8.2.2.2. Hazard and cum hazard plotting

8. Assessing Product Reliability
8.2. Assumptions/Prerequisites
8.2.2. How do you plot reliability data?

8.2.2.2. Hazard and cum hazard plotting

Another kind of plotting, called Cum Hazard Plotting, has the same purpose as probability plotting

Just commercial probability paper is available for most life distribution models for probability plotting of reliability data, there are also special Cum Hazard Plotting papers available for many life distribution models. These papers plot estimates for the Cum Hazard H(t_i)vs the time t_i of the i-th failure. As with probability plots, the plotting positions are calculated independently of the model or paper used and a reasonable straight-line fit to the points confirms that the chosen model and the data are consistent.

Advantages of Cum Hazard Plotting

It is much easier to calculate plotting positions for multicensored data using cum hazard plotting techniques.
Linear graph paper can be used for exponential data and log-log paper can be used for Weibull data.

Disadvantages of Cum Hazard Plotting

Commercial Cum Hazard paper may be difficult to find.
It is less intuitively clear just what is being plotted. Cum percent failed (i.e., probability plots) is meaningful and the resulting straight-line fit can be used to read off times when desired percents of the population will have failed. Percent cumulative hazard increases beyond 100% and is harder to interpret.
Normal cum hazard plotting techniques require exact times of failure and running times.
With computers to calculate the K-M estimates for probability plotting, the main advantage of cum hazard plotting goes away.

Since probability plots are generally more useful, we will only give a brief description of hazard plotting.

How to Make Cum Hazard Plots

Order the failure times and running times for each of the n units on test in ascending order from 1 to n. The order is called the rank of the unit. Calculate the reverse rank for each unit (reverse rank = n- rank +1).
Calculate a Hazard "value" for every failed unit (do this only for the failed units). The Hazard value for the failed unit with reverse rank k is just 1/k.
Calculate the cumulative hazard values for each failed unit. The cumulative hazard value corresponding to a particular failed unit is the sum of all the hazard values for failed units with ranks up to and including that failed unit.
Plot the time of fail vs the cumulative hazard value on special Cum Hazard paper (or construct your own paper as covered below for the exponential and the Weibull model).

A life test cum hazard plotting example

Example: Ten units were tested at high stress test for up to 250 hours. Six failures occurred at 37, 73, 132, 195, 222 and 248 hours. Four units were taken off test without failing at the following run times: 50, 100, 200 and 250 hours. Cum hazard values were computed in the following table:

(1) Time of Event	(2) 1= failure 0=runtime	(3) Rank	(4) Reverse Rank	(5) Haz Val (2) x 1/(4)	(6) Cum Hazard Value
37	1	1	10	1/10	.10
50	0	2	9
73	1	3	8	1/8	.225
100	0	4	7
132	1	5	6	1/6	.391
195	1	6	5	1/5	.591
200	0	7	4
222	1	8	3	1/3	.924
248	1	9	2	1/2	1.424
250	0	10	1

Next ignore the rows with no cum hazard value and plot column (1) vs column (6).

As with probability plotting, you can make your own special hazard plotting paper for many models

Exponential and Weibull "Homemade" Hazard Paper

The cum hazard for the exponential is just H(t) = t, which is linear in t with a 0 intercept. So a simple linear graph paper plot of y = col (6) vs x = col (1) should line up as approximately a straight line going through the origin with slope lambda if the exponential model is appropriate. The Dataplot commands and graph of this are shown below:

LET X = DATA 37 73 132 195 222 248
LET Y = DATA .1 .225 .391 .591 .924 1.424
PLOT Y X

The cum Hazard for the Weibull is , so a plot of y vs x on log log paper should resemble a straight line with slope , if the Weibull model is appropriate. The Dataplot commands and graph of this are shown below:

XLOG ON
YLOG ON
PLOT Y X
Plot of Weibull cumulative hazard

The equation of the least squares line fit through these points can be found from

LET YY = LOG10(Y)
LET XX = LOG10(X)
FIT Y X

The Weibull fit looks better, although the slope estimate is 1.27, which is not far from an exponential model slope of 1. Of course, with a sample of just 10, and only 6 failures, it is difficult to pick a model from the data alone.