Life Data Classification

Data & Data Types

Statistical models rely extensively on data to make predictions. In our case, the models are the statistical distributions and the data are the life dataor times-to-failure data of our product. The accuracy of any prediction is directly proportional to the quality, accuracy and completeness of the supplied data. Good data, along with the appropriate model choice, usually results in good predictions. Bad, or insufficient data, will almost always result in bad predictions.

In the analysis of life data, we want to use all available data which sometimes is incomplete or includes uncertainty as to when a failure occurred. To accomplish this, we separate life data into two categories: complete (all information is available) or censored (some of the information is missing). This chapter details these data classification methods.

Data Classification

Most types of non-life data, as well as some life data, are what we term as complete data. Complete data means that the value of each sample unit is observed or known. In many cases, life data contains uncertainty as to when exactly an event happened (i.e.when the unit failed). Data containing such uncertainty as to exactly when the event happened is termed as censored data.

Data & Data Types

Statistical models rely extensively on data to make predictions. In our case, the models are the statistical distributions and the data are the life dataor times-to-failure data of our product. The accuracy of any prediction is directly proportional to the quality, accuracy and completeness of the supplied data. Good data, along with the appropriate model choice, usually results in good predictions. Bad, or insufficient data, will almost always result in bad predictions.

In the analysis of life data, we want to use all available data which sometimes is incomplete or includes uncertainty as to when a failure occurred. To accomplish this, we separate life data into two categories: complete (all information is available) or censored (some of the information is missing). This chapter details these data classification methods.

Data Classification

Most types of non-life data, as well as some life data, are what we term as complete data. Complete data means that the value of each sample unit is observed or known. In many cases, life data contains uncertainty as to when exactly an event happened (i.e.when the unit failed). Data containing such uncertainty as to exactly when the event happened is termed as censored data.

Censored Data

In many cases when life data are analyzed, all of the units in the sample may not have failed (i.e. the event of interest was not observed) or the exact times-to-failure of all the units are not known. This type of data is commonly called censored data. There are three types of possible censoring schemes, right censored (also called suspended data), interval censored and left censored.

Right Censored (Suspended)

The most common case of censoring is what is referred to as right censored data, or suspended data. In the case of life data, these data sets are composed of units that did not fail. For example, if we tested five units and only three had failed by the end of the test, we would have suspended data (or right censored data) for the two unfailed units. The term right censored implies that the event of interest (i.e. the time-to-failure) is to the right of our data point. In other words, if the units were to keep on operating, the failure would occur at some time after our data point (or to the right on the time scale).

Graphical representation of right censored data.

Interval Censored

The second type of censoring is commonly called interval censored data. Interval censored data reflects uncertainty as to the exact times the units failed within an interval. This type of data frequently comes from tests or situations where the objects of interest are not constantly monitored. If we are running a test on five units and inspecting them every 100 hours, we only know that a unit failed or did not fail between inspections. More specifically, if we inspect a certain unit at 100 hours and find it is operating and then perform another inspection at 200 hours to find that the unit is no longer operating, we know that a failure occurred in the interval between 100 and 200 hours. In other words, the only information we have is that it failed in a certain interval of time. This is also called inspection data by some authors.

Left Censored

The third type of censoring is similar to the interval censoring and is called left censored data. In left censored data, a failure time is only known to be before a certain time. For instance, we may know that a certain unit failed sometime before 100 hours but not exactly when. In other words, it could have failed any time between 0 and 100 hours. This is identical to interval censored datain which the starting time for the interval is zero.

Data Types and Weibull++

Weibull++ allows you to use all of the above data types in a single data set. In other words, a data set can contain complete data, right censored data, interval censored data and left censored data. An overview of this is presented in this section.

Grouped Data and Weibull++

All of the previously mentioned data types can also be put into groups. This is simply a way of collecting units with identical failure or censoring times. If ten units were put on test with the first four units failing at 10, 20, 30 and 40 hours respectively, and then the test were terminated after the fourth failure, you can group the last six units as a group of six suspensions at 40 hours. Weibull++ allows you to enter all types of data as groups, as shown in the following figure.

Depending on the analysis method chosen, i.e. regression or maximum likelihood, Weibull++ treats grouped data differently. This was done by design to allow for more options and flexibility. Appendix B describes how Weibull++ treats grouped data.

Classifying Data in Weibull++

In Weibull++, data classifications are specified using data types. A single data set can contain any or all of the mentioned censoring schemes. Weibull++, through the use of the New Project Wizard and the New Data Sheet Wizard features, simplifies the choice of the appropriate data type for your data. Weibull++ uses the logic tree shown in Fig. 4-1 in deciding which is the appropriate data type for your data.

Analysis & Parameter Estimation Methods for Censored Data

In Chapter 3 we discussed parameter estimation methods for complete data. We will expand on that approach in this section by including estimation methods for the different types of censoring. The basic methods are still based on the same principles covered in Chapter 3, but modified to take into account the fact that some of the data points are censored. For example, assume that you were asked to find the mean (average) of 10, 20, a value that is between 25 and 40, a value that is greater than 30 and a value that is less than 50. In this case, the familiar method of determining the average is no longer applicable and special methods will need to be employed to handle the censored data in this data set.

Analysis of Right Censored (Suspended) Data

Background

All available data should be considered in the analysis of times-to-failure data. This includes the case when a particular unit in a sample has been removed from the test prior to failure. An item, or unit, which is removed from a reliability test prior to failure, or a unit which is in the field and is still operating at the time the reliability of these units is to be determined, is called a suspended item or right censored observation or right censored data point. Suspended items analysis would also be considered when:

1. We need to make an analysis of the available results before test completion.
2. The failure modes which are occurring are different than those anticipated and such units are withdrawn from the test.
3. We need to analyze a single mode and the actual data set is comprised of multiple modes.
4. A warranty analysis is to be made of all units in the field (non-failed and failed units). The non-failed units are considered to be suspended items (or right censored).

Probability Plotting and Rank Regression Analysis of Right Censored Data

When using the probability plotting or rank regression method to accommodate the fact that units in the data set did not fail, or were suspended, we need to adjust their probability of failure, or unreliability. As discussed in Chapter 3, estimates of the unreliability for complete data are obtained using the median ranks approach. The following methodology illustrates how adjusted median ranks are computed to account for right censored data. To better illustrate the methodology, consider the following example [20] where five items are tested resulting in three failures and two suspensions.

