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exp[(
xY
)
]
n
i
i
L
(,)
(1)
1exp(
x
)
i
1
i
ˆ
(, )
and the maximum likelihood estimate (MLE) of (
, that
maximizes this function. Once the MLE is available, the influence of the covariates can be
assessed by the magnitude (relative to their standard error) of the individual slopes. In
particular, it can be ascertained which attributes of customer service have a substantial effect
on making the probability of a 'Yes' response high.
,
)
is the value, say
2.2 Multinomial logistic regression
Suppose now that the outcome variable has more than two categories. For example, suppose
the responses
{
i
Y
are measured on the 11-point NPS scale. The multinomial logistic
model seeks to represent the probability of
each
response as a function of the covariates.
Since each response is an ordinal categorical variable taking on values in the set {0, 1, , 10
,
we consider a multinomial logistic regression model with 10 cumulative link functions:
1
exp(
x
)
j
i
Pr(
Yj
)
,
j
0,1,
, 9
(2)
i
1exp(
x
)
j
i
9
where
is again a vector of slope parameters. In
order to affect the required non-decreasing behavior (relative to
j
) of the right hand sides of
the link functions, the constraint
{
jj
are intercept parameters and
0
is imposed on the intercept parameters.
0
2
9
Starting with Pr(
Y
0)
Pr(
Y
0)
, and then differencing the above expressions in (2) as in
i
i
j
, we obtain expressions for the individual
probabilities of the response as a function of the intercept and slope parameters. Defining
the responses
Pr(
Yj
)
Pr(
Yj
)
Pr(
Yj
,
1)
1,
,10
i
i
i
10
{}
ijj
Y
to be one (zero) if and only if
i
Y
, the likelihood function for this
0
model is
n
10
Y
(3)
L
(,)
P Y
(
j
)
ij
i
i
10
j
ˆ
and the MLE of (
, that maximizes this function. Once the MLE
is available, the magnitudes of the slope estimates (relative to their standard errors) can be
used to identify the covariates that push the distribution of the response towards 9s and 10s.
We note that the constraint on the intercepts is a standard constraint. In the next section, we
will discuss an additional and novel constraint that can optionally be imposed on the slope
parameters.
,
)
is the value, say
(, )
3. Case study
Carestream Health, Inc. (CSH) was formed in 2007 when Onex Corporation of Toronto,
Canada purchased Eastman Kodak Company's Health Group and renamed the business as
Carestream Health. They are an annual $2.5B company and a world leader in medical
imaging (digital and film), healthcare information systems, dental imaging and dental
practice management software, molecular imaging and non-destructive testing. Their
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