
Rating Data
From a strictly statistical perspective, there is no reason why rating data cannot be used
for a conjoint analysis. All that is required is that a derived ranking be created
from the ratings data for each of the hypothetical products that would be explicitly
ranked in a typical conjoint analysis. Once done, the statistical algorithms of a conjoint
analysis can be run on these data.
How can ratings data be used to create the derived rankings in the transposed matrix
for input into a conjoint analysis? To discuss this, we introduce a hypothetical project
involving tea where the three underlying attributes of the tested products are binary
factors and where, because the fullycrossed design is only a 2 × 2 × 2 with eight
unique combinations, all possible combinations are used in the study. The three factors
and the associated levels include:

Temperature (Hot versus Cold)

Strength (Very Strong versus Very Weak)

Sweetness (No Sugar versus Two Teaspoons Sugar)
If each respondent is asked to rate the bipolar factors on some preference scale, say
18, where 1 indicates a preference for one end of the continuum (e.g., Hot) and 8
indicates a preference for the other end of the continuum (Cold), then developing the
derived ranking for that subject for each combination simply requires a comparison
function f(X) that evaluates each subject's score on each of the three underlying
attributes in comparison to the ideal point for that combination, and sums across the
three attributes to develop an overall derived ranking. A simple form of the comparison
function f(X) is the absolute value of the differences between observed and ideal, summed
across the attributes. To illustrate, assume that a respondent presents the following
pattern of ratings:
The blue number, i.e., response 5 for all three ratings, indicates the selected
response for this subject.
The eight profiles yield the following derived rankings. Note that this example
precisely illustrates one of the most significant problems with using rating scales in
that a substantial number of ties occur in the data.
Attribute Combination 
Derived Preference Score 
 
HotVery StrongNo Sugar 
12 
HotVery StrongTwo Teaspoons 
11 
HotVery WeakNo Sugar 
11 
HotVery WeakTwo Teaspoons 
10 
ColdVery StrongNo Sugar 
11 
ColdVery StrongTwo Teaspoons 
10 
ColdVery WeakNo Sugar 
10 
ColdVery WeakTwo Teaspoons 
9 
Note that low scores indicate a higher preference with this scoring system. The lowest
score logically matches to the most preferred combination, and the highest score is
logically the least preferred. Obviously, however, no single combination is ranked 1
because this subject does not have extreme opinions. This is not a crucial problem with
estimating a conjoint solution. What is important, however, is the use of relatively few
ranks, with three 10 scores and three 11 scores capturing the sole variation between the
most and least preferred options.
With some justification, we could argue that any person who holds such mushy opinions
should appropriately be treated as having only very weak preferences for one product
configuration versus another. For the statistical properties of the conjoint, however, the
presence of the large number of ties is disconcerting in that degenerate solutions may
well occur. A preferred pattern with middleoftheroad responses such as these would be a
pattern with scores of, let's say, 9.0, 9.2, 9.7, 10.2, 10.7, 11.1, 11.7, and 12.
The presence of ties is the prior example is exacerbated by the fact that identical
scores occur on all three ratings. Ties can well occur with other, more diverse and
extreme ratings. Consider the following pattern of responses.
Using the absolute value of the simple differences model, this score profile yields
derived rankings as follows:
Attribute Combination 
Derived Preference Score 
 
HotVery StrongNo Sugar 
9 
HotVery StrongTwo Teaspoons 
14 
HotVery WeakNo Sugar 
4 
HotVery WeakTwo Teaspoons 
9 
ColdVery StrongNo Sugar 
12 
ColdVery StrongTwo Teaspoons 
17 
ColdVery WeakNo Sugar 
7 
ColdVery WeakTwo Teaspoons 
12 
While the problem with ties is diminished in this example because of the presence of
more diverse ratings, it is not eliminated. The derived preference scores are logically
appropriate at the extremes, i.e., the two most and least preferred options, but the
middle four while separated into two logical groups of two continue to show ties.
The continuing problem with ties can be further addressed by adding a more limited ranking
task to the data collection effort. Rather than ranking all eight product combinations as
in the typical conjoint problem, subjects can be asked to rank the intensity of their
preference for their stated rating.
Assume that the respondent in the immediately prior set of data stated that the most
important issue in this set of ratings was a preference for a weak taste, followed by
their preference for no sugar, followed by the preference for hot tea. Using rankings of
13 for these scores, a new set of derived rankings using a weighted simple difference
model yields the following scores:
Attribute Combination 
Derived Preference Score 
 
HotVery StrongNo Sugar 
14 
HotVery StrongTwo Teaspoons 
24 
HotVery WeakNo Sugar 
9 
HotVery WeakTwo Teaspoons 
19 
ColdVery StrongNo Sugar 
23 
ColdVery StrongTwo Teaspoons 
33 
ColdVery WeakNo Sugar 
18 
ColdVery WeakTwo Teaspoons 
28 
No ties exist, and a pure 18 ranking of the different combinations could be developed
by simply rank ordering the derived preference scores.
Simply using weights for the ranks in conjunction with the ratings does not completely
eliminate any problems with ties, however. Using the same weighting system, but applying
these weights to the first sample ratings profile that included the subject who responded
with a score of 5 to each question (see page 4, above) yields the following distribution
of derived preference scores:
Attribute Combination 
Derived Preference Score 
 
HotVery StrongNo Sugar 
24 
HotVery StrongTwo Teaspoons 
22 
HotVery WeakNo Sugar 
23 
HotVery WeakTwo Teaspoons 
21 
ColdVery StrongNo Sugar 
21 
ColdVery StrongTwo Teaspoons 
19 
ColdVery WeakNo Sugar 
20 
ColdVery WeakTwo Teaspoons 
18 
Note that the range of the derived preference scores is, under this contrived example,
very restricted. This is expected given the ties in the ratings. Using the weights derived
from the attribute rankings, however, almost eliminates the problems with ties in the
derived preference ranking score. Only in the exact middle of the distribution at a
derived score of 21 does a tie exist.
These examples illustrate the general logic of the alternative conjoint method.
By virtue of having only three underlying attributes, however, these examples mask the
fact that continued problems with ties will occur as the number of binary attributes
increases, even with the weighting function. This problem can be minimized by using a more
complex comparison function f(X) and a more complex weighting function.
We consider the exact comparison function and weighting function that we have developed
to be proprietary intellectual property, and we will not disclose the exact nature of this
information. We will, however, add the following pieces of information to help further
describe the transformations that are used:

Unlike the absolute difference comparison function presented above, the model we have
developed is not a linear comparison function, nor does it yield integer values. It is,
however, a monotonic function.

The weighting function for the attribute rankings we have developed is not a linear
weighting function. It is monotonic. The nonlinear function captures what we believe
characterizes the way in which most individuals make decisions regarding sorting or
ranking of multiple options.
These modifications to the simplistic example using absolute value differences as the
comparison function and simple integer weights for the weighting function serve to
minimize the problem with ties. At this point, we intend to conduct analyses of the data
where the derived preference scores are converted into true ranks and also analyzed
directly to determine the correspondence of the utilities that are obtained from using two
different approaches.
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