Maximum likelihood (ML) and restricted maximum likelihood (REML) are two statistical methods used for estimating parameters in linear mixed models (LMMs). Both methods are widely used in various fields, including biology, social sciences, and engineering. While both ML and REML have their advantages and disadvantages, it’s important to understand the differences between them to choose the appropriate method for a given analysis. In this article, we’ll discuss the key differences between ML and REML, their applications, and the factors to consider when choosing between them.

## Overview of Maximum Likelihood (ML) and REML

### What is Maximum Likelihood (ML)?

Maximum likelihood (ML) is a statistical method used to estimate the parameters of a statistical model by maximizing the likelihood function. The likelihood function is a function of the parameters that measures how well the model fits the data. The ML method finds the values of the parameters that maximize the likelihood function, or equivalently, minimize the negative log-likelihood function.

### What is Restricted Maximum Likelihood (REML)?

Restricted maximum likelihood (REML) is a variant of the ML method that is used to estimate the parameters of a mixed-effects model. Unlike ML, which estimates the fixed and random effects simultaneously, REML estimates only the variance components of the random effects, assuming that the fixed effects are known. By removing the fixed effects from the model, REML can provide unbiased estimates of the variance components of the random effects.

## Differences between Maximum Likelihood (ML) and REML

### Estimation of fixed and random effects

The main difference between ML and REML is in the estimation of the fixed and random effects of the model. ML estimates both the fixed and random effects simultaneously, whereas REML estimates only the variance components of the random effects, assuming that the fixed effects are known.

### Bias

ML tends to produce biased estimates of the variance components of the random effects due to the fact that it estimates the fixed effects simultaneously. On the other hand, REML provides unbiased estimates of the variance components of the random effects.

### Likelihood function

The likelihood function used in ML is based on the assumption that the fixed effects are the true values of the parameters. In contrast, the likelihood function used in REML is based on the assumption that the fixed effects are estimated from the data.

### Degrees of freedom

ML uses the full likelihood function, which includes the fixed and random effects, to estimate the degrees of freedom of the model. In contrast, REML uses a restricted likelihood function, which excludes the fixed effects, to estimate the degrees of freedom of the model.

## Applications of Maximum Likelihood (ML) and REML

### Maximum Likelihood (ML)

ML is commonly used in linear regression models and generalized linear models, where the parameters of the model can be estimated using the likelihood function. It’s also used in mixed-effects models to estimate the fixed and random effects of the model simultaneously.

### Restricted Maximum Likelihood (REML)

REML is commonly used in linear mixed models, where the variance components of the random effects need to be estimated. It’s also used in meta-analysis, where the variance components of the random effects can be used to estimate the heterogeneity of the effect sizes.

## Factors to Consider when Choosing between Maximum Likelihood (ML) and REML

### Sample size

ML is more appropriate for large sample sizes, while REML is more appropriate for small to moderate sample sizes.

### Complexity of the model

ML is more appropriate for simple models, while REML is more appropriate for complex models with many random effects.

### Goal of the analysis

The choice between ML and REML depends on the goal of the analysis. If the goal is to estimate the fixed and random effects of the model simultaneously, then ML is the appropriate method. On the other hand, if the goal is to estimate the variance components of the random effects, then REML is the appropriate method.

### Assumptions of the model

The choice between ML and REML also depends on the assumptions of the model. If the model assumptions are violated, then the estimates obtained from ML may be biased. In contrast, the estimates obtained from REML may be less affected by violations of the model assumptions.

## Conclusion

In summary, Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) are two statistical methods used for estimating parameters in linear mixed models (LMMs). While both methods have their advantages and disadvantages, it’s important to understand the differences between them to choose the appropriate method for a given analysis. ML estimates both the fixed and random effects simultaneously, while REML estimates only the variance components of the random effects, assuming that the fixed effects are known. The choice between ML and REML depends on various factors, including the sample size, complexity of the model, and the goal of the analysis.

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