Item Number(Position)	State*"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	5,100
2	[math]\displaystyle{ S_1 }[/math]	9,500
3	[math]\displaystyle{ F_2 }[/math]	15,000
4	[math]\displaystyle{ S_2 }[/math]	22,000
5	[math]\displaystyle{ F_3 }[/math]	40,000

The methodology for plotting suspended items involves adjusting the rank positions and plotting the data based on new positions, determined by the location of the suspensions. If we consider these five units, the following methodology would be used: The first item must be the first failure; hence, it is assigned failure order number [math]\displaystyle{ j=1 }[/math]. The actual failure order number (or position) of the second failure, [math]\displaystyle{ {{F}_{2}}, }[/math] is in doubt. It could be either in position 2 or in position 3. Had [math]\displaystyle{ {{S}_{1}} }[/math] not been withdrawn from the test at 9,500 hours it could have operated successfully past 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 2. Alternatively [math]\displaystyle{ {{S}_{1}} }[/math] could also have failed before 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 3. In this case, the failure order number for [math]\displaystyle{ {{F}_{2}} }[/math] will be some number between 2 and 3. To determine this number, consider the following:

We can find the number of ways the second failure can occur in either order number 2 (position 2) or order number 3 (position 3). The possible ways are listed next.

[math]\displaystyle{ F_2 }[/math] in Position 2						OR	[math]\displaystyle{ F_2 }[/math] in Position 3
1	2	3	4	5	6		1	2
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]

It can be seen that [math]\displaystyle{ {{F}_{2}} }[/math] can occur in the second position six ways and in the third position two ways. The most probable position is the average of these possible ways, or the mean order number ( [math]\displaystyle{ MON }[/math] ), given by:

[math]\displaystyle{ {{F}_{2}}=MO{{N}_{2}}=\frac{(6\times 2)+(2\times 3)}{6+2}=2.25 }[/math]

Using the same logic on the third failure, it can be located in position numbers 3, 4 and 5 in the possible ways listed next.

[math]\displaystyle{ F_3 \text{in Position 3} }[/math]		OR	[math]\displaystyle{ F_3 \text{in Position 4} }[/math]			OR	[math]\displaystyle{ F_3 \text{in Position 5} }[/math]
1	2		1	2	3		1	2	3
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]

Then, the mean order number for the third failure, (Item 5) is:

[math]\displaystyle{ MO{{N}_{3}}=\frac{(2\times 3)+(3\times 4)+(3\times 5)}{2+3+3}=4.125 }[/math]

Once the mean order number for each failure has been established, we obtain the median rank positions for these failures at their mean order number. Specifically, we obtain the median rank of the order numbers 1, 2.25 and 4.125 out of a sample size of 5, as given next.

Plotting Positions for the Failures(Sample Size=5)
Failure Number	MON	Median Rank Position(%)
1:[math]\displaystyle{ F_1 }[/math]	1	13%
2:[math]\displaystyle{ F_2 }[/math]	2.25	36%
3:[math]\displaystyle{ F_3 }[/math]	4.125	71%

Once the median rank values have been obtained, the probability plotting analysis is identical to that presented before. As you might have noticed, this methodology is rather laborious. Other techniques and shortcuts have been developed over the years to streamline this procedure. For more details on this method, see Kececioglu [20].

Shortfalls of this Rank Adjustment Method

Even though the rank adjustment method is the most widely used method for performing suspended items analysis, we would like to point out the following shortcoming. As you may have noticed from this analysis of suspended items, only the position where the failure occurred is taken into account, and not the exact time-to-suspension. For example, this methodology would yield the exact same results for the next two cases.

Case 1			Case 2
Item Number	State*"F" or "S"	Life of an item, hr	Item number	State*,"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	1,000	1	[math]\displaystyle{ F_1 }[/math]	1,000
2	[math]\displaystyle{ S_1 }[/math]	1,100	2	[math]\displaystyle{ S_1 }[/math]	9,700
3	[math]\displaystyle{ S_2 }[/math]	1,200	3	[math]\displaystyle{ S_2 }[/math]	9,800
4	[math]\displaystyle{ S_3 }[/math]	1,300	4	[math]\displaystyle{ S_3 }[/math]	9,900
5	[math]\displaystyle{ F_2 }[/math]	10,000	5	[math]\displaystyle{ F_2 }[/math]	10,000
[math]\displaystyle{ *F-Failed, S-Suspended }[/math]			[math]\displaystyle{ *F-Failed, S-Suspended }[/math]

This shortfall is significant when the number of failures is small and the number of suspensions is large and not spread uniformly between failures, as with these data. In cases like this, it is highly recommended that one use maximum likelihood estimation (MLE) to estimate the parameters instead of using least squares, since maximum likelihood does not look at ranks or plotting positions, but rather considers each unique time-to-failure or suspension. For the data given above the results are as follows: The estimated parameters using the method just described are the same for both cases (1 and 2):

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{0}\text{.81} \\ \widehat{\eta }= & \text{11,417 hr} \\ \end{array} }[/math]

However, the MLE results for Case 1 are:

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{1}\text{.33} \\ \widehat{\eta }= & \text{6,900 hr} \\ \end{array} }[/math]

and the MLE results for Case 2 are:

As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression. The results for both cases are identical when using the regression estimation technique, as regression considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of [math]\displaystyle{ \eta }[/math], which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression, considers the values of the suspensions when estimating the parameters. This is illustrated in the following section.

