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\textbf{PROJECT SUPERVISOR }\\
INTRODUCTION. Cholera is an infection of the small intestine. It is caused by eating food or drinking water contaminated with a bacterium called Vibrio cholerae. It causes severe watery diarrhea and vomiting, which can lead to dehydration and even death if untreated. Every year, there are an estimated 3 to 5 million cholera cases and 100 000 to 120 000 deaths due to cholera. The short incubation period of two hours to five days, enhances the potentially explosive pattern of outbreaks (WHO, 2014). About 75\% of people infected with Vibrio. cholerae do not develop any symptoms, although the bacteria are present in their faeces for 7 to 14 days after infection and are shed back into the environment, potentially infecting other people. Among people who develop symptoms, 80\% have mild or moderate symptoms, while around 20\% develop acute watery diarrhea with severe dehydration. This can lead to death if untreated (WHO, 2014). Cholera was first identified in early 1800 in Asia. \\ The first pandemic occurred in the Bengal region of India starting in 1817 through 1824. The disease dispersed from India to Southeast Asia, China, Japan, the Middle East, and southern Russia. The disease is most common in places with poor sanitation, crowding, war, and famine. Common locations include parts of Africa, south Asia, and Latin America. \\ In this chapter, an overview of the cox Proportional hazard model would be given; a brief description of the problem statement of the thesis is also presented together with the objectives, the methodology, the justification and the organization of the thesis. Survival analysis is the modern name given to the collection of statistical procedures which accommodate time-to-event censored data. It is concerned with studying the time between entry to a study and a subsequent event (such as death). Survival Analysis typically focuses on time to event data. In the most general sense, it consists of techniques for positive valued random variable, such as \begin{itemize} \item Time to death We may be interested in characterizing the distribution of “time to event” for a given population as well as comparing this “time to event” among different groups ( e . g ., treatment vs. control in a clinical trial or an observational study), or modeling the relationship of “time to event” to other covariates (sometimes called prognostic factors or predictors) (). Typically, in biomedical applications the data are collected over a finite period of time and consequently the “time to event” may not be observed for all the individuals in our study population (sample). This results in what is called censored data. That is, the “time to event” for those individuals who have not experienced the event under study is censored (by the end of study).\\ It is also common that the amount of follow-up for the individuals in a sample vary from subject to subject. Survival analysis examines and models the time it takes for events to occur. The prototypical such event is death, from which the name ''survival analysis'' and much of its terminology derives, but the ambit of application of survival analysis is much broader. Essentially the same methods are employed in a variety of disciplines under various rubrics – for example, ‘event-history analyses in sociology. In this appendix, therefore, terms such as survival are to be understood generically.\\ Survival analysis focuses on the distribution of survival times. Although there are well known methods for The Cox model is based on a modeling approach to the analysis of survival data. The purpose of the model is to simultaneously explore the effects of several variables on survival. The Cox model is a well-recognized statistical technique for analyzing survival data. When it is used to analyses the survival of patients in a clinical trial, the model allows us to isolate the effects of treatment from the effects of other variables. The model can also be used, a priori, if it is known that there are other variables besides treatment that influence patient survival and these variables cannot be easily controlled in a clinical trial. Using the model may improve the estimate of treatment effect by narrowing the confidence interval. Survival times now often refer to the development of a particular symptom or to relapse after remission of a disease, as well as to the time to death
