Survival Analysis Based Framework For Early Prediction Of .

els because of neglecting available information while, on theother hand, treating censoring time as the actual event timecauses underestimation of the model. Another importantthing to point out is that, unlike machine learning and datamining techniques, which normally provide single outcomeprediction, survival analysis estimates the survival (failure)as a function of time. In survival analysis, subjects are usually followed over a specified time period and the focus is onthe time at which the event of interest occurs [25].In spite of the success of survival analysis methods in otherdomains such as healthcare, engineering, etc., there is only alimited attempt of using these methods in student retentionproblem [32, 19]. In this paper, we propose a survival analysis based framework that uses pre-enrollment and semesterwise information to address the problem of student attritionin the presence of censored information. For this purpose,we implement Cox and time-dependent Cox, (TD-Cox ) tomodel the student retention problem. The fundamental ideais that we can utilize the survival analysis method at an earlystage of college study to predict student dropouts. Thus, themain contributions of this paper are summarized as follows: Rigorously define the student attrition problem and createimportant variables that influence this problem. Propose a novel student retention prediction frameworkto simultaneously deal with both problems, namely, “whois going to dropout” and “when the dropout will occur”. Using survival analysis methodology to study the temporal nature of student retention by focusing on dropoutinformation as the outcome of interest. Demonstrate the performance of the proposed method using Wayne State University student data and comparewith the existing state-of-the-art methods.The rest of the paper is organized as follows. In Section 2,we describe the literature relevant to the student retentionproblem. After defining the notations and definitions thatwill be used throughout the paper, we describe the proposedCox and TD-Cox methods for student dropout prediction inSection 3. In Section 4, we first describe the data sourcesthat will be used in this study and then show the performance of our method on Wayne State University studentdata. Finally, Section 5 concludes the work and offers directions for future research.2.RELATED WORKStudent retention is one of the most widely studied areasin higher education [39]. There are many institutions, consulting firms and businesses focusing on student retention.In the past decades, comprehensive models have been developed to address the college student attrition problem. Mostof the earlier studies try to understand the reasons behindstudent dropout by developing theoretical models [38]. Formany years, statistical methods have been used widely topredict student dropout and also to find the important factors that has an effect on the problem [44, 22]. Regression isone of the primary techniques that has been applied in thisarea [11]. Logistic regression is another statistical methodthat was frequently used in this domain [26, 9]. [27] usedlogistic regression, discriminant analysis and regression treeto address this issue. In another work, logistic regressionmethod is developed to identify freshman at risk of attritionwithin few weeks after freshman orientation [16]. However,these models cannot incorporate information from censoredinformation and are likely to produce suboptimal results.While predictive analytics has been used in other industries for many years [13], higher education is a relativelylate adopter of these approaches as a tool to support makingdecision [40]. Recently, researchers in the area of machinelearning and data mining, tried to address the student retention phenomenon [10, 42, 34]. Genetic algorithms for selecting feature subset and artificial neural networks for performance modeling have been developed to give better prediction of first year high risk students to dropout at VirginiaCommonwealth University [1]. Several classification algorithms including Bayes classifier [30, 3],decision tree [18, 31,43], boosting methods and support vector machines [45, 23]have been developed to predict student attrition with higheraccuracy compared to the traditional statistical methods.A slightly more complex relevant modeling technique issurvival analysis [14]. Survival analysis is a subfield of statistics which aims at modeling longitudinal data where theoutcome variable is the time until an occurrence of event[24]. In this type of regression, both components, (i) if anevent (i.e. dropout) occurs or not and (ii) when the eventwill occur, can be incorporated [28]. Thus, the benefit ofusing survival analysis over logistic regression or other datamining methods is the ability to add the time componentinto the model and also effectively handle censored data.However, the literature in this area is limited. The use ofsurvival analysis modeling to study both student retentionand student dropout has been developed in [32, 21, 19, 20].Among those, only [19] developed an event history modelto assessing attrition behaviour among first-generation students using pre-enrollment attributes. They assume thattime to dropout follows exponential distribution. However,this assumption may not be valid in many situations wheretime to event has more complex distribution [5].Despite the fact that survival model have more flexibilityto handle the student retention problem, there were few efforts in the literature in this student education domain. Itis evident that there is considerable room for improvementin the current state-of-the-art. In this paper, we relax someof the previous assumptions including linear dropout rateof student by implementing a more rigorous survival modelsuch as Cox proportional hazard model and also utilize timevarying features such as semester-wise GPA in more comprehensive manner by developing time-dependent Cox model.Therefore, this paper will further improve the existing ability to predict student success by showing an in-depth application of survival algorithms on student data and comparethe result with other statistical and machine learning approaches which, to the best of our knowledge, has not beendone anywhere else in the past.3.PROPOSED METHODThe primary goal of this work is to develop a time-dependentmodel to predict student dropout based on both pre-enrollmentand semester-wise information. We also build a survivalanalysis framework to estimate the semester of dropout onlybased on pre-enrollment attributes. We begin by presentingthe basic concepts and notations required to comprehendthis problem. Table 1 describes the notations used in thispaper.