MLE Analysis of Right Censored Data

When performing maximum likelihood analysis on data with suspended items, the likelihood function needs to be expanded to take into account the suspended items. The overall estimation technique does not change, but another term is added to the likelihood function to account for the suspended items. Beyond that, the method of solving for the parameter estimates remains the same. For example, consider a distribution where [math]\displaystyle{ x }[/math] is a continuous random variable with [math]\displaystyle{ pdf }[/math] and [math]\displaystyle{ cdf }[/math]:

[math]\displaystyle{ \begin{align} & f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & F(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \end{align} }[/math]

where [math]\displaystyle{ {{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}} }[/math] are the unknown parameters which need to be estimated from [math]\displaystyle{ R }[/math] observed failures at [math]\displaystyle{ {{T}_{1}},{{T}_{2}}...{{T}_{R}}, }[/math] and [math]\displaystyle{ M }[/math] observed suspensions at [math]\displaystyle{ {{S}_{1}},{{S}_{2}} }[/math] ... [math]\displaystyle{ {{S}_{M}}, }[/math] then the likelihood function is formulated as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R,}}{{S}_{1}},...,{{S}_{M}})= & \underset{i=1}{\overset{R}{\mathop \prod }}\,f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \cdot \underset{j=1}{\overset{M}{\mathop \prod }}\,[1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})] \end{align} }[/math]

The parameters are solved by maximizing this equation, as described in Chapter 3. In most cases, no closed-form solution exists for this maximum or for the parameters. Solutions specific to each distribution utilizing MLE are presented in Appendix C.

MLE Analysis of Interval and Left Censored Data

The inclusion of left and interval censored data in an MLE solution for parameter estimates involves adding a term to the likelihood equation to account for the data types in question. When using interval data, it is assumed that the failures occurred in an interval, i.e. in the interval from time [math]\displaystyle{ A }[/math] to time [math]\displaystyle{ B }[/math] (or from time [math]\displaystyle{ 0 }[/math] to time [math]\displaystyle{ B }[/math] if left censored), where [math]\displaystyle{ A\lt B }[/math]. In the case of interval data, and given [math]\displaystyle{ P }[/math] interval observations, the likelihood function is modified by multiplying the likelihood function with an additional term as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}|{{x}_{1}},{{x}_{2}},...,{{x}_{P}})= & \underset{i=1}{\overset{P}{\mathop \prod }}\,\{F({{x}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \ \ -F({{x}_{i-1}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})\} \end{align} }[/math]

Note that if only interval data are present, this term will represent the entire likelihood function for the MLE solution. The next section gives a formulation of the complete likelihood function for all possible censoring schemes.

ReliaSoft's Alternate Rank Method (RRM)

Difficulties arise when attempting the probability plotting or rank regression analysis of interval or left censored data, especially when an overlap on the intervals is present. This difficulty arises when attempting to estimate the exact time within the interval when the failure actually occurs. The standard regression method (SRM) is not applicable when dealing with interval data; thus ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. See Appendix B for a step-by-step example of this method.

The Complete Likelihood Function

We have now seen that obtaining MLE parameter estimates for different types of data involves incorporating different terms in the likelihood function to account for complete data, right censored data, and left, interval censored data. After including the terms for the different types of data, the likelihood function can now be expressed in its complete form or,

[math]\displaystyle{ \begin{array}{*{35}{l}} L= & \underset{i=1}{\mathop{\overset{R}{\mathop{\prod }}\,}}\,f({{T}_{i}};{{\theta }_{1}},...,{{\theta }_{k}})\cdot \underset{j=1}{\mathop{\overset{M}{\mathop{\prod }}\,}}\,[1-F({{S}_{j}};{{\theta }_{1}},...,{{\theta }_{k}})] \\ & \cdot \underset{l=1}{\mathop{\overset{P}{\mathop{\prod }}\,}}\,\left\{ F({{I}_{{{l}_{U}}}};{{\theta }_{1}},...,{{\theta }_{k}})-F({{I}_{{{l}_{L}}}};{{\theta }_{1}},...,{{\theta }_{k}}) \right\} \\ \end{array} }[/math]

where

[math]\displaystyle{ L\to L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R}},{{S}_{1}},...,{{S}_{M}},{{I}_{1}},...{{I}_{P}}), }[/math]

and

1. [math]\displaystyle{ R }[/math] is the number of units with exact failures
2. [math]\displaystyle{ M }[/math] is the number of suspended units
3. [math]\displaystyle{ P }[/math] is the number of units with left censored or interval times-to-failure
4. [math]\displaystyle{ {{\theta }_{k}} }[/math] are the parameters of the distribution
5. [math]\displaystyle{ {{T}_{i}} }[/math] is the [math]\displaystyle{ {{i}^{th}} }[/math] time to failure
6. [math]\displaystyle{ {{S}_{j}} }[/math] is the [math]\displaystyle{ {{j}^{th}} }[/math] time of suspension
7. [math]\displaystyle{ {{I}_{{{l}_{U}}}} }[/math] is the ending of the time interval of the [math]\displaystyle{ {{l}^{th}} }[/math] group
8. [math]\displaystyle{ {{I}_{{{l}_{L}}}} }[/math] is the beginning of the time interval of the [math]\displaystyle{ {l^{th}} }[/math] group

The total number of units is [math]\displaystyle{ N=R+M+P }[/math]. It should be noted that in this formulation if either [math]\displaystyle{ R }[/math], [math]\displaystyle{ M }[/math] or [math]\displaystyle{ P }[/math] is zero the product term associated with them is assumed to be one and not zero.

Censored Data

In many cases when life data are analyzed, all of the units in the sample may not have failed (i.e. the event of interest was not observed) or the exact times-to-failure of all the units are not known. This type of data is commonly called censored data. There are three types of possible censoring schemes, right censored (also called suspended data), interval censored and left censored.

Right Censored (Suspended)

The most common case of censoring is what is referred to as right censored data, or suspended data. In the case of life data, these data sets are composed of units that did not fail. For example, if we tested five units and only three had failed by the end of the test, we would have suspended data (or right censored data) for the two unfailed units. The term right censored implies that the event of interest (i.e. the time-to-failure) is to the right of our data point. In other words, if the units were to keep on operating, the failure would occur at some time after our data point (or to the right on the time scale).

Graphical representation of right censored data.

Data & Data Types

Statistical models rely extensively on data to make predictions. In our case, the models are the statistical distributions and the data are the life dataor times-to-failure data of our product. The accuracy of any prediction is directly proportional to the quality, accuracy and completeness of the supplied data. Good data, along with the appropriate model choice, usually results in good predictions. Bad, or insufficient data, will almost always result in bad predictions.

In the analysis of life data, we want to use all available data which sometimes is incomplete or includes uncertainty as to when a failure occurred. To accomplish this, we separate life data into two categories: complete (all information is available) or censored (some of the information is missing). This chapter details these data classification methods.

