A Simulation Study on Comparing General Class of Semiparametric Transformation Models for Survival Outcome with Time-Varying Coefficients and Covariates

The consideration of the time-varying covariate and time-varying coefficient effect in survival models are plausible and robust techniques. Such kind of analysis can be carried out with a general class of semiparametric transformation models. The aim of this article is to develop modified estimating equations under semiparametric transformation models of survival time with time-varying coefficient effect and time-varying continuous covariates. For this, it is important to organize the data in a counting process style and transform the time with standard transformation classes which shall be applied in this article. In the situation when the effect of coefficient and covariates change over time, the widely used maximum likelihood estimation method becomes more complex and burdensome in estimating consistent estimates. To overcome this problem, alternatively, the modified estimating equations were applied to estimate the unknown parameters and unspecified monotone transformation functions. The estimating equations were modified to incorporate the time-varying effect in both coefficient and covariates. The performance of the proposed methods is tested through a simulation study. To sum up the study, the effect of possibly time-varying covariates and time-varying coefficients was evaluated in some special cases of semiparametric transformation models. Finally, the results have shown that the role of the time-varying covariate in the semiparametric transformation models was plausible and credible. How to cite this paper: Fissuh, Y.H., Woldu, T.G., Ahmed, I.A.I. and Kebebe, A.Z. (2019) A Simulation Study on Comparing General Class of Semiparametric Transformation Models for Survival Outcome with Time-Varying Coefficients and Covariates. Open Journal of Statistics, 9, 169-180. https://doi.org/10.4236/ojs.2019.92013 Received: February 11, 2019 Accepted: March 30, 2019 Published: April 2, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
In many experimental and observational studies such as randomized clinical trials, agricultural experiments, and engineering and industrial production commonly we obtain time-to-end outcomes so-called survival time or failure time.
In biomedical researches, the main concern is usually on the survival time, which is a time from defined origin until the defined endpoint or outcome [1].
The survival data have missing value raised through the censoring mechanisms.
Censoring is the problem of not finding the exact time of an event during the experimental or observational studies, which makes the analysis much more complex.
Central to the entire discipline of survival analysis, mostly right censoring exists.Besides, a time-varying covariate is a classical problem in modeling survival time.The semiparametric transformation models which have been attracted by several authors have been an important concept in the study of right censored survival time.The another important concept in analysing survival data is proportionality assumption.Sometimes, in our experimental study, we have no warrantee of the fulfillment of this assumption.Because the effect of covariate may vary over time breaking the proportionality assumption for Cox proportional hazards model of [2].In this situation, we need to consider the time-varying coefficient to our model.Due to this, the time-dependent effect and time-dependent covariates have been given attentions these days.Generally, someone may need to extend this model to more general model that can incorporate both time-varying covariate and time-varying effect.Thus the combination brings more general version.
A key role of semiparametric transformation models (STM) is that the model provides a framework for deriving the effect of time-varying covariates and the effect of time-varying coefficients on failure time.In this model, since the model consists of different special cases inside, the failure of proportionality assumption might not be much problem.
The remaining part of this paper is organized as follows.Section 2 introduces the methods and model framework which are going to be used in the whole paper and proposes a modified estimating equation for robust semiparametric transformation models.Section 3 presents a large sample theory and regularity conditions for the consistency and asymptotic properties of the proposed estimators.Section 4 devotes simulation studies to check the performance of the proposed techniques.Finally, the conclusion is presented in Section 5.

Methods and Model Framework
Here we start with some basic notations that are used throughout this paper.Where the covariate is allowed to vary over time, possibly the furthermost instant tactic is to use the step-function as follows.
( ) where r T is transaction time for change.
Whenever the covariate only changes once at fixed time point and do not change after that, the step function is used.However, in some situations it is common to have covariate that change over time continuously and frequently at a time with the only requirement that the intervals of the observation need not be contiguous.Therefore, in this situation a simple way to code time-dependent covariates is using intervals of time and recorded in to two columns as the start, stop or time 1, time 2, entry, end and so forth.The "tmerge" package in R can do this arrangement in the survival library.
When the censoring time is denoted by C, the failure or censor time represented by Y  is the minimum of failure time of censoring time and failure time; i.e.; ( ) . We write ( ) for the event indicator.Finally, the summarized n independent random vectors of observations are formulated as ( ) { } , , .

