Hypertabastic survival models

Consider a sample of survival times of n individuals ${\displaystyle t_{1},t_{2},\ldots ,t_{n}}$ with associated p-dimensional covariate vectors ${\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}}$ and an unknown parameter vector ${\displaystyle \mathbf {\theta } =(\theta _{0},\theta _{1},\ldots ,\theta _{p})}$. Let ${\displaystyle f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } ),F(t_{i}|\mathbf {x} _{i},\theta ),S(t_{i}|\mathbf {x} _{i},)}$ and ${\displaystyle h(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )}$ stand for the corresponding probability density function, cumulative distribution function, survival function and hazard function respectively. In the absence of censoring (censoring normally occurs when the failure time of some individuals cannot be observed), the likelihood function is

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )}

and the log-likelihood ${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })} is

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}ln{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}}

For the right censored data, the likelihood function is

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{\delta _{i}}[S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\delta _{i}}}}

or equivalently,

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}{[h(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{\delta _{i}}S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )}} ,

and the log-likelihood function is

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}(\delta _{i}ln{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}+(1-\delta _{i})ln{[S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]})}

or equivalently,

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}(\delta _{i}ln{[h(t_{i}|\mathbf {x} _{i},\mathbf {\theta } ]}+ln{[S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]})}

where

${\displaystyle \delta _{i}={\begin{cases}0&t_{i}{\text{is a right censored observation}}\\1&otherwise\end{cases}}}$,

In the presence of left censored data, the likelihood function is

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{\gamma _{i}}[F(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\gamma _{i}}}

and the corresponding log-likelihood function is

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}(\gamma _{i}ln{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}+(1-\gamma _{i})ln{[F(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]})}

where

${\displaystyle \gamma _{i}={\begin{cases}0&t_{i}{\text{is a left censored observation}}\\1&otherwise\end{cases}}}$,

In the presence of interval censored data, the likelihood function is

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}([f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{\xi _{i}}[F(v_{i}|\mathbf {x} _{i},\mathbf {\theta } )-F(u_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\xi _{i}})}

and the log-likelihood function is

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}(\xi _{i}ln{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}+(1-\xi _{i})ln{[F(v_{i}|\mathbf {x} _{i},\mathbf {\theta } )-F(u_{i}|\mathbf {x} _{i},\mathbf {\theta } )]})}

where ${\displaystyle u_{i}\leq t_{i}\leq v_{i}}$ for all interval censored observations and

${\displaystyle \xi _{i}={\begin{cases}0&t_{i}{\text{is an interval censored observation}}\\1&otherwise\end{cases}}}$,

If the intended sample consists of all types of censored data (right censored, left censored and interval censored), then its likelihood function takes the following form

${\displaystyle L(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\prod _{i=1}^{n}([S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\delta _{i}}[F(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\gamma _{i}}[F(v_{i}|\mathbf {x} _{i},\mathbf {\theta } )-F(u_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{1-\xi _{i}}[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]^{\delta _{i}+\gamma _{i}+\xi _{i}-2})}

and its corresponding log-likelihood function is given by

${\displaystyle LL(\mathbf {\theta } ,\alpha ,\beta$ :{\mathbf {x} })=\sum _{i=1}^{n}{(1-\delta _{i})ln{[S(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}+(1-\gamma _{i})ln{[F(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}(1-\xi _{i})ln{[F(v_{i}|\mathbf {x} _{i},\mathbf {\theta } )-F(u_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}+(\delta _{i}+\gamma _{i}+\xi _{i}-2)ln{[f(t_{i}|\mathbf {x} _{i},\mathbf {\theta } )]}}}

The Hypertabastic Accelerated Failure Time model was used to analyze a total of 27,532 patients regarding the impact of histology on the survival of patients with cutaneous or mucosal melanoma. Understanding patients' histological subtypes and their failure rate assessment would enable clinicians and healthcare providers to perform individualized treatment, resulting in a lower risk of complication and higher survivability of patients.^[7]

The quantities of 49 locations of the same area of an oil field was examined to identify its underlying distribution. Using generalized chi-squared, the distribution of oil field quantities was represented by the Hyperbolastic distribution and compared with the lognormal (LN), log-logistic (LL), Birnbaum-Saunders (BS) and inverse Gaussian (IG) distributions.^[8]

The times of remission from clinical trial for acute leukemia of children study were used to analyze the remission duration of acute leukemia data for two groups of patients controlling for log of white blood cell counts. The Hypertabastic accelerated failure time model was used to analyze the remission duration of acute leukemia patient.^[8]

A randomized clinical trial comparing two chemotherapy regimens for 447 individuals with malignant glioma. A total of 293 patients died within a five-year time period and the median survival time was about 11 months. The overall model fit, in comparison with other parametric distributions, was performed using the generalized chi-square test statistics and proportional hazards model.^[8]

The Hypertabastic proportional hazard model was used to analyze numerous breast cancer data including the survival of breast cancer patients by exploring the role of a metastasis variable in combination with clinical and gene expression variables.^[9][10]