Data Classification

Most types of non-life data, as well as some life data, are what we term as complete data. Complete data means that the value of each sample unit is observed or known. In many cases, life data contains uncertainty as to when exactly an event happened (i.e.when the unit failed). Data containing such uncertainty as to exactly when the event happened is termed as censored data.

Censored Data

In many cases when life data are analyzed, all of the units in the sample may not have failed (i.e. the event of interest was not observed) or the exact times-to-failure of all the units are not known. This type of data is commonly called censored data. There are three types of possible censoring schemes, right censored (also called suspended data), interval censored and left censored.

Right Censored (Suspended)

The most common case of censoring is what is referred to as right censored data, or suspended data. In the case of life data, these data sets are composed of units that did not fail. For example, if we tested five units and only three had failed by the end of the test, we would have suspended data (or right censored data) for the two unfailed units. The term right censored implies that the event of interest (i.e. the time-to-failure) is to the right of our data point. In other words, if the units were to keep on operating, the failure would occur at some time after our data point (or to the right on the time scale).

Graphical representation of right censored data.

Interval Censored

The second type of censoring is commonly called interval censored data. Interval censored data reflects uncertainty as to the exact times the units failed within an interval. This type of data frequently comes from tests or situations where the objects of interest are not constantly monitored. If we are running a test on five units and inspecting them every 100 hours, we only know that a unit failed or did not fail between inspections. More specifically, if we inspect a certain unit at 100 hours and find it is operating and then perform another inspection at 200 hours to find that the unit is no longer operating, we know that a failure occurred in the interval between 100 and 200 hours. In other words, the only information we have is that it failed in a certain interval of time. This is also called inspection data by some authors.

Left Censored

The third type of censoring is similar to the interval censoring and is called left censored data. In left censored data, a failure time is only known to be before a certain time. For instance, we may know that a certain unit failed sometime before 100 hours but not exactly when. In other words, it could have failed any time between 0 and 100 hours. This is identical to interval censored datain which the starting time for the interval is zero.

Data Types and Weibull++

Weibull++ allows you to use all of the above data types in a single data set. In other words, a data set can contain complete data, right censored data, interval censored data and left censored data. An overview of this is presented in this section.

Grouped Data and Weibull++

All of the previously mentioned data types can also be put into groups. This is simply a way of collecting units with identical failure or censoring times. If ten units were put on test with the first four units failing at 10, 20, 30 and 40 hours respectively, and then the test were terminated after the fourth failure, you can group the last six units as a group of six suspensions at 40 hours. Weibull++ allows you to enter all types of data as groups, as shown in the following figure.

Depending on the analysis method chosen, i.e. regression or maximum likelihood, Weibull++ treats grouped data differently. This was done by design to allow for more options and flexibility. Appendix B describes how Weibull++ treats grouped data.

Classifying Data in Weibull++

In Weibull++, data classifications are specified using data types. A single data set can contain any or all of the mentioned censoring schemes. Weibull++, through the use of the New Project Wizard and the New Data Sheet Wizard features, simplifies the choice of the appropriate data type for your data. Weibull++ uses the logic tree shown in Fig. 4-1 in deciding which is the appropriate data type for your data.

Analysis & Parameter Estimation Methods for Censored Data

In Chapter 3 we discussed parameter estimation methods for complete data. We will expand on that approach in this section by including estimation methods for the different types of censoring. The basic methods are still based on the same principles covered in Chapter 3, but modified to take into account the fact that some of the data points are censored. For example, assume that you were asked to find the mean (average) of 10, 20, a value that is between 25 and 40, a value that is greater than 30 and a value that is less than 50. In this case, the familiar method of determining the average is no longer applicable and special methods will need to be employed to handle the censored data in this data set.

Analysis of Right Censored (Suspended) Data

Background

All available data should be considered in the analysis of times-to-failure data. This includes the case when a particular unit in a sample has been removed from the test prior to failure. An item, or unit, which is removed from a reliability test prior to failure, or a unit which is in the field and is still operating at the time the reliability of these units is to be determined, is called a suspended item or right censored observation or right censored data point. Suspended items analysis would also be considered when:

1. We need to make an analysis of the available results before test completion.
2. The failure modes which are occurring are different than those anticipated and such units are withdrawn from the test.
3. We need to analyze a single mode and the actual data set is comprised of multiple modes.
4. A warranty analysis is to be made of all units in the field (non-failed and failed units). The non-failed units are considered to be suspended items (or right censored).

Probability Plotting and Rank Regression Analysis of Right Censored Data

When using the probability plotting or rank regression method to accommodate the fact that units in the data set did not fail, or were suspended, we need to adjust their probability of failure, or unreliability. As discussed in Chapter 3, estimates of the unreliability for complete data are obtained using the median ranks approach. The following methodology illustrates how adjusted median ranks are computed to account for right censored data. To better illustrate the methodology, consider the following example [20] where five items are tested resulting in three failures and two suspensions.

Item Number(Position)	State*"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	5,100
2	[math]\displaystyle{ S_1 }[/math]	9,500
3	[math]\displaystyle{ F_2 }[/math]	15,000
4	[math]\displaystyle{ S_2 }[/math]	22,000
5	[math]\displaystyle{ F_3 }[/math]	40,000

The methodology for plotting suspended items involves adjusting the rank positions and plotting the data based on new positions, determined by the location of the suspensions. If we consider these five units, the following methodology would be used: The first item must be the first failure; hence, it is assigned failure order number [math]\displaystyle{ j=1 }[/math]. The actual failure order number (or position) of the second failure, [math]\displaystyle{ {{F}_{2}}, }[/math] is in doubt. It could be either in position 2 or in position 3. Had [math]\displaystyle{ {{S}_{1}} }[/math] not been withdrawn from the test at 9,500 hours it could have operated successfully past 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 2. Alternatively [math]\displaystyle{ {{S}_{1}} }[/math] could also have failed before 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 3. In this case, the failure order number for [math]\displaystyle{ {{F}_{2}} }[/math] will be some number between 2 and 3. To determine this number, consider the following:

We can find the number of ways the second failure can occur in either order number 2 (position 2) or order number 3 (position 3). The possible ways are listed next.

