SOCI 620: Quantitative methods 2

Agenda

Intro to logistic regression

  1. Administrative
  2. Cocaine use among adolescents
  3. The inverse logit transformation
  4. Starting simple: intercept-only
    logistic regression
  5. Hands on:
    Estimating logistic regression
    using MCMC in R

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Cocaine
use among adolescents
(The trouble with binary outcomes)

A still of a young James Spader in a white suite and a shirt open to expose his whole chest, hair feathered in true mid-1980s wealthy youth style.

Cocaine use among adolescents

Why not use a
standard linear
regression?

Gaussian model for binary data?

Why not use a
standard linear
regression?

Wrong support

Normal distribution has a support of (-∞,∞), but we know the outcome variable takes on only two values.

Bad intuitive “fit”

Interpretation

Under some circumstances, results can be interpreted as proportions or probabilities, but this can lead to predicted values less than zero or more than one.

Gaussian model for binary data?

Why not use a
standard linear
regression?

Gaussian vs. Bernoulli

Gaussian (normal) distribution

Bernoulli distribution

Logistic regression model


Logistic regression model

sepia photographic portrait of a young, early-20th-century elite man. He has specatactles and some sort of medal pinned to his chest.

Ronald Fisher
statistician and eugenicist

black and white photographic portrait of a mid-century man in a suit.

Joseph Berkson
statistician and tobacco apologist

The
inverse
logit transfor­mation

A still from that Apple ad where they joyfully crushed the tools people use to make art in a hydraulic press. The still shows a toy face being squished between two large metal plates so its eyes bulge out in apparent distress.

Inverse logit transformation

Inverse Logit transformation

Inverse logit transformation

Logit function

Takes values between 0 and 1, and turns
them into values between -∞ and ∞.

Inverse logit function

(aka ‘logistic’)

Takes values between -∞ and ∞, and turns
them into values between 0 and 1.

⇔

Inverse logit transformation

x logit–1(x)
-40.018
-20.119
-10.269
00.500
10.731
20.881
40.982

Intercept-only logistic model

Why this model instead of the model we built in the first week of class?

Logistic regression allows us to include explanatory covariates.

Priors in logistic regression

Priors in logistic regression

Pr(α)

logit–1(Pr(α))

Priors in logistic regression

Priors in logistic regression

Priors in logistic regression

Intercept-only logistic regression

Median 95% C.I.

-3.34 (-3.48, -3.20)

0.036 (0.031, 0.041)

0.034 (0.030, 0.039)

Image credit

Figures by Peter McMahan (source code)

A still of a young James Spader in a white suite and a shirt open to expose his whole chest, hair feathered in true mid-1980s wealthy youth style.

Promotional image for Pretty in Pink (1987)

sepia photographic portrait of a young, early-20th-century elite man. He has specatactles and some sort of medal pinned to his chest.

Ronald Fisher, via The Science History Institute

black and white photographic portrait of a mid-century man in a suit.

Joseph Berkson, via The Rochester Epidemiology Project

James Spader in Pretty in Pink

James Spader in Pretty in Pink