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Generalized loglogistic proportional hazard model with applications in survival analysis
 Shahedul A. Khan^{1}Email authorView ORCID ID profile and
 Saima K. Khosa^{1}
https://doi.org/10.1186/s404880160054z
© The Author(s) 2016
 Received: 12 May 2016
 Accepted: 17 November 2016
 Published: 29 November 2016
Abstract
Proportional hazard (PH) models can be formulated with or without assuming a probability distribution for survival times. The former assumption leads to parametric models, whereas the latter leads to the semiparametric Cox model which is by far the most popular in survival analysis. However, a parametric model may lead to more efficient estimates than the Cox model under certain conditions. Only a few parametric models are closed under the PH assumption, the most common of which is the Weibull that accommodates only monotone hazard functions. We propose a generalization of the loglogistic distribution that belongs to the PH family. It has properties similar to those of loglogistic, and approaches the Weibull in the limit. These features enable it to handle both monotone and nonmonotone hazard functions. Application to four data sets and a simulation study revealed that the model could potentially be very useful in adequately describing different types of timetoevent data.
Keywords
 Cox PH
 Loglogistic distribution
 Parametric model
 Proportional hazard
 Semiparametric model
 Timetoevent data
 Weibull distribution
AMS Subject Classification
 Primary 62N01; Secondary 62P10
Introduction
Proportional hazard (PH) models play a vital role in analyzing timetoevent data. A key assumption in the PH model is that the hazard ratio comparing any two specifications of covariates is constant over time (commonly known as PH assumption). Although the PH assumption may not hold for one or more covariates over the entire study period, it may hold in shorter time intervals. Therefore, violation of the PH assumption may be handled using timedependent covariates (Kleinbaum and Klein 2012). One of the appealing features of PH models is that the regression coefficients have relative risk interpretation, which is preferred by many clinicians.
The Cox PH model (Cox 1972) is the most popular in survival analysis mainly because of two reasons: (a) no assumption is required about the probability distribution of survival times (i.e., a semiparametric model), and (b) it usually fits the data well no matter which parametric model is appropriate. In contrast, distributional assumption is required for a fully parametric PH model (Kalbfleisch and Prentice 2002; Lawless 2002). This also leads to the added requirement of checking the appropriateness of the chosen distribution. Nevertheless, as demonstrated by Efron (1977) and Oakes (1977), parametric models lead to more efficient estimates than Cox’s model under certain conditions. More specifically, if the distributional assumption is valid, a parametric model leads to smaller standard errors of the estimates than would be in the absence of a distributional assumption (Collett 2003). Moreover, the use of Cox PH in joint modeling of timetoevent and longitudinal data (Wulfsohn and Tsiatis 1997) usually leads to an underestimation of the standard errors of the parameter estimates (Hsieh et al. 2006; Rizopoulos 2012), and therefore most methods for joint modeling are based on parametric response distributions (Hwang and Pennell 2014). Regarding the choice between a parametric and Cox’s PH model, Nardi and Schemper (2003) suggested to use a richer parametric model or simply the Cox’s model in case of an unsatisfactory fit of the chosen probability distribution.
The most commonly used parametric timetoevent models are the Weibull, loglogistic and lognormal distributions. The loglogistic and lognormal distributions belong to the accelerated failure time (AFT) family, and are useful in modeling nonmonotone hazard rates (Lawless 2002). Note that the loglogistic also accommodates decreasing hazard functions. Only a few parametric models are closed under PH assumption, the most common of which is the Weibull that accommodates only monotone hazard functions. In fact, Weibull is the only distribution that is closed under both AFT and PH families (Kalbfleisch and Prentice 2002). Mudholkar et al. (1996) proposed a generalization of the Weibull distribution which permits parametric PH regression modeling. It is a threeparameter distribution and is capable of modeling both monotone and nonmonotone hazard functions. One difficulty with this model is that it is nonregular (the support depends on some parameters) in the case of increasing hazard functions, and therefore the standard maximum likelihood asymptotics do not hold. In this paper, we propose a simple extension of the loglogistic model which is closed under the PH relationship. The proposed generalized loglogistic model is a threeparameter distribution, and has characteristics similar to those of the loglogistic model. Moreover, it approaches the Weibull in the limit. These features enable it to satisfactorily handle both monotone (increasing and decreasing) and nonmonotone (unimodal) hazard functions. In Section 1, we introduce the generalized loglogistic model and discuss estimation and testing of the parameters using the maximum likelihood method. The proposed method is then illustrated with applications to four data sets, one of which involves joint modeling of timetoevent and longitudinal data (Section Examples). In Section Simulations, a simulation study is presented to evaluate the performance of generalized loglogistic in comparison with other commonly used PH models to describe different types of timetoevent data. We conclude in Section Conclusion by summarizing our findings.