[math]\displaystyle{ F_2 }[/math] in Position 2						OR	[math]\displaystyle{ F_2 }[/math] in Position 3
1	2	3	4	5	6		1	2
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]

It can be seen that [math]\displaystyle{ {{F}_{2}} }[/math] can occur in the second position six ways and in the third position two ways. The most probable position is the average of these possible ways, or the mean order number ( [math]\displaystyle{ MON }[/math] ), given by:

[math]\displaystyle{ {{F}_{2}}=MO{{N}_{2}}=\frac{(6\times 2)+(2\times 3)}{6+2}=2.25 }[/math]

Using the same logic on the third failure, it can be located in position numbers 3, 4 and 5 in the possible ways listed next.

[math]\displaystyle{ F_3 \text{in Position 3} }[/math]		OR	[math]\displaystyle{ F_3 \text{in Position 4} }[/math]			OR	[math]\displaystyle{ F_3 \text{in Position 5} }[/math]
1	2		1	2	3		1	2	3
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]

Then, the mean order number for the third failure, (Item 5) is:

[math]\displaystyle{ MO{{N}_{3}}=\frac{(2\times 3)+(3\times 4)+(3\times 5)}{2+3+3}=4.125 }[/math]

Once the mean order number for each failure has been established, we obtain the median rank positions for these failures at their mean order number. Specifically, we obtain the median rank of the order numbers 1, 2.25 and 4.125 out of a sample size of 5, as given next.

Plotting Positions for the Failures(Sample Size=5)
Failure Number	MON	Median Rank Position(%)
1:[math]\displaystyle{ F_1 }[/math]	1	13%
2:[math]\displaystyle{ F_2 }[/math]	2.25	36%
3:[math]\displaystyle{ F_3 }[/math]	4.125	71%

Once the median rank values have been obtained, the probability plotting analysis is identical to that presented before. As you might have noticed, this methodology is rather laborious. Other techniques and shortcuts have been developed over the years to streamline this procedure. For more details on this method, see Kececioglu [20].

Shortfalls of this Rank Adjustment Method

Even though the rank adjustment method is the most widely used method for performing suspended items analysis, we would like to point out the following shortcoming. As you may have noticed from this analysis of suspended items, only the position where the failure occurred is taken into account, and not the exact time-to-suspension. For example, this methodology would yield the exact same results for the next two cases.

Case 1			Case 2
Item Number	State*"F" or "S"	Life of an item, hr	Item number	State*,"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	1,000	1	[math]\displaystyle{ F_1 }[/math]	1,000
2	[math]\displaystyle{ S_1 }[/math]	1,100	2	[math]\displaystyle{ S_1 }[/math]	9,700
3	[math]\displaystyle{ S_2 }[/math]	1,200	3	[math]\displaystyle{ S_2 }[/math]	9,800
4	[math]\displaystyle{ S_3 }[/math]	1,300	4	[math]\displaystyle{ S_3 }[/math]	9,900
5	[math]\displaystyle{ F_2 }[/math]	10,000	5	[math]\displaystyle{ F_2 }[/math]	10,000
[math]\displaystyle{ *F-Failed, S-Suspended }[/math]			[math]\displaystyle{ *F-Failed, S-Suspended }[/math]

This shortfall is significant when the number of failures is small and the number of suspensions is large and not spread uniformly between failures, as with these data. In cases like this, it is highly recommended that one use maximum likelihood estimation (MLE) to estimate the parameters instead of using least squares, since maximum likelihood does not look at ranks or plotting positions, but rather considers each unique time-to-failure or suspension. For the data given above the results are as follows: The estimated parameters using the method just described are the same for both cases (1 and 2):

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{0}\text{.81} \\ \widehat{\eta }= & \text{11,417 hr} \\ \end{array} }[/math]

However, the MLE results for Case 1 are:

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{1}\text{.33} \\ \widehat{\eta }= & \text{6,900 hr} \\ \end{array} }[/math]

and the MLE results for Case 2 are:

As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression. The results for both cases are identical when using the regression estimation technique, as regression considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of [math]\displaystyle{ \eta }[/math], which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression, considers the values of the suspensions when estimating the parameters. This is illustrated in the following section.

MLE Analysis of Right Censored Data

When performing maximum likelihood analysis on data with suspended items, the likelihood function needs to be expanded to take into account the suspended items. The overall estimation technique does not change, but another term is added to the likelihood function to account for the suspended items. Beyond that, the method of solving for the parameter estimates remains the same. For example, consider a distribution where [math]\displaystyle{ x }[/math] is a continuous random variable with [math]\displaystyle{ pdf }[/math] and [math]\displaystyle{ cdf }[/math]:

[math]\displaystyle{ \begin{align} & f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & F(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \end{align} }[/math]

where [math]\displaystyle{ {{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}} }[/math] are the unknown parameters which need to be estimated from [math]\displaystyle{ R }[/math] observed failures at [math]\displaystyle{ {{T}_{1}},{{T}_{2}}...{{T}_{R}}, }[/math] and [math]\displaystyle{ M }[/math] observed suspensions at [math]\displaystyle{ {{S}_{1}},{{S}_{2}} }[/math] ... [math]\displaystyle{ {{S}_{M}}, }[/math] then the likelihood function is formulated as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R,}}{{S}_{1}},...,{{S}_{M}})= & \underset{i=1}{\overset{R}{\mathop \prod }}\,f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \cdot \underset{j=1}{\overset{M}{\mathop \prod }}\,[1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})] \end{align} }[/math]

The parameters are solved by maximizing this equation, as described in Chapter 3. In most cases, no closed-form solution exists for this maximum or for the parameters. Solutions specific to each distribution utilizing MLE are presented in Appendix C.

MLE Analysis of Interval and Left Censored Data

The inclusion of left and interval censored data in an MLE solution for parameter estimates involves adding a term to the likelihood equation to account for the data types in question. When using interval data, it is assumed that the failures occurred in an interval, i.e. in the interval from time [math]\displaystyle{ A }[/math] to time [math]\displaystyle{ B }[/math] (or from time [math]\displaystyle{ 0 }[/math] to time [math]\displaystyle{ B }[/math] if left censored), where [math]\displaystyle{ A\lt B }[/math]. In the case of interval data, and given [math]\displaystyle{ P }[/math] interval observations, the likelihood function is modified by multiplying the likelihood function with an additional term as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}|{{x}_{1}},{{x}_{2}},...,{{x}_{P}})= & \underset{i=1}{\overset{P}{\mathop \prod }}\,\{F({{x}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \ \ -F({{x}_{i-1}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})\} \end{align} }[/math]

Note that if only interval data are present, this term will represent the entire likelihood function for the MLE solution. The next section gives a formulation of the complete likelihood function for all possible censoring schemes.

