Delivered by Andrew Gelman (Statistics Professor, Columbia University) at the 2016 New York R Conference on April 8th and 9th at Work-Bench.

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Data on putts in pro golf
Distance from hole (feet)
Probabilityofsuccess
1346/1443
577/694
337/455
208/353
149/272
136/256
111/240
69/217
67/200
75/237
52/202
46/192
54/174
28/167
27/201
31/195
33/191
20/147
24/152
1/33

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What's the probability of making a golf putt?
Distance from hole (feet)
Probabilityofsuccess
Logistic regression,
a = 2.2, b = −0.3
2/33

3. Geometry-based model
x
R
r
−2σ 0 2σ
3/33

4. Stan code
data {
int J;
int n[J];
real x[J];
int y[J];
real r;
real R;
}
parameters {
real<lower=0> sigma;
}
model {
real p[J];
for (j in 1:J)
p[j] <- 2*Phi(asin((R-r)/x[j]) / sigma) - 1;
y ~ binomial(n, p);
}

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0.00.20.40.60.81.0
What's the probability of making a golf putt?
Distance from hole (feet)
Probabilityofsuccess
Geometry−based model,
sigma = 1.5
5/33

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Two models fit to the golf putting data
Distance from hole (feet)
Probabilityofsuccess
Logistic regression,
a = 2.2, b = −0.3
Geometry−based model,
sigma = 1.5
6/33

7. A tougher model
1. Sum of declining exponentials:
y = a1e−b1x
+ a2e−b2x
2. Statistical version:
yi = (a1e−b1xi
+ a2e−b2xi
) · i
7/33

8. Stan code
data {
int N;
vector[N] x;
vector[N] y;}
parameters {
vector[2] log_a;
ordered[2] log_b;
real<lower=0> sigma;}
transformed parameters {
vector<lower=0>[2] a;
vector<lower=0>[2] b;
a <- exp(log_a);
b <- exp(log_b);}
model {
vector[N] ypred;
ypred <- a[1]*exp(-b[1]*x) + a[2]*exp(-b[2]*x);
y ~ lognormal(log(ypred), sigma);
}

9. Simulate fake data in R
# Set up the true parameter values
a <- c(.8, 1)
b <- c(2, .1)
sigma <- .2
# Simulate data
x <- (1:1000)/100
N <- length(x)
ypred <- a[1]*exp(-b[1]*x) + a[2]*exp(-b[2]*x)
y <- ypred*exp(rnorm(N, 0, sigma))
9/33

10. Fit the model in Stan
Inference for Stan model: exponentials.
4 chains, each with iter=1000; warmup=500; thin=1;
post-warmup draws per chain=500, total post-warmup draws=2000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
a[1] 1.00 0.00 0.03 0.95 0.99 1.00 1.02 1.05 494 1
a[2] 0.70 0.00 0.08 0.56 0.65 0.69 0.75 0.87 620 1
b[1] 0.10 0.00 0.00 0.09 0.10 0.10 0.10 0.11 532 1
b[2] 1.71 0.02 0.34 1.15 1.48 1.67 1.90 2.49 498 1
sigma 0.19 0.00 0.00 0.19 0.19 0.19 0.20 0.20 952 1
10/33

11. Try again with new parameter values
mean se_mean sd 2.5% 25% 50% 75% 9
a[1] 1.33e+00 0.54 0.77 0.00 1.28 1.77e+00 1.79e+00 1.82
a[2] 2.46e+294 Inf Inf 0.00 0.00 0.00e+00 1.77e+00 2.66e
b[1] 1.00e-01 0.04 0.06 0.00 0.10 1.30e-01 1.30e-01 1.40
b[2] 3.09e+305 Inf Inf 0.13 0.50 1.15e+109 4.77e+212 5.28e
sigma 2.00e-01 0.00 0.00 0.19 0.19 2.00e-01 2.00e-01 2.00
11/33

