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Eduard Ponarin Veronica Kostenko Boris Sokolov Multilevel binary logistic regression Lecture 3

Multilevel Binary Logistic Regression

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Page 1: Multilevel Binary Logistic Regression

Eduard Ponarin Veronica Kostenko

Boris Sokolov

Multilevel binary logistic regression

Lecture 3

Page 2: Multilevel Binary Logistic Regression

The basic logistic regression

• X on Y in case of a binary outcome.

• For example, if a candidate won or not during the elections, Y is either 0 or 1). Here X stands for the money spent on the campaign, Y – the outcome.

Page 3: Multilevel Binary Logistic Regression

Plotting X against proportion of successes

Where ni stands for the number of observations at X = h.

Page 4: Multilevel Binary Logistic Regression

Why not a linear model for probabilities?

• Linear approximation is problematic in this case because:

a) Residuals are non-randomly distributed

b) 0.2 < p < 0.8 is distributed otherwise then the tails of the function (p < 0.2; p > 0.8)

c) Regression line should fall into the interval between 0 and 1 which is hard to fit for a linear model

• Estimated probabilities should be transformed into logits

Page 5: Multilevel Binary Logistic Regression

Transformation of probabilities into logits

Page 6: Multilevel Binary Logistic Regression

Plotting logit functions

Increasing logit function Decreasing logit function

Page 7: Multilevel Binary Logistic Regression

Plotting probabilities for a single level logistic regression

Page 8: Multilevel Binary Logistic Regression

Multilevel logistic regression formula

logit (Pr (Yi=1)) = αj + εi = γ00 + η0j + εi

logit (Pr (Yi=1)) = αj + βgender * gender + βage * age + εi.

αj = γ00 + η0j

Page 9: Multilevel Binary Logistic Regression

Script for a simple model

• M1 <- glmer(y ~ female + age + (1|country), family=binomial(link="logit"))

• display (M1)

Page 10: Multilevel Binary Logistic Regression

Output for a logistic multilevel regression

• Coefficients shouldn’t be interpreted as in linear models, they should be transformed (exponential or divided-by-4 rule)

• Signs of the coefficients stay the same

• Coefficients can be compared with each other

Page 11: Multilevel Binary Logistic Regression

Output for a simple model

Page 12: Multilevel Binary Logistic Regression

Summary (more informative)

Page 13: Multilevel Binary Logistic Regression

Adding 1st level interaction

• M2 <- glmer (relig ~ age + gender +

age: gender + (1|country), family = binomial(link = "logit"))

• display (M2)

• summary(M2)

Page 14: Multilevel Binary Logistic Regression

Summary with interaction

Page 15: Multilevel Binary Logistic Regression

Varying intercepts and slopes without group – level predictor

• M3 <- glmer (relig ~ gender + age + (1 + gender|country), family = binomial(link = "logit"))

• summary (M3)

Page 16: Multilevel Binary Logistic Regression

Summary with varying slope

Page 17: Multilevel Binary Logistic Regression

Adding a group-level predictor

• M4 <- glmer (relig ~ gender +

+ gdp + (1+ gender|country), family = binomial(link = "logit"))

• display (M4)

• summary(M4)

Page 18: Multilevel Binary Logistic Regression

A model with between-level interaction