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to impose the monotone constraint  
i i ik onto the parameter space. Adding these
constraints when finding the MLEs complicates the required maximization of the
likelihoods in (1) and (3). In this section, however, we will show how this can be done using
SAS with PROC NLP.
In order to simplify our use of PROC NLP, it is convenient to work with a full-rank
parameterization of the logistic regression model. Because countries are nested within
regions, a linear dependency exists between the dummy variables corresponding to regions
and countries within regions. We can eliminate the linear dependency by removing region
from the model and specifying country to be non-nested factor. The result of this model
reparameterization is that instead of 6 degrees of freedom in the model for regions and 16
degrees of freedom for countries nested within regions, we equivalently have 22 degrees of
freedom for countries. For the same purpose, we also redefine the dummy variable coding
used for other categorical and ordinal covariates by using a full rank parameterization
scheme. In particular, we use k -1 dummy variables (rather than k ) to represent a k -level
categorical or ordinal variable. With the full rank parameterization, the highest level of
customer satisfaction has a slope parameter that is fixed to be 0. Lines 3-10 in the SAS code
shown in Appendix B are used to set up the full rank parameterization of the logistic
regression model.
1
2
3.00
q82a
2.50
q82b
q82d
q82f
2.00
q79
1.50
1.00
0.50
0.00
0
1
2
3
4
5
6
7
8
Response Level
Fig. 2. Constrained MLEs of Slopes for 7-Point Likert Scale Customer Satisfaction Covariates
Beginning with line 12 in the SAS code, PROC NLP is used to derive the MLEs of the
parameters under the constrained parameter space. The 'max' statement (line 13) indicates
the objective function is the log-likelihood function of the model and that it is to be
maximized. The maximization is carried out using a Newton-Raphson algorithm, and the
'parms' statement (line 14) specifies initial values for the intercept and slope parameters.
The SAS variables bq j , ba j, bb j, bd j and bf j are used to symbolize the slope parameters
corresponding to the j -th response level of the customer satisfaction covariates q79, q82a,
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