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Parameter Estimation and Hypothesis Testing of The Bivariate Polynomial Ordinal Logistic Regression Model

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  • Marisa Rifada

    (Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia
    Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya 60115, Indonesia)

  • Vita Ratnasari

    (Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia)

  • Purhadi Purhadi

    (Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia)

Abstract

Logistic regression is one of statistical methods that used to analyze the correlation between categorical response variables and predictor variables which are categorical or continuous. Many studies on logistic regression have been carried out by assuming that the predictor variable and its logit link function have a linear relationship. However, in several cases it was found that the relationship was not always linear, but could be quadratic, cubic, or in the form of other curves, so that the assumption of linearity was incorrect. Therefore, this study will develop a bivariate polynomial ordinal logistic regression (BPOLR) model which is an extension of ordinal logistic regression, with two correlated response variables in which the relationship between the continuous predictor variable and its logit is modeled as a polynomial form. There are commonly two correlated response variables in scientific research. In this study, each response variable used consisted of three categories. This study aims to obtain parameter estimators of the BPOLR model using the maximum likelihood estimation (MLE) method, obtain test statistics of parameters using the maximum likelihood ratio test (MLRT) method, and obtain algorithms of estimating and hypothesis testing for parameters of the BPOLR model. The results of the first partial derivatives are not closed-form, thus, a numerical optimization such as the Berndt–Hall–Hall–Hausman (BHHH) method is needed to obtain the maximum likelihood estimator. The distribution statistically test is followed the Chi-square limit distribution, asymptotically.

Suggested Citation

  • Marisa Rifada & Vita Ratnasari & Purhadi Purhadi, 2023. "Parameter Estimation and Hypothesis Testing of The Bivariate Polynomial Ordinal Logistic Regression Model," Mathematics, MDPI, vol. 11(3), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:579-:d:1043596
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    References listed on IDEAS

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    1. W. Sauerbrei & P. Royston, 1999. "Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 162(1), pages 71-94.
    2. Patrick Royston & Gareth Ambler, 1999. "Multivariable fractional polynomials," Stata Technical Bulletin, StataCorp LP, vol. 8(43).
    3. Patrick Royston & Douglas G. Altman, 1994. "Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(3), pages 429-453, September.
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