Regression models attempt to explain the variation in an outcome (dependent variable) through its relationship with a set of one or more predictors (independent variables).
Although a regression coefficient quantifies the strength of the relationship between outcome and predictor (effect size), it is expressed in units specific to the predictor, so direct comparison to other predictors’ coefficients is not straightforward.
Standardized effect sizes, like Cohen’s \(d\) and \(R^2\), are unitless and thus permit direct comparison between predictors measured in different units, even acros different studies.
Multilevel regression models are commonly used to model data that are hierarchically structured, which are usually characterized by non-independence of observations collected within aggregate units.
However, currently no set of standardized effect size measures have been widely adopted by researchers using multilevel models.
This workshop discusses a new set of \(R^2\) measures introduced by Rights and Sterba in their 2019 article ” Quantifying Explained Variance in Multilevel Models: An Integrative Framework for Defining R-Squared Measures”.
The authors of the r2mlm
package have themselves said
that using their new \(R^2\) measures
requires a solid understanding of multilevel models (Shaw et al.,
2023).
To this end, we will discuss:
r2mlm
packageStatistical models like linear regression use a set of predictors to explain the variation in the outcome.
Later in this workshop we will learn how to quantify how much variation has been explained, but first we need to understand how variation is measured.
Imagine we have an outcome, \(y_i\), measured for units \(i=1,...,N\).
One common way to quantify variation is to measure deviations from the mean, \(y_i - \bar{y}\), where
\[\bar{y}=\frac{1}{n}\sum_{i=1}^{n}{y_i}\]