 Reliability HotWire

Issue 11, January 2002

Tool Tips What is the difference between "upper" and "lower" confidence bounds, and "top" and "bottom" confidence bounds?

When a reference to "upper" and "lower" confidence bounds appears in ReliaSoft software applications, this is the conventional use of these terms. A lower bound defines a point on a distribution that a specified percentage of the population is greater than. Similarly, an upper bound defines a point on a distribution that a specified percentage of the population is less than.

The terms "top" and "bottom" are used in ReliaSoft software only for the plots, in reference to the location of the bounds on the plots. For example, the top bound on a probability plot is the one that actually appears above the model line, even though it represents the lower bound on reliability (this is because the y-axis scale for most probability plots represents unreliability). Similarly, the top bound on a linear reliability vs. time plot represents the upper bound on reliability, since the y-axis represents reliability. The terms "top" and "bottom" are used in reference to the location of the bounds on all plots, regardless of the y-axis, and are not necessarily interchangeable with the terms "upper" and "lower" confidence bounds. In Weibull++, what does "PNZ" stand for and how does it work?

"PNZ" stands for "percent non-zero" and is used by Weibull++ for calculations with data sets that contain zero-value data points. While not often encountered in life data analysis, these data sets may be found in other QRD (quality, reliability, dependability) related analyses. The option to show this measurement in the results area of Weibull++ can be set on the Data Folio tab of the User Setup. A common example of this is the inclusion of data points with a value of zero. This is commonly found in data from customer usage measurement programs of level crossings (measurement of a number of events) or time in band (measurement of time spent in a given state). For example, an automotive manufacturer is interested in how customers use their vehicles' engines with respect to engine speed. In order to measure this, they develop a direct measurement program that captures the amount of time the engines spend in certain RPM bands (0 to 500 RPM, 500 to 100 RPM, etc.). A problem occurs when measuring the higher RPM bands because some drivers might never spend time in higher RPM bands. This results in data sets that contain a certain number of zero data points. Following is a table of time in band for 5000 to 5500 RPM, from an automotive direct measurement program.

 Source Time in 5000 - 5500 RPM band (minutes) Driver #1 9 Driver #2 0 Driver #3 42 Driver #4 141 Driver #5 94 Driver #6 427 Driver #7 0 Driver #8 1216 Driver #9 185 Driver #10 45 Driver #11 0 Driver #12 152 Driver #13 156 Driver #14 0 Driver #15 14 Driver #16 7 Driver #17 0 Driver #18 3480 Driver #19 45 Driver #20 125

The problem in analyzing this data involves dealing with the zero-value data points. ReliaSoft's Weibull++ software can deal with these type of data sets with the percent non-zero (PNZ) calculation. These calculations are performed by first analyzing the data without the zeroes and determining the parameter estimates for this data subset. In this example, the parameter estimates for a two-parameter Weibull distribution for the non-zero data in this set, beta (β) is 0.71 and eta (η) is 212.3. These parameters will be used for the full data set, but any subsequent reliability calculations for the full data set will be multiplied by the proportion of non-zero data points, or PNZ:

R'(t) = PNZ*[R(t)]

where R'(t) is the reliability function for the whole data set and R(t) is the reliability function for the non-zero data subset. One result of the mathematics is that the fitted line on a probability plot is not straight, but curves so that it would intercept the y = 0 point at a percentage equal to the percentage of zero data points (1-PNZ). This is illustrated in the following figure.  