Publicado

2022-07-01

FLAT LIKELIHOODS: SIR-POISSON MODEL CASE

VEROSIMILITUDES PLANAS: CASO DEL MODELO SIR-POISSON

DOI:

https://doi.org/10.15446/rev.fac.cienc.v11n2.100986

Palabras clave:

Flat likelihood function, ODE model, SIR model, basic reproductive number, likelihood contours, profile likelihood function (en)
Función de verosimilitud plana, modelo EDO, modelo SIR, número reproductivo básico, contornos de verosimilitud, función de verosimilitud perfil (es)

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Autores/as

  • José A. Montoya Universidad de Sonora, México
  • Gudelia Figueroa-Preciado Universidad de Sonora, México
  • Mayra Rosalia Tocto-Erazo Universidad de Sonora, México

Systems of differential equations are used as the basis to define mathematical structures for moments, like the mean and variance, of random variables probability distributions.  Nevertheless, the integration of a deterministic model and a probabilistic one, with the aim of describing a random phenomenon, and take advantage of the observed data for making inferences on certain population dynamic characteristics, can lead to parameter identifiability problems. Furthermore, approaches to deal with those problems are usually inappropriate. In this paper, the shape of the likelihood function of a SIR-Poisson model is used to describe the relationship between flat likelihoods and the identifiability parameter problem.   In particular, we show how a flattened shape for the profile likelihood of the basic reproductive number R0, arises as the observed sample (over time) becomes smaller, causing ambiguity regarding the shape of the average model behavior.  We conducted some simulation studies to analyze the flatness severity of the R0 likelihood, and the coverage frequency of the likelihood-confidence regions for the model parameters. Finally, we describe some approaches to deal the practical identifiability problem, showing the impact those can have on inferences.  We believe this work can help to raise awareness on the way statistical inferences can be affected by a priori parameter assumptions and the underlying relationship between them, as well as by model reparameterizations and incorrect model assumptions.

Sistemas de ecuaciones diferenciales son usados como base para definir estructuras matemáticas para momentos, como la media y la varianza, de distribuciones de probabilidad de variables aleatorias. Sin embargo, integrar un modelo determinista con uno de probabilidad, para representar un fenómenos aleatorio y aprovechar datos observados de dicho fenómeno para hacer inferencia sobre características de la dinámica poblacional, puede producir problemas de identificabilidad de parámetros para la muestra observada. Más aún, dichos problemas suelen ser tratados de forma inapropiada. En este artículo utilizamos la forma de la función de verosimilitud del modelo SIR-Poisson para describir la relación entre funciones de verosimilitud planas y el problema de identificabilidad práctica de parámetros. En particular, mostramos cómo la forma aplanada de la función de verosimilitud perfil del número reproductivo básico R0 aparece a medida que la muestra observada (en el tiempo) se reduce y produce ambigüedad en la forma del comportamiento medio del modelo. Efectuamos diversos estudios de simulación con el fin de analizar la severidad de la planura de la verosimilitud de R0 y la frecuencia de cobertura de las regiones de verosimilitud-confianza de los parámetros del modelo. Finalmente, describimos algunos tratamientos que se han utilizado para abordar el problema de identificabilidad práctica y mostramos el impacto que éstos pueden tener en las inferencias. Consideramos que este trabajo ayudará a tomar conciencia sobre cómo las inferencias estadísticas se ven afectadas por suposiciones a priori sobre los parámetros y la relación subyacente entre ellos, así como por reparametrizaciones del modelo y por suposiciones incorrectas sobre el mismo

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Cómo citar

APA

Montoya, J. A., Figueroa-Preciado, G. . y Tocto-Erazo , M. R. . (2022). FLAT LIKELIHOODS: SIR-POISSON MODEL CASE. Revista de la Facultad de Ciencias, 11(2), 74–99. https://doi.org/10.15446/rev.fac.cienc.v11n2.100986

ACM

[1]
Montoya, J.A., Figueroa-Preciado, G. y Tocto-Erazo , M.R. 2022. FLAT LIKELIHOODS: SIR-POISSON MODEL CASE. Revista de la Facultad de Ciencias. 11, 2 (ago. 2022), 74–99. DOI:https://doi.org/10.15446/rev.fac.cienc.v11n2.100986.

ACS

(1)
Montoya, J. A.; Figueroa-Preciado, G. .; Tocto-Erazo , M. R. . FLAT LIKELIHOODS: SIR-POISSON MODEL CASE. Rev. Fac. Cienc. 2022, 11, 74-99.

ABNT

MONTOYA, J. A.; FIGUEROA-PRECIADO, G. .; TOCTO-ERAZO , M. R. . FLAT LIKELIHOODS: SIR-POISSON MODEL CASE. Revista de la Facultad de Ciencias, [S. l.], v. 11, n. 2, p. 74–99, 2022. DOI: 10.15446/rev.fac.cienc.v11n2.100986. Disponível em: https://revistas.unal.edu.co/index.php/rfc/article/view/100986. Acesso em: 7 jul. 2024.

Chicago

Montoya, José A., Gudelia Figueroa-Preciado, y Mayra Rosalia Tocto-Erazo. 2022. «FLAT LIKELIHOODS: SIR-POISSON MODEL CASE». Revista De La Facultad De Ciencias 11 (2):74-99. https://doi.org/10.15446/rev.fac.cienc.v11n2.100986.

Harvard

Montoya, J. A., Figueroa-Preciado, G. . y Tocto-Erazo , M. R. . (2022) «FLAT LIKELIHOODS: SIR-POISSON MODEL CASE», Revista de la Facultad de Ciencias, 11(2), pp. 74–99. doi: 10.15446/rev.fac.cienc.v11n2.100986.

IEEE

[1]
J. A. Montoya, G. . Figueroa-Preciado, y M. R. . Tocto-Erazo, «FLAT LIKELIHOODS: SIR-POISSON MODEL CASE», Rev. Fac. Cienc., vol. 11, n.º 2, pp. 74–99, ago. 2022.

MLA

Montoya, J. A., G. . Figueroa-Preciado, y M. R. . Tocto-Erazo. «FLAT LIKELIHOODS: SIR-POISSON MODEL CASE». Revista de la Facultad de Ciencias, vol. 11, n.º 2, agosto de 2022, pp. 74-99, doi:10.15446/rev.fac.cienc.v11n2.100986.

Turabian

Montoya, José A., Gudelia Figueroa-Preciado, y Mayra Rosalia Tocto-Erazo. «FLAT LIKELIHOODS: SIR-POISSON MODEL CASE». Revista de la Facultad de Ciencias 11, no. 2 (agosto 4, 2022): 74–99. Accedido julio 7, 2024. https://revistas.unal.edu.co/index.php/rfc/article/view/100986.

Vancouver

1.
Montoya JA, Figueroa-Preciado G, Tocto-Erazo MR. FLAT LIKELIHOODS: SIR-POISSON MODEL CASE. Rev. Fac. Cienc. [Internet]. 4 de agosto de 2022 [citado 7 de julio de 2024];11(2):74-99. Disponible en: https://revistas.unal.edu.co/index.php/rfc/article/view/100986

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