12. Informative prior distribution
log_a ~ normal(0, 1);
log_b ~ normal(0, 1);
12/33

13. Happy ending
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
a[1] 1.56 0.09 0.32 0.65 1.52 1.72 1.75 1.79 13 1.25
a[2] 0.32 0.08 0.28 0.06 0.14 0.22 0.37 1.13 13 1.20
b[1] 0.13 0.00 0.01 0.10 0.12 0.13 0.13 0.14 22 1.14
b[2] 1.94 0.20 2.29 0.14 0.22 1.26 3.00 7.34 127 1.05
sigma 0.20 0.00 0.00 0.19 0.19 0.20 0.20 0.21 656 1.00
13/33

14. The political impact of social penumbras
Andrew Gelman and Yotam Margalit
Department of Statistics and Department of Political Science
Columbia University, New York
8 Apr 2016
14/33

16. Sen. Rob
Portman
(R-Ohio)

17. 1996: co-sponsored the Defense of Marriage Act,
defining marriage as one man and one woman
Sen. Rob
Portman
(R-Ohio)

18. 1996: co-sponsored the Defense of Marriage Act,
defining marriage as one man and one woman
1999: voted for a measure prohibiting same-sex
couples from adopting children.
Sen. Rob
Portman
(R-Ohio)

19. 1996: co-sponsored the Defense of Marriage Act,
defining marriage as one man and one woman
1999: voted for a measure prohibiting same-sex
couples from adopting children.
2011: “openly hostile” record on gay rights led to
a mass protest of students at the University of
Michigan against Portman speaking at the
graduation ceremony
Sen. Rob
Portman
(R-Ohio)

20. 1996: co-sponsored the Defense of Marriage Act,
defining marriage as one man and one woman
1999: voted for a measure prohibiting same-sex
couples from adopting children.
2011: “openly hostile” record on gay rights led to
a mass protest of students at the University of
Michigan against Portman speaking at the
graduation ceremony
“Rob believes marriage is a sacred bond between
one man and one woman”—Portman spokesman
to Cleveland Plain Dealer at the time of the
protest
Sen. Rob
Portman
(R-Ohio)

21. Sen. Rob
Portman
(R-Ohio)
16/33

22. 2013: “I’m announcing today a change of heart
[for] gay marriage. . . My son came to Jane, my
wife, and I, told us that he was gay, and that it
was not a choice, and that it’s just part of who
he is, and that’s who he’d been that way for as
long as he could remember.”
Sen. Rob
Portman
(R-Ohio)
16/33

23. 2013: “I’m announcing today a change of heart
[for] gay marriage. . . My son came to Jane, my
wife, and I, told us that he was gay, and that it
was not a choice, and that it’s just part of who
he is, and that’s who he’d been that way for as
long as he could remember.”
“I’ve come to the conclusion that . . . [marriage] is
something that we should allow people to do
. . . and to have the joy and stability of marriage
that I’ve had for over 26 years.”
Sen. Rob
Portman
(R-Ohio)
16/33

24. 2013: “I’m announcing today a change of heart
[for] gay marriage. . . My son came to Jane, my
wife, and I, told us that he was gay, and that it
was not a choice, and that it’s just part of who
he is, and that’s who he’d been that way for as
long as he could remember.”
“I’ve come to the conclusion that . . . [marriage] is
something that we should allow people to do
. . . and to have the joy and stability of marriage
that I’ve had for over 26 years.”
Portman also mentioned his consultation with
former Vice President Dick Cheney, whose own
daughter’s coming out: “forced him to re-think
the issue too, and over time, he changed his view
on it”
Sen. Rob
Portman
(R-Ohio)
16/33

25. Research questions
1. To what extent does familiarity with members of a certain
social group influence people’s views on public policies related
to that group?
17/33

26. Penumbra
18/33

27. Social penumbra
Size and the shape of the penumbra can relate to the political
salience and influence of a social group
19/33

28. Research questions
1. To what extent does familiarity with members of a certain
social group influence people’s views on public policies related
to that group?
2. To what extent does the social penumbra of a group—its
shape and size—matter for explaining public preferences on
key policies?
20/33

29. Penumbras and change in public opinion?
2004: Karl Rove puts gay marriage on the ballot in key states
to help mobilize mass support in favor of Bush
2012: Karl Rove criticizes the Democrats for “using the issue
of gay marriage for political gain”
21/33

