Statistical Rethinking 2023 - Lecture 09

Statistical Rethinking
9. Modeling Events
2023

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Lecture 8

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FLOW

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Flow forward
Everything comes all at once:
Scienti c modeling, research design,
statistical modeling, coding,
interpretation, communication, the
price of eggs
Don’t have to understand it all at once
Nobody ever does

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UC Berkeley Admissions
4526 graduate school
applications for 1973 UC
Berkeley
Strati ed by department and
gender of applicant
Gender discrimination by
admission o cers?
dept admit reject applications gender
1 A 512 313 825 male
2 A 89 19 108 female
3 B 353 207 560 male
4 B 17 8 25 female
5 C 120 205 325 male
6 C 202 391 593 female
7 D 138 279 417 male
8 D 131 244 375 female
9 E 53 138 191 male
10 E 94 299 393 female
11 F 22 351 373 male
12 F 24 317 341 female
See ?UCBAdmissions for citation

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https://arxiv.org/abs/2207.13665

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Modeling events
Events: Discrete, unordered outcomes
Observations are counts
Unknowns are probabilities, odds
Everything interacts always
everywhere
A beast known as “log-odds”

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Admissions: Drawing the Owl
(1) Estimand(s)
(2) Scienti c model(s)
(3) Statistical model(s)
(4) Analyze

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Admissions
G A
gender admission
Was there gender discrimination in
graduate admissions?

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Admissions
G
D
A
gender
department
admission

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G
D
A
What can the “causal e ect of gender” mean?
R
Referee
G*
Perceived
gender

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G
D
A
R
Referee
G*
Perceived
gender
blinding

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G
D
A
R
Referee
G*
Perceived
gender

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Context & discrimination
X
Z
Y
status
context
event

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Wage discrimination
X
Z
Y
status
occupation
wage

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G
D
A
gender
department
admission
Which path is “discrimination”?

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G
D
A
gender
department
admission
Direct discrimination
(status-based or taste-based discrimination)

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G
D
A
gender
department
admission
Indirect discrimination
(structural discrimination)

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G
D
A
gender
department
admission
Total discrimination
(experienced discrimination)

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G
D
A
G
D
A G
D
A
Total Indirect Direct
Requires mild
assumptions
Requires strong
assumptions
Requires strong
assumptions

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G
D
A
Confounds!
U person feature
Will ignore for now, but confounds will return

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(1) Estimand(s)
(4) Analyze

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Generative model
How can choice of
department create
structural discrimination?
When departments vary in
baseline admission rates.
# generative model, basic mediator scenario
N # even gender distribution
G # gender 1 tends to apply to department 1, 2 to 2
D # matrix of acceptance rates [dept,gender]
accept_rate # simulate acceptance
A

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Generative model
A G
D
A

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Generative model
A > table(G,A)
A
G 0 1
1 421 101
2 350 128
Accept rates
Gender 1: 19%
Gender 2: 27%
> table(G,D)
D
G 1 2
1 361 161
2 99 379

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Generative model
A > table(G,A)
A
G 0 1
1 473 46
2 404 77
Accept rates
Gender 1: 9%
Gender 2: 16%
> table(G,D)
D
G 1 2
1 355 164
2 95 386
> accept_rate
[,1] [,2]
[1,] 0.05 0.1
[2,] 0.20 0.3
same pattern as absence of
discrimination

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Generative model
Could do a lot better
Admission rate depends upon size
of applicant pool, distribution of
quali cations
Should sample applicant pool and
then sort to select admissions
Rates are conditional on structure of
applicant pool
> table(G,A)
A
G 0 1
1 473 46
2 404 77
Accept rates
Gender 1: 9%
Gender 2: 16%
> table(G,D)
D
G 1 2
1 355 164
2 95 386

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PAUSE

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(1) Estimand(s)
(4) Analyze

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Modeling events
We observe: Counts of events
We estimate:
Probability (or log-odds) of events
Like the globe tossing model, but
need “proportion of water”
strati ed by other variables

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Generalized Linear Models
Linear Models: Expected value is
additive (“linear”) combination of
parameters
Generalized Linear Models:
Expected value is some function of
an additive combination of
parameters
Y
i
∼ Normal(μ
i
, σ)
μ
i
= α + β
X
X
i
+ β
Z
Z
i

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-3 -2 -1 0 1 2 3
-0.5 0.0 0.5 1.0 1.5
x
a+b*x
-3 -2 -1 0 1 2 3
-0.5 0.0 0.5 1.0 1.5
x
f^-1(a+b*x)

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Linear Models: Expected value is
additive (“linear”) combination of
parameters
Generalized Linear Models:
Expected value is some function of
an additive combination of
parameters
Y
i
∼ Normal(μ
i
, σ)
μ
i
= α + β
X
X
i
+ β
Z
Z
i
Y
i
∼ Bernoulli(p
i
)
f(p
i
) = α + β
X
X
i
+ β
Z
Z
i
f(p
i
)