The generalized loglogistic model
where ρ>0, κ>0 and γ>0 are parameters and α=(κ,γ,ρ)^{′}. If γ depends on ρ via γ=ρ and γ=ρ η ^{−1/κ } with η>0, then (1) reduces to the hazard function of the loglogistic (Lawless 2002) and Burr XII (Wang et al. 2008) distributions, respectively. Taking γ not dependent on ρ, it is easy to verify that (1) is closed under PH relationship (see below). The hazard function is monotone decreasing when κ≤1, and unimodal when κ>1 (i.e., h(t;α)=0 at t=0, increases to a maximum at t=[(κ−1)/γ ^{ κ }]^{1/κ }, and then approaches zero monotonically as t→∞). Note that (1) approaches the Weibull hazard function as γ ^{ κ }→0. This particular feature of the generalized loglogistic model enables it to handle monotone increasing hazard satisfactorily via κ>1 and γ small (close to zero).
In particular, the mean is \(E(T)=\frac {\rho ^{\kappa }}{\gamma ^{\kappa }} \frac {\Gamma \left (\frac {\rho ^{\kappa }}{\gamma ^{\kappa }}\frac {1}{\kappa }\right) \Gamma \left (\frac {1}{\kappa }+1\right)}{\Gamma \left (\frac {\rho ^{\kappa }}{\gamma ^{\kappa }}+1\right)}\) provided \(\frac {\kappa \rho ^{\kappa }}{\gamma ^{\kappa }}>1\).
where \(\phantom {\dot {i}\!}\rho ^{*}=e^{\mathbf {z}'\boldsymbol {\beta }/\kappa }\). Thus the generalized loglogistic is closed under proportionality of hazards. Another widely used parametric PH family is the Weibull, for which h _{0}(t;α)=κ ρ(ρ t)^{ κ }. Note that the Cox PH model is semiparametric, for which the baseline hazard function in (5) is left arbitrary and is denoted by h _{0}(t).
Estimation
Many software packages have reliable optimization procedures to maximize loglikelihood functions. We wrote our computer code in R (R Core Team 2016), and used the function nlminb for optimization (see the Additional file 1).
Initial values
We may use Weibull, loglogistic and Cox PH fits to generate initial values in solving the equations ∂ ℓ(θ ^{∗})/∂ κ ^{∗}=0, ∂ ℓ(θ ^{∗})/∂ γ ^{∗}=0, ∂ ℓ(θ ^{∗})/∂ ρ ^{∗}=0 and ∂ ℓ(θ ^{∗})/∂ β _{ j }=0. Let \(\hat {\kappa }_{1}\) and \(\hat {\rho }_{1}\) be the maximum likelihood estimates of the Weibull shape and scale parameters, respectively, \(\hat {\kappa }_{2}\) and \(\hat {\rho }_{2}\) the maximum likelihood estimates of the loglogistic shape and scale parameters, respectively, and \(\hat {\boldsymbol {\beta }^{*}}\) the estimates of the regression coefficients for the Cox PH model. Note that maximum likelihood methods for the Weibull, loglogistic and Cox PH models are available in many statistical softwares, including R (R Core Team 2016). We propose to use \(\log {\hat {\kappa }_{1}}\), \(\log {\hat {\kappa }_{1}\hat {\kappa }_{2}}\), \(\log {\hat {\rho }_{1}}\) and \(\hat {\boldsymbol {\beta }^{*}}\) as initial values for κ ^{∗}, γ ^{∗}, ρ ^{∗} and β, respectively. If convergence is not achieved with these initial values, we propose to replace \(\log {\hat {\kappa }_{1}}\) and \(\log {\hat {\rho }_{1}}\) by \(\log {\hat {\kappa }_{2}}\) and \(\log {\hat {\rho }_{2}}\), respectively. In fitting the generalized loglogistic model to many data sets, we have not experienced any difficulty in obtaining convergence with this technique.
