Life Data Analysis with Zero-Time
In many circumstances, it is possible for failures to occur immediately at the beginning of the life of the product. These failures are sometimes referred to as out-of-the-box or short-term failures and might be attributed to problems in manufacturing, insufficient quality control and pre-shipping inspection, poor packaging or shipping damages. The failure times for such units are zero. Zero values can also be found in a variety of other data set types, such as level crossings in customer usage measurement programs (measurement of a number of events) or time in band (measurement of time spent in a given state). This article examines the effect of the zero values on data analysis and presents a warranty data analysis example containing out-of-the-box failures data using Weibull++ 7.
1 - Analysis Method
Let us consider a data set with the following failure times: 0, 0, 0, 65, 65, 92, 120, 124, 141 and 171. Assuming a Weibull distribution, and using rank regression on X (RRX) as the analysis method, the probability plots looks like the one shown next.
You might wonder why the probability plot is curved! When analyzing data containing zero values, Weibull++ 7 first calculates the parameters for the subset of non-zero data points in the data set. These parameters define a reliability function, R(t), for the subset. In order to make predictions for the entire data set, a new reliability function, R'(t), is defined as follows:
Where PNZ is the percentage of the population with non-zero failure times (note that zero-value suspensions are ignored in the calculations). In the previous example, 7 out of 10 units failed after time zero and 3 units failed at t = 0. Therefore, PNZ = 0.7. It is because of the multiplication of the R(t) function with PNZ that the probability plot curves.
More reliability-related functions are listed in Section 3 of this article.
2 - Example
The sales data and returns data are shown in the next figures. The product is assumed to follow a Weibull distribution and the maximum likelihood estimation (MLE) method is chosen for the analysis because of the large number of suspended units. These figures also show the estimated Weibull distribution parameters (β = 4.5135 and η = 3.4370 months) and the PNZ value, PNZ = 0.9615.
Note: The option to display the PNZ value was chosen on the Data Sheet page of the User Setup.
The probability plot is shown in the next figure. The curvature of the plot is due to the non-zero failure times in the data. The curve crosses the y-axis at 1-PNZ, or the percentage of units failed at time equal to zero.
To further clarify the analysis method, let us extract the times-to-failure data set used for the analysis and export it to a Standard Folio in Weibull++ 7. This can be done by selecting Transfer Life Data to New Folio tool from the Data menu. The times-to-failure data set looks as follows:
The parameters β and η were calculated using rows 2 through 7, then the reliability function for the entire data set was adjusted using the PNZ value.
To compare the effect of the non-zero data on the life data analysis functions, a new folio was created without the non-zero subset, as shown in the next figure. Notice that the β and η for the original data set and the non-zero subset are identical, while the PNZ is 1 in the non-zero subset, indicating that all the failures happened after time equal to zero.
The next set of figures shows the probability, reliability, and failure rate functions for the entire data set (in black) and the non-zero subset (in blue). The effect of the PNZ is seen on the probability and reliability plots. Notice that the difference between the curves on these plots is largest for small values of time, and decreases as time increases. The last plot provides verification that the zero-value data points have no effect on failure rate, since the curves are indistinguishable.
3 - Summary of Reliability Related Functions for Data
Sets with Zero-Times
In life data analysis, we examine units that either fail or succeed. That is, we have two mutually exclusive outcomes. The probability of success is called the reliability function and the probability of failure is the unreliability function. Since all units must either fail or succeed, the sum of the reliability and the unreliability must be unity. Thus:
Substituting for the reliability function of the entire data set, we obtain the following expression for the unreliability of the entire data set:
This can also be written in terms of the unreliability of the non-zero subset as:
The probability density function, or pdf, can be obtained from the unreliability function as follows:
Substituting for the unreliability function yields:
Simplifying gives the pdf of the entire data set in terms of the reliability function as:
and in terms of the pdf of the non-zero subset as:
An interesting side note is that the equation that is used to obtain the unreliability function from the pdf for distributions that are defined from time zero to infinity (such as the Weibull distribution) is not the following equation, as one might expect:
The difference arises because the above formulation assumes that there is a zero probability of failure up to and including time equal to zero. For the case where zero-value data points are included in the analysis, the probability of failure at time equal to zero is greater than zero. Thus, our equation must be rewritten to account for the zero-value failures as follows:
where Q(0) is 1-PNZ. In other words, the unreliability at time t is the sum of the unreliability of the population at time equal to zero plus the area under the pdf until time equal to t.
The conditional reliability is given by:
Substituting for the reliability functions of the entire data set yields:
Canceling the PNZ in the numerator with the PNZ in the denominator gives the following expression that does not depend on PNZ:
The right side can be written in terms of the conditional reliability for the non-zero subset of the data. Therefore:
Notice that the conditional reliability function is unchanged by the addition of the zero-value failures. A similar analysis of the failure rate function shows that the failure rate is unchanged by the addition of the zero-value failures as well. Thus:
Copyright 2008 ReliaSoft Corporation, ALL RIGHTS RESERVED