Eduard Ponarin- Higher School of Economics, Russia Veronica Kostenko- The National Research University ERF Training Workshop on Opinion Poll Data Analysis Using Multilevel Models Beirut, Lebanon August 22-23, 2016 www.erf.org.eg

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

2. 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.

3. Plotting X against proportion of
successes
Where ni stands for the number of
observations at X = h.

4. 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

5. Transformation of probabilities into
logits

6. Plotting logit functions
Increasing logit function Decreasing logit function

7. Plotting probabilities for a single level
logistic regression

8. 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

9. Script for a simple model
• M1 <- glmer(y ~ female + age + (1|country),
family=binomial(link="logit"))
• display (M1)

10. 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

11. Output for a simple model

12. Summary (more informative)

13. Adding 1st level interaction
• M2 <- glmer (relig ~ age + gender +
age: gender + (1|country), family = binomial(link
= "logit"))
• display (M2)
• summary(M2)

14. Summary with interaction

15. Varying intercepts and slopes without
group – level predictor
• M3 <- glmer (relig ~ gender + age + (1 +
gender|country), family = binomial(link =
"logit"))
• summary (M3)

16. Summary with varying slope

17. Adding a group-level predictor
• M4 <- glmer (relig ~ gender +
+ gdp + (1+ gender|country), family =
binomial(link = "logit"))
• display (M4)
• summary(M4)

18. A model with between-level
interaction