ReliaSoft's Alternate Rank Method (RRM)

Difficulties arise when attempting the probability plotting or rank regression analysis of interval or left censored data, especially when an overlap on the intervals is present. This difficulty arises when attempting to estimate the exact time within the interval when the failure actually occurs. The standard regression method (SRM) is not applicable when dealing with interval data; thus ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. See Appendix B for a step-by-step example of this method.

The Complete Likelihood Function

We have now seen that obtaining MLE parameter estimates for different types of data involves incorporating different terms in the likelihood function to account for complete data, right censored data, and left, interval censored data. After including the terms for the different types of data, the likelihood function can now be expressed in its complete form or,

[math]\displaystyle{ \begin{array}{*{35}{l}} L= & \underset{i=1}{\mathop{\overset{R}{\mathop{\prod }}\,}}\,f({{T}_{i}};{{\theta }_{1}},...,{{\theta }_{k}})\cdot \underset{j=1}{\mathop{\overset{M}{\mathop{\prod }}\,}}\,[1-F({{S}_{j}};{{\theta }_{1}},...,{{\theta }_{k}})] \\ & \cdot \underset{l=1}{\mathop{\overset{P}{\mathop{\prod }}\,}}\,\left\{ F({{I}_{{{l}_{U}}}};{{\theta }_{1}},...,{{\theta }_{k}})-F({{I}_{{{l}_{L}}}};{{\theta }_{1}},...,{{\theta }_{k}}) \right\} \\ \end{array} }[/math]

where

[math]\displaystyle{ L\to L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R}},{{S}_{1}},...,{{S}_{M}},{{I}_{1}},...{{I}_{P}}), }[/math]

and

1. [math]\displaystyle{ R }[/math] is the number of units with exact failures
2. [math]\displaystyle{ M }[/math] is the number of suspended units
3. [math]\displaystyle{ P }[/math] is the number of units with left censored or interval times-to-failure
4. [math]\displaystyle{ {{\theta }_{k}} }[/math] are the parameters of the distribution
5. [math]\displaystyle{ {{T}_{i}} }[/math] is the [math]\displaystyle{ {{i}^{th}} }[/math] time to failure
6. [math]\displaystyle{ {{S}_{j}} }[/math] is the [math]\displaystyle{ {{j}^{th}} }[/math] time of suspension
7. [math]\displaystyle{ {{I}_{{{l}_{U}}}} }[/math] is the ending of the time interval of the [math]\displaystyle{ {{l}^{th}} }[/math] group
8. [math]\displaystyle{ {{I}_{{{l}_{L}}}} }[/math] is the beginning of the time interval of the [math]\displaystyle{ {l^{th}} }[/math] group

The total number of units is [math]\displaystyle{ N=R+M+P }[/math]. It should be noted that in this formulation if either [math]\displaystyle{ R }[/math], [math]\displaystyle{ M }[/math] or [math]\displaystyle{ P }[/math] is zero the product term associated with them is assumed to be one and not zero.

Interval Censored

The second type of censoring is commonly called interval censored data. Interval censored data reflects uncertainty as to the exact times the units failed within an interval. This type of data frequently comes from tests or situations where the objects of interest are not constantly monitored. If we are running a test on five units and inspecting them every 100 hours, we only know that a unit failed or did not fail between inspections. More specifically, if we inspect a certain unit at 100 hours and find it is operating and then perform another inspection at 200 hours to find that the unit is no longer operating, we know that a failure occurred in the interval between 100 and 200 hours. In other words, the only information we have is that it failed in a certain interval of time. This is also called inspection data by some authors.

Left Censored

The third type of censoring is similar to the interval censoring and is called left censored data. In left censored data, a failure time is only known to be before a certain time. For instance, we may know that a certain unit failed sometime before 100 hours but not exactly when. In other words, it could have failed any time between 0 and 100 hours. This is identical to interval censored datain which the starting time for the interval is zero.

Data Types and Weibull++

Weibull++ allows you to use all of the above data types in a single data set. In other words, a data set can contain complete data, right censored data, interval censored data and left censored data. An overview of this is presented in this section.

Grouped Data and Weibull++

All of the previously mentioned data types can also be put into groups. This is simply a way of collecting units with identical failure or censoring times. If ten units were put on test with the first four units failing at 10, 20, 30 and 40 hours respectively, and then the test were terminated after the fourth failure, you can group the last six units as a group of six suspensions at 40 hours. Weibull++ allows you to enter all types of data as groups, as shown in the following figure.

Depending on the analysis method chosen, i.e. regression or maximum likelihood, Weibull++ treats grouped data differently. This was done by design to allow for more options and flexibility. Appendix B describes how Weibull++ treats grouped data.

Classifying Data in Weibull++

In Weibull++, data classifications are specified using data types. A single data set can contain any or all of the mentioned censoring schemes. Weibull++, through the use of the New Project Wizard and the New Data Sheet Wizard features, simplifies the choice of the appropriate data type for your data. Weibull++ uses the logic tree shown in Fig. 4-1 in deciding which is the appropriate data type for your data.

Analysis & Parameter Estimation Methods for Censored Data

In Chapter 3 we discussed parameter estimation methods for complete data. We will expand on that approach in this section by including estimation methods for the different types of censoring. The basic methods are still based on the same principles covered in Chapter 3, but modified to take into account the fact that some of the data points are censored. For example, assume that you were asked to find the mean (average) of 10, 20, a value that is between 25 and 40, a value that is greater than 30 and a value that is less than 50. In this case, the familiar method of determining the average is no longer applicable and special methods will need to be employed to handle the censored data in this data set.

Analysis of Right Censored (Suspended) Data

Background

All available data should be considered in the analysis of times-to-failure data. This includes the case when a particular unit in a sample has been removed from the test prior to failure. An item, or unit, which is removed from a reliability test prior to failure, or a unit which is in the field and is still operating at the time the reliability of these units is to be determined, is called a suspended item or right censored observation or right censored data point. Suspended items analysis would also be considered when:

1. We need to make an analysis of the available results before test completion.
2. The failure modes which are occurring are different than those anticipated and such units are withdrawn from the test.
3. We need to analyze a single mode and the actual data set is comprised of multiple modes.
4. A warranty analysis is to be made of all units in the field (non-failed and failed units). The non-failed units are considered to be suspended items (or right censored).