The cox-ph model in survival analysis will help us model this problem mathematically using some explanatory variables which are the factors that can affect the death of a patient. It will also help us access multiplicative effect of each variable on the hazard and to access the probability that an individual will experience an event (for example, death) within a small time interval, given that the individual has survived up to the beginning of the interval. The study would help to reduce further death of cholera patients should one be infected by identifying the main factor or variable that has the lowest multiplicative effect on the hazard of survival of a patient. And ways to reduce or manage such a variable. This will also help the ministry of health and the government for that matter to properly allocate resource that are to be used to address the issue, so that unnecessary allocations will not be made. And thereby reducing the expenditure of government on cholera. Due to limited funds and time constraints, this thesis focuses on three selected hospitals; Korle Bu Polyclinic, Mamprobi Polyclinic and Tema General Hospital all from Accra.\\ \newpage \begin{center} \textit{(King et al., 1979)}. Compared three diets’ abilities to keep the rats tumor-free. They were interested in the relationship between diet and the development of tumors and therefore divided 90 rats into three groups and fed them low-fat, saturated fat, and unsaturated fat diets, respectively. The rats were of the same age and species and were in similar physical condition. An identical amount of tumor cells were injected into a foot pad of each rat. The rats were observed for 200 days. Many developed a recognizable tumor early in the study period. Some were tumor-free at the end of the 200 days. Rat 16 in the low-fat group and rat 24 in the saturated group died accidentally after 140 days and 170 days, respectively, with no evidence of tumor. Fifteen of the 30 rats on the low-fat diet developed a tumor before the experiment was terminated. The rat that died had a tumor-free time of at least 140 days. The other 14 rats did not develop any tumor by the end of the experiment; their tumor-free times were at least 200 days. Among the 30 rats in the saturated fat diet group, 23 developed a tumor, one died tumor-free after 170 days, and six were tumor-free at the end of the experiment. All 30 rats in the unsaturated fat diet group developed tumors within 200 days. The two early deaths can be considered losses to follow-up. The data are singly censored if the two early deaths are excluded.\\ Stephen J Walters School of Health and Related Research (ScHARR), University of Sheffield ( 2009) carried out a cox ph analysis on the data from a randomized trial comparing the effect of low-dose adjuvant interferon alfa-2a therapy with that of no further treatment in patients with malignant melanoma at high risk of recurrence. Malignant melanoma is a serious type of skin cancer, characterized by uncontrolled growth of pigment cells called melanocytes. In his trial, 674 patients with a radically resected malignant melanoma (who were at high risk of disease recurrence) were randomly assigned to one of two treatment groups: interferon (3 megaunits of interferon alfa-2a three times a week until recurrence of cancer, or for two years – whichever occurred first) or no further treatment. His primary aim of this multicentre study was to determine the effects of interferon on overall survival. Patients were followed for up to eight years from randomization. The final Cox model included two demographic (age and gender) and one baseline clinical variable (histology) as independent prognostic factors, plus a treatment variable. \textit{Masaaki Tsujitani et al. (2012)} discussed a flexible method for modeling survival data using penalized smoothing splines when the values of covariates change for the duration of the study. The Cox proportional hazards model has been widely used for the analysis of treatment and prognostic effects with censored survival data. However, a number of theoretical problems with respect to the baseline survival function remain unsolved. We use the generalized additive models (GAMs) with B splines to estimate the survival function and select the optimum smoothing parameters based on a variant multifold cross-validation (CV) method. The methods are compared with the generalized cross-validation (GCV) method using data from a long-term study of patients with primary biliary cirrhosis (PBC).\\ In total, 54,519 people from the placebo clusters were assembled. The incidence of cholera (1.30/1000/year) was significantly higher than that of V. parahaemolyticus diarrhea (0.63/1000/year). Cholera incidence was inversely related to age, whereas the risk of V. parahaemolyticus diarrhea was age-independent. The seasonality of diarrhea due to the two Vibrio species was similar. Cholera was distinguished by a higher frequency of severe dehydration, and V. parahaemolyticus diarrhea was by abdominal pain. Hindus and those who live in household not using boiled or treated water were more likely to have V. parahaemolyticus diarrhea. Young age, low socioeconomic status, and living closer to a project healthcare facility were associated with an increased risk for cholera. The high risk area for cholera differed from the high risk area for V. parahaemolyticus diarrhea. They report coexistence of the two vibrios in the slums of Kolkata. The two etiologies of diarrhea had a similar seasonality but had distinguishing clinical features. The risk factors and the high risk areas for the two diseases differ from one another suggesting different modes of transmission of these two pathogens.