no d.f. are available
for the estimation of 2. In the 2 there was
Tukeys test for non-additivity, which relied on
the assumption that the interactions were of a
certain mathematically simple but statistically du-
bious form (even more so for 2). A more
common remedy is to not even try to estimate
certain e ects - usually higher order interactions
- and use the d.f. released in this way to estimate
error.
A graphical way of identifying the important ef-
fects which must be in the model, and those which
can be dropped to facilitate error estimation, is a
normal probability plot of the absolute values of
the e ect estimates - a half-normal plot. Those
e ects which deviate signicantly from the qqline
tend to be the important ones.
Example. Data in Table 6-10. A chemical prod-
uct is produced using two levels each of tempera-
ture (A), pressure (B), concentration of formalde-
hyde (C) and rate (D) at which the product is
stirred. Response (Y) is the ltration rate. 137
A B C D y
1 -1 -1 -1 -1 45
2 1 -1 -1 -1 71
3 -1 1 -1 -1 48
4 1 1 -1 -1 65
5 -1 -1 1 -1 68
6 1 -1 1 -1 60
7 -1 1 1 -1 80
8 1 1 1 -1 65
9 -1 -1 -1 1 43
10 1 -1 -1 1 100
11 -1 1 -1 1 45
12 1 1 -1 1 104
13 -1 -1 1 1 75
14 1 -1 1 1 86
15 -1 1 1 1 70
16 1 1 1 1 96 138
> g anova(g)
Df Sum Sq Mean Sq F value Pr(>F)
A 1 1870.56 1870.56
B 1 39.06 39.06
C 1 390.06 390.06
D 1 855.56 855.56
A:B 1 0.06 0.06
A:C 1 1314.06 1314.06
A:D 1 1105.56 1105.56
B:C 1 22.56 22.56
B:D 1 0.56 0.56
C:D 1 5.06 5.06
A:B:C 1 14.06 14.06
A:B:D 1 68.06 68.06
A:C:D 1 10.56 10.56
B:C:D 1 27.56 27.56
A:B:C:D 1 7.56 7.56
Residuals 0 0.00 139
> g$effects
# Note that these are twice as large in absolute value as those
in the text, and the signs sometimes di er. This is because of
Rs denition of e ect, and makes no di erence for comparing
their absolute values.
(Intercept) A1 B1 C1
-280.25 -43.25 -6.25 19.75
etc.
A1:B1:D1 A1:C1:D1 B1:C1:D1 A1:B1:C1:D1
-8.25 3.25 5.25 2.75
> effects qq

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