Tests and confidence intervals
Generalized loglogistic distribution in joint modeling
Joint models are used to quantify association between an internal timedependent covariate and time until an event of interest occurs (Wulfsohn and Tsiatis 1997). It involves two separate models: a model that takes into account measurement error in the timedependent covariate to estimate its true values (longitudinal model), and another model that uses these estimated values to quantify the association between this covariate and the time to the occurrence of the event (timetoevent model). The idea behind the joint modeling technique is to couple the timetoevent model with the longitudinal model. The general framework of the maximum likelihood method and large sample theory can be found in Rizopoulos (2012). Maximization of the loglikelihood function for joint modeling is computationally challenging, as it involves evaluating multiple integrals that do not have an analytical solution, except in very special cases. The R package JM has been developed by Rizopoulos (2010) to fit joint models using Weibull baseline hazard, piecewiseconstant baseline hazard, spline approximation of the baseline hazard and unspecified baseline hazard functions. We have modified the source codes for Weibull to fit joint models using the generalized loglogistic baseline hazard function. The application of the generalized loglogistic distribution in joint modeling is illustrated with an example in Section 1.
Goodness of fit
The nonparametric estimates are useful for assessing the quality of fit of a particular parametric timetoevent model (Lawless 2002). For a model without covariate, we use the approach to simultaneously examine plots of parametric and nonparametric estimates of the survival function, superimposed on the same graph. Let \(S(t;\hat {\boldsymbol {\theta }})\) and \(\hat {S}(t)\) be the estimates of the survivor functions based on the parametric model of interest and the KaplanMeier method (Kaplan and Meier 1958), respectively. The estimates \(S(t;\hat {\boldsymbol {\theta }})\) as a function of t should be close to \(\hat {S}(t)\) if the parametric model is adequate. For a model with covariates, we consider residual diagnostic plots, where the residuals are defined based on the cumulative hazard function H(t;θ). If \(\hat {S}(H(t;\boldsymbol {\hat {\theta }}))\) is the KaplanMeier estimate of \(H(t;\boldsymbol {\hat {\theta }})\), then a plot of \(\log \hat {S}(H(t;\boldsymbol {\hat {\theta }}))\) versus \(H(t;\boldsymbol {\hat {\theta }})\) should be roughly a straight line with unit slope when the model is adequate (Lawless 2002).
Examples
Three data sets are taken from the literature to demonstrate the ability of the generalized loglogistic distribution in modeling timetoevent data. The application of the generalized loglogistic PH in joint modeling is illustrated using another data set of AIDS patients. We first use the scaled TTT transform of failure times to detect the shape of the hazard function (Mudholkar et al. 1996). The scaled TTT transform is given by \(\phi (v/n)=\left [\sum _{i=1}^{v}{T_{(i)}}+(nv)T_{(v)}\right ]/\left (\sum _{i=1}^{n}{T_{(i)}}\right)\), where T _{(i)} represent the order statistics of the sample, and v=1,2,…,n. The hazard function is increasing, decreasing and unimodal if the plot of (v/n,ϕ(v/n)) is concave, convex, and concave followed by convex, respectively. For the first three examples (Sections Example 1: Head and neck cancer dataExample 3: Vaginal cancer mortality in rats), we first fit the generalized loglogistic, Weibull and loglogistic models (without covariate) and check the appropriateness of the distributional assumption using diagnostic plots. Then, we analyze the data using regression models, and compare the fits via residual plots. Note that the regression model based on the loglogistic distribution is given by logT=β _{0}+β _{1} z _{1}+…+β _{ p } z _{ p }+τ W where τ=1/κ, β _{0}=− logρ and W has the logistic distribution with density f(w)=e ^{ w }/(1+e ^{ w })^{2}, – ∞<w<∞. This model has an accelerated life interpretation (Lawless 2002), whereas the generalized loglogistic and Weibull PH models have relative risk interpretation. In the fourth example (Section Example 4: AIDS data), we consider joint models based on the generalized loglogistic, Weibull and piecewiseconstant baseline hazard functions.
