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Related Concept Videos

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.
When independent and dependent variables are plotted on a scatter plot, the slope of a line is a value that describes the rate of change between the two...
Multiple Regression01:25

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.
Farmers can use multiple regression to determine the crop yield based on more than one factor, such as water availability, fertilizer, soil properties, etc. Here, the crop yield is the response or dependent variable as it depends on the other independent variables. The analysis requires the construction of a scatter plot...
Calculating and Interpreting the Linear Correlation Coefficient01:11

Calculating and Interpreting the Linear Correlation Coefficient

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable, x, and the dependent variable, y. Hence, it is also known as the Pearson product-moment correlation coefficient. It can be calculated using the following equation:
Coefficient of Correlation01:12

Coefficient of Correlation

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the strength of the linear...
Calibration Curves: Correlation Coefficient01:10

Calibration Curves: Correlation Coefficient

In a linear calibration curve, there is a value called the calibration coefficient, denoted by 'r,' which measures the strength and the direction of association between two variables. The correlation coefficient value ranges from −1 to +1. A value of +1 indicates a perfect positive linear correlation, −1 denotes a perfect negative correlation, and 0 implies no correlation between the two variables. A positive correlation value establishes that as one variable increases, the other increases, and...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...

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Related Experiment Video

Updated: Jun 18, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

Coefficients of Determination for Mixed-Effects Models.

Dabao Zhang1

  • 1Department of Statistics, Purdue University.

Journal of Agricultural, Biological, and Environmental Statistics
|June 17, 2026
PubMed
Summary

We extended the coefficient of determination for mixed-effects models, offering new measures for explained variation. These methods account for individual heterogeneity in agricultural and ecological research.

Keywords:
Exponential family distributionGeneralized linear mixed modelLinear mixed modelQuasi-modelR2Variance function

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Area of Science:

  • Statistics
  • Ecology
  • Agricultural Science

Background:

  • The coefficient of determination is standard for linear models but lacks a clear extension for mixed-effects models.
  • Mixed-effects models are crucial in agricultural, biological, and ecological research, necessitating robust measures of model fit.
  • Existing methods do not adequately capture variation explained by fixed and random effects separately, nor do they fully account for individual heterogeneity.

Purpose of the Study:

  • To extend the coefficient of determination for both linear mixed models and generalized linear mixed models.
  • To develop measures quantifying the proportion of variation explained by the overall model, fixed effects, and random effects.
  • To propose methods that preserve individual heterogeneity by calculating unexplained variation conditional on effects.

Main Methods:

  • Revisiting the definition of the coefficient of determination for extension to mixed-effects models.
  • Defining measures for proportions of variation explained by the whole model, fixed effects only, and random effects only.
  • Proposing calculation of unexplained variations conditional on individual random and/or fixed effects.
  • Adapting measures for generalized linear mixed models using a distance along the variance function to handle heteroscedasticity.

Main Results:

  • The proposed methods provide robust measures for the coefficient of determination in mixed-effects models.
  • The measures effectively quantify variation explained by different model components (total, fixed, random).
  • The approach successfully accounts for individual heterogeneity and heteroscedasticity in generalized linear mixed models.
  • Demonstrated utility through simulations and real-world applications in agricultural and ecological data.

Conclusions:

  • The extended coefficient of determination offers valuable tools for assessing mixed-effects models in various scientific fields.
  • The proposed measures enhance the interpretability of model performance by dissecting explained variation.
  • These methods are applicable to both linear and generalized linear mixed models, broadening their utility.
  • The techniques are particularly beneficial for agricultural and ecological research requiring nuanced understanding of complex data structures.