Biomedical Engineering Reference
In-Depth Information
regression model is its ability to accommodate a wide range of over-dispersion
degrees. An NBD distribution with mean
l
p
RX
and over-dispersion parameter α
RX
is
represented by:
RX
a
k
RX
l
al
Γ
(
a
RX
+
k
)
a
al
RX
pt
RX RX
Pr(
RX
=
k
|
la
)
=
(7.6)
pt
pt
RX
RX
RX
RX
RX
Γ
(
a
)
Γ
(
k
+
1
)
+
+
pt
pt
This flexible count model has been used in several studies investigating physi-
cians' prescribing behavior (e.g., Manchanda et al.
2005
; Stremersch et al.
2013
;
Venkataraman and Stremersch
2007
). In these studies, the most common specifica-
tion for the conditional mean of the number of prescriptions is a log-link function
that specifies the log of the mean of the conditional distribution as linear in the
parameters.
In the case of new pharmaceutical drugs, time dynamics in the adoption process
can be integrated into the NBD regression model through the specification of the
conditional mean. Specifically, the mean number of prescriptions can be modeled as
a function of the number of time periods,
t
, since the introduction of the new drug,
as follows:
Rx
Rx
ln(
l
)
=++
b bg
t X
+
z
(7.7)
pt
0
p
1
p p t
pt
where
X
pt
includes a set of time-varying physician-level covariates such as the
volume of detailing to the physician. Moreover, time in this specification can also
take a nonlinear form.
Count models can be used for prediction purposes in at least two ways. First, they
allow extension of the horizon for the prescriptions a physician writes. Once the
model parameters are estimated on past data, one can calculate predictions of future
states for each physician in the data set given the physician's past behavior. For
instance, the number of detailing visits the firm expects the physician to receive can
be used to predict the expected number of prescriptions for that physician, on the basis
of the estimated model parameters. In other words, knowing the prescription history
of a given physician for periods 1 through
T
(but not for any time after
T
), researchers
can develop probabilities for some future time period
T
+
t
. Once these individual
predictions are aggregated across all physicians in the data set for each time period in
the investigated timeframe, they provide a predicted pattern of the total number of
prescriptions. The aggregated predictions can serve as a diagnostic tool that depicts
not only to what extent a new drug is expected to be prescribed across physicians, but
also the differential effects that various factors, such as marketing efforts or the pre-
scription volume of other physicians (e.g., word-of-mouth) or opinion leaders, have
on this process. Second, “analogical” count models can be used in a similar fashion as
“analogical” diffusion models, discussed above. In essence, given two drugs (drug A
and drug B) that are similar in terms of category, administration method, etc., one
could use response parameters retrieved for drug A (e.g., the responsiveness of new
drug prescriptions at the physician level to detailing over time) to forecast physicians'
responsiveness to drug B. This forecasting strategy works better for “follow-on” drugs
(non-bioequivalent drugs in the same therapeutic category) than for radically new
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