multilevel binary logistic regression

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  • 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)) = j + i = 00 + 0j + i

    logit (Pr (Yi=1)) = j + gender * gender + age * age + i.

    j = 00 + 0j

  • Script for a simple model

    M1

  • Output for a logistic multilevel regression

    Coefficients shouldnt 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

  • Summary with interaction

  • Varying intercepts and slopes without group level predictor

    M3

  • Summary with varying slope

  • Adding a group-level predictor

    M4

  • A model with between-level interaction