Hazard and cum hazard plotting http://itl. nist. gov/div898/handbook/apr/section2/apr222. htm 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(ti)vs the time ti 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 1. It is much easier to calculate plotting positions for multicensored data using cum hazard plotting techniques. 2. Linear graph paper can be used for exponential data and log-log paper can be used for Weibull data. Disadvantages of Cum Hazard Plotting 1. 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. 3. Normal cum hazard plotting techniques require exact times of failure and running times. 4. 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 1. 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). 2. 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. 3. Calculate the cumulative hazard values for each failed unit. The 1 van 4 17/12/2011 0:33 8. 2. 2. 2. Hazard and cum hazard plotting http://itl. nist. gov/div898/handbook/apr/section2/apr222. htm umulative 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. 4. 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 37 50 73 100 132 195 200 222 248 250 (2) (3) 1= failure Rank 0=runtime 1 0 1 0 1 1 0 1 1 0 1 2 3 4 5 6 7 8 9 10 (4) Reverse Rank 10 9 8 7 6 5 4 3 2 1 1/3 1/2 . 924 1. 424 1/6 1/5 . 391 . 591 1/8 . 225 (5) Haz Val (2) x 1/(4) 1/10 (6) Cum Hazard Value . 10 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 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 2 van 4 17/12/2011 0:33 8. 2. 2. 2. Hazard and cum hazard plotting http://itl. nist. gov/div898/handbook/apr/section2/apr222. htm 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 The equation of the least squares line fit through these points can be found from 3 van 4 17/12/2011 0:33 8. 2. 2. 2. Hazard and cum hazard plotting http://itl. nist. gov/div898/handbook/apr/section2/apr222. htm 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. 4 van 4 17/12/2011 0:33