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Related Concept Videos

Friedman Two-way Analysis of Variance by Ranks01:21

Friedman Two-way Analysis of Variance by Ranks

Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures from...
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Multiple Regression

Multiple regression assesses a linear relationship between one response or dependent variable and two or more independent variables. It has many practical applications.
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Bias01:22

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Bias refers to any tendency that prevents a question from being considered unprejudiced. In research, bias occurs when one outcome or answer is selected or encouraged over others in sampling or testing. Bias can occur during any research phase, including study design, data collection, analysis, and publication.
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Bias in Epidemiological Studies01:29

Bias in Epidemiological Studies

Biases can arise at various stages of research, from study design and data collection to analysis and interpretation. Recognizing and addressing these biases is essential to ensure the validity and reliability of epidemiological findings.Broadly speaking, biases in epidemiology fall into three main categories: selection bias, information bias, and confounding. A more detailed description of possible biases is:
Variation01:19

Variation

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation, which is the square root of variance.
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Related Experiment Video

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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Published on: July 3, 2020

Extending bias adjustments for R-squared to multilevel models.

Yingchi Guo1, Jason D Rights1

  • 1Department of Psychology, University of British Columbia.

Psychological Methods
|May 18, 2026
PubMed
Summary

Researchers often use R-squared to explain variance in regression models. This study introduces adjusted R-squared for multilevel models to correct upward bias, improving model accuracy.

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Area of Science:

  • Statistics
  • Multilevel Modeling
  • Regression Analysis

Background:

  • R-squared is widely used to assess model fit by quantifying explained variance.
  • Classic R-squared estimators in single-level regression are known to be upwardly biased.
  • Adjusted R-squared is common in single-level models but rarely used in multilevel contexts due to complexity and lack of attention to bias.

Purpose of the Study:

  • To provide a pedagogical overview of adjusted R-squared logic and its extension to multilevel models.
  • To evaluate the susceptibility of multilevel R-squared measures to upward bias using analytical and simulation methods.
  • To propose and assess adjustments for correcting bias in multilevel R-squared measures.

Main Methods:

  • Extension of single-level adjusted R-squared logic to multilevel regression.
  • Analytical evaluation using expectation algebra to assess bias in R-squared measures.
  • Empirical evaluation through simulation studies to quantify bias and the effectiveness of adjustments.

Main Results:

  • Proposed adjustments effectively reduce upward bias in multilevel R-squared measures compared to unadjusted versions.
  • Identified factors influencing the discrepancy between adjusted and unadjusted R-squared values.
  • Demonstrated the practical computation of adjusted measures with software and an empirical example.

Conclusions:

  • Adjusted R-squared measures offer a less biased estimation of explained variance in multilevel models.
  • Understanding factors affecting bias is crucial for accurate model interpretation.
  • The proposed adjustments and computational guidance facilitate improved statistical practice in multilevel research.