Example 1: Head and neck cancer data
Data description, hazard shape and distributional assumption
Regression analysis
Generalized loglogistic, Weibull and loglogistic fits for the head and neck cancer data
Generalized loglogistic PH  Weibull PH  Loglogistic AFT  

(AIC =1053.39)  (AIC =1082.52)  (AIC =1067.25)  
Parameter  Estimate  SE  Estimate  SE  Estimate  SE 
β  0.5459  0.2382  0.6686  0.2415  −0.5549  0.2779 
logκ  0.9790  0.1986  −0.1619  0.0921  0.2764  0.0971 
logρ  −5.2692  0.1844  −6.8248  0.2112  −6.0492  0.2128 
logγ  −4.6497  0.1755 
Example 2: Autologous and allogeneic bone marrow transplants
Data description, hazard shape and distributional assumption
Regression analysis
Generalized loglogistic, Weibull and loglogistic fits for the bone marrow transplants data
Generalized loglogistic PH  Weibull PH  Loglogistic AFT  

(AIC =444.64)  (AIC =450.08)  (AIC =446.46)  
Parameter  Estimate  SE  Estimate  SE  Estimate  SE 
β  0.1981  0.2854  0.2535  0.2854  −0.0808  0.4481 
logκ  0.2148  0.2376  −0.3878  0.1229  −0.1694  0.1213 
logρ  −2.4055  0.4917  −3.9683  0.3300  −3.1847  0.3474 
logγ  −1.3188  0.6253 
Example 3: Vaginal cancer mortality in rats
Data description, hazard shape and distributional assumption
Regression analysis
Generalized loglogistic, Weibull and loglogistic fits for the vaginal cancer mortality data
Generalized loglogistic PH  Weibull PH  Loglogistic AFT  

(AIC =391.35)  (AIC =389.87)  (AIC =391.89)  
Parameter  Estimate  SE  Estimate  SE  Estimate  SE 
β  0.6254  0.3485  0.6599  0.3474  −0.1861  0.1203 
logκ  1.2568  0.2168  1.1308  0.1300  1.5077  0.1429 
logρ  −5.3516  0.5889  −5.0864  0.0754  −4.9301  0.0846 
logγ  −5.0190  0.1154 
Example 4: AIDS data
Data description and hazard shape
This example illustrates the use of the generalized loglogistic distribution in joint modeling. Rizopoulos (2012) described a study involving 467 human immunodeficiency virus (HIV) infected patients who had failed or were intolerant to zidovudine therapy (ZT). The main objective was to compare two antiretroviral drugs to prevent the progression of HIV infections: didanosine (ddI) and zalcitabine (ddC). Patients were randomly assigned to receive either ddI or ddC and followed until death or the end of the study, resulting in 188 complete and 279 censored observations. It was also of interest to quantify the association between CD4 cell counts (internal timedependent covariate) measured at t=0, 2, 6, 12 and 18 months, and time to death. The TTT plot in Fig. 1(d) indicates an increasing hazard shape.
Regression analysis
AIDS data: estimates and standard errors for the timetoevent process of joint models
Generalized loglogistic  Weibull  Piecewiseconstant  

(AIC =8699.61)  (AIC =8699.26)  (AIC =8711.61)  
Parameter  Estimate  SE  Estimate  SE  Estimate  SE 
β _{0}  −3.1615  0.4411  −2.9477  0.3898  −  − 
β _{1}  0.3690  0.1575  0.3727  0.1576  0.3647  0.1573 
β _{2}  −0.3647  0.2583  −0.3619  0.2591  −0.3364  0.2585 
β _{3}  0.3372  0.1556  0.3455  0.1555  0.3329  0.1555 
β _{4}  −0.2824  0.0382  −0.2784  0.0378  −0.2860  0.0382 
logκ  0.3838  0.7709  0.2377  0.0732  
logγ  −2.8874  0.1333 
Simulations

Scenario 1: Decreasing hazard. Lifetimes were generated from generalized Weibull with κ=0.5, γ=−0.1 and ρ=0.1, and censoring times were generated from the exponential distribution with rate parameter λ=0.045.

Scenario 2: Increasing hazard. Lifetimes were generated from generalized Weibull with κ=2, γ=0.1 and ρ=0.1, and censoring times were generated from the exponential distribution with rate parameter λ=0.060.

Scenario 3: Unimodal hazard. Lifetimes were generated from generalized Weibull with κ=2, γ=−0.1 and ρ=0.1, and censoring times were generated from the exponential distribution with rate parameter λ=0.060.
