Issue 17, July 2002

When displaying confidence bounds on a plot there are many decisions to be made, such as whether to use one-sided or two-sided confidence bounds, what confidence level should be used, and which confidence interval method is to be implemented. But when creating a probability plot you must also decide whether to display confidence bounds on time (Type I) or on reliability (Type II). A probability plot contains both an axis for time (x-axis) and an axis for reliability (y-axis), although the y-axis is usually displayed in terms of probability of failure. Both types of confidence bounds can be shown on a probability plot, so which one should you use? This article will explain the differences between the two types of confidence bounds and offer suggestions to help you decide which type to use.

For the purposes of this article, all of the analysis will be conducted using the 2-parameter Weibull distribution. The process can then be expanded to be used with other distributions. The confidence bounds will be estimated using the Fisher Matrix method.

Confidence Bounds on Time (Type I)

The bounds on time are estimated by first solving the reliability equation for time:

or:

where u = ln(T).

The upper and lower confidence bounds on u are estimated from:

where:

The upper and lower confidence bounds on time are then found by:

Confidence Bounds on Reliability (Type II)

The bounds on reliability can be derived easily by first looking at the general extreme value distribution (EVD). Its reliability function is given by:

By transforming t = ln(T) and converting p₁ = ln(η), p₂ = 1/β, the above equation becomes the Weibull reliability function:

Set u = β(ln(T) - ln(η)) and the reliability function now becomes:

The upper and lower confidence bounds on u are estimated from:

where:

The upper and lower confidence bounds on reliability are then found by:

Which Type (I or II) to Use

OK, so now that a bunch of equations have been thrown at you, but what does it mean? How can you  determine whether to use confidence bounds on time or on reliability? In general, you would display confidence bounds on the value that you do not know (the value that you are trying to estimate). For example, if you want to determine the time by which 10% of the units have failed (i.e. 90% reliability) then you would use confidence bounds on time (Type I). The unknown in this case is time. Figure 1 displays a probability plot with confidence bounds on time.

Figure 1: Probability Plot with Confidence Bounds on Time (Type I)

To read the confidence bounds on time, start on the y-axis with the given 10% unreliability and follow the 10% line until it reaches the first line. This is the lower bound on time and is represented with the blue arrow. The upper bound on time is the line farthest to the right and is represented with the red arrow. You can read the time value from the lower and upper confidence bound lines by simply reading the time at each point. Therefore, from Figure 1 the lower and upper confidence bounds at 10% unreliability are approximately 950 and 1500, respectively. Greater accuracy in the results could be achieved by calculating these values directly. One of the disadvantages of reading values directly from the plot is the lack of accuracy that can be achieved.

So what about bounds on reliability? Let's assume that you want to determine the probability of failure at 1000 hours. This represents a scenario in which you would display confidence bounds on reliability, since probability of failure (or reliability) is what you are trying to determine. Figure 2 displays a probability plot with confidence bounds on reliability.

Figure 2: Probability Plot with Confidence Bounds on Reliability (Type II)

The first thing you may notice is that the bounds on reliability are different than the bounds on time. This can be seen easily by comparing the plots in Figure 1 and Figure 2. Given Figure 2, you can read the confidence bounds values by starting on the x-axis at time equals 1000 and follow the 1000 line until it reaches the first confidence bound line. This is the lower bound on reliability and is represented with the blue arrow. The upper bound on reliability is the line farthest to the left and is represented with the red arrow. Therefore, from Figure 2 the lower and upper confidence bounds on probability of failure at 1000 hours are approximately 2.5% and 13%, respectively.

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