Background: 
Gastric cancer is a malignant tumor of the stomach and it can develop in any part of the stomach. It is also called stomach cancer [22]. Gastric cancer is one of the leading causes of death in the world and represents a tremendous burden on patients, families, and societies [25]. Based on Global burden of cancer 2018 data, GC is the 5th most common neoplasm and the 3rd most deadly cancer, with an estimated 783,000 deaths in 2018. Over a million new cases of GC are diagnosed, worldwide, each year [2]. Despite the universal decline in GC incidence and mortality, it is still the second most common cancer worldwide [14].   

In this study, the event of interest was the time to first recurrence of GC after treatment. PH model popularized by Cox is the classical model for this kind of data. However, the correct inference based on Cox's models needs identically and independently distributed samples. Often, subjects may be exposed to different risk levels, even after controlling for known risk factors. This is because the covariates that are relevant to the researcher are often unavailable or even unknown. In current study, shared frailty models explored assuming that patients within the same cluster (region) shares similar risk factors, which would take care of the frailty term at region level. The study aimed to model time to first recurrence of gastric cancer patients in TASH.

This thesis considered parametric frailty models to investigate the relationship between different potential covariates (demographic, clinical and environmental factors) and time to recurrence of GC for clustered survival data with random right censoring. The choice of distribution for the hazard is very important than the choice of frailty distribution [6]. The advantage of parametric method over the semiparametric method shows that having distribution may calculate the quantiles, simplicity and completeness are reasons for the popularity of parametric distributions [13]. Hence, in this study Weibull, log-logistic and lognormal baseline hazard functions used. On the other hand, among frailty distribution we have assumed gamma and inverse Gaussian distributions to fit GC data set. Gamma and inverse Gaussian are the two most common choices of frailty distributions due to their mathematical tractability. For comparison of different models, the AIC criteria used

Methods:

Study setting and design:

A retrospective study was conducted on GC patients under follow- up at Tikur Anbesa Specialized Hospital, Oncology center, Addis Ababa, from 1 January 2015 through 31 December 2019. The total number of patients considered in the study was 409 who were patient from all nine regions and two city administrations of Ethiopia. Regions that contribute single patients were omitted. Therefore, a total of 407 GC patients were considered in this study. For analysis of the data, R statistical software has been used.

Variables in the Study:

The response variable is time to first recurrence of GC from registry time to study ends. The explanatory variables considered in this study were: Age (in years),  Gender of patients, Residence, Marital status, Smoking history, Helicobacter pylori infection, Family history, Obesity status, Tumor location, Stage of GC, Initial Treatment,  Vascular invasion, Tumor size and Histology type. These were categorized as follows:

Age were categorized as (≤49,  50-69 and ≥70), Gender(Male and Female), Residence(Urban,  married, divorced and widowed), Smoking history(No, yes), Vascular invasion(Absent, Present), Obesity (Normal, Underweight, Overweight), Family history(Negative, Positive), Tumor location(Non-gastro intestinal and Gastro intestinal), Stage(I, II, III and IV), Treatment taken (Surgery alone, Chemotherapy, Radiotherapy and Combination of ≥2), Helicobacter pylori infection(Absent and Present), Tumor size(<5cm and ≥5cm) and Histology type(Well-differentiated tumors, Poorly differentiated tumors and Signet ring cell cancer). 

Shared frailty models:

A shared frailty model is a random effects model where the frailties are common (or shared) among groups of individuals or spells and are randomly distributed across groups. They are conditional independence model in which frailty is common to all subjects in a cluster. It is also known as a mixture model because the frailties in each cluster are assumed to be random [8].

Conditional on the random term, called the frailty denoted by wi, the survival times in cluster i (1≤i≤ n) are assumed to be independent and an accelerated failure time frailty model which assumes:

hij(t/Xij,wi)=ho(ϕt)exp(β'Xij+wi)

Where