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Y
i
∼ Bernoulli(p
i
)
f(p
i
) = α + β
X
X
i
+ β
Z
Z
i
f(p
i
)
0/1 outcome probability of event
can take any real value
f() maps probability scale
to linear model scale

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Links and inverse links
Y
i
∼ Bernoulli(p
i
)
f(p
i
) = α + β
X
X
i
+ β
Z
Z
i
f(p
i
)
is the link function
Links parameters of distribution to
linear model
f
f−1 is the inverse link function p
i
= f−1(α + β
X
X
i
+ β
Z
Z
i
)
f−1

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Example inverse function
f(a) = a2

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Example inverse function
f(a) = a2
f−1(a2) = a2 = a

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Distributions and link functions
Distributions: Relative number of ways to observe data, given
assumptions about rates, probabilities, slopes, etc.
Distributions are matched to constraints on observed variables
Link functions are matched to distributions

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sum
large mean
count events
low rate
count
events
low probability
many trials
dnorm
dgamma
dpois dbinom
dexp
Z ∼ /PSNBM(µ, σ)
Z ∼ #JOPNJBM(O, Q)
Z ∼ 1PJTTPO(λ)
Z ∼ (BNNB(λ, L)
Z ∼ &YQPOFOUJBM(λ)
Figure 9.5

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sum
large mean
count events
low rate
count
events
low probability
many trials
dnorm
dgamma
dpois dbinom
dexp
Z ∼ 1PJTTPO(λ)
Z ∼ (BNNB(λ, L)
Figure 9.5
Distributional assumptions
are assumptions about
constraints on observations
You cannot “test” if your
data are “normal”

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Distributions and link functions
Distributions: Relative number
of ways to observe data, given
assumptions
Distributions are matched to
constraints on observed
variables
Link functions are matched to
distributions
(&/&3"-*;&% -*/&"3 .0%&-4
sum
large mean
count events
low rate
count
events
low probability
many trials
dnorm
dgamma
dpois dbinom
dexp
Z ∼ 1PJTTPO(λ)
Z ∼ (BNNB(λ, L)
'ĶĴłĿĲ Ƒƍ 4PNF PG UIF FYQPOFOUJBM GBNJMZ EJTUSJCVUJPOT UIFJS OPUBUJPO
BOE TPNF PG UIFJS SFMBUJPOTIJQT $FOUFS FYQPOFOUJBM EJTUSJCVUJPO $MPDL

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Overthinking: Binomial maximum entropy. The usual way to derive a maximum entropy distribu-
tion is to state the constraints and then use a mathematical device called the Lagrangian to solve for
the probability assignments that maximize entropy. But instead we’ll extend the strategy used in the
Overthinking box on page 306. As a bonus, this strategy will allow us to derive the constraints that
are necessary for a distribution, in this case the binomial, to be a maximum entropy distribution.
Let p be the binomial distribution, and let pi
be the probability of a sequence of observations i
with number of successes xi
and number of failures n − xi
. Let q be some other discrete distribution
defined over the same set of observable sequences. As before, KL divergence tells us that:
−H(q, p) ≥ H(q) =⇒ −
i
qi
log pi ≥ −
i
qi
log qi
What we’re going to do now is work with H(q, p) and simplify it until we can isolate the constraint
that defines the class of distributions for which p has maximum entropy. Let λ = i
pi
xi
be the
expected value of p. Then from the definition of H(q, p):
−H(q, p) = −
i
qi
log λ
n
xi
1 −
λ
n
n−xi
= −
i
qi
xi
log λ
n + (n − xi) log 1 −
λ
n
After some algebra:
−H(q, p) = −
i
qi
xi
log λ
n − λ
+ n log
n − λ
n = −n log
n − λ
n − log λ
n − λ
i
qi
xi
¯
q
The term on the far right labeled ¯
q is the expected value of the distribution q. If we knew it, we could
complete the calculation, because no other term depends upon qi
. This means that expected value is
the constraint that defines the class of distributions for which the binomial p has maximum entropy.
If we now set the expected value of q equal to λ, then H(q) = H(p). For any other expected value of
q, H(p) > H(q).
Finally, notice the term log[λ/(n − λ)]. This term is the log of the ratio of the expected number
of successes to the expected number of failures. That ratio is the “odds” of a success, and its logarithm
is called “log odds.” This quantity will feature prominently in models we construct from the binomial
distribution, in Chapter 11.
Page 312

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work with H(q, p) and simplify it until we can isolate the constraint
butions for which p has maximum entropy. Let λ = i
pi
xi
be the
m the definition of H(q, p):
xi
1 −
λ
n
n−xi
= −
i
qi
xi
log λ
n + (n − xi) log 1 −
λ
n
λ
n − λ
+ n log
n − λ
n = −n log
n − λ
n − log λ
n − λ
i
qi
xi
¯
q
d ¯
q is the expected value of the distribution q. If we knew it, we could
use no other term depends upon qi
. This means that expected value is
class of distributions for which the binomial p has maximum entropy.