Table 1: Notations used in this paperNotationDescriptionnpqXiZi (t)YCTδdiS0 (t)S(t X, Z(t))h0 (t)h(t X, Z(t))βL(β)number of data pointsnumber of static featuresnumber of time-dependent features1 p matrix of features for student i1 q matrix of time-dependent features for student in 1 vector of actual event timen 1 vector of last follow-up timen 1 vector of observed time which is min(Y, C)n 1 binary vector of censored statusnumber of events occurred at time tibaseline survival probabilityconditional survival probability at time tbase hazard rateconditional hazard probabilityvector of Cox regression coefficientsmaximum likelihood function for βWe will first define some of the terms that will be used inthis paper.Figure 1: An illustration to demonstrate the problem ofstudent retention. In this example, students A, B and Ddropped out after 4, 1 and 2 semesters, respectively. Students C and E did not drop out in their first 6 semestersand therefore they are censored.where f (u) is a probability density function and F (t) is acumulative distribution function. An alternative characterization of the distribution of T is given by the hazard function, or instantaneous rate of occurrence of the event whichis defined as Dropout Student: It is defined as a student who does notregister in a semester or whose semester GPA is zero. Event: Student dropout before his graduation is our eventof interest. Censored : If student does not dropout within the first 6semesters or by a cut-off timepoint, then it is defined ascensored data.3.1Survival AnalysisSurvival analysis is defined as a collection of statisticalmethods which contains time of a particular event of interest as the outcome variable to be estimated. In manysurvival applications, it is common to see that the observation period of interest is incomplete for some subjects andsuch data is considered to be censored [33]. Let Dn (t) {Xi , Zi (t), Ti , δi (t); i 1, ., n} denote a sample from datasetD at time t, where Xi represents a (1 p) covariate vectorfor subject i when there are p static variables in the data,Zi (t) represents (1 q) vector of time-dependent covariatesat time t and Ti denotes the observed event time. Let ussuppose that Yi is the survival time, but this may not beobserved and we instead observe Ti min(Yi , Ci ), whereCi is the censored time or the last follow-up time. We doknow if the data has been censored, and together with Yi wehave the indicator variable 1 Yi C iδi 0 Yi C iSo, for individual i, if δi 0, it is censored and if δi 1it is uncensored. Figure 1 illustrates the student retentionproblem using survival analysis in which students A, B andD drop out before the 6th semester and students C, E andF remain at school even at the end of the 6th semester or inother words they are censored at semester 6 (shown by ‘X’).Considering the duration to be a continuous random variable T , the survival function, S(t) is the probability thatthe time of event occurs later than a certain specified timet, which is defined as S(t) Pr(T t) tf (u) du 1 F (t)(1)h(t) limdt 0Pr(t T t dt)dt(2)In other words, h(t) is the event rate at time t conditionalon survival until time t or later. The numerator of thisexpression is the conditional probability that the event willoccur within the interval [t; t dt) given that it has notoccurred before t, and the denominator is the width of theinterval. Dividing one by the other, we obtain a rate of eventoccurrence per unit of time. Taking the limit, as the widthof the interval goes down to zero, we obtain an instantaneousrate of occurrence.3.2Cox Proportional Hazard RegressionOne of the popular methods in survival analysis is the Coxproportional hazard model [6]. The Cox regression model isa semi-parametric technique which has fewer assumptionsthan typical parametric methods [4]. In particular, and incontrast with parametric models, it makes no assumptionsabout the shape of the baseline hazard function [8]. TheCox model