h(t) = f(t) ÷ S(t) - The hazard function is the PDF divided by the survival function

hi I'm Eric hi the chemical statistician and today I'm going to provide a quick explanation of why the hazard function can be written as the probability density function divided by the survival function in an earlier video I define the hazard function in the context of survival analysis for events that are defined on a continuous time scale so if you haven't watched that menu yet or view if you don't know what the hazard function is I encourage you to watch that first now in that video I defined the hazard function as the limit as delta T goes to zero of the probability that the event X happens in some interval between T and T plus delta T conditions on the event that X is bigger than T all divided by delta T that is a big mouthful and my goal in this video is to show you that this complicated dynamic expression is exactly equal to that fraction down there let's start with that numerator that is a conditional probability so we can rewrite that and by definition it is equal to the probability of the intersection between the first and the second events divided by the probability of that second event so let's do that now I hope that you can recognize that those two events can be written on the number line here's what I mean here's the number line with the times T and T plus delta T and this event right here is the event X is bigger than T notice how I used an open circle to the note that X is strictly bigger than T now here is the event T is between T is less than X which is less than T plus delta T notice how use a closed circle to denote that X is less than or equal to T plus delta T now I hope that you can recognize that this event down here is a subset of this event if you are between T and T plus delta T if you're bigger than T for less than or equal to T plus delta T that you're definitely bigger than T so to illustrate this you can draw on that diagram for yourself so here's the event X is between T and T plus delta T I will draw an oval around that and in this rectangle we have given X is bigger than T so if the joint probability of these two events sorry if the this event is a subset of this event then this joint probability of these two events is simply the probability of this first event so we can simplify that numerator now this probability down here has a special name it's called the survival and it is usually denoted by SMT I hope that you can recognize that the survival function is simply 1 minus the cumulative distribution function this relationship will be very useful in a later video when I talk about the relationships between any combination of two of those three quantities but until next time for now let's simplify that numerator well let's elaborate on let's talk about that numerator even further that numerator can be written as the probability that X is less or equal to t plus delta t minus the probability that X is less or equal to T now hopefully you recognize that these two quantities in the numerator or simply the CDF evaluated at T plus delta T and T if you notice this whole French this whole expression right here excluding the survival function is quite simply the definition of the derivative of the CDF it is the limit as delta T goes to zero of the CDF evaluated at T plus Delta t minus the CDF NT all divided by delta T if you go back to the first or calculus on the definition of the derivative that is exactly the definition of the derivative applied to the CDF which of course is the probability density function or PDF and so that is why the hazard function can be written as the PDF divided by the survival function so I hope that that explanation was useful to you as always you can visit my blog the chemical statistician to get your daily lesson on statistics machine learning and chemistry on weekdays and once a week I will provide my very popular full length in-depth explanations of a statistical machine learning or chemical concept often with our programming examples or an advice column about professional development in science and statistics as well you can follow me on twitter at chemists at eric thank you for watching this video and I hope that you learn something useful today

Channel: Eric Cai

Views: 22,692

Rating: 5 out of 5

Keywords: hazard function, probability density function, PDF, survival function, survival analysis, statistics, math, biostatistics, probability

Id: o2DWODTs3xo

Length: 9min 2sec (542 seconds)

Published: Sat Mar 08 2014