The Semiparametric Transformation Models
The flexibility extended general class of semiparametric transformation models with the effect of time-varying coefficients is formulated where X is a set of covariates, the set of time-varying regression coefficients or parameters ( ) ( ) However, the model (3) does not applicable for time-varying covariate.Then, with the extension of time-varying covariates, the special cases of the transformation models consider proportional hazards (PH) model and proportional Open Journal of Statistics odds (PO) model.These special models are based on the given distribution of random error term ε corresponds to extreme value distribution and the stan- dard logistic distribution respectively [4] [5] [6].

Let
( ) i t  be the counting process recording the number of events that have occurred by time t and let ( ) X t be a set of predictors which contains a vector of possibly time-varying covariates.We specify that the cumulative intensity function for and therefore, equivalent formulation of model ( 3) can be expressed as where and the class of logarithmic transformation ( ) ( ) Therefore, the choice of ( ) Remark: Specifying the function Φ while leaving the function 0 Λ unspeci- fied is equivalent to specifying the distribution of ε while leaving the function  unspecified.Non-identifiability arises if both Φ and 0 Λ (or both  and ε ) are unspecified and 0 ϑ = ([3], p. 169) which was quoted by [8].

The Modified Estimating Equations
Before developing estimating equations, let us impose on the following two unignorable assumptions., estimating equations of [5] which has been lately used by several authors for example [9] [10] [11] and [12] is modified for the effect of time-varying coeffcients and time-varying covariates.
In this paper we suppose,  for the r failure times among the n observations.Furthermore, we suppose ( ) , are an at-risk indicator process and the distinct ordered uncensored failure times Thus, the martingale decomposition can minimize the complexity of the estimation of equations by constructing the following easily tractable formula.
for complete σ-field  since where  ( ) Thus, slightly modified estimating equations of [5] are proposed by making possibly time-varying covariate under consideration.The two modified estimating equations are where is the intensity function for . Therefore, this requirement in turn ensures that for any finite number k, For the special case when we assume the Cox's proportional hazards model of [5] in which ( ) ( ) ( ) therefore, by plugging this in (12) we simply obtain Someone may use computationally easiest alternative versions of ( 12) which were first mentioned by [5] and lately by [11].
Finally, the survival function of T given possibly time-varying covariates ( ) can easily be derived from the model (5) as follows.
( ) Therefore, the cumulative hazard function is given by ( . Thus, the true induced intensity (hazard) function for failure time T given possibly time-varying covariates ( ) X t is the derivative of the true cumulative intensity function of Equation (18) which is defined as therefore, to ease the notations without lose of truth, here we propose some representations where where ( ) ( ) Now, we set a zero-mean martingale process with respective filtration  of complete Thus, by imposing at Lemma 1, we modify the estimating Equation ( 12) and Equation ( 13) as

Large Sample Theory and Conditions
Some regularity conditions are necessarily imposed here.Theorem 1: Under some suitable regularity conditions C1-C6 in order to ensure CLT for counting process martingale holds, ( ) Thus, similar to [5] [9] [12] and others, the asymptotic variance of estimator for any fixed ( ] . The following theo-Open Journal of Statistics function.Finally, semiparametric transformation models are applied for the simulated data.The different models were compared based on their performance in precision.

