Avoiding Linear Dependency

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CVA's OLS regression utility calculator deletes one level of each attribute from the utility computation.  That is done to avoid linear dependence among attribute levels, which would result in indeterminacy in the solution.  The purpose of this section is to explain why that precaution is required, when it is appropriate, and when it is useful but not essential.


We consider two popular types of conjoint designs:


Full-profile single concept

Paired comparison


In each case we will consider a design matrix with a row for each question and a column for each attribute level.  Suppose that the total number of attributes is n and the total number of levels (columns) is N.




Full-profile Single Concept


In these designs, each question presents a single concept.  Each concept has a specified level of each attribute.  Each row of the design matrix has 1's in the columns corresponding to the attribute levels in that concept, and other elements of 0 (zero).  Since each row contains exactly n 1's and N -1 zeroes, the sum of each row is n.


CVA computes an intercept term, which is equivalent to adding an imaginary N + first column to the design matrix in which every element is a 1.  That column would be linearly dependent on the sum of the N existing columns, which would cause the solution to be indeterminate.


However, the problem is greater than that.  Consider just the subset of columns corresponding to levels of one attribute:  Every row will have a single 1 and other elements of 0.  Therefore, within that attribute the sum of every row is equal to 1.  This problem generalized to every attribute.  The most popular solution is to delete one level from each attribute, which is equivalent to assuming that that level has a utility of 0.  If this is done, then the rows no longer have equal sums and the indeterminacy is resolved.  CVA arbitrarily deletes the first level of each attribute.




Paired Comparisons


In these designs each question presents two concepts and the respondent is asked to express a preference.  Suppose one concept is presented on the "left" and one on the "right" and that the response scale is arranged so that larger responses indicate greater preference for the concept on the right.  Also, we assume that if a particular attribute appears in one of the concepts, it will also appear (probably with a different level) in the other concept.


The design matrix has a row for each concept and N columns.  Each element is 1 if that attribute level appears in the concept on the right, -1 if it appears in the concept on the left, or 0 if that attribute level does not appear in either concept.  (If an attribute level should happen to appear in both concepts it should not have an effect on respondent preference, and is given a value of zero.)


Consider just those columns corresponding to levels of one attribute.  If an attribute appears in either concept, it will also be represented in the other concept.  Therefore, each value of 1 has a corresponding value of -1, and the sum of values will be 0 for every row within the columns corresponding to each attribute.  Thus paired comparison designs (even those that present concepts specified on only subsets of attributes) are subject to the same indeterminacy as full-profile single concept designs.  CVA handles the problem of indeterminacy in exactly the same way as with full-profile single concept designs, by deleting the first level of each attribute and assuming its utility to be 0.


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