\textit{ Kanungo et al. (2012)}.\\ \textit{Ali M et al (2012)} evaluated the herd protection conferred by an oral cholera vaccine using 2 approaches:cluster design and geographic information system (GIS) design. Residents living in 3933 dwellings (clusters) in Kolkata, India, were cluster-randomized to receive either cholera vaccine or oral placebo. Nonpregnant residents $ aged ≥1 $ year were invited to participate in the trial. Only the first episode of cholera detected for a subject between 14 and 1095 days after a second dose was considered. In the cluster design, indirect protection was assessed by comparing the incidence of cholera among onparticipants in vaccine clusters vs those in placebo clusters. In the GIS analysis, herd protection was assessed by evaluating association between vaccine coverage among the population residing within 250 m of the household and the occurrence of cholera in that population. Result s. Among 107 347 eligible residents, 66 990 received 2 doses of either cholera vaccine or placebo. In the cluster design, the 3-year data showed significant total protection $ (66\% protection, 95\% confidence interval [CI], 50\%–78\%, P < .01) $ but no evidence of indirect protection. With the GIS approach, the risk of cholera among placebo recipients was inversely related to neighborhood-level vaccine coverage, and the trend was highly signifi- A sample of 432 inmates released from Maryland state prisons was followed for one year after. The event of interest was the first rearrest. The aim was to determine how the occurrence and timing of arrests depended on several covariates (predictor variables). Some of these covariates (like race, age at release, and number of previous convictions) remained constant over the one-year interval. Others (like marital status and employment status) could change at any time during the follow-up period release. It was observed that fully 75 percent of the cases were not arrested during the first year after release which shows in particular that, someone who is jailed after an Efficacy and safety of a two-dose regimen of bivalent killed whole-cell oral cholera vaccine (Shantha Biotechnics, Hyderabad, India) to 3 years is established, but long-term efficacy is not. We aimed to assess protective efficacy up to 5 years in a slum area of Kolkata, India. In their double-blind, cluster-randomized, placebo-controlled trial, they assessed incidence of cholera in non-pregnant individuals older than 1 year residing in 3933 dwellings (clusters) in Kolkata, India. They randomly allocated participants, by dwelling, to receive two oral doses of modified killed bivalent whole-cell cholera vaccine or heat-killed Escherichia coli K12 placebo, 14 days apart. Randomization was done by use of a computer-generated sequence in blocks of four. The primary endpoint was prevention of episodes of culture-confirmed Vibrio cholerae O1 diarrhea severe enough for patients to seek treatment in a health-care facility. They identified culture-confirmed cholera cases among participants seeking treatment for diarrhea at a study clinic or government hospital between 14 days and 1825 days after receipt of the second dose. They assessed vaccine protection in a per-protocol population of participants who had completely ingested two doses of assigned study treatment. They observed that, 69 of 31 932 recipients of vaccine and 219 of 34 968 recipients of placebo developed cholera during 5 year follow-up (incidence 2•2 per 1000 in the vaccine group and 6•3 per 1000 in the placebo group). Cumulative protective efficacy of the vaccine at 5 years was $ 65\% (95\% CI 52–74; p<0•0001) $, and point estimates by year of follow-up suggested no evidence of decline in protective efficacy. Interpretation Sustained protection for 5 years at the level we reported has not been noted previously with other oral cholera vaccines. Established long-term efficacy of this vaccine could assist policy makers formulate rational vaccination strategies to reduce overall cholera burden in endemic settings. \textit{Dipika Sur et al. (2011)}\\ \textit{Tiago dos Santos Ferreira et al. (2012)} Infection increases the morbidity and mortality in liver cirrhosis patients. The aim of their study was to investigate the impact of infection related to survival and risk factors for death in adult patients with liver cirrhosis in a university hospital. Methods: In a retrospective cohort study of Brazilian hospitalized cirrhotic patients, medical records data were analysed, and all patients who have had one or more confirmed bacterial infection during admission were se-ected for the study. Also, some data as biochemical investigation, Child score, MELD estimation, and evolution and death event were included. Statistical analysis: chi-square, Fisher and Mann-Whitney tests were used. Uni and multivariate analysis were performed, according to Cox regression model. The significant statistical level was p 2.5 mg/dl had increased the risk of death of 4.1, 3.2 and 3.2, respectively. Conclusion: Bacterial infections in hospitalized irrhotic patients deserve special care, mainly spontaneous bacterial peritonitis, and also patients whose hiponatremia, upper gastrointestinal bleeding, high levels of cre-atinine and MELD high score are found.