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Regression correlation coefficient for a Poisson regression model

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  • Takahashi, Akihito
  • Kurosawa, Takeshi

Abstract

This study examines measures of predictive power for a generalized linear model (GLM). Although many measures of predictive power for GLMs have been proposed, most have limitations. Hence, we focus on the regression correlation coefficient (RCC) (Zheng and Agresti, 2000), which satisfies the four requirements of (i) interpretability, (ii) applicability, (iii) consistency, and (iv) affinity. The RCC is a population value that is defined by the correlation between a response variable and the conditional expectation of the response variable. Its sample value is defined by the sample correlation between the observed response values and estimated values of the response variable. For an arbitrary GLM, we do not always have an explicit form of the RCC. However, for a Poisson regression model, assuming that the predictor variables have a multivariate normal distribution, we can find the explicit form of the RCC (true value). Therefore, it is possible to compare the estimators (sample values) of the RCC in terms of bias and RMSE (root of the mean square error) by using the true value. Furthermore, by using the explicit form, we propose a new estimator of the RCC for the Poisson regression model. We then compare the new estimator with the sample correlation estimator, the jack-knife estimator, and the leave-one-out cross validation estimator in terms of bias and RMSE. The leave-one-out cross validation estimator has large negative bias and large RMSE. Although the remaining three estimators show similar behavior for a large sample size, for a small sample size the new estimator shows the best behavior in terms of bias and RMSE.

Suggested Citation

  • Takahashi, Akihito & Kurosawa, Takeshi, 2016. "Regression correlation coefficient for a Poisson regression model," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 71-78.
  • Handle: RePEc:eee:csdana:v:98:y:2016:i:c:p:71-78
    DOI: 10.1016/j.csda.2015.12.012
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    References listed on IDEAS

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    1. Eshima, Nobuoki & Tabata, Minoru, 2007. "Entropy correlation coefficient for measuring predictive power of generalized linear models," Statistics & Probability Letters, Elsevier, vol. 77(6), pages 588-593, March.
    2. Eshima, Nobuoki & Tabata, Minoru, 2010. "Entropy coefficient of determination for generalized linear models," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1381-1389, May.
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    Cited by:

    1. Chen, Xi & Yang, Hongxing, 2017. "A multi-stage optimization of passively designed high-rise residential buildings in multiple building operation scenarios," Applied Energy, Elsevier, vol. 206(C), pages 541-557.
    2. Takeshi Kurosawa & Francis K.C. Hui & A.H. Welsh & Kousuke Shinmura & Nobuoki Eshima, 2020. "On goodness‐of‐fit measures for Poisson regression models," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 62(3), pages 340-366, September.
    3. Otilia Vanessa Cordero-Ahiman & Jorge Leonardo Vanegas & Christian Franco-Crespo & Pablo Beltrán-Romero & María Elena Quinde-Lituma, 2021. "Factors That Determine the Dietary Diversity Score in Rural Households: The Case of the Paute River Basin of Azuay Province, Ecuador," IJERPH, MDPI, vol. 18(4), pages 1-16, February.

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