Probability Plotting and Rank Regression Analysis of Right Censored Data

When using the probability plotting or rank regression method to accommodate the fact that units in the data set did not fail, or were suspended, we need to adjust their probability of failure, or unreliability. As discussed in Chapter 3, estimates of the unreliability for complete data are obtained using the median ranks approach. The following methodology illustrates how adjusted median ranks are computed to account for right censored data. To better illustrate the methodology, consider the following example [20] where five items are tested resulting in three failures and two suspensions.

Item Number(Position)	State*"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	5,100
2	[math]\displaystyle{ S_1 }[/math]	9,500
3	[math]\displaystyle{ F_2 }[/math]	15,000
4	[math]\displaystyle{ S_2 }[/math]	22,000
5	[math]\displaystyle{ F_3 }[/math]	40,000

The methodology for plotting suspended items involves adjusting the rank positions and plotting the data based on new positions, determined by the location of the suspensions. If we consider these five units, the following methodology would be used: The first item must be the first failure; hence, it is assigned failure order number [math]\displaystyle{ j=1 }[/math]. The actual failure order number (or position) of the second failure, [math]\displaystyle{ {{F}_{2}}, }[/math] is in doubt. It could be either in position 2 or in position 3. Had [math]\displaystyle{ {{S}_{1}} }[/math] not been withdrawn from the test at 9,500 hours it could have operated successfully past 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 2. Alternatively [math]\displaystyle{ {{S}_{1}} }[/math] could also have failed before 15,000 hours, thus placing [math]\displaystyle{ {{F}_{2}} }[/math] in position 3. In this case, the failure order number for [math]\displaystyle{ {{F}_{2}} }[/math] will be some number between 2 and 3. To determine this number, consider the following:

We can find the number of ways the second failure can occur in either order number 2 (position 2) or order number 3 (position 3). The possible ways are listed next.

[math]\displaystyle{ F_2 }[/math] in Position 2						OR	[math]\displaystyle{ F_2 }[/math] in Position 3
1	2	3	4	5	6		1	2
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ F_3 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ S_2 }[/math]

It can be seen that [math]\displaystyle{ {{F}_{2}} }[/math] can occur in the second position six ways and in the third position two ways. The most probable position is the average of these possible ways, or the mean order number ( [math]\displaystyle{ MON }[/math] ), given by:

[math]\displaystyle{ {{F}_{2}}=MO{{N}_{2}}=\frac{(6\times 2)+(2\times 3)}{6+2}=2.25 }[/math]

Using the same logic on the third failure, it can be located in position numbers 3, 4 and 5 in the possible ways listed next.

[math]\displaystyle{ F_3 \text{in Position 3} }[/math]		OR	[math]\displaystyle{ F_3 \text{in Position 4} }[/math]			OR	[math]\displaystyle{ F_3 \text{in Position 5} }[/math]
1	2		1	2	3		1	2	3
[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]		[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]	[math]\displaystyle{ F_1 }[/math]
[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]		[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ F_2 }[/math]
[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_2 }[/math]	[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]
[math]\displaystyle{ S_1 }[/math]	[math]\displaystyle{ S_2 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]
[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_2 }[/math]	[math]\displaystyle{ S_1 }[/math]		[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]	[math]\displaystyle{ F_3 }[/math]

Then, the mean order number for the third failure, (Item 5) is:

[math]\displaystyle{ MO{{N}_{3}}=\frac{(2\times 3)+(3\times 4)+(3\times 5)}{2+3+3}=4.125 }[/math]

Once the mean order number for each failure has been established, we obtain the median rank positions for these failures at their mean order number. Specifically, we obtain the median rank of the order numbers 1, 2.25 and 4.125 out of a sample size of 5, as given next.

Plotting Positions for the Failures(Sample Size=5)
Failure Number	MON	Median Rank Position(%)
1:[math]\displaystyle{ F_1 }[/math]	1	13%
2:[math]\displaystyle{ F_2 }[/math]	2.25	36%
3:[math]\displaystyle{ F_3 }[/math]	4.125	71%

Once the median rank values have been obtained, the probability plotting analysis is identical to that presented before. As you might have noticed, this methodology is rather laborious. Other techniques and shortcuts have been developed over the years to streamline this procedure. For more details on this method, see Kececioglu [20].

Shortfalls of this Rank Adjustment Method

Even though the rank adjustment method is the most widely used method for performing suspended items analysis, we would like to point out the following shortcoming. As you may have noticed from this analysis of suspended items, only the position where the failure occurred is taken into account, and not the exact time-to-suspension. For example, this methodology would yield the exact same results for the next two cases.

Case 1			Case 2
Item Number	State*"F" or "S"	Life of an item, hr	Item number	State*,"F" or "S"	Life of item, hr
1	[math]\displaystyle{ F_1 }[/math]	1,000	1	[math]\displaystyle{ F_1 }[/math]	1,000
2	[math]\displaystyle{ S_1 }[/math]	1,100	2	[math]\displaystyle{ S_1 }[/math]	9,700
3	[math]\displaystyle{ S_2 }[/math]	1,200	3	[math]\displaystyle{ S_2 }[/math]	9,800
4	[math]\displaystyle{ S_3 }[/math]	1,300	4	[math]\displaystyle{ S_3 }[/math]	9,900
5	[math]\displaystyle{ F_2 }[/math]	10,000	5	[math]\displaystyle{ F_2 }[/math]	10,000
[math]\displaystyle{ *F-Failed, S-Suspended }[/math]			[math]\displaystyle{ *F-Failed, S-Suspended }[/math]

This shortfall is significant when the number of failures is small and the number of suspensions is large and not spread uniformly between failures, as with these data. In cases like this, it is highly recommended that one use maximum likelihood estimation (MLE) to estimate the parameters instead of using least squares, since maximum likelihood does not look at ranks or plotting positions, but rather considers each unique time-to-failure or suspension. For the data given above the results are as follows: The estimated parameters using the method just described are the same for both cases (1 and 2):

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{0}\text{.81} \\ \widehat{\eta }= & \text{11,417 hr} \\ \end{array} }[/math]

However, the MLE results for Case 1 are:

[math]\displaystyle{ \begin{array}{*{35}{l}} \widehat{\beta }= & \text{1}\text{.33} \\ \widehat{\eta }= & \text{6,900 hr} \\ \end{array} }[/math]

and the MLE results for Case 2 are:

As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression. The results for both cases are identical when using the regression estimation technique, as regression considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of [math]\displaystyle{ \eta }[/math], which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression, considers the values of the suspensions when estimating the parameters. This is illustrated in the following section.

