Sampling Clarification

2/11/2016
Stu -Good question. I think it is easiest to answer this by describing the model(s) we
will estimate. The crux of the matter is that the unit of observation in our analyses is an
individual respondent. Seen in this way, our sample size is sufficient for all the models
we propose.
Take the cancellation question for example. After viewing a wind farm offshore
in a photo simulation, respondents are asked if it would cause them to cancel a trip. Each
respondent answers three questions in this format at three different offshore distances.
The distances shown to a respondent are randomly drawn from the set 2.5, 5, 7.5, 10,
12.5, 15, and 20 miles offshore.
This gives us dichotomous response data for cancellation – 1 for cancel; 0 for not
cancel. We anticipate having roughly 1925 beachgoers in our sample, so that gives us
5775 (3 * 1925) responses to the cancellation question. Since the distances are randomly
assigned, at each distance we have 825 responses (5775/3).
Because we are working with dichotomous response data, we intend to estimate a
binary choice model (logit or probit regression). A simple linear version of the model is
 𝒑𝒓(𝒄𝒂𝒏𝒄𝒆𝒍)𝒊𝒋 = 𝒇(𝜷𝒅𝒊𝒋 + 𝜶𝒃𝒊𝒋 )
𝑝𝑟 𝑐𝑎𝑛𝑐𝑒𝑙

!"

= 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦  𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑡  𝑖  𝑐𝑎𝑛𝑐𝑒𝑙𝑠  𝑡𝑟𝑖𝑝  𝑡𝑜  𝑏𝑒𝑎𝑐ℎ  𝑗 ,

!.!
𝑑!" = 𝑣𝑒𝑐𝑡𝑜𝑟  𝑜𝑓  𝑑𝑢𝑚𝑚𝑦  𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠  𝑓𝑜𝑟  𝑒𝑎𝑐ℎ  𝑜𝑓  𝑡ℎ𝑒  7  𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠  (𝑑!"
=
!
               1  𝑖𝑓  𝑤𝑖𝑛𝑑  𝑓𝑎𝑟𝑚  𝑖𝑠  𝑎𝑡  2.5  𝑚𝑖𝑙𝑒𝑠  𝑜𝑓𝑓𝑠ℎ𝑜𝑟𝑒  𝑓𝑜𝑟  𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑡  𝑖  𝑎𝑡  𝑏𝑒𝑎𝑐ℎ  𝑗, 0  𝑖𝑓  𝑛𝑜𝑡;  𝑑!"
=
               1  𝑖𝑓  𝑤𝑖𝑛𝑑  𝑓𝑎𝑟𝑚  𝑖𝑠  𝑎𝑡  5  𝑚𝑖𝑙𝑒𝑠  𝑜𝑓𝑓𝑠ℎ𝑜𝑟𝑒  𝑓𝑜𝑟  𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑡  𝑖  𝑎𝑡  𝑏𝑒𝑎𝑐ℎ  𝑗  𝑎𝑛𝑑  0  𝑖𝑓  𝑛𝑜𝑡; 𝑒𝑡𝑐. ),

𝑏!" = 𝑣𝑒𝑐𝑡𝑜𝑟  𝑜𝑓  𝑏𝑒𝑎𝑐ℎ  𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠  𝑓𝑜𝑟  𝑏𝑒𝑎𝑐ℎ  𝑗  𝑣𝑖𝑠𝑖𝑡𝑒𝑑  𝑏𝑦  𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑡  𝑖,
𝑓 ∙  𝑖𝑠  𝑎  𝑐𝑜𝑚𝑚𝑜𝑛  𝑙𝑜𝑔𝑖𝑡  𝑜𝑟  𝑝𝑟𝑜𝑏𝑖𝑡  𝑓𝑜𝑟𝑚, 𝑤ℎ𝑖𝑐ℎ  𝐼  𝑤𝑖𝑙𝑙  𝑛𝑜𝑡  𝑤𝑟𝑖𝑡𝑒  𝑜𝑢𝑡  ℎ𝑒𝑟𝑒,
𝛽 = 𝑎  𝑣𝑒𝑐𝑡𝑜𝑟  𝑜𝑟  𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠  𝑎𝑡  𝑡ℎ𝑒  7  𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠:  𝛽!.! , 𝛽! , 𝛽!.! , 𝛽!" , 𝛽!".! , 𝛽!" , 𝛽!" , and
𝛼 = 𝑎  𝑣𝑒𝑐𝑡𝑜𝑟  𝑜𝑓  𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠  𝑓𝑜𝑟  𝑡ℎ𝑒  𝑏𝑒𝑎𝑐ℎ  𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑡𝑖𝑐𝑠
Estimates of the parameters 𝛽 tell us how people respond to wind farms at
different distances offshore. We expect 𝛽!.! , for example, to be the largest of the
coefficients indicating that the probability of cancelling is highest at the closest distance.
As the wind farms get further offshore we expect the probability to decline and hence the

parameters 𝛽 to fall as distance increases. We will also try a simple continuous measure
for distance – entering distance as an explanatory variable instead of a series of dummies.
The estimates for 𝛼 tell us how different beaches may get different responses to
wind farms. For example, some argue that wind farms on developed beaches will have
less impact than they would on more natural (park) beaches. Also, there may be different
responses in different states. These propositions can be explored in the estimation results
for parameters in 𝛼. We will also explore individual characteristics in the model, such as
income, frequency of visiting the beach, etc. To simplify here, I have exclude these
arguments.
With the model above we can predict the impacts on any beach in our model
(beaches from MA to SC), by plugging in the relevant beach characteristics and offshore
distance. This will be BOEM’s model for analysis of wind farms in different locations.
What is important for your comment is that the number of observation we have is
more than sufficient to estimate this model and pick up a signal on the distance effects on
cancellation. With a sample size well above 5000 and never more than a twenty
parameters to estimate, we easily have >5000 degrees of freedom. We have estimated
models like this with much less data and found significance over distance variables.1
Several other models will be estimated in similar fashion. For example, the
response to whether or not wind farms make your experience better or worse will be
estimated using an ordered logit model since the response format has five outcomes
which are ordered – better, somewhat better, no effect, somewhat worse, worse. Also,
there is a question about whether or not a “special trip” would be taken (more or less a
curiosity trip) to see a wind farm. All of the models will be estimated in much the same
way as the cancellation model is described above and, for the same reason, there is
sufficient data.
I hope that helps explain the study design and gives you reasoning why we
believe our sample size is sufficient to answer the questions we have posed.

-- George Parsons, University of Delaware

1

We recognize that the three response values from each person for cancellation should not be treated as independent of
one another. This will be dealt with in the econometrics using fixed-effects. Still, the number of observations is more
than enough for the analysis.