Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data

Abstract

In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum-Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum-likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real-world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.

Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data

“…We use the notation U ∼BS( μ , δ ) and its PDF is given by Some appealing properties of the BS distribution are the following. If U ∼BS( μ , δ ), then (i) c U ∼BS( c μ , δ ), with c > 0, which means that the BS distribution is closed under scalar multiplication (proportionality); (ii) 1/ U ∼BS( μ ⋆ , δ ), where μ ⋆ = ( δ + 1)/( δ μ ), implying that the BS distribution is closed under reciprocation; (iii) the median of the distribution of U is ( δ /( δ + 1)) μ , which can be directly obtained when q = 0.5 from its quantile function given by where z ( q ) is the q × 100th percentile (or quantile function) of the standard normal distribution and is the inverse of the cumulative distribution function (CDF) F U ; (iv) the BS distribution has different shapes for its PDF, which cover high, medium, and low asymmetry; (v) the new parameterization of the BS distribution based on the mean permits us to analyze data in their original scale, avoiding problems of interpretation in models based on logged data; (vi) in frailty models, the BS distribution is highly competitive in terms of fitting…”

“…The BS distribution has valid theoretical arguments to describe medical data as detailed in Section 2.4. Furthermore, it has shown empirically to be a good alternative to model this type of data …”

“…Furthermore, it has shown empirically to be a good alternative to model this type of data. [25][26][27][28][29] The objective of this paper is to propose a new frailty-based cure rate model. Its inference and diagnostics are performed with maximum likelihood (ML) methods.…”

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

“…The third period (2011 to the present) is characterized by a new inventiveness, breaking the link with lifetime data analysis and hence extended application in new areas such as: biology, crop yield assessment, econometrics, energy production, forestry, industry, informatics, insurance, inventory management, medicine, psychology, neurology, pollution monitoring, quality control, sociology and seismology; see, for example, Bhatti (2010); Kotz et al (2010); Balakrishnan et al (2011); Leiva et al (2010Leiva et al ( , 2011Leiva et al ( , 2012; Vilca et al (2010); Villegas et al (2011); Azevedo et al (2012); Ferreira et al (2012); Paula et al (2012); Santos-Neto et al (2012; Marchant et al (2013; Saulo et al (2013Saulo et al ( , 2018; Barros et al (2014); Rojas et al (2015); Wanke and Leiva (2015); Bourguignon et al (2017); Garcia-Papani et al (2017); ; Lillo et al (2018) and the references therein. In addition, risk and hazard analysis applications, in engineering and medicine, using the BS distribution were performed by Bebbington et al (2008); Kundu et al (2008); Azevedo et al (2012); Athayde (2017); Leão et al (2017Leão et al ( , 2018aLeão et al ( , 2018b; Athayde et al (2018) and Desousa et al (2018). Furthermore, the issue of robust parameter estimation has been considered, for example, by Wang et al (2013Wang et al ( , 2015 and Lemonte (2016).…”

Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data

“…The BS distribution originates from material fatigue and has interesting properties, doing it widely studied. Some of its recent applications range across fields different to engineering, such as business, environment, finance, industry and medicine, which have been conducted by an international, transdisciplinary group of researchers; see Jin and Kawczak (2003), Bhatti (2010), Lio et al (2010), Castillo et al (2011), Saulo et al (2013), Leiva et al (2015Leiva et al ( , 2016bLeiva et al ( , 2017, Wanke and Leiva (2015), Garcia-Papani et al (2016), Marchant et al (2016b), and Leao et al (2017). The BSACD model proposed by Bhatti (2010) is constructed in terms of a conditional median duration, rather than an ACD model in the sense of Engle and Russell (1998) based on the mean.…”