for all t in the range, where dimeans the number of events at time ti.
The Product-limit estimator is a right continuous step function with jumps at the observed event times and it provides an efficient means of estimating the survival function for right-censored data. An alternative non-parametric estimator is suggested by Nelson (1972). It has better small-sample-size performance estimator based on the Product-limit estimator.
2.2 Cox’s Proportional Hazard Regression Model
The Cox-PH regression model [Cox DR, 1972] is widely used in epidemiological research to examine the association between an exposure and a health outcome. In a typical approach to the analysis of epidemiological data with a continuous exposure variable, the exposure is transformed to an ordinal or nominal variable and relative risk (RR) is modeled as a step function of the exposure. The Cox-PH model is used to analyze censored data. Suppose the observed data are the triples (ti, zi, ci) where ti is the possibly censored survival time, zithe scalar predictor variable, and ci the event indicator, taking values of 1 if the event of interest occurred and 0 if it did not. Survival analysis model that has fully parametric regression structure, but leaves its dependence on time unspecified is known as semi-parametric regression model. Cox PH model is a semi-parametric regression model. Then, the Cox’s PH model takes the form:
where h0(t) reflects how hazard function changes with survival time, and Z(t)′β characterizes how hazard function changes with covariates. Cox has proposed exponential function the function of covariate x, and parameter β, say it f(x, β), and the hazard function formulated as:
[Due to technical limitations, this equation is only available as a download in the supplemental files section.]
when x changes from x 0 to x 1, the hazard ratio is computed as
[Due to technical limitations, this equation is only available as a download in the supplemental files section.]
this model in literature is termed as Cox proportional hazard model Cox DR [6]. In this model researchers are mainly interested on the parameter β, interpreted as changing rate of hazard when the covariate changed by unit value of (x1 − x0). Remembering that the given baseline hazard function h0(t) remains known, because of this is that the model is called semi-parametric model [11].
In cox proportional hazard regression model there are basic concepts we have to know, these are: (i) the baseline hazard λ0(t) depends on t but not on the covariates x1, x2, …, xn; (ii) the hazard ratio exp(βX), depends on the covariates x1, x2, …, xnbut not on time t; and also, (iii) the covariates x1, x2, …, xndo not depend on the time t. These concepts are assumptions of Cox PH model. To check these Cox PH assumptions we can develop diagnostic plots and formal tests. We have been developed diagnostic plots of selected graphical representations and used formal tests to test time-varying coefficients using proportionality test by including a covariate and time interaction terms (time-dependent covariates) associated with the graphical ones.
Allison et.al 1995) and Hosmer, Lemeshow (1999) presented two different things about the PH model assumptions. Allison et.al 1995) was presented that violation of PH assumptions is not that much serious. On the contrary, Hosmer, Lemeshow (1999) put their ideas as violation of PH assumptions should be taken into account and appropriate modification of the model should be used for the concise interpretation of the model and covariates. Partial likelihood used for the cox model allows using time-dependent covariates. A covariate is time-dependent mean that if the covariate value is change over-time for an individual.
2.3 Accelerated Failure Time (AFT) Model
The AFT model is an alternative model when the proportional hazards assumption does not fulfill. For the past twenty years the Cox proportional hazards model has been used widely to study the covariate effects on the hazard function for the failure time variable. On the other hand, the AFT model, which simply regresses the logarithm of the survival time over the covariates, has rarely been utilized in the analysis of censored survival data. The AFT model has an intuitive physical interpretation and would be an alternative method to the Cox model in survival analysis.
The AFT model treats the logarithm of survival time as the response variable and includes an error term that is assumed to follow a particular some distribution. Equation below shows the log-linear representation of the AFT model for the ithindividual, where log(Ti) is the log-transformed survival time, Tisurvival-time, X1, X2, …, Xpare explanatory variables with coefficients β 1, β 2, …, βp, respectively, and ϵi represents residual or unexplained variation in the log-transformed survival times, while β0 and σ are intercept and scale parameters, respectively [5]. The main reason we use logarithm of Tiis that to consider the truth behind the survival times are always positive with probability of 1.
[Due to technical limitations, this equation is only available as a download in the supplemental files section.]
In the absence of censored data, we can estimate the data by using ordinary least square (OLS) estimators. By assigning a new variable, Y = log (T), using the linear regression model with Y as the response variable. But, survival data have at least some censored observations and these are difficult to manage in OLS. Instead we have to use the MLE with different assumptions on ϵ. For each the distribution of ϵ, there should be a corresponding distribution of survival time, T.
In the presence of only a single explanatory variable, X1was defined as a 0–1 indicator variable, and the deceleration factor (cˆ) was calculated from the coefficient estimate associated with X1 (i.e, cˆ = exp(βˆ1)). When there are many covariates were available, additional terms β 2x 2 … βpxpwere included in the model, where the added variables represented factors such as gender, age, marital status, weight, WHO Stage, functional status, regimen class.