Computational Algorithm
Since we have more than one unknown items to be estimated, it is necessary to apply some sophisticated iterative algorithms to handle the iteration problem.
Thus, in this paper expectation-maximization (EM) algorithm is proposed to estimate unknown true parameter ( ) 0 t ϑ and nondecreasing monotone function In this concept, it is necessary to fix one of them and estimate the another one and in terms of the fixed one and vice versa.Therefore, as it was done in [5], it is not difficult to show the unique solution of ( 12), ( 13) in H, for every fixed value of ( ) . Consequently, Equation (3) and Equation ( 5) logically suggest the following iterative algorithms for computing ( ; Step 0: Opt an initial value of ϑ , denoted by ( ) 0 ϑ .
Step 1: For each k t , obtain ( ) ( ) by solving Equation (12) and Equation ( 13) by setting  , one-by-one by solving the equation Step 2: Then obtain new estimate of ( )

Numerical Results
This subsection explores the numerical results based on simulation studies through figures and numerical analysis.This numerical result is expected to evaluate the performance of the proposed model.with small standard errors have high precision.In these simulations, the effect of time-varying coefficient did not improve the model performance.However, the effect of time-varying covariates did improve the performance of the model.

Conclusions
The study is basically concerned on comparisons of the semiparametric transformation models with and without the effect of the time on covariates and coefficients.The summary review of other works was done and the result of simulation was included to come up with reasonable review of the study.The data were generated in four different cases under the "sim.survdata()"function of R package called "coxed".Then the results of semiparametric transformation models for four types of simulation studied were compared based.Three special cases of semiparametric transformation models such as PH, PO and model when r = 0.5 were considered.
The results have shown that the semiparametric transformation models with time-dependent covariates did relatively better perform with small standard errors.
However, the effect of time-varying coefficient did not improve the performance of the semiparametric transformation models in our simulation studies.

Y
. H. Fissuh et al.DOI: 10.4236/ojs.2019.92013171 Open Journal of Statistics time-varying covariates, where p is number covariates included in the model.
)log T is a natural logarithm or logarithm of base e 2.71828 =  and the unspecified continuously differentiable monotone arbitrary transformation function

(
. Here the independent identically distributed (i.i.d) random variable  with known distribution is unobservable positive noise associated to random biological features.For strictly increasing transformation function ( ).Φ , the class of Box-Cox transformations which was recently used by[7] is also considered here.For the two special cases of transformation model classes namely Proportional hazards (PH) and proportional odds (PO) models, we reflect on the Box-Cox transformation functions the special case of transformation model indeed yields PH model for survival data.Equivalently, the choice of 1 r = , the special case of transformation model indeed yields PO model for survival data.
hazard and cumulative hazard functions of ε , respectively.Let us propose the true values of the usual counting process notation, let the mean of a martingale process with respect to  is zero.Lemma 1: The mean of the derivative of regular martingale process is zero.( )

C1:
The covariate vectors are bounded in the sense that() the possibly time-varying covariate ( )X t has a uniformly bounded variation on [ ] 0,  and its left limit exists with any t where  is the maximum follow-up time.C2: The true value of ( ) superscript dot always refers derivatives.C6:Both the variance covariance matrices Ψ and Σ  are nonsingular.

Figure 1
Figure 1 illustrates about the baseline characteristics of survival data.The top panel of the figure refers the feature of probability density function, cumulative distribution function, hazard function, and cumulative hazard function of failure time.The bottom panel shows the feature of simulated duration in terms of histogram of failure time or duration, linear predictor and exponentiated linear predictors respectively.The left panel of Figure 1 is when the survival data are

Figure 1 .
Figure 1.Plots of baseline feature of simulated survival data.(a) Plot with 25% censoring rate; (b) Plot with 45% censoring rate.

Table 1 .
Estimates of Regression Coefficients with their respective standard errors in the brackets for Semiparametric Transformation models for n = 200.TCV and TVbeta refers time-varying covariates and time-varying coefficients.cases such as the semiparametric transformation models with time-varying covariates and both time-varying covariates and time-varying coefficients have shown better performance.Therefore, we can give the general conclusion that when the proportionality assumption fails to fulfill, incorporating the time-varying coefficient effect in the model is advisable.Considering only baseline covariate may not be always true; because there is the time when the covariate changes throughout the time.Therefore, incorporating time-varying covariate in the model may help us to get reasonable results.Sometimes it can be happened that both covariate and coefficient effect changes over time.Thus, incorporating both time-varying covariates and time-varying coefficients shall give us more reasonable results.