Transcript
Eduard Ponarin Veronica Kostenko
Boris Sokolov
Multilevel binary logistic regression
Lecture 3
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.
Plotting X against proportion of successes
Where ni stands for the number of observations at X = h.
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
Transformation of probabilities into logits
Plotting logit functions
Increasing logit function Decreasing logit function
Plotting probabilities for a single level 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
Script for a simple model
• M1 <- glmer(y ~ female + age + (1|country), family=binomial(link="logit"))
• display (M1)
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
Output for a simple model
Summary (more informative)
Adding 1st level interaction
• M2 <- glmer (relig ~ age + gender +
age: gender + (1|country), family = binomial(link = "logit"))
• display (M2)
• summary(M2)
Summary with interaction
Varying intercepts and slopes without group – level predictor
• M3 <- glmer (relig ~ gender + age + (1 + gender|country), family = binomial(link = "logit"))
• summary (M3)
Summary with varying slope
Adding a group-level predictor
• M4 <- glmer (relig ~ gender +
+ gdp + (1+ gender|country), family = binomial(link = "logit"))
• display (M4)
• summary(M4)
A model with between-level interaction
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