30. Our data
Panel design
Two surveys administered by YouGov 12 months apart
3,000 respondents in wave 1; 2,106 re-interviewed in wave 2
Penumbra membership in 14 social groups and 8 names
Attitude questions on 12 related policies
Also GSS in 2006 and 2014

31. Active Military
q
Immigrant in Past 5 Yrs NRA Member Abortion in Past 5 Yrs
Lost Job Last Year Muslim Gay/Lesbian Currently Unemployed
Mortgage Underwater No Health Insurance Care for Elder Gun Owner
Receive Govt Welfare Serious Health Problem
Group size (area of red circle) and penumbras (areas of black circles indicate close family, close family + close friends,
and close family + close friends + others known) compared to the population of American adults (light gray circle)
23/33

32. Walter
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Rose
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Bruce
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Tina
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Kyle
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Emily
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Jose
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Maria
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Walter+Rose+...+Emily
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Jose + Maria
q
All 8 names
Number of people with each name (area of red circle) and penumbra (areas of black circle indicates number of people
who know at least one person with this name) compared to the population of American adults (light gray circle)
24/33

33. Know any Mortgage Underwater?
sd 9%
Know any Gun Owner?
sd 7%
Know any Gay/Lesbian?
sd 6%
Know any Immigrant in Past 5 Yrs?
sd 5%
Know any NRA Member?
sd 5%
Know any Active Military?
sd 5%
Know any Muslim?
sd 4%
Know any Receive Govt Welfare?
sd 4%
Know any No Health Insurance?
sd 4%
Know any Currently Unemployed?
sd 3%
Know any Lost Job Last Year?
sd 3%
Know any Care for Elder?
sd 2%
Know any Serious Health Problem?
sd 2%
Know any Abortion in Past 5 Yrs?
sd 1%
25/33

34. Are you conservative?
sd 5%
Do you attend church regularly?
sd 8%
Know any Rose?
sd 2%
Know any Emily?
sd 2%
Know any Walter?
sd 3%
Know any Bruce?
sd 3%
Know any Tina?
sd 3%
Know any Kyle?
sd 5%
Know any Maria?
sd 6%
Know any Jose?
sd 10%
Shades of blue: People in these states are more likely to know someone of that group
Shades of tan: Less likely to know someone of that group
26/33

35. Family OtherFriends
012
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Active Military
Family OtherFriends
012
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Immigrant in Past 5 Yrs
Family OtherFriends
012
q
q
qq
q
q
q
q
q
NRA Member
Family OtherFriends
012
q q q
q q
q
q q q
Abortion in Past 5 Yrs
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Lost Job Last Year
Family OtherFriends
012
q
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q
q
q
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Muslim
Family OtherFriends
012
q
q
q
q
q
q
q
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q
Gay/Lesbian
Family OtherFriends
012
q
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q
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Currently Unemployed
Family OtherFriends
012
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q
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q
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Mortgage Underwater
Family OtherFriends
012
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q
q
q
q
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q
No Health Insurance
Family OtherFriends
012
q
q
qq
q
qq q
q
Care for Elder
Family OtherFriends
012345
q q
q
q
q
q
q
q
q
Gun Owner
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Receive Govt Welfare
Family OtherFriends
012
q
q
q
q
q
q
q q
q
Serious Health Problem
Average number of people known in each group, among Democrats (blue), Independents (purple), and Republicans (red)
27/33

36. Family OtherFriends
012
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q
q
q
q
q
q
q
Active Military
Family OtherFriends
012
q
q
q
q q
q
q
q
q
Immigrant in Past 5 Yrs
Family OtherFriends
012
q q
q
q
q
q
q
q
q
NRA Member
Family OtherFriends
012
q q q
q q qq
q
q
Abortion in Past 5 Yrs
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Lost Job Last Year
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Muslim
Family OtherFriends
0123
q
q
q
q
q
q
q
q
q
Gay/Lesbian
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Currently Unemployed
Family OtherFriends
012
q q
q
q q
q
q
q
q
Mortgage Underwater
Family OtherFriends
012
q q
q
q q
q
q
q
q
No Health Insurance
Family OtherFriends
012
q
q
q
q
q q
q
q
q
Care for Elder
Family OtherFriends
012345
q q
q
q
q
q
q
q
q
Gun Owner
Family OtherFriends
012
q q
q
q q
q
q
q
q
Receive Govt Welfare
Family OtherFriends
012
q
q
q
q
q
q
q
q
q
Serious Health Problem
Average number of people known in each group, among low, middle, and high−income respondents
(indicated by thin, intermediate, and thick lines)
28/33