Model performance and comparison using simulation study (n=100) with about 40% censored observations
Generalized  

Parameter  loglogistic PH  Weibull PH  Cox PH  
Scenarios  True  Mean  AB  MSE  Mean  AB  MSE  Mean  AB  MSE  
Scenario 1  β _{1}  0.50  0.524  0.024  0.024  0.531  0.031  0.024  0.524  0.024  0.025  
(True model:  β _{2}  −0.50  −0.538  0.038  0.032  −0.545  0.045  0.033  −0.536  0.036  0.032  
generalized  β _{3}  0.75  0.798  0.048  0.088  0.808  0.058  0.090  0.795  0.045  0.091  
Weibull)  β _{4}  −0.75  −0.780  0.030  0.081  −0.792  0.042  0.083  −0.782  0.032  0.083  
ρ  0.10  0.148  0.103  
κ  0.50  0.550  0.508  
γ  −0.10  0.073  
Scenario 2  β _{1}  0.50  0.516  0.016  0.027  0.518  0.018  0.027  0.532  0.032  0.031  
(True model:  β _{2}  −0.50  −0.522  0.022  0.035  −0.523  0.023  0.035  −0.533  0.033  0.039  
generalized  β _{3}  0.75  0.785  0.035  0.099  0.788  0.038  0.099  0.813  0.063  0.112  
Weibull)  β _{4}  −0.75  −0.765  0.015  0.082  −0.768  0.018  0.082  −0.796  0.046  0.092  
ρ  0.10  0.107  0.106  
κ  2.00  2.269  2.249  
γ  0.10  0.006  
Scenario 3  β _{1}  0.50  0.519  0.019  0.025  0.530  0.030  0.026  0.516  0.016  0.026  
(True model:  β _{2}  −0.50  −0.548  0.048  0.038  −0.557  0.057  0.039  −0.547  0.047  0.039  
generalized  β _{3}  0.75  0.791  0.041  0.099  0.811  0.061  0.103  0.791  0.041  0.102  
Weibull)  β _{4}  −0.75  −0.790  0.040  0.089  −0.811  0.061  0.093  −0.792  0.042  0.095  
ρ  0.10  0.103  0.098  
κ  2.00  2.130  2.016  
γ  −0.10  0.024 
Results for scenario 1. For the continuous covariates (z _{1} and z _{2}), all three models produced estimates with similar MSE, whereas for the binary covariates (z _{3} and z _{4}), the generalized loglogistic demonstrated the smallest MSE. In terms of bias, generalized loglogistic and Cox PH were roughly equivalent, and both were superior to Weibull.
Results for scenario 2. For the regression coefficients, the generalized loglogistic produced estimates with the smallest bias. We also see that the generalized loglogistic and Weibull produced estimates with similar MSE, and both were superior to the Cox PH model. Note that the generalized loglogistic estimates for κ and γ were 2.269 and 0.006, respectively (i.e., the estimate of γ ^{ κ } is close to zero), supporting the fact that the hazard function is monotone increasing.
Results for scenario 3. In terms of bias, the generalized loglogistic and Cox PH produced comparable estimates of the regression coefficients. However, the generalized loglogistic produced the most accurate estimates in terms of MSE, mostly as a consequence of smaller standard deviations of the estimates. As expected, the Weibull produced the least accurate estimates in terms of both bias and MSE for Scenario 3 (i.e., unimodal hazard).
A simulation study with about 20% censored observations per data set also led to similar conclusions (data not shown). In summary, our simulation study has demonstrated that the generalized loglogistic could potentially be a very useful parametric model to adequately describe different types of timetoevent data.
Conclusion
In this paper, we proposed a simple extension of the loglogistic distribution to a PH model by appending an additional parameter. As described in Section 1, the proposed model naturally accommodates decreasing and unimodal hazard functions. The loglogistic distribution is known to be useful to describe unimodal hazard functions (Lawless 2002). As demonstrated in Examples 1 and 2, it turns out that the generalized loglogistic may provide better fits in describing unimodal hazard functions compared to the loglogistic distribution. Moreover, our simulation study revealed that the generalized loglogistic could produce more accurate results compared to the Weibull and Cox PH models in describing monotone decreasing and unimodal hazard functions. In summary, the flexibility provided by the generalized loglogistic model could be very useful in adequately describing different types of timetoevent data.