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Logit link
Y
i
∼ Bernoulli(p
i
)
logit(p
i
) = α + β
X
X
i
+ β
Z
Z
i
logit(p
i
)
logit(p
i
) = log
p
i
1 − pi
p
i
= logit−1(α + β
X
X
i
+ β
Z
Z
i
)
logit−1
Bernoulli/Binomial models
most o en use logit link

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Logit link
logit(p
i
) = log
p
i
1 − pi
Bernoulli/Binomial models
most o en use logit link
“log odds” odds
Y
i
∼ Bernoulli(p
i
)
logit(p
i
) = α + β
X
X
i
+ β
Z
Z
i
logit(p
i
)
p
i
= logit−1(α + β
X
X
i
+ β
Z
Z
i
)
logit−1
“logistic”

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From link to inverse link
logit(p
i
) = log
p
i
1 − pi
= q
i
logit−1(q
i
) = ?

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logit(p
i
) = log
p
i
1 − pi
= q
i
logit−1(q
i
) = ? = p
i

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logit(p
i
) = log
p
i
1 − pi
= q
i
logit−1(q
i
) = ? = p
i
log
p
i
1 − pi
= q
i
p
i
=
exp(q
i
)
1 + exp(qi
)
logit−1

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logit link is a harsh transform
“log-odds scale”: e value of
the linear model
logit(p)=0, p=0.5
logit(p)=4, p=nearly always
logit(p)=–4, p=hardly ever
-6 -4 -2 0 2 4 6
logit(p)
p
0 0.5 1

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logit(p
i
) = α + βx
i

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Logistic priors
e logit link compresses parameter
distributions
Anything above +4 = almost always
Anything below –4 = almost never
logit(p
i
) = α
-30 -20 -10 0 10 20 30
0.00 0.02 0.04
alpha
Density
-30 -20 -10 0 10 20 30
0.00 0.10 0.20
alpha
Density
-30 -20 -10 0 10 20 30
0.0 0.2 0.4
alpha
Density
α ∼ Normal(0,10)
α ∼ Normal(0,1.5)
α ∼ Normal(0,1)
α
p
i

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0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.4 0.8
probability of event
Density
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4
Density
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5
Density
α ∼ Normal(0,10)
α ∼ Normal(0,1)
-30 -20 -10 0 10 20 30
0.00 0.02 0.04
alpha
Density
-30 -20 -10 0 10 20 30
0.00 0.10 0.20
alpha
Density
-30 -20 -10 0 10 20 30
0.0 0.2 0.4
alpha
Density
logit(p
i
) = α
α
p
i

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logit(p
i
) = α + βx
i
β ∼ Normal(0,10)
α ∼ Normal(0,10)
-2 -1 0 1 2
0.0 0.4 0.8
x value
probability
a b xseq p inv_logit(a+b*x) )
plot( NULL , xlim=c(-2.5,2.5) , ylim=c(0,1) ,
xlab="x value" , ylab="probability" )
for ( i in 1:10 ) lines( xseq , p[i,] , lwd=3 ,
col=2 )

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β ∼ Normal(0,10)
α ∼ Normal(0,10)
-2 -1 0 1 2
0.0 0.4 0.8
x value
probability
-2 -1 0 1 2
0.0 0.4 0.8
x value
probability
β ∼ Normal(0,0.5)

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A statistical model
A G
D
A

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Estimand: Total e ect of G
A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
]
G
D
A
α = [α
1
, α
2]
Genders
Pr(A
i
= 1) = p
i
p
i
=
exp(α[G
i
])
1 + exp(α[Gi
])

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Estimand: Direct e ect of G
A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
, D
i
]
G
D
A
α =
[
α
1,1
α
1,2
α
2,1
α
2,2]
Departments
Genders

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A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
]
A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
, D
i
]
α
j
∼ Normal(0,1) α
j,k
∼ Normal(0,1)
Total e ect Direct e ect(s)

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A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
]
A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
, D
i
]
α
j
∼ Normal(0,1) α
j,k
∼ Normal(0,1)
dat_sim m1 alist(
A ~ bernoulli(p),
logit(p) a[G] ~ normal(0,1)
), data=dat_sim , chains=4 , cores=4 )
dat_sim m2 alist(
A ~ bernoulli(p),
logit(p) matrix[G,D]:a ~ normal(0,1)

