Likelihood and its use in Parameter Estimation and Model Comparison

Abstract

Parameter estimation and model fitting underlie many statistical procedures. Whether the objective is to examine central tendency or the slope of a regression line, an estimation method must be used. Likelihood is the basis for parameter estimation, for determining the best relative fit among several statistical models, and for significance testing. In this review, the concept of Likelihood is explained and applied computation examples are given. The examples provided serve to illustrate how likelihood is relevant, and related to, the most frequently applied test statistics (Student’s t-test, ANOVA). Additional examples illustrate the computation of Likelihood(s) using common population model assumptions (e.g., normality) and alternative assumptions for cases where data are non-normal. To further describe the interconnectedness of Likelihood and the Likelihood Ratio with modern test statistics, the relationship between Likelihood, Least Squares Modeling, and Bayesian Inference are discussed. Finally, the advantages and limitations of Likelihood methods are listed, alternatives to Likelihood are briefly reviewed, and R code to compute each of the examples in the text is provided.L’estimation de paramètres et l’ajustement de modèles est au coeur de toutes procédures statistiques. Que l’objectif soit d’examiner la tendance centrale ou une pente de régression, une méthode d’estimation est nécessaire. La fonction de vraisemblance est la pierre angulaire sur laquelle repose l’estimation de paramètres, les tests d’hypothèses et la comparaison de modèles. Cet article présente le concept de vraisemblance et les tests statistiques communément utilisés (tests t, ANOVA). Certains exemples présentent le calcul de la fonction de vraisemblance lorsque le postulat de normalité est présent et lorsqu’il n’est pas adéquat. Les liens entre vraisemblance, rapport de vraisemblance, méthodes des moindres carrés et bayésienne sont discutés. Finalement, les forces et les faiblesses des méthodes basées sur la vraisemblance sont énumérées et des méthodes alternatives sont mentionnées. Des instructions en R sont données pour tester les exemples du texte.A estimativa de parâmetros e o ajustamento de modelos está no cerne de todos os procedimentos estatísticos. Se o objetivo é analisar a tendência central ou uma inclinação de regressão, é necessário um método de estimativa. A função de verossimilhança é a pedra angular sobre a qual assentam a estimativa de parâmetros, os testes de hipóteses e a comparação de modelos. Este artigo introduz o conceito de verosimilhança e os testes estatísticos vulgarmente utilizados (testes t, ANOVA). Alguns exemplos mostram o cálculo da função de verossimilhança quando o pressuposto de normalidade está presente e sempre que não é adequado. Discutem-se as ligações entre a verosimilhança, razão de verossimilhança, os métodos dos mínimos quadrados e o bayesianismo. Por fim, são enumeradas as forças e as fraquezas dos métodos baseados na verosimilhança e são mencionados os métodos alternativos. As instruções em R s...

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“…Typical tests of fit based on likelihood includes the likelihood ratio test (an exact test for nested models; Wilks, 1938) or AIC, BIC and the like (for non-nested or unrelated models). See Hélie (2006) and Cousineau and Allen (2015) for more on this topic.…”

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“…We compared the performance of the full model to all possible alternative model combinations with different fixed effect structures. For model comparisons, parameters were estimated with maximum‐likelihood approximation, whereas for the best‐fitted model, restricted maximum likelihood was used (Cousineau and Allan 2015). We used pseudo ‐ R 2 as goodness of fit statistic (Nakagawa and Schielzeth 2013) where marginal R 2 (m) indicates the variance explained by the fixed effects and conditional R 2 (c) the variance explained by both fixed and random effects.…”