One hundred five Nigerian patients who were diagnosed with hypertension from January 2013 to July 2018 were included in this study, where death was the event of interest. Six parametric models such as; exponential, Weibull, lognormal, log-logistic, Gompertz and Hypertabastic distributions were fitted to the data using goodness of fit tests such as S.E., AIC, and BIC to determine the best fit model. The parametric models were considered because they are all lifetime distributions. S.E., AIC, and BIC measures were used to compare these parametric models.^[11]

Stress fractures in older individuals are very important due to the growing number of elderly. Fatigue tests on 23 female bone samples from three individuals were analyzed. Hypertabastic survival and hazard functions of the normalized stress level and age were developed using previously published bone fatigue stress data. The event of interest was the number of cycles until the bone gets microscopic fracture. Furthermore, Hypertabastic proportional hazard models were used to investigate tensile fatigue and cycle-to-fatigue for cortical bone data.^[12]

Hypertabastic survival models have been used in the analysis of unemployment data and its comparison with the cox regression model.^[13]

Using National Cancer Institute data from 1975 to 2016, the impact of histological subtypes on the survival probability of 134,150 kidney carcinoma patients were examined. The study variables were a race/ethnicity, age, sex, tumor grade, type of surgery, geographical location of patient and stage of disease. The Hypertabastic proportional hazards model was used to analyze the survival time of patients diagnosed with kidney carcinoma to explore the effect of histological subtypes on their survival probability and assess the relationship between the histological subtypes, tumor stage, tumor grade, and type of surgery. ^[14]

Sample code in SAS:

Proc nlp data=sasuser.KidneyCarcinoma tech=quanew cov=2 vardef=n pcov phes maxiter=250; 
  /* Hypertabastic Proportional Hazards Model with Log Time */ 
  title1 'Kidney Carcinoma';
  max logf; 
  
  /* Model Parameter Initial Values for Explanatory Variables */
  parms a=0.01,b=0.1, 
  c=0.01, /* Age */ /* Continuous */
  d=-.01, /* Male */ /* reference: Female */
  r1=.071, /* Hispanic */
  r2=.044, /* Asian */
  r3=.134, /* Black */ /* reference: White */
  h1=.205, /* Adeno Carcinoma w/ Mixed Subtypes */
  h2=.505, /* Papillary Adeno Carcinoma NOS */
  h3=.537, /* Clear Cell Adeno Carcinoma */
  h4=.316, /* Renal Cell Adeno Carcinoma */
  h5=1.15, /* Chromophobe Renal Cell Carcinoma */
  h6=-.21, /* Sarcomatoid Renal Cell Carcinoma */
  h7=.378, /* Granular Cell Carcinoma */ /* reference: Other */
  g1=.03, /* East */ 
  g2=.088, /* Northern Plains */ 
  g3=.06, /* Pacific Coast */ /* reference: Southwest */
  s1=1.2, /* Localized */
  s2=-1.3, /* Distant */ /* reference: Regional */
  gr1=1.169, /* Well Differentiated */
  gr2=.99, /* Moderately Differentiated */
  gr3=.413, /* Poorly Differentiated */ /* reference: Undifferentiated */
  su1=-.945, /* No Surgery */
  su2=.84, /* Cryocergery */
  su3=.56, /* Thermal Ablation */
  su4=.574, /* Cryosurgery */
  su5=1.173, /* Partial Nephrectomy or Partial Uretterectomy */
  su6=.25, /* Complete Nephrectomy */
  su7=.073, /* Radical Nephrectomy */
  su8=-.096, /* Any Nephrectomy */
  su9=.028; /* Nephrectomy, Urectomy */ /* reference: Other */
  
  /* Log-Likelihood Function */ 
  in6=exp(-(c*Age+
  d*Gender+
  r1*Race1+r2*Race2+r3*Race3+
  h1*Hist1+h2*Hist2+h3*Hist3+h4*Hist4+h5*Hist5+h6*Hist6+h7*Hist7+
  g1*Geo1+g2*Geo2+g3*Geo3+
  s1*Stage1+s2*Stage2+
  gr1*Grade1+gr2*Grade2+gr3*Grade3+
  su1*Surgery1+su2*Surgery2+su3*Surgery3+su4*Surgery4+su5*Surgery5+su6*Surgery6+su7*Surgery7+su8*Surgery8+su9*Surgery9)); /* covariates */ 
  
  s = log(1/cosh(a*(1-(time**b)/tanh(time**b))/b))*in6+Status*log(((a*time**(-1+2*b)/sinh(time**b)**2-
  a*time**(-1+b)/tanh(time**b))*tanh(a*(1-time**b/tanh(time**b))/b))*in6);
  
  logf=s;
run;