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dat_sim m1 alist(
A ~ bernoulli(p),
dat_sim m2 alist(
A ~ bernoulli(p),
precis(m1,depth=2)
mean sd 5.5% 94.5% n_eff Rhat4
a[1] -1.80 0.13 -2.01 -1.60 1549 1
a[2] -1.09 0.10 -1.25 -0.93 1159 1

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dat_sim m1 alist(
A ~ bernoulli(p),
dat_sim m2 alist(
A ~ bernoulli(p),
precis(m1,depth=2) precis(m2,depth=3)
a[1] -1.80 0.13 -2.01 -1.60 1549 1
a[2] -1.09 0.10 -1.25 -0.93 1159 1
a[1,1] -2.31 0.18 -2.60 -2.04 2529 1
a[1,2] -0.92 0.19 -1.23 -0.62 2216 1
a[2,1] -1.93 0.31 -2.45 -1.44 2214 1
a[2,2] -0.93 0.11 -1.11 -0.75 2055 1

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dat_sim m1 alist(
A ~ bernoulli(p),
dat_sim m2 alist(
A ~ bernoulli(p),
precis(m1,depth=2)
precis(m2,depth=3)
a[1] -1.80 0.13 -2.01 -1.60 1549 1
a[2] -1.09 0.10 -1.25 -0.93 1159 1
a[1,1] -2.31 0.18 -2.60 -2.04 2529 1
a[1,2] -0.92 0.19 -1.23 -0.62 2216 1
a[2,1] -1.93 0.31 -2.45 -1.44 2214 1
a[2,2] -0.93 0.11 -1.11 -0.75 2055 1
> inv_logit(coef(m2))
a[1,1] a[1,2] a[2,1] a[2,2]
0.06296434 0.21109945 0.08253890 0.20003819

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PAUSE

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(1) Estimand(s)
(4) Analyze

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UC Berkeley Admissions
4526 graduate school
applications for 1973 UC
Berkeley
Strati ed by department and
gender of applicant
Evidence of gender
discrimination?
dept admit reject applications gender
1 A 512 313 825 male
2 A 89 19 108 female
3 B 353 207 560 male
4 B 17 8 25 female
5 C 120 205 325 male
6 C 202 391 593 female
7 D 138 279 417 male
8 D 131 244 375 female
9 E 53 138 191 male
10 E 94 299 393 female
11 F 22 351 373 male
12 F 24 317 341 female
See ?UCBAdmissions for citation

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Logistic & Binomial Regression
Logistic regression:
Binary [0,1] outcome and logit link
Binomial regression:
Count [0,N] outcome and logit link
A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
, D
i
]
A
i
∼ Binomial(N
i
, p
i
)
logit(p
i
) = α[G
i
, D
i
]
Completely equivalent for inference

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A G D
1 0 2 1
2 0 1 1
3 1 2 2
4 1 2 2
5 0 2 2
6 0 1 1
7 0 2 2
8 0 2 2
9 0 2 2
10 0 2 2
11 0 2 2
12 1 2 2
13 0 2 2
14 0 1 1
15 0 2 2
16 0 1 2
17 0 1 1
18 0 1 1
19 0 1 1
20 0 1 1
G D A N
1 1 1 30 355
2 2 1 10 92
3 1 2 38 135
4 2 2 117 418
dat_sim2 dat_sim2$N LOOOOOOOONG
Aggregated

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A
i
∼ Bernoulli(p
i
)
logit(p
i
) = α[G
i
, D
i
]
A
i
∼ Binomial(N
i
, p
i
)
logit(p
i
) = α[G
i
, D
i
]
m2 alist(
A ~ bernoulli(p),
m2_bin alist(
A ~ binomial(N,p),
), data=dat_sim2 , chains=4 , cores=4 )

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logit(p
i
) = α[G
i
, D
i
]
A
i
∼ Binomial(N
i
, p
i
)
logit(p
i
) = α[G
i
, D
i
]
m2 alist(
A ~ binomial(1,p),
m2_bin alist(
A ~ binomial(N,p),
), data=dat_sim2 , chains=4 , cores=4 )
A
i
∼ Binomial(1,p
i
)

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data(UCBadmit)
d dat A = d$admit,
N = d$applications,
G = ifelse(d$applicant.gender=="female",1,2),
D = as.integer(d$dept)
)
# total effect gender
mG alist(
A ~ binomial(N,p),
), data=dat , chains=4 , cores=4 )
Total e ect

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data(UCBadmit)
d dat A = d$admit,
N = d$applications,
G = ifelse(d$applicant.gender=="female",1,2),
D = as.integer(d$dept)
)
mG alist(
A ~ binomial(N,p),
# direct effects
mGD alist(
A ~ binomial(N,p),

