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Homework 4: Diffusion of Tetracycline
We continue examining the diffusion of tetracycline among doctors in Illinois in the early 1950s, building on
our work in lab 6. You will need the data sets ckm_nodes.csv and ckm_network.dat from the labs.
1. Clean the data to eliminate doctors for whom we have no adoption-date information, as in the labs.
Only use this cleaned data in the rest of the assignment.
2. Create a new data frame which records, for every doctor, for every month, whether that doctor began
prescribing tetracycline that month, whether they had adopted tetracycline before that month, the
number of their contacts who began prescribing strictly before that month, and the number of their
contacts who began prescribing in that month or earlier. Explain why the dataframe should have 6
columns, and 2125 rows. Try not to use any loops.
3. Let
pk = Pr(A doctor starts prescribing tetracycline this month | Number of doctor’s contacts prescribing before this month = k)
and
qk = Pr(A doctor starts prescribing tetracycline this month | Number of doctor’s contacts prescribing this month = k)
We suppose that pk and qk are the same for all months.
a. Explain why there should be no more than 21 values of k for which we can estimate pk and qk
directly from the data.
b. Create a vector of estimated pk probabilities, using the data frame from (2). Plot the probabilities
against the number of prior-adoptee contacts k.
c. Create a vector of estimated qk probabilities, using the data frame from (2). Plot the probabilities
against the number of prior-or-contemporary-adoptee contacts k.
4. Because it only conditions on information from the previous month, pk is a little easier to interpret than
qk. It is the probability per month that a doctor adopts tetracycline, if they have exactly k contacts
who had already adopted tetracycline.
a. Suppose pk = a + bk. This would mean that each friend who adopts the new drug increases the
probability of adoption by an equal amount. Estimate this model by least squares, using the values
you constructed in (3b). Report the parameter estimates.
b. Suppose pk = e
a+bk/(1 + e
a+bk). Explain, in words, what this model would imply about the
impact of adding one more adoptee friend on a given doctor’s probability of adoption. (You can
suppose that b > 0, if that makes it easier.) Estimate the model by least squares, using the values
you constructed in (3b).
c. Plot the values from (3b) along with the estimated curves from (4a) and (4b). (You should have
one plot, with k on the horizontal axis, and probabilities on the vertical axis .) Which model do
you prefer, and why?
For quibblers, pedants, and idle hands itching for work to do: The pk values from problem 3 aren’t all equally
precise, because they come from different numbers of observations. Also, if each doctor with k adoptee
contacts is independently deciding whether or not to adopt with probability pk, then the variance in the
number of adoptees will depend on pk. Say that the actual proportion who decide to adopt is pˆk. A little
probability (exercise!) shows that in this situation, E[pˆk] = pk, but that Var[pˆk] = pk(1 − pk)/nk, where nk is
the number of doctors in that situation. (We estimate probabilities more precisely when they’re really extreme
[close to 0 or 1], and/or we have lots of observations.) We can estimate that variance as Vˆ
k = pˆk(1 − pˆk)/nk.
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Find the Vˆ
k, and then re-do the estimation in (4a) and (4b) where the squared error for pk is divided by Vˆ
k.
How much do the parameter estimates change? How much do the plotted curves in (4c) change?
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