Although survival analysis tools and techniques have been widely used in medical and biomedical applications over the last few decades, their applications to engineering problems have been more sporadic and limited. The probabilistic assessment of service life of a wide variety of engineering systems, from small mechanical components to large bridge structures,^[15] can substantially benefit from the well-established survival analysis techniques. Modeling of time-to-event phenomena in engineering applications can be performed under the influence of numerical and categorical covariates using observational or test data. The "survival" of an engineering component or system is synonymous with the more commonly used term "reliability". The term "hazard rate" or "conditional failure rate" (defined as probability of survival per unit time assuming survival up to that time) is an important measure of the change in the rate of failure over time. In this context, failure is defined as reaching the target event in the time-to-event process. This could be defined as reaching a particular serviceability condition state, localized/partial structural failure, or global/catastrophic failure^[16] applied the Hypertabastic parametric accelerated failure time survival model to develop probabilistic models of bridge deck service life for Wisconsin. Bridge decks are typically concrete slabs on which traffic rides as seen in the Marquette Interchange bridge. The authors used the National Bridge Inventory (NBI) dataset to obtain the needed data for their study. NBI records include discrete numerical ratings for bridge decks (and other bridge components) as well as other basic information such as Average Daily Traffic (ADT) and deck surface area (obtained by multiplying the provided bridge length with bridge deck width). The numerical ratings range from 0 to 9 with 9 corresponding to brand new condition and 0 being complete failure. A deck condition rating of 5 was selected as the effective end of service life of bridge deck. The numerical covariates used were the ADT and deck surface area, while the categorical covariate was the superstructure material (structural steel or concrete).

The hypertabastic Proportional Hazards and Accelerated Failure Time models are useful techniques in analyzing bridge-related structures due to its flexibility of hazard curves, which can be monotonically increasing or decreasing with upward or downward concavity. It can also take the shape of a single mound curve.^[16][1][17] This flexibility in modeling various hazard shapes makes the model suitable for a wide variety of engineering problems.^[16]

Tabatabai et al. extended the Hypertabastic bridge deck models developed for Wisconsin bridges to bridges in six northern US states Nabizadeh, A. (2015). Reliability of Bridge Superstructures in Wisconsin. Master's Thesis (Thesis). UWM Digital Commons. and then to all 50 US states.^[18] The study of bridge decks in all 50 states indicated important differences in reliability of bridge decks in different states and regions. Stevens et al. ^[19] discuss the importance of survival analyses in identifying key bridge performance indicators and discuss the use of Hypertabastic survival models for bridges.^[20] and Nabizadeh et al.^[21] further extended the use of Hypertabastic survival models to bridge superstructures. The covariates used were ADT, maximum bridge span length and superstructure type. The survival function can be used to determine the expected life using the following equation (area under the entire survival curve)

${\displaystyle {EL}_{0}=\int _{0}^{\infty }S(t)dt}$

It is important to note that both the survival function and the expected life would change as the time passes by. The conditional survival function ${\displaystyle C_{S}}$ is a function of time ${\displaystyle t}$ and survival time ${\displaystyle t_{s}}$ and is defined as^[22]

${\displaystyle CS(t,t_{s})={\begin{cases}1&0\leq t\leq t_{s}\\{\frac {S(t)}{S(t_{s})}}&t>t_{s}\end{cases}}}$,

Nabizadeh et al.^[22] used the Hypertabastic survival functions developed for Wisconsin to analyze conditional survival functions and conditional expected service lives ${\displaystyle (EL_{c}(t_{s}))}$

${\displaystyle {EL}_{c}(t_{s})=\int _{0}^{\infty }CS(t)dt=t_{s}+\int _{t_{s}}^{\infty }{CS(t)dt=t_{s}+\int _{t_{s}}^{\infty }{\frac {S(t)}{S(t_{s})}}}dt}$

The conditional expected life would continue to increase as the survival time ${\displaystyle t_{s}}$ increases. Nabizadeh et al. term this additional expected life as "survival dividend."

An important mode of failure in bridge engineering is metal fatigue, which can result from repetitive applications of stress cycles to various details and connections in the structure. As the number of cycles ${\displaystyle (n_{c})}$ increase, the probability of fatigue failure increases. An important factor in fatigue life ${\displaystyle (N_{c})}$ is the stress range (Sr)(maximum minus minimum stress in a cycle). The probabilistic engineering fatigue problem can be treated as a "time"-to-event survival analysis problem if the number of cycles ${\displaystyle (n_{c})}$ is treated as a fictitious time variable ${\displaystyle (t)}$^[23]

This would facilitate the application of well-established survival analysis techniques to engineering fatigue problems^[23] and Tabatabai et al.^[24] The survival function ${\displaystyle S(n_{c})}$, probability density function ${\displaystyle f(n_{c})}$, hazard rate ${\displaystyle h(n_{c})}$, and cumulative probability of failure ${\displaystyle F(n_{c})}$ can then be defined as

${\displaystyle S(n_{c})=P(N_{c}>n_{c})=1-F(n_{c})}$

${\displaystyle f(n_{c})=\lim _{\delta n_{c}\to 0}{\frac {P(n_{c}<N_{c}<n_{c}+\delta n_{c})}{\delta n_{c}}}}$

${\displaystyle h(n_{c})=\lim _{\delta n_{c}\to 0}{\frac {P(n_{c}<N_{c}<n_{c}+\delta n_{c}|N_{c}>n_{c})}{\delta n_{c}}}}$

The hypertabastic accelerated failure time model was used to analyze probabilistic fatigue life for various detailed categories in steel bridges.^[23]