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0 200 400 600 800 1000
0.0 1.0 2.0 3.0
n_eff = 2548
a[1,1]
0 200 400 600 800 1000
0.2 0.6 1.0
n_eff = 2669
a[2,1]
0 200 400 600 800 1000
-0.5 0.5 1.5
n_eff = 2859
a[1,2]
0 200 400 600 800 1000
0.2 0.5 0.8
n_eff = 2590
a[2,2]
0 200 400 600 800 1000
-1.0 -0.7 -0.4
n_eff = 2612
a[1,3]
0 200 400 600 800 1000
-1.0 0.0
n_eff = 2617
a[2,3]
0 200 400 600 800 1000
-1.2 -0.8 -0.4
n_eff = 2598
a[1,4]
0 200 400 600 800 1000
-1.2 -0.8 -0.4
n_eff = 2433
a[2,4]
0 200 400 600 800 1000
-1.6 -1.2 -0.8
n_eff = 2520
a[1,5]
0 200 400 600 800 1000
-1.5 -0.5
n_eff = 3003
a[2,5]
0 200 400 600 800 1000
-3.0 -2.0 -1.0
n_eff = 2346
a[1,6]
0 200 400 600 800 1000
-3.5 -2.5 -1.5
n_eff = 2384
a[2,6]

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n_eff = 2548
a[1,1] n_eff = 2669
a[2,1] n_eff = 2859
a[1,2]
n_eff = 2590
a[2,2] n_eff = 2612
a[1,3] n_eff = 2617
a[2,3]
n_eff = 2598
a[1,4] n_eff = 2433
a[2,4] n_eff = 2520
a[1,5]
n_eff = 3003
a[2,5] n_eff = 2346
a[1,6] n_eff = 2384
a[2,6]

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mG alist(
A ~ binomial(N,p),
# direct effects
mGD alist(
A ~ binomial(N,p),
a[1] -0.83 0.05 -0.91 -0.75 1487 1
a[2] -0.22 0.04 -0.28 -0.16 1499 1
a[1,1] 1.48 0.24 1.11 1.87 2548 1
a[1,2] 0.66 0.40 0.03 1.32 2859 1
a[1,3] -0.65 0.08 -0.79 -0.52 2612 1
a[1,4] -0.62 0.11 -0.79 -0.45 2598 1
a[1,5] -1.15 0.12 -1.34 -0.96 2520 1
a[1,6] -2.50 0.20 -2.81 -2.18 2346 1
a[2,1] 0.49 0.07 0.38 0.60 2669 1
a[2,2] 0.53 0.08 0.40 0.67 2590 1
a[2,3] -0.53 0.11 -0.72 -0.35 2617 1
a[2,4] -0.70 0.10 -0.87 -0.54 2433 1
a[2,5] -0.94 0.16 -1.20 -0.69 3003 1
a[2,6] -2.67 0.21 -3.00 -2.34 2384 1

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Total e ect
post1 PrA_G1 PrA_G2 diff_prob dens(diff_prob,lwd=4,col=2,xlab="Gender
contrast (probability)")
-0.18 -0.14 -0.10
0 10 25
Gender contrast (probability)
Density
men advantaged

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post1 PrA_G1 PrA_G2 diff_prob dens(diff_prob,lwd=4,col=2,xlab="Gender
contrast (probability)")
post2 PrA diff_prob_D_ PrA[,1,i] - PrA[,2,i] )
plot(NULL,xlim=c(-0.2,0.3),ylim=c(0,25),xlab="G
ender contrast (probability)",ylab="Density")
for ( i in 1:6 ) dens( diff_prob_D_[,i] , lwd=4
, col=1+i , add=TRUE )
-0.18 -0.14 -0.10
0 10 25
Density
-0.2 -0.1 0.0 0.1 0.2 0.3
0 5 15 25
Density
men adv women adv
men advantaged

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What is the average direct e ect of gender across departments?
Depends upon distribution of applications, probability
woman/man applies to each department
G
D
A
What is the
intervention actually?

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Depends upon distribution of applications, probability
woman/man applies to each department
G
D
A
gender
G*
perceived
gender
What is the average direct e ect of gender across departments?
What is the

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To calculate causal e ect of G*, must average
(marginalize) over departments
Easy to do it as a simulation
NB: We are still assuming no confounds!
What is the
G
D
A
G*

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# number of applicatons to simulate
total_apps # number of applications per department
apps_per_dept sum(dat$N[dat$D==i]) )
# simulate as if all apps from women
p_G1 D=rep(1:6,times=apps_per_dept),
N=rep(1,total_apps),
G=rep(1,total_apps)))
# simulate as if all apps from men
# summarize
dens( p_G1 - p_G2 , lwd=4 , col=2 ,
xlab="effect of gender perception" )
men adv women adv
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0 2 4 6 8
effect of gender perception
Density

