Missing R Squared and ICC for Multilevel Binary Logistic Regression I am currently assessing if allocation to a rejection scenario and degree of negative beliefs predicts the odds of engaging in a series of behavioural outcomes (e.g. avoiding the person).
My first IV (Rejection Condition) is a repeated measures 3 level categorical variable (control, ambiguous rejection, and rejection conditions). My second IV is a continuous variable (overall negative belief score). I am also looking at the moderation between rejection condition and negative beliefs.
To assess this I have used a multilevel binary logistic regression model with participant id acting as the clustering variable and a random intercept. I have used the software Jamovi to estimate the model with an N=163 (i.e. 489 observations).
I must admit, my understanding of these kinds of models is mostly rudimentary. One thing I cannot work out is why my model has excluded the conditional and marginal r-squared values and estimation of the inter-correlation coefficient. Can anyone explain why this might be occurring and how to remedy it?
 A: Unlike in linear regression, there are multiple defendable variants of $R^2$ in logistic regression.
For instance, the logistic regression can be evaluated using square loss, same as in an OLS linear regression. To calculate this, take the numerator to be the square loss (Brier score) of the predictions from your model, and take the denominator to be the square loss (Brier score) of the naïve model that always guesses the prior probability (proportion of observations belonging to each class).
$$
R^2_{\text{SquareLoss}}=
1-\dfrac{
\text{Your Model’s Brier Score
}}{
\text{
Naïve Model’s Brier Score
}}
$$
Similarly, you can do this with a comparison of how your model performs on log loss (the standard loss function in logistic regression, equivalent to maximum likelihood estimation) vs the naïve model that always guesses the prior probabilities. This sometimes goes by “McFadden’s $R^2$”.
$$
R^2_{McFadden}=1-\dfrac{
\text{Your Model’s Log Loss
}}{
\text{Naïve Model’s Log Loss
}}
$$
UCLA has a page that describes many other options.
Your software does not want to commit to one of these definitions, so it omits any notion of $R^2$. This is common. For instance, the summary of a logistic glm in R does not mention any $R^2$ values, unlike linear regression as implemented in lm.
The way around this is to pick a definition (or several, if you want several) and calculate $R^2$ yourself.
