Modelling the dropout patterns of MOOC learners

We conduct a survival analysis for the viewing durations of massive open online courses. The hazard function of the empirical duration data presents as a bathtub curve with the Lindy effect in its tail. To understand the evolutionary mechanisms underlying these features, we categorize learners into two classes based on their different distributions of viewing durations, namely log normal distribution and power law with exponential cutoff. Two random differential equations are provided to describe the growth patterns of viewing durations for the two classes respectively. The expected duration change rate of the learners featured by log normal distribution is supposed to be dependent on their past duration, and that of the remainder of learners is supposed to be inversely proportional to time. Solutions to the equations predict the features of viewing duration distributions, and those of the hazard function. The equations also reveal the features of memory and memorylessness for the respective viewing behaviors of the two classes.