Y603
Lectures Online
Lecture
#6
1. Orthogonal Planned
Comparisons
--Orthogonal planned
comparisons are advantageous over the overall F test
because you can test those comparisons of interest
directly and specifically; the alpha level is
controlled at each contrast level instead of at the
experimental level; results are directly generalizable
to the corresponding contrasts of means in the
population.
--The number of orthogonal
planned comparisons permitted is determined from the
degrees of freedom of the effect. So for a one-way 4
groups ANOVA design, the degrees of freedom associated
with the independent variable is 3. Therefore, the
number of orthogonal planned comparisons is capped at
3.
--To construct orthogonal
planned comparisons, first ask yourself what are the
research questions that you are interested in
investigating, then making sure that these comparisons
of means are orthogonal (or independent) from each
other. A pair of comparisons are said to be orthogonal
if their coefficients for the corresponding means are
summed to zero after cross-multiplying with each other
for the same mean. Below are an example of two
orthogonal contrasts:
If group sizes (nís)
are not equal, then the requirement for orthogonality
is modified into
Can you think other pairs of
orthogonal comparisons which can be derived from a 4
group one-way ANOVA design?
--The conceptual
unit of alpha (or Type I error) control for
orthogonal planned comparisons is at the contrast
level and it is applied to each contrast directly
and precisely. Other kinds of Type I error rates
are discussed in Section 4.1 of
Kirk.
2. To test if an
orthogonal planned comparison is significant, you may
follow these steps:
Step 1 Preset an
alpha level for the contrast (say, .05 or .01); this
level may be called a
pc.
Step2 Compute either a
t-ratio, [or an F-ratio (=
)] or
the Minimum Significant
Difference (MSD).
,
F=
and
MSD = 
.
Step 3 Compare the
t-ratio with a t critical value at the
a
pc level or compare MSD with the obtained
value to see if the obtained value, in absolute
values, exceeds the MSD value.
3. Perform the orthogonal
planned comparisons via SAS:
Use PROC GLM and
the CONTRAST statements in the SAS program.
Refer to the "P171ORTHO.LIS" printout
and its SAS program distributed in class. Details of
programming the CONTRAST statement are found in the
attached handout or in SAS/STAT-Vol
2.
4.
Assignments:
(1) Review Sections
4.1-4.2 in Kirk
(2) Do questions 2, 3, 4, 5,
6, 7, 8, 9(a), 9(b), 10(a), and 10(b) of Chapter 4 in
Kirk.
For questions 9(a) and 10(a),
a
=a
pc.
(3) Preview Sections 4.3-4.4
in Kirk.
* SAS PROCEDURE for
ANALYSIS OF VARIANCE *
PROC
GLM;
PROC
GLM
1. Purpose: compare
means obtained from an ANOVA design
2. Syntax:
PROC GLM
data=data set name ;
CLASS vars
;
MODEL dep
var=effects ;
MEANS
effects/options ;
TEST H=effects
E=error ;
MANOVA H=effects
E=error ;
REPEATED vars
;
BY classification
vars
;
3. Simple Main
Effects
PROC
GLM;
CLASS A B
C;
MODEL Y=A B
C;
4. Interaction and Main
Effects
PROC
GLM;
CLASS A B
C;
MODEL Y=A B C
A*B
B*C A*C
A*B*C;
(or MODEL
Y=A|B|C;)
5. Nested Effects
PROC
GLM;
CLASS A B
C;
MODEL Y=A B C(A
B);
6. MEANS
effects/options;
Options are :
TUKEY = HSD or WSD
procedure
SCHEFFE = Scheff
procedure
BON = Bonferroni
t-test
SIDAK = Dunn-Sidak
pairwise procedure
DUNNETT('j')= The Dunnette
procedure where j=control group
SNK = Newman-Keuls
procedures
LSD = least significant
differences
CLDIFF = 95% Confidence
interval
ALPHA = significance
level
(default =
.05)
For Example,
MEANS A B A*B/
SNK ALPHA = 10;
will perform the
Newman-Keuls test, at a
= 10%, of group mean differences on A and B main
factors. The A*B interaction generates all of cell
means without actually testing
them.
7. The PROC GLM also allows
us to test planned <orthogonal or nonorthogonal>
contrasts by using the CONTRAST statement. The
general syntax is
CONTRAST 'label'
effect coefficients;
For example,
CONTRAST 'A LINEAR &
QUADRATIC'
A -2 -1 0 1
2,
A 2 -1 -2 -1
2;
will carry out two
trend analyses, one on the linear trend of Factor A
and the other on the quadratic trend of Factor
A.
In order to carry out a
test on the interaction term, such as A*B, you need
at least four cell means. The following are two
orthogonal contrasts tested by the CONTRAST
statements:
CONTRAST 'The first and
the second contrasts'
|
A*B
|
1
|
-1
|
0
|
-1
|
1
|
0,
|
|
A*B
|
.5
|
.5
|
-1
|
-.5
|
-.5
|
1;
|
CONTRAST
statement should be placed anywhere after PROC GLM,
CLASS, and MODEL
statements.