MLE Analysis of Right Censored Data

When performing maximum likelihood analysis on data with suspended items, the likelihood function needs to be expanded to take into account the suspended items. The overall estimation technique does not change, but another term is added to the likelihood function to account for the suspended items. Beyond that, the method of solving for the parameter estimates remains the same. For example, consider a distribution where [math]\displaystyle{ x }[/math] is a continuous random variable with [math]\displaystyle{ pdf }[/math] and [math]\displaystyle{ cdf }[/math]:

[math]\displaystyle{ \begin{align} & f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & F(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \end{align} }[/math]

where [math]\displaystyle{ {{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}} }[/math] are the unknown parameters which need to be estimated from [math]\displaystyle{ R }[/math] observed failures at [math]\displaystyle{ {{T}_{1}},{{T}_{2}}...{{T}_{R}}, }[/math] and [math]\displaystyle{ M }[/math] observed suspensions at [math]\displaystyle{ {{S}_{1}},{{S}_{2}} }[/math] ... [math]\displaystyle{ {{S}_{M}}, }[/math] then the likelihood function is formulated as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R,}}{{S}_{1}},...,{{S}_{M}})= & \underset{i=1}{\overset{R}{\mathop \prod }}\,f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \cdot \underset{j=1}{\overset{M}{\mathop \prod }}\,[1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})] \end{align} }[/math]

The parameters are solved by maximizing this equation, as described in Chapter 3. In most cases, no closed-form solution exists for this maximum or for the parameters. Solutions specific to each distribution utilizing MLE are presented in Appendix C.

MLE Analysis of Interval and Left Censored Data

The inclusion of left and interval censored data in an MLE solution for parameter estimates involves adding a term to the likelihood equation to account for the data types in question. When using interval data, it is assumed that the failures occurred in an interval, i.e. in the interval from time [math]\displaystyle{ A }[/math] to time [math]\displaystyle{ B }[/math] (or from time [math]\displaystyle{ 0 }[/math] to time [math]\displaystyle{ B }[/math] if left censored), where [math]\displaystyle{ A\lt B }[/math]. In the case of interval data, and given [math]\displaystyle{ P }[/math] interval observations, the likelihood function is modified by multiplying the likelihood function with an additional term as follows:

[math]\displaystyle{ \begin{align} L({{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}|{{x}_{1}},{{x}_{2}},...,{{x}_{P}})= & \underset{i=1}{\overset{P}{\mathop \prod }}\,\{F({{x}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \ \ -F({{x}_{i-1}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})\} \end{align} }[/math]

Note that if only interval data are present, this term will represent the entire likelihood function for the MLE solution. The next section gives a formulation of the complete likelihood function for all possible censoring schemes.

ReliaSoft's Alternate Rank Method (RRM)

Difficulties arise when attempting the probability plotting or rank regression analysis of interval or left censored data, especially when an overlap on the intervals is present. This difficulty arises when attempting to estimate the exact time within the interval when the failure actually occurs. The standard regression method (SRM) is not applicable when dealing with interval data; thus ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. See Appendix B for a step-by-step example of this method.

The Complete Likelihood Function

We have now seen that obtaining MLE parameter estimates for different types of data involves incorporating different terms in the likelihood function to account for complete data, right censored data, and left, interval censored data. After including the terms for the different types of data, the likelihood function can now be expressed in its complete form or,

[math]\displaystyle{ \begin{array}{*{35}{l}} L= & \underset{i=1}{\mathop{\overset{R}{\mathop{\prod }}\,}}\,f({{T}_{i}};{{\theta }_{1}},...,{{\theta }_{k}})\cdot \underset{j=1}{\mathop{\overset{M}{\mathop{\prod }}\,}}\,[1-F({{S}_{j}};{{\theta }_{1}},...,{{\theta }_{k}})] \\ & \cdot \underset{l=1}{\mathop{\overset{P}{\mathop{\prod }}\,}}\,\left\{ F({{I}_{{{l}_{U}}}};{{\theta }_{1}},...,{{\theta }_{k}})-F({{I}_{{{l}_{L}}}};{{\theta }_{1}},...,{{\theta }_{k}}) \right\} \\ \end{array} }[/math]

where

[math]\displaystyle{ L\to L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R}},{{S}_{1}},...,{{S}_{M}},{{I}_{1}},...{{I}_{P}}), }[/math]

and

1. [math]\displaystyle{ R }[/math] is the number of units with exact failures
2. [math]\displaystyle{ M }[/math] is the number of suspended units
3. [math]\displaystyle{ P }[/math] is the number of units with left censored or interval times-to-failure
4. [math]\displaystyle{ {{\theta }_{k}} }[/math] are the parameters of the distribution
5. [math]\displaystyle{ {{T}_{i}} }[/math] is the [math]\displaystyle{ {{i}^{th}} }[/math] time to failure
6. [math]\displaystyle{ {{S}_{j}} }[/math] is the [math]\displaystyle{ {{j}^{th}} }[/math] time of suspension
7. [math]\displaystyle{ {{I}_{{{l}_{U}}}} }[/math] is the ending of the time interval of the [math]\displaystyle{ {{l}^{th}} }[/math] group
8. [math]\displaystyle{ {{I}_{{{l}_{L}}}} }[/math] is the beginning of the time interval of the [math]\displaystyle{ {l^{th}} }[/math] group

The total number of units is [math]\displaystyle{ N=R+M+P }[/math]. It should be noted that in this formulation if either [math]\displaystyle{ R }[/math], [math]\displaystyle{ M }[/math] or [math]\displaystyle{ P }[/math] is zero the product term associated with them is assumed to be one and not zero.