MY RESPONSES IN CAPS DIRECTLY FOLLOWING YOUR COMMENTS/QUESTIONS. --BRYAN
Orthogonality means each pair of levels appears equally across all pairs of attributes within the design.
YES, THAT IS TRUE ONLY AS LONG AS EACH ATTRIBUTE HAS THE SAME NUMBER OF LEVELS. WHEN ATTRIBUTES HAVE DIFFERENT NUMBERS OF LEVELS, ORTHOGONAL DESIGNS OFTEN HAVE TO OVERREPRESENT SOME LEVELS WITHIN AN ATTRIBUTE COMPARED TO OTHER LEVELS WITHIN THE SAME ATTRIBUTE. THUS, EACH PAIRING OF LEVELS WOULD NOT NECESSARILY BE REPRESENTED AN EQUAL NUMBER OF TIMES. RATHER, ORTHOGONAL DESIGNS WOULD SHOW PAIRINGS OF LEVELS PROPORTIONALLY OFTEN.
Therefore, orthogonality is not only relevant for alternative design. It is also relevant for choice sets design. YES, ORTHOGONALITY IS RELEVANT TO TRADITIONAL(CARD-SORT) CONJOINT DESIGNS AS WELL AS DISCRETE CHOICE EXPERIMENTS (ALSO KNOWN AS CBC).
Moreover, according to definition of orthogonality a full factorial design is perfectly orthogonal that can capture interaction also. TRUE, IT’S ORTHOGONAL. TRUE, FULL FACTORIAL CAN MEASURE ALL INTERACTION EFFECTS.
Fractional factorial design can also be orthogonal and at the same time measure some interactions. TRUE. A SPECIFIC SUBSET OF FRACTIONAL FACTORIAL DESIGNS CAN MEASURE CERTAIN INTERACTION EFFECTS.
Therefore, orthogonality in general does not mean that interaction are ignored. NOR DOES IT MEAN THAT THEY WILL BE CAPTURED. ORTHOGONALITY AND THE ABILITY OF A DESIGN TO MEASURE INTERACTIONS ARE SEPARATE CONCEPTS. OFTEN, ORTHOGONAL FRACTIONAL FACTORIAL DESIGNS CANNOT MEASURE ALL POTENTIAL INTERACTIONS.
In fact OMEP is a specific kind of orthogonal design that measure only main effect not interaction. Correct? YES.
If this is correct, how do you interpret definition of orthogonality in Orme's book glossary "A statistical term that, when applied to conjoint analysis experimental designs, refers to experiments in which the attributes are uncorrelated across product concepts. In more technical terms the columns in the design matrix have zero correlation between levels of different attributes"
According to this definition full factorial design that inherently includes interaction terms should not be orthogonal. I’M SORRY, I’M CONFUSED WITH YOUR LOGIC. A FULL FACTORIAL DESIGN INDEED IS ORTHOGONAL IN TERMS OF ALL POTENTIAL MAIN EFFECTS AND ALL POTENTIAL INTERACTION TERMS THAT THE RESEARCHER WOULD WANT TO INCLUDE IN THE MODEL.
I think above definition should be related to orthogonal main effect plan (OMEP) that is an specific kind of orthogonal design. Correct?
NO, THE DEFINITION IN THE ORME BOOK SHOULD ALSO APPLY TO PLANS WHERE MAIN EFFECTS ARE ORTHOGONAL AND INTERACTION EFFECTS ARE ALSO ORTHOGONAL – MEANING THAT THE CORRELATIONS IN THE DESIGN MATRIX BETWEEN DIFFERENT ATTRIBUTES BOTH FOR MAIN EFFECTS AND INTERACTION EFFECTS ARE ORTHOGONAL. --BRYAN