Discussion on selecting the number of breaks in the pattern of spread of COVID-19 (a reply to Zhao and Liang)

Abstract

Kim et al., 2020Kim Y. Seo M.H. Yeom H. Estimating a breakpoint in the spread pattern of COVID-19 in South Korea.IJID. 2020; 97: 360-364https://doi.org/10.1016/j.ijid.2020.06.055Abstract Full Text Full Text PDF PubMed Scopus (11) Google Scholar highlighted the importance of allowing for a structural break in fitting the SIR model to capture the impact on the transmission of the policy responses and the behavioral changes. A natural extension of Kim et al., 2020Kim Y. Seo M.H. Yeom H. Estimating a breakpoint in the spread pattern of COVID-19 in South Korea.IJID. 2020; 97: 360-364https://doi.org/10.1016/j.ijid.2020.06.055Abstract Full Text Full Text PDF PubMed Scopus (11) Google Scholar is to develop a formal statistical procedure to determine the number of breaks. This is important in the face of resurfacing COVID-19. A few common statistical approaches for this task may include: a sequence of testing for the presence of a structural break; a model selection procedure such as the AIC or BIC; and a regularized estimation to select the number of breaks and simultaneously estimate the coefficients, such as Tibshirani, 1996Tibshirani R. Regression shrinkage and selection via the lasso.J R Stat Soc Series B Stat Methodol. 1996; 58: 267-288https://doi.org/10.1111/j.2517-6161.1996.tb02080.xCrossref Google Scholar lasso, in the linear regression context. To make these statistical developments, one needs to extend the classic deterministic SIR model to a suitable stochastic version and then analyze the various time series properties of the generated series, which might contain a non-linear deterministic or stochastic time trend. Zhao and Liang, 2020Zhao S. Liang X. A re-analysis to identify the structural breaks in the COVID-19 transmissibility during the early phase of the outbreak in South Korea.IJID. 2020; Google Scholar letter proposes an approach to apply the AIC. To that end, it proposes a continuously time-varying parameter model, where the variation of the parameter follows a linear time trend with kinks. Under an additional parametric assumption on the multiplicative noise (a Gamma process), the authors estimate the number of kinks based on the AIC. This letter discusses a few issues with their comments. First is an issue related to the data. The transmission rate in the baseline SIR model captures the rate of change in the number of infected, not that of the newly infected. However, they used the number of the newly infected to estimate the parameters, which invalidated their claim about the number of kinks at the transmission parameter in the SIR model. Second is an issue with the statistical methodology. Specifically, the penalty term embedded in the AIC criterion may need to be more carefully justified. For instance, Ng and Perron, 2001Ng S. Perron P. Lag length selection and the construction of unit root tests with good size and power.Econometrica. 2001; 69: 1519-1554https://doi.org/10.1111/1468-0262.00256Crossref Scopus (2280) Google Scholar highlighted the importance of adjusting the magnitude of the penalty according to the time series property of the data in the context of the lag length selection for the unit root testing. Since Zhao and Liang, 2020Zhao S. Liang X. A re-analysis to identify the structural breaks in the COVID-19 transmissibility during the early phase of the outbreak in South Korea.IJID. 2020; Google Scholar) propose a model with a nonlinear time trend, the time series is not stationary and thus some concerns noted by Ng and Perron, 2001Ng S. Perron P. Lag length selection and the construction of unit root tests with good size and power.Econometrica. 2001; 69: 1519-1554https://doi.org/10.1111/1468-0262.00256Crossref Scopus (2280) Google Scholar) may occur. In this vein, we recommend the letter’s authors and readers to refer to the work by Lee et al., 2020Lee S. Liao Y. Seo M.H. Shin Y. Sparse HP filter: finding kinks in the COVID-19 contact rate.arXiv. 2020; (preprint arXiv. 2006; 10555.)https://arxiv.org/pdf/2006.10555.pdfGoogle Scholar). They propose various ways to fit the time-varying transmission rate by the linear time trend with kinks, as in Zhao and Liang, 2020Zhao S. Liang X. A re-analysis to identify the structural breaks in the COVID-19 transmissibility during the early phase of the outbreak in South Korea.IJID. 2020; Google Scholar), through several different regularization methods and relate them to the well-known HP-filtering in the time series literature. They also provide certain statistical guarantees. The authors declare no conflict of interest.