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# simulate as if all apps from women
# simulate as if all apps from men
men adv women adv
men adv women adv
# show each dept density with weight as in
population
w plot(NULL,xlim=c(-0.2,0.3),ylim=c(0,25),xlab="
Gender contrast (probability)",ylab="Density")
for ( i in 1:6 ) dens( diff_prob_D_[,i] ,
lwd=2+20*w[i] , col=1+i , add=TRUE )
abline(v=0,lty=3)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0 2 4 6 8
Density
-0.2 -0.1 0.0 0.1 0.2 0.3
0 5 10 15 20 25
Density

View Slide

Post-strati cation
Description, prediction & causal
inference o en require post-strati cation
Post-strati cation: Re-weighting
estimates for target population
At a di erent university, distribution of
applications will di er => predicted
consequence of intervention di erent
men adv women adv
-0.2 -0.1 0.0 0.1 0.2 0.3
0 5 10 15 20 25
Density

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Admissions so far
Evidence for discrimination?
Big structural e ects, but
(1) Distribution of applications can
be a consequence of discrimination
(data do not speak to this)
(2) Confounds likely
men adv women adv
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0 2 4 6 8
Density

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View Slide

BONUS

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Survival Analysis
Counts o en modeled as time-to-event
Tricky, because cannot ignore censored cases
Le -censored: Don’t know when time started
Right-censored: Observation ended before event
Ignoring censored cases leads to inferential error
Imagine estimating time-to-PhD: Time in
program before dropping out is info about rate

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Survival Analysis
Example: Cat adoptions
data(AustinCats)
20-thousand cats
Estimand: Adoption rates of black and non-black cats
Events: (1) adopted or (2) something else
Something else could be: death, escape, censored

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Austin Cats
Outcome variable: days_to_event
What is a good distribution?
Basic choices: Exponential or gamma
Maximum entropy distributions for
waiting times
Generative story: X things required
before event
$&/403*/( "/% 4637*7"-
3 DP

ʚǶ Ǐ
3 ʚǶ - +'$/ ǿ ǎ ǒ Ǣ ($)ǿ-0)$!ǿǢǎǢǎǍǍȀȀ Ȁ
*G ZPV QMPU UIF EFOTJUZ PG UIF 3 WBMVFT ZPVMM TFF UIF CMBDL EFOTJUZ JO UIF MFę QMPU CFMPX "T UIF OVNCFS
PG QBSUT JODSFBTFT UIF EFOTJUZ DPOWFSHFT UP BO FYQPOFOUJBM :0V EPOU OFFE WFSZ NBOZ #Z / =
UIF CMVF EFOTJUZ CFMPX JU JT BMNPTU FYBDUMZ FYQPOFOUJBM
0 20 40 60 80 100
0.00 0.02 0.04
day
Density
2 parts
5 parts
Single part failure: Exponential
0 20 40 60 80 100
0.00 0.02 0.04
day
Density
2/10 parts
5/10 parts
Multiple part failure: Gamma
ćF QMPU PO UIF SJHIU BCPWF JT GPS UIF HBNNB EJTUSJCVUJPO XIJDI BSJTFT XIFO UIF NBDIJOF CSFBLT POMZ
UIF CMVF EFOTJUZ CFMPX JU JT BMNPTU FYBDUMZ FYQPOFOUJBM
0 20 40 60 80 100
0.00 0.02 0.04
day
Density
2 parts
5 parts
Single part failure: Exponential
0.00 0.02 0.04
Density
ćF QMPU PO UIF SJHIU BCPWF JT GPS UIF HBNNB EJTUSJCVUJPO X
BęFS . PG JUT QBSUT IBWF CSPLFO ćJT DPEF XJMM MFU ZPV TJN
ʚǶ ǎǍ
ʚǶ Ǐ
3 ʚǶ - +'$/ ǿ ǎ ǒ Ǣ .*-/ǿ-0)$!ǿǢǎǢǎǍǍȀȀȁȂ Ȁ
ćJT QSPEVDFT UIF CMBDL EFOTJUZ JO UIF SJHIU IBOE QMPU ć
UP CSFBL /BUVSBMMZ UIF FYQPOFOUJBM JT B KVTU B TQFDJBM DBT
UIJT JT ZFU BOPUIFS XBZ UP HFU BO BQQSPYJNBUF (BVTTJBO EJT
JU MPPLT NPSF (BVTTJBO ćF CMVF EFOTJUZ BCPWF JT BMSFBEZ
IBQQFO UP DBVTF TPNF FWFOU UIFO XBJUJOH UJNFT DBO FOE V

