Balancing level combinations in an orthogonal design

[Ben Beardmore]

Design
;alts = alt1, alt2
;rows = 19
;orth = sim
;model:
U(alt1) =
b1 +
b2.dummy[0|0|0] * A[0,1,2,3] +
b3.dummy[0|0|0] * B[0,1,2,3]+
b4.dummy[0|0|0] * C[0,1,2,3] +
b5.dummy[0|0|0] * D[0,1,2,3] +
b6.dummy[0|0|0] * E[0,1,2,3] +
b7.dummy[0|0|0] * F[0,1,2,3]
$

Provides the following design:

Choice situation alt1.a alt1.b alt1.c alt1.d alt1.e alt1.f
1 1 3 2 0 3 2
2 2 2 2 2 1 3
3 3 1 3 2 0 1
4 2 0 3 0 2 2
5 0 1 0 0 3 1
6 2 2 1 1 2 1
7 0 1 1 3 1 3
8 1 0 2 3 0 0
9 3 3 0 1 0 2
10 1 2 1 3 3 0
11 0 3 3 2 2 0
12 1 2 1 0 0 3
13 3 1 1 0 1 0
14 1 0 0 2 2 3
15 2 2 0 2 1 0
16 3 0 2 1 3 1
17 2 1 0 3 2 2
18 0 0 3 1 1 2
19 0 3 2 1 0 1
20 3 3 3 3 3 3

Where alt1.a =3, I have bolded it along with alt1.b. Note: the following combinations {a,b} appear {3,0},{3,1},{3,3}. Strangely, {3,2} does not appear in the design. When the syntax does not specify dummying the attributes, the design (in 16 rows) only gives combinations {3,1} and {3,2}.

My understanding of orthogonal designs is that the levels should be balanced, appearing an equal number of times for each combination. I noticed the issue because there are a few conditions that need to be specified - {3,3} is an impossible combination - and I was expecting there to be equal representation of all level combinations.

Being pretty new to all this, perhaps my understanding is wrong, and any clarification you can offer would be most appreciated.

How should I be specifying my syntax?

[Ben Beardmore]

Here is the syntax and design without specifying dummy[0|0|0]

Design
;alts = alt1, alt2
;rows = 13
;orth = sim
;model:
U(alt1) =
b1 +
b2* A[0,1,2,3] +
b3* B[0,1,2,3]+
b4* C[0,1,2,3] +
b5* D[0,1,2,3] +
b6* E[0,1,2,3] +
b7* F[0,1,2,3] +
$

It produced a 16 row design. As you can see for attributes a and b, the combination {a,b} for a=3 only ever appears as {3,1} or {3,2}.

Choice situation alt1.a alt1.b alt1.c alt1.d alt1.e alt1.f
1 2 1 0 1 1 3
2 3 2 1 1 0 1
3 0 3 3 2 0 1
4 3 1 3 0 2 3
5 1 3 1 0 1 2
6 3 1 2 3 0 0
7 3 2 0 2 2 2
8 2 2 2 0 3 1
9 2 2 3 3 1 2
10 0 3 2 1 2 2
11 1 0 2 2 1 3
12 0 0 1 3 2 3
13 2 1 1 2 3 0
14 1 3 0 3 3 1
15 1 0 3 1 3 0
16 0 0 0 0 0 0

[Ben Beardmore]

With 32 rows, the problem is solved but as I've indicated in another thread, there seems to be a problem with estimating 6 4-level attributes in 32 rows. The first attribute gets assigned only 3 levels. I had to go to a 7X4 in 32 and drop one attribute.

Interestingly the 10att X 4level in 32 suffers from the same problem that I initially brought up here. each level pairs with only 2 levels in the next attribute. I have a 10X4 design from another source that does not have this issue, so I know that it is possible...

Again, if I'm doing something wrong, please let me know.

[Michiel Bliemer]

I do not see any problem with certain combinations not occuring. An attribute level balanced design means that each attribute level (in each column) appears an equal amount of times, and that is the case with the designs you present. The 0, 1, 2, and 3 all appear 5 times in the design with 20 rows, so it is attribute level balanced. Only in a full factorial design all combinations will appear an equal number of times, but for a fractional factorial orthogonal design, which you have, there I no reason why this should happen. It is definitely not a defintion of orthogonality. The design is perfectly orthogonal and attribute level balanced.