# Hazard function, survival function, and retention rate

What is the difference between the hazard function, survival function, and retention rate in survival analysis?

• What does your book say about these things? – Dilip Sarwate Apr 26 '15 at 20:22
• I don't have a book. my professors are not fully explaining. is there a book you can recommend? – Amanda Christina Washington Apr 26 '15 at 23:47

Hazard Function/Hazard Rate,say, $\lambda_i$:- probability of life not survive in the $i^{th}$ time interval($t_i-t_{i-1}$).

Survival Function,$S(t_i)$:- probability of life survive upto $t_i$.

Retention Rate,$1-\lambda_i$:- probability of Life survive in $i_{th}$ time interval($t_i-t_{i-1}$).

Relations:-

Relation between hazard rate and retention rate remains same(Retention Rate=1- Hazard Rate)

But relation with Survival Function may differ, like in Kaplan-Meier method $S(t_i)=\prod_i (1-\lambda_i)$ and in Nelson-Aalen $S(t_i)=exp(-\sum_i\lambda_i)$

Using testing light bulbs on a daily basis for example. (You count how many fail a day)

Hazard rate ---- 'Failures'/'number of current good bulbs' by day.

Survival rate ---- Proportion of the ones the remain OK by day.

Retention rate ---- 1 - 'hazard rate'

Plot the rates v.s. time(days) you get the functions.