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Un-censored observations
For observed adoptions, just need:
HFU BO BQQSPYJNBUF (BVTTJBO EJTUSJCVUJPO "T UIF HBNNBT NFBO JODSFBTFT
ćF CMVF EFOTJUZ BCPWF JT BMSFBEZ SBUIFS (BVTTJBO *G TFWFSBM UIJOHT OFFE UP
FOU UIFO XBJUJOH UJNFT DBO FOE VQ MPPLJOH WFSZ (BVTTJBO
Y
UJPOT QSPCBCJMJUZ PG PCTFSWFE XBJUJOH UJNF JT TJNQMZ
%J ∼ &YQPOFOUJBM(λJ)
Q(%J|λJ) = λJ
FYQ(−λJ
%J)
BUT UIBU BSF USJDLZ *G TPNFUIJOH FMTF IBQQFOFE CFGPSF B DBU DPVME CF
IBTOU CFFO BEPQUFE ZFU UIFO XF OFFE UIF QSPCBCJMJUZ PG OPU CFJOH
O UIF PCTFSWBUJPO UJNF TP GBS 0OF XBZ UP NPUJWBUF UIJT JT UP JNBHF
M KPJOJOH UIF TIFMUFS PO UIF TBNF EBZ *G IBMG IBWF CFFO BEPQUFE BęFS
BCJMJUZ PG XBJUJOH EBZT BOE TUJMM OPU CFJOH BEPQUFE JT *G BęFS
0 1 2 3 4 5
0.0 0.2 0.4 0.6 0.8 1.0
x
dexp(x)
probability event happens at time x
λ = 0.5
λ = 1.0
OFOUJBM JT B KVTU B TQFDJBM DBTF PG UIF HBNNB XIFO . = /PUF UIBU
BO BQQSPYJNBUF (BVTTJBO EJTUSJCVUJPO "T UIF HBNNBT NFBO JODSFBTFT
CMVF EFOTJUZ BCPWF JT BMSFBEZ SBUIFS (BVTTJBO *G TFWFSBM UIJOHT OFFE UP
UIFO XBJUJOH UJNFT DBO FOE VQ MPPLJOH WFSZ (BVTTJBO
OT QSPCBCJMJUZ PG PCTFSWFE XBJUJOH UJNF JT TJNQMZ
%J ∼ &YQPOFOUJBM(λJ)
Q(%J|λJ) = λJ
FYQ(−λJ
%J)
IBU BSF USJDLZ *G TPNFUIJOH FMTF IBQQFOFE CFGPSF B DBU DPVME CF
OU CFFO BEPQUFE ZFU UIFO XF OFFE UIF QSPCBCJMJUZ PG OPU CFJOH
IF PCTFSWBUJPO UJNF TP GBS 0OF XBZ UP NPUJWBUF UIJT JT UP JNBHF
OJOH UIF TIFMUFS PO UIF TBNF EBZ *G IBMG IBWF CFFO BEPQUFE BęFS
JUZ PG XBJUJOH EBZT BOE TUJMM OPU CFJOH BEPQUFE JT *G BęFS

View Slide

Censored cats
0 1 2 3 4 5
0.0 0.2 0.4 0.6 0.8 1.0
x
pexp(x)
probability event before-or-at time x
0 1 2 3 4 5
0.0 0.2 0.4 0.6 0.8 1.0
x
exp(-x)
probability not-event before-or-at time x
λ = 0.5
λ = 1.0
CDF CCDF
HJWFT UIF QSPQPSUJPO PG DBUT BEPQUFE CFGPSF PS BU B DFSUBJO OVNCFS PG
IF DVNVMBUJWF EJTUSJCVUJPO HJWFT UIF QSPCBCJMJUZ B DBU JT OPU BEPQUFE
PG EBZT ćBU JT UIF QSPCBCJMJUZ UIBU XF OFFE ćJT EJTUSJCVUJPOPOF
F QSPCBCJMJUZ EJTUSJCVUJPOJT DBMMFE UIF İļĺĽĹĲĺĲĻŁĮĿņ İłĺłĹĮ
ĶŀŁĿĶįłŁĶļĻ *O UIF DBTF PG UIF FYQPOFOUJBM EJTUSJCVUJPO UIF DVNVMB
1S(%J|λJ) = − FYQ(−λJ
%J)
KVTU
1S(%J|λJ) = FYQ(−λJ
%J)
E JO PVS NPEFM TJODF JU JT QSPCBCJMJUZ PG XBJUJOH %J
EBZT XJUIPVU CFJOH
ɶ*0/Ǿ 1 )/ʙʙǫ*+/$*)ǫ Ǣ ǎ Ǣ Ǎ Ȁ
ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǎ Ȁ
NJOVT UIF DVNVMBUJWF QSPCBCJMJUZ EJTUSJCVUJPOJT DBMMFE UIF İļĺĽĹĲĺĲĻŁĮĿņ İłĺ
ŁĶŃĲ ĽĿļįĮįĶĹĶŁņ ıĶŀŁĿĶįłŁĶļĻ *O UIF DBTF PG UIF FYQPOFOUJBM EJTUSJCVUJPO UIF DV
UJWF JT
1S(%J|λJ) = − FYQ(−λJ
%J)
4P UIF DPNQMFNFOU JT KVTU
%J)
4P UIBUT XIBU XF OFFE JO PVS NPEFM TJODF JU JT QSPCBCJMJUZ PG XBJUJOH %J
EBZT XJUIPVU
BEPQUFE ZFU
3 DPEF
'$--4ǿ- /#$)&$)"Ȁ
/ǿ0./$)/.Ȁ
ʚǶ 0./$)/.
ɶ*+/ ʚǶ $! '. ǿ ɶ*0/Ǿ 1 )/ʙʙǫ*+/$*)ǫ Ǣ ǎ Ǣ Ǎ Ȁ
ɶ'& ʚǶ $! '. ǿ ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǎ Ȁ
/ ʚǶ '$./ǿ
4.Ǿ/*Ǿ 1 )/ ʙ .ǡ)0( -$ǿ ɶ4.Ǿ/*Ǿ 1 )/ ȀǢ
'& ʙ .ǡ$)/ " -ǿ ɶ'& Ȁ Ǣ
*+/ ʙ .ǡ$)/ " -ǿ ɶ*+/ Ȁ
Cumulative distribution
(CDF)
Complementary cumulative
distribution (CCDF)
Event happened Event didn’t happen yet