\\ \textit{Durham et al. (1998)} estimated the efficacy of killed whole-cell-only (WC) and B subunit killed whole-cell (BS-WC) oral cholera vaccines over 4 1/2 years of a vaccine trial in rural Matlab, Bangladesh . The placebo was a killed Escherichia coli strain. The trial was randomized and double-blinded among 89,596 subjects aged 2-1 5 years (male and female) and greater than 15 years (females only). They restrict our analyses to subjects that received three doses of vaccine or placebo (i.e., the full vaccination regimen) before May 1, 1985. There were 20,837, 20,743, and 20,705 such subjects in the placebo, WC, and BS-WC arms of the trial, respectively. The events of interest are reported, confirmed cases
since cox Ph is a form of regression (Semi- Parametric), we will first discus few things about Regression.\\
$ $cov(\varepsilon_{i},\varepsilon_{j})= 0\hspace{1cm} \mbox{for} \: i\ne j
\underline{\textbf{Multiple Regression}}\\ where:\\
Survival analysis is the modern name given to the collection of statistical procedures which accommodate time-to-event censored data. It is concerned with studying the time between entry to a study and a subsequent event (such as death). Survival Analysis typically focuses on time to event data. In the most general sense, it consists of techniques for positive valued random variable, such as \ • Time to death \underline{\textbf{The Hazard Function}}\\ The hazard function is the probability that an individual will experience an event (for example, death) within a small time interval, given that the individual has survived up to the beginning of the interval $ $ If we now integrate from 0 to t and introduce the boundary condition $S(0) =1$ (since the event is sure not to have occurred by duration 0), we can solve Discrete random Variables: $ \underline{\textbf{MODELING SURVIVAL DATA WITH SOME PARAMETRIC REGRESSION MODELS. }} \textbf{ THE EXPONENTIAL DISTRIBUTION}\\ $ f(t)=\lambda(e^{-\lambda t})$ for $t \geq 0$\\ $ S(t)= \int_{t}^{\infty}f(u)du $\\ $ S(t)=e^{-\lambda t} $\\ $\lambda(t)=\frac{f(t)}{S(t)} $ $\Lambda(t)= \int_{0}^{t} \lambda(u) du $\\ $\Lambda(t)= \int_{0}^{t} \lambda du$\\ $\Lambda(t) = \lambda t$\\
Let $ \lambda $ be the scale parameter\\ $ S(t)= e^{- \lambda t^{k}} $\\ $f(t)=\frac{-d}{dt}S(t)= k \lambda t^{k-1}e^{- \lambda t^{k}}$\\ $ \lambda (t) = k \lambda t^{k-1} $\\ $\Lambda (t)\int_{0}^{t} \lambda (u)du = \lambda t^{k} $\\ The weibull distribution is convenient because of its simple form. it includes several hazard shapes:\\ $k=1$ for constant hazard\\
The gamma distribution with parameters $\lambda $ and $k$, denoted $\Gamma (\lambda, k)$, has density\\ $f(t) =\frac{\lambda(\lambda t)^{k-1}e^{-\lambda t}}{\Gamma(k)} $\\ and survivor function \\ where $I_{k}(x)= \int_{0}^{x} \lambda^{k-1}e^{-x}dx/ \Gamma(k) $\\ there is no closed-form expression for the survival function, but there are excellent algorithms for its computation. thee is no explicit formula for the hazard either, but this may be computed easily as the ratio of the density to the survival function, $\lambda(t)= f(t)/S(t)$.\\ \underline{\textbf{NON-PARAMETRIC ESTIMATION}}\\ \textbf{KAPLAN- MEIER ESTIMATE.}\\ \textbf{Greenwood's formula}\\ If $ Now, instead of dealing with $ \hat{S}(t)$ directly, we will look at the log of it\\ Now, $\hat{S}(t)= exp[log[\hat{S}(t)]]$.\\ \textbf{hence the Greenwood's formula:}\\
A Cox proportional hazard model is a well-recognized statistical technique for exploring the relationship between the survival of a patient and several explanatory variables. A Cox model provides an estimate of the treatment effect on survival after adjustment for other explanatory variables. It allows us to estimate the hazard (or risk) of death, or other event of interest, for individuals, given their prognostic variables. \textbf{THE LIKELIHOOD}\\ Let $Yi$ denote the observed time (either censoring time or event time) for subject $i$, and let $Ci $ be the indicator that the time corresponds to an event (i.e. if $Ci = 1 $ the event occurred and if $ Ci = 0 $ the time is a censoring time). \\ The hazard function for the Cox proportional hazard model has the form: This expression gives the hazard at time t for an individual with covariate vector (explanatory variables) $X $. Based on this hazard function $$ \underline{\textbf{The partial likelihood}}\\ The hazards included in the denominator are only those individuals who are at risk at the ith event (or censoring) time. The entire likelihood function can be expressed very concisely as | |
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