On the basis of the reported preference values for each stimulus conjoint analysis determines the part-worths. Conjoint analysis uses an additive model of the form
airbag | ||||
1 | 2 | |||
50 kW | 1 | 1 | 3 | |
engine | 70 kW | 2 | 2 | 6 |
90 kW | 3 | 4 | 5 |
There are preferences altogether. Suppose that the stimuli have been sorted so that
corresponds to engine level 1 and
airbag level 1,
corresponds to engine level 1 and airbag level 2, and so on.
Then model (16.1) reads:
Now we would like to estimate the part-worths .
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If we order the stimuli as follows:
Our aim is to estimate the part-worths
as well as possible from a collection of
tables like Table 16.5 that have been generated by a sample of
test persons.
First, the so-called metric solution to this problem is discussed
and then a non-metric solution.
The problem of conjoint measurement analysis can be solved by the technique of Analysis of Variance. An important assumption underlying this technique is that the ``distance'' between any two adjacent preference orderings corresponds to the same difference in utility. That is, the difference in utility between the products ranked 1st and 2nd is the same as the difference in utility between the products ranked 4th and 5th. Put differently, we treat the ranking of the products--which is a cardinal variable--as if it were a metric variable.
Introducing a mean utility equation (16.1) can be rewritten.
The mean utility in the above Example 16.3 is
.
In order to check the deviations of the utilities
from this mean, we enlarge Table 16.5
by the mean utility
, given a certain level of the other factor.
The metric solution for the car example is given in Table 16.6:
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In the margarine example the resulting part-worths for are
The coefficients are computed
as
, where
is the average preference
ordering for each factor level.
For instance,
.
The fit can be evaluated by calculating the deviations
of the fitted values to the observed preference orderings.
In the rightmost column of Table 16.8
the quadratic deviations between the observed rankings (utilities)
and the estimated utilities
are listed.
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The technique described that generated Table 16.7 is in fact the solution to a least squares problem.
The conjoint measurement problem (16.1) may be rewritten as a linear regression model (with error ):
and define the design matrix as
The least squares solution to this problem is the technique used for Table 16.7.
In practice we have more than one person to answer the utility rank question for the
different factor levels. The design matrix is then obtained by stacking the above design
matrix times. Hence, for
persons we have as a final design matrix:
Solving (16.7) we have:
In fact, we obtain the same estimated part-worths as in Table 16.7. The stimulus
corresponds to adding up
and
(see (16.3)). Adding
and
gives:
If we drop the assumption that utilities are measured on a metric scale, we have to use (16.1)
to estimate the coefficients from an adjusted set of estimated
utilities. More precisely, we may use the monotone ANOVA as developed by Kruskal (1965).
The procedure works as follows. First, one estimates
model (16.1) with the ANOVA technique described above.
Then one applies a monotone transformation
to the estimated stimulus utilities.
The monotone transformation
is used because the
fitted values
from (16.2) of the
reported preference orderings
may not be monotone. The transformation
is introduced to guarantee
monotonicity of preference orderings. For the car example
the reported
values were
.
The estimated values are computed as:
If we make a plot of the estimated preference orderings against the revealed ones, we obtain Figure 16.1.
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We see that the estimated
is below the estimated
and thus an inconsistency in ranking the utilities occurrs.
The monotone transformation
is introduced to make
the relationship in Figure 16.1 monotone. A very simple procedure
consists of averaging the ``violators''
and
to obtain
. The relationship is then monotone
but the model (16.1) may now be violated.
The idea is therefore to iterate these two steps.
This procedure is iterated until the stress measure
(see Chapter 15)