Lecture slides stats1.13.l20.air

Statistics One
Lecture 20
Binary Logistic Regression
1

Two segments
•  Overview
•  Example

2

Lecture 20 ~ Segment 1
Overview
3

Binary logistic regression
•  Appropriate when predicting a binary
categorical outcome variable from a set of
predictor variables that may be continuous
and/or categorical
–  Same logic as multiple regression but outcome
variable is categorical and binary

•  When outcome has two levels
–  Binary logistic regression

•  When outcome has multiple levels
–  Multinomial regression

Multiple regression
•  Ŷ= B0 + Σ(BkXk)
Ŷ = predicted value on the outcome variable Y
B0 = predicted value on Y when all X = 0
Xk = predictor variables
Bk = unstandardized regression coefficients
(Y – Ŷ) = residual (prediction error)
k = the number of predictor variables
6

•  ln(Ŷ / (1 - Ŷ)) = B0 + Σ(BkXk)
Ŷ = predicted value on the outcome variable Y
B0 = predicted value on Y when all X = 0
Xk = predictor variables
Bk = unstandardized regression coefficients
(Y – Ŷ) = residual (prediction error)
k = the number of predictor variables
7

•  Why ln(Ŷ / (1 – Ŷ))?
•  Predicted score must fall between 0 and 1

8

•  Why not P(outcome) = B0 + Σ(BkXk) ???
•  There is no guarantee that the linear
combination of predictors will produce a
score between 0 and 1
•  A transformation is therefore applied
10

•  Odds = P(outcome) / (1 – P(outcome))
•  For example, what are the odds a flipped coin will land
heads? Odds = .5 / .5 = 1
•  Then take the natural log of the odds, which is called the
log-odds or logit
•  Logit = ln(P(outcome) / (1 – P(outcome))
•  Logit = ln(Ŷ / (1 – Ŷ))
11

•  Logit = ln(Ŷ / (1 – Ŷ))
•  Ŷ = P(outcome)

•  P(outcome) = odds / (1 + odds)
•  Odds = P(outcome) / P(~outcome)
•  For example,
•  If P = .50 then Odds = 1 and Logit = 0

•  Example

•  Outcome variable = Faculty Promotion to tenure
•  Predictor variable = Publications (Pubs)
•  Logit(Promotion) = B0 + B1(Pubs)
•  Logit(Promotion) = 0.00 + .39(Pubs)
•  For every one unit increase in Pubs, the Logit
increases .39

•  Logit = ln(P(outcome) / (1 – P(outcome))
•  Odds = P(outcome) / (1 – P(outcome))
•  Logit = .39 translates to an odds ratio of 1.48
–  This means that the odds of promotion are
multiplied by 1.48 for each increment in Pubs

•  Thus, if the odds of Promotion with 16 publications
is 1.27 then the Odds of Promotion with 17
publications is 1.27*1.48 = 1.88
•  This can also be presented in terms of probability
•  Pubs = 17 means P(Promotion) = .65 because
P(Promotion) = Odds / (1 + Odds) = 1.88/2.88 = .65

•  Hypothesis tests

•  Is an individual predictor variable significant?
•  Is the overall model significant?
•  Is Model A significantly better than Model B?

•  To test each predictor variable
•  Regression coefficient
•  Odds ratio
•  Wald test

•  Tests the model vs. the model without the predictor

•  To test the overall model

•  Compare the chi-square for the model to the chi-square
of a model with no predictors (the null model)
•  And/or compare multiple models
•  Also, does the model classify cases correctly?

Segment summary
•  Binary logistic regression is appropriate
when predicting a binary categorical
outcome variable from a set of predictor
variables that may be continuous and/or
categorical

Segment summary
•  Main components of the output are
–  Regression coefficients
–  Odds ratios
–  Wald tests
–  Model chi-square
–  Classification success

Lecture 20 ~ Segment 2
Example
23

•  This example is based on “mock jury” research by
Diamond & Casper (1992)

–  People (mock jurors) watched a video of the sentencing
phase of a murder trial in which the defendant had already
been found guilty
–  The issue for the jurors to decide was whether the
defendant deserved the death penalty

•  This example is based on “mock jury” research by
Diamond & Casper (1992)

–  Assume the data were collected “pre-deliberation”, which
means that each juror was asked to provide his or her vote
on the death penalty verdict before the jurors met as a group
to decide the overall jury verdict

•  Outcome variable (Y)
•  Verdict

•  1 = Voted for the death penalty
•  0 = Voted against the death penalty

•  Predictors (Xs)
• 
• 
• 
• 
• 
• 

Danger
Rehab
Punish
Gendet
Specdet
Incap

•  All measured on a scale of 0 – 10

•  Danger (Dangerousness)

•  Individual’s beliefs as to the future dangerousness of the
defendant

•  Rehab (Rehabilitation)

•  Individual’s beliefs as to the importance of rehabilitation as a
goal of criminal sentencing

•  Punish (Punishment)

•  Individual’s beliefs as to the importance of punishment as a
goal of criminal sentencing

•  Gendet (General deterrence)

•  Individual’s beliefs as to the importance of general deterrence as a
goal of criminal sentencing (sentencing should deter the general
public)

•  Specdet (Specific deterrence)

•  Individual’s beliefs as to the importance of specific deterrence as a
goal of criminal sentencing (sentencing should deter the specific
defendant)

•  Incap (Incapacitation)

•  Individual’s beliefs as to the importance of punishment as a goal of
criminal sentencing

•  The General Linear Model will not guarantee a
predicted outcome score between 0 and 1
•  The Logit transformation is a feature of an even more
“general” mathematical framework in regression
•  The Generalized Linear Model

•  Allows for non-linear relationships between predictors and
the outcome variable (see Lecture 23)

•  Evaluation of individual predictors
–  Odds ratios

•  For a one unit increase in X, the predicted change in odds
•  Can also report confidence intervals for odds

–  Wald test

•  A function of the regression coefficient. A Wald tests is
calculated for each predictor variable and compares the fit of
the model to the fit of the model without the predictor.

•  Evaluation of the model
–  Model chi-square

–  Compares the fit of the model to the fit of the null model

–  Classification success

•  Percentage of cases classified correctly

•  More than 2 categories on the outcome
–  Multinomial logistic regression
•  A-1 logistic regression equations are formed
–  Where A = # of groups
–  One group serves as reference group

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