Appendix: Derivatives of the loglikelihood function

∙
$$ {\log{(\gamma t_{i})}= \frac{\log{b_{i}}}{\kappa},} $$(15) 
∙
$$ {(\gamma t_{i})^{\kappa} \log{(\gamma t_{i})}= \frac{b_{i}\log{b_{i}}}{\kappa},} $$(16) 
∙
$$ {\frac{\partial b_{i}}{\partial \kappa}=\frac{\partial}{\partial \kappa} (\gamma t_{i})^{\kappa} = (\gamma t_{i})^{\kappa} \log{(\gamma t_{i})}= \frac{b_{i}\log{b_{i}}}{\kappa},} $$(17) 
∙
$$ {\frac{\partial \log{b_{i}}}{\partial \kappa}= \frac{\log{b_{i}}}{\kappa},} $$(18) 
∙
$$ {\frac{\partial \log{(1+b_{i})}}{\partial \kappa}= \frac{b_{i}\log{b_{i}}}{\kappa(1+b_{i})}=\frac{b_{i}c_{i}}{\kappa},} $$(19) 
∙
$$ {\frac{\partial b_{i}\log{b_{i}}}{\partial \kappa}=\frac{b_{i}\log{b_{i}}}{\kappa}\log{b_{i}}+b_{i}\frac{\log{b_{i}}}{\kappa}=\frac{b_{i}(\log{b_{i}})(1+\log{b_{i}})}{\kappa},} $$(20) 
∙
$$ {\frac{\partial c_{i}}{\partial \kappa}=\frac{\partial}{\partial \kappa} \frac{\log{b_{i}}}{1+b_{i}} =\frac{\log{b_{i}}}{\kappa(1+b_{i})}\left(1\frac{b_{i}\log{b_{i}}}{1+b_{i}}\right) =\frac{c_{i}(1b_{i}c_{i})}{\kappa},} $$(21) 
∙
$$ {\frac{\partial b_{i}c_{i}}{\partial \kappa}=\frac{b_{i}c_{i}(1b_{i}c_{i}+\log{b_{i}})}{\kappa} = \frac{b_{i}c_{i}(1+c_{i})}{\kappa},} $$(22) 
∙
$$ {\frac{\partial d_{i}}{\partial \kappa}=\frac{\partial}{\partial \kappa}\frac{b_{i}}{1+b_{i}}=\frac{b_{i}\log{b_{i}}}{\kappa(1+b_{i})}\left(1\frac{b_{i}}{1+b_{i}}\right)= \frac{c_{i}d_{i}}{\kappa},} $$(23) 
∙
$$ {\frac{\partial \log{(1d_{i})}}{\partial \kappa}= \frac{\partial}{\partial \kappa} \log{(1+b_{i})^{1}}=\frac{\partial}{\partial \kappa} \log{(1+b_{i})}=\frac{b_{i}c_{i}}{\kappa},} $$(24) 
∙
$$ {\frac{\partial b_{i}}{\partial \gamma}=\frac{\partial}{\partial \gamma} (\gamma t_{i})^{\kappa} = \kappa\gamma^{\kappa1}t_{i}^{\kappa}=\frac{\kappa}{\gamma}b_{i},} $$(25) 
∙
$$ {\frac{\partial d_{i}}{\partial \gamma}=\frac{\partial}{\partial \gamma} \frac{b_{i}}{1+b_{i}}=\frac{\kappa}{\gamma}\frac{b_{i}}{1+b_{i}}\left(1\frac{b_{i}}{1+b_{i}}\right)=\frac{\kappa}{\gamma}~d_{i}(1d_{i}),} $$(26) 
∙$$ {\frac{\partial}{\partial \gamma} \log{(1d_{i})} =\frac{\partial}{\partial \gamma}\log{(1+b_{i})}=\frac{\kappa}{\gamma}~\frac{b_{i}}{1+b_{i}} = \frac{\kappa}{\gamma}~d_{i}.} $$(27)
Declarations
Acknowledgements
The authors acknowledge the comments and suggestions of the editor and the reviewers. This work was partially supported by NSERC through Discovery Grant (#368532) to SA Khan, and the University of Saskatchewan through New Faculty Startup Operating Fund to SA Khan.
Authors’ contributions
SAK, the principal investigator, conceptually developed the proposed distribution with related mathematical results and R code for computation, and drafted the manuscript. SKK contributed in developing mathematical results and writing Sections 13. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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