provides a useful and easy way to interpret information regarding the relationship of the hazard functionto predictors. The hazard function for the Cox proportionalhazard model has the formh(t X) h0 (t) exp(β1 X1 · · · βp Xp ) h0 (t)e(βX) (3)where h0 (t) eα(t) is the baseline hazard function at time tand exp(β1 X1 · · · βp Xp ) is the risk associated with the covariate values. Therefore, the survival probability functionfor Cox model can be formulated asS(t X) S0 (t)exp(βX)whereS0 (t) e t0h0 (x)dx(4)(5)Parameters of the Cox regression model are estimated bymaximizing the partial likelihood [7]. Based on Cox regression formula, a partial likelihood can be constructed fromthe dataset as follows: θ i(6)L(β) j:tj ti θji:δi 1

where θi exp(βXi ) and (X1 , ., Xn ) are the covariatevectors for the n independently sampled individuals in thedataset. By solving L(β) 0, the covariate coefficient can βbe estimated as β̂. To obtain the baseline hazard function,in full likelihood function, β should be replaced by β̂. Thus,h0 (ti ) can be obtained ĥ0 t(i) 3.31j R(t(i) )(7)θjTime-Dependent Cox (TD-Cox)The Cox proportional hazard regression has an assumption that covariates are independent of time. In anotherwords, when covariates do not change over time or when datais only collected for the covariates at one time point, it is appropriate to use static variables to explain the outcome. Onthe other hand, there are many situations (such as our student retention problem) where covariates change over timeand the above assumption does not hold. Thus, it is moreappropriate to use time-dependent covariates which result inmore accurate estimates of the outcomes [15]. Consequently,we can define time-dependent variables that can change invalue over the course of the observation period. Extensionsto time-dependent variables can be incorporated using thecounting process based formulation [2]. Essentially, in counting process, data are expanded from one record-per-studentto one record-per-interval between each event time for eachstudent. In order to have a better understanding of thecounting process, we provide an illustrative example. Table 2 shows the data in record-per-student format. In thisexample, for each student we record the time of dropout,status and semester-wise GPA. If status is 1 it means student dropout and if it is 0, it means student did not dropoutuntil the observed time.Table 2: Example of survival data (GPA(1) refers to GPAfor the first semester).Student IDID 1ID 2ID s, for time-dependent survival analysis, we need tochange the format using counting process. Using the firstpart of Algorithm 1, data is changed to record-per-intervalbetween each event time (Table 3) per student. Basically,we consider time interval by adding t0 column and for eachinterval, GPA is calculated independently. Other static variables such as demographic information which do not changeover different intervals for a given student can also be appended to the same row.Table 3: Example of survival data after counting processbased reformatting.Student IDID 1ID 2ID 2ID 3ID 3ID 3t0001012t112123Status101000GPA23.21.8443.5In this paper, we develop Time Dependent Cox regression, namely TD-Cox, which can simultaneously handleboth static and time-dependent covariates. Thus, the hazard function can be defined ash(t X, Z(t)) h0 (t)eβ(X Z(t))(8)Consequently, the survival probability function for TD-Coxmodel can be formulated asS(t X, Z(t)) S0 (t)exp(β(X Z(t)))(9)where S0 (t) can be estimated using Eq. (5). Algorithm 1summarizes the TD-Cox method. First, TD-Cox parameters are learnt using the training data based on maximumlikelihood function. Then, for

Survival Analysis based Framework for Early Prediction of Student Dropouts Sattar Ameri Wayne State University Detroit, MI - 48202 ameri@wayne.edu Mahtab J. Fard Wayne State University Detroit, MI - 48202 fard@wayne.edu Ratna B. Chinnam Wayne State University Detroit, MI - 48202 ratna.chinn