View Slide

%J)
4P UIBUT XIBU XF OFFE JO PVS NPEFM TJODF JU JT UIF QSPCBCJMJUZ PG XBJUJOH %J
EBZT XJUIPVU
CFJOH BEPQUFE ZFU
ćF NPEFM
%J|"J = ∼ &YQPOFOUJBM(λJ)
%J|"J = ∼ &YQPOFOUJBM$$%'(λJ)
λJ = /µJ
MPH µJ = αİĶı[J]
3 DPEF
'$--4ǿ- /#$)&$)"Ȁ
/ǿ0./$)/.Ȁ
ʚǶ 0./$)/.
/ ʚǶ '$./ǿ

View Slide

%J)
4P UIBUT XIBU XF OFFE JO PVS NPEFM TJODF JU JT UIF QSPCBCJMJUZ PG XBJUJOH %J
EBZT XJUIPVU
CFJOH BEPQUFE ZFU
ćF NPEFM
λJ = /µJ
3 DPEF
'$--4ǿ- /#$)&$)"Ȁ
/ǿ0./$)/.Ȁ
ʚǶ 0./$)/.
/ ʚǶ '$./ǿ
Observed adoptions
Not-yet-adoptions
log average time
to adoption

View Slide

λJ = /µJ
3 DPEF
'$--4ǿ- /#$)&$)"Ȁ
/ǿ0./$)/.Ȁ
ʚǶ 0./$)/.
/ ʚǶ '$./ǿ
4.Ǿ/*Ǿ 1 )/ ʙ .ǡ)0( -$ǿ ɶ4.Ǿ/*Ǿ 1 )/ ȀǢ
*'*-Ǿ$ ʙ $! '. ǿ ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǐ Ȁ Ǣ
*+/ ʙ ɶ*+/
Ȁ
(ǎǎǡǎǑ ʚǶ 0'(ǿ
'$./ǿ
library(rethinking)
data(AustinCats)
d dat days = d$days_to_event,
adopted = ifelse( d$out_event=="Adoption" , 1 , 0 ),
color_id = ifelse( d$color=="Black" , 1 , 2 ) )
meow alist(
days|adopted==1 ~ exponential(lambda),
days|adopted==0 ~ custom(exponential_lccdf(!Y|lambda)),
lambda log(mu) a[color_id] ~ normal(0,1)
) , data=dat , chains=4 , cores=4 )

View Slide

Black cats
Other cats
45 50 55 60 65
0.0 0.2 0.4 0.6
waiting time
Density

View Slide

Black cats
Other cats
0 20 40 60 80 100
0.0 0.2 0.4 0.6 0.8 1.0
days until adoption
fraction
post blank2()
plot(NULL,xlim=c(0,100),ylim=c(0,1),xlab=
"days until adoption",ylab="fraction")
for ( i in 1:50 ) curve(exp(-x/
exp(post$a[i,1])),add=TRUE,col=1)
for ( i in 1:50 ) curve(exp(-x/
exp(post$a[i,2])),add=TRUE,col=2)

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Statistical Rethinking 2023 - Lecture 09

Statistical Rethinking 2023 - Lecture 09

Richard McElreath

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