37. Is penumbra membership associated with group-relevant
attitudes?
Do you think the number of immigrants from foreign countries
who are permitted to come to the United States to live should
be increased, decreased, or left the same as it is now?
Do you agree or disagree: Homosexual couples should have the
right to marry one another.
Some are calling for the government to provide tax breaks for
family expenditures on care provision for elderly family
members. Others worry that such tax breaks would add to the
national debt. Do you support or oppose providing tax breaks
for care provision to the elderly?

38. More policy questions related to core groups
Would you support or oppose a law requiring a nationwide ban
on the sale of assault weapons?
Which comes closest to your point of view: As a general rule,
do you think the United States should be willing to use
military force around the world, or the United States should be
very reluctant to use military force?
Do you agree or disagree with the following statement: “It is
the responsibility of the federal government to make sure that
all Americans have healthcare coverage.”
Etc.

39. 2.03.04.0
Active Military
Penumbra score
Avgattitudeon
MilitaryForce
0 1 2 3
2.03.04.0
Immigrant in Past 5 Yrs
Penumbra score
Avgattitudeon
AllowIllegalImmigrant
0 1 2 3
2.03.04.0
NRA Member
Penumbra score
Avgattitudeon
NOTWeaponBan
0 1 2 3
2.03.04.0
Abortion in Past 5 Yrs
Penumbra score
Avgattitudeon
LegalAbortion
0 1 2 3
2.03.04.0
Lost Job Last Year
Penumbra score
Avgattitudeon
UnemployedBenefits
0 1 2 3
2.03.04.0
Muslim
Penumbra score
Avgattitudeon
NOTMuslimairportscreen
0 1 2 3
2.03.04.0
Gay/Lesbian
Penumbra score
Avgattitudeon
HomesexualMarriage
0 1 2 3
2.03.04.0
Currently Unemployed
Penumbra score
Avgattitudeon
UnemployedBenefits
0 1 2 3
2.03.04.0
Mortgage Underwater
Penumbra score
Avgattitudeon
PreventForceclosure
0 1 2 3
2.03.04.0
No Health Insurance
Penumbra score
Avgattitudeon
GovtHealthcare
0 1 2 3
2.03.04.0
Care for Elder
Penumbra score
Avgattitudeon
Taxbreak
0 1 2 3
2.03.04.0
Gun Owner
Penumbra score
Avgattitudeon
NOTWeaponBan
0 1 2 3
2.03.04.0
Receive Govt Welfare
Penumbra score
Avgattitudeon
PublicAssistance
0 1 2 3
2.03.04.0
Serious Health Problem
Penumbra score
Avgattitudeon
GovtHealthcare
0 1 2 3
31/33

40. Coefs predicting change in attitude,
given entrance into the penumbra
−0.2 −0.1 0.0 0.1 0.2
Serious Health Problem
Receive Govt Welfare
Gun Owner
Care for Elder
No Health Insurance
Mortgage Underwater
Currently Unemployed
Gay/Lesbian
Muslim
Muslim
Muslim
Lost Job Last Year
Abortion in Past 5 Yrs
NRA Member
Immigrant in Past 5 Yrs
Immigrant in Past 5 Yrs
Immigrant in Past 5 Yrs
Active Military
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Small but nonzero effects
No effects with cross-predictions, reverse-time predictions
32/33

41. Summary
Social penumbras are a politically meaningful concept
Penumbra membership is associated with group-related policy
views.
Entering a penumbra can lead to attitude shifts
Penumbra’s shape might help explain speed and patterns of
change in public opinion