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

8.2.2.1.

Probability plotting

Use probability plots to see your data and visually check model assumptions

Probability plots are simple visual ways of summarizing reliability data by plotting CDF estimates vs time on specially constructed probability paper. 

Commercial papers are available for all the typical life distribution models. One axis (some papers use the y-axis and others the x-axis, so you have to check carefully) is labeled "Time" and the other axis is labeled "Cum Percent" or "Percentile". There are rules, independent of the model or type of paper, for calculating plotting positions from the reliability data. These only depend on the type of censoring in the data and whether exact times of failure are recorded or only readout times. 

Plot each failure mode separately

Remember that different failure modes can and should be separated out and individually analyzed. When analyzing failure mode A, for example, treat failure times from failure modes B, C, etc., as censored run times. Then repeat for failure mode B, and so on. 

Data points line up roughly on a straight line when the model chosen is reasonable

When the points are plotted, the analyst fits a straight line through them (either by eye, or with the aid of a least squares fitting program). Every straight line on, say, Weibull paper uniquely corresponds to a particular Weibull life distribution model and the same is true for lognormal or exponential paper. If the points follow the line reasonably well, then the model is consistent with the data. If it was your previously chosen model, there is no reason to question the choice. Depending on the type of paper, there will be a simple way to find the parameter estimates that correspond to the fitted straight line.

Plotting positions on probability paper depend on the type of data censoring

Plotting Positions: Censored Data (Type I or Type II)

At the time t_i of the i-th failure, we need an estimate of the CDF (or the Cum. Population Percent Failure). The simplest and most obvious estimate is just 100 × i/n (with a total of n units on test). This, however, is generally an overestimate (i.e. biased). Various texts recommend corrections such as 100 × (i-.5)/n or 100 × i/(n+1). Here, we recommend what are known as (approximate) median rank estimates: 

Corresponding to the time t_i of the i-th failure, use a CDF or Percentile estimate of 100 × (i - .3)/(n + .4) 

Plotting Positions: Readout Data

Let the readout times be T₁, T₂, ..., T_k and let the corresponding new failures recorded at each readout be r₁, r₂, ..., r_k. Again, there are n units on test.

Corresponding to the readout time T_j, use a CDF or Percentile estimate of

Plotting Positions: Multicensored Data

The calculations are more complicated for multicensored data. K-M estimates (described in a preceding section) can be used to obtain plotting positions at every failure time. The more precise Modified K-M Estimates are recommended. They reduce to the Censored Type I or the Censored Type II median rank estimates when the data consist of only failures, without any removals except possibly at the end of the test. 

How Special Papers Work

It is not difficult to do probability plotting for many reliability models even without specially constructed graph paper

The general idea is to take the model CDF equation and write it in such a way that a function of F(t) is a linear equation of a function of t. This will be clear after a few examples. In the formulas that follow, "ln" always means "natural logarithm", while "log" always means "base 10 logarithm". 

a) Exponential Model: Take the exponential CDF and rewrite it as

If we let y = 1/{1 - F(t)} and x = t, then log (y) is linear in x with slope /ln10. This shows we can make our own special exponential probability paper by using standard semi log paper (with a logarithmic y-axis). Use the plotting position estimates for F(t_i) described above (without the 100 × multiplier) to calculate pairs of (x_i,y_i) points to plot. 

If the data are consistent with an exponential model, the resulting plot will have points that line up almost as a straight line going through the origin with slope /ln10. 

b) Weibull Model: Take the Weibull CDF and rewrite it as

If we let y = ln [1/{1-F(t)}] and x = t, then log (y) is linear in log(x) with slope . This shows we can make our own Weibull probability paper by using log log paper. Use the usual plotting position estimates for F(t_i) (without the 100 × multiplier) to calculate pairs of (x_i,y_i) points to plot. 

If the data are consistent with a Weibull model, the resulting plot will have points that line up roughly on a straight line with slope. This line will cross the log x-axis at time t = and the log y axis (i.e., the intercept) at -log.

c) Lognormal Model: Take the lognormal cdf and rewrite it as

with  denoting the inverse function for the standard normal distribution (taking a probability as an argument and returning the corresponding "z" value).

If we let y = t and x = {F(t)}, then log y is linear in x with slope /ln10 and intercept (when F(t) = .5) of log T₅₀. We can make our own lognormal probability paper by using semi log paper (with a logarithmic y-axis). Use the usual plotting position estimates for F(t_i) (without the 100 × multiplier) to calculate pairs of (x_i,y_i) points to plot. 

If the data are consistent with a lognormal model, the resulting plot will have points that line up roughly on a straight line with slope /ln10 and intercept T₅₀ on the y-axis.

d) Extreme Value Distribution (Type I - for minimum): Take the extreme value distribution CDF and rewrite it as 

If we let y = -ln(1 - F(x)), then ln y is linear in x with slope 1/ and intercept -µ /. We can use semi log paper (with a logarithmic y-axis) and plot y vs x. The points should follow a straight line with a slope of 1/ln10 and an intercept of -ln10. The ln 10 factors are needed because commercial log paper uses base 10 logarithms. 

DATAPLOT Example

A Dataplot Weibull example of probability plotting

Using the Dataplot commands to generate Weibull random failure times, we generate 20 Weibull failure times with a shape parameter of  g  = 1.5 and a  = 500. Assuming a test time of T = 500 hours, only 10 of these failure times would have been observed. They are, to the nearest hour: 54, 187, 216, 240, 244, 335, 361, 373, 375, and 386. First we will compute plotting position CDF estimates based on these failure times, and then a probability plot using the "make our own paper" method. 

( 1) Fail # = i	( 2)  Time of Fail  (x)	(3) F(ti) estimate  (i-.3)/20.4	(4) ln{1/(1-F(ti)}  (y)
1	54	.034	.035
2	187	.083	.087
3	216	.132	.142
4	240	.181	.200
5	244	.230	.262
6	335	.279	.328
7	361	.328	.398
8	373	.377	.474
9	375	.426	.556
10	386	.475	.645

Of course, with commercial Weibull paper we would plot pairs of points from column (2) and column (3). With ordinary log log paper we plot (2) vs (4).

The Dataplot sequence of commands and resulting plot follow: 

LET X = DATA 54 187 216 240 244 335 361 373 375 386 
LET Y = DATA .035 .087 .142 .2 .262 .328 .398 .474 .556 .645
XLOG ON
YLOG ON
XLABEL LOG TIME
YLABEL LOG LN (1/(1-F))
PLOT Y X

Note that the configuration of points appears to have some curvature. This is mostly due to the very first point on the plot (the earliest time of failure). The first few points on a probability plot have more variability than points in the central range and less attention should be paid to them when visually testing for "straightness". 

Use of least squares (regression) technique to fit a line through the points on probability paper

We could use Dataplot to fit a straight line through the points via the commands

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

This would give a slope estimate of 1.46, which is close to the 1.5 value used in the simulation.
The intercept is -4.114 and setting this equal to -log we estimate  = 657 (the "true" value used in the simulation was 500).

Dataplot has a special Weibull probability paper function for complete data samples (no censoring)

Finally, we note that Dataplot has a built-in Weibull probability paper command that can be used whenever we have a complete sample (i.e., no censoring and exact failure times). First you have to run PPCC to obtain an estimate of  = GAMMA. This is stored under SHAPE. The full sequence of commands (with XLOG and YLOG both set to OFF) is

SET MINMAX = 1
WEIBULL PPCC PLOT SAMPLE
SET GAMMA = SHAPE
WEIBULL PLOT SAMPLE