Analysis & Parameter Estimation Methods for Censored Data

In the Statistical Background chapter 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 the Statistical Background chapter, 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 In this case, the familiar method of determining the average is no longer applicable and special methods will need to be employed to handle the censored data in this data set. (Note: Assuming a normal distribution and using MLE in Weibull++, the mean of this data set is calculated to be 24.563.)

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:

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 the Statistical Background chapter, 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	F1	5,100
2	S1	9,500
3	F2	15,000
4	S2	22,000
5	F3	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 j = 1.

The actual failure order number (or position) of the second failure, F2, is in doubt. It could be either in position 2 or in position 3. Had S1 not been withdrawn from the test at 9,500 hours it could have operated successfully past 15,000 hours, thus placing F2 in position 2. Alternatively S1 could also have failed before 15,000 hours, thus placing F2 in position 3. In this case, the failure order number for F2 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.

F2 in Position 2
1	2	3	4	5	6
F1	F1	F1	F1	F1	F1
F2	F2	F2	F2	F2	F2
S1	S2	F3	S1	S2	F3
S2	S1	S1	F3	F3	S2
F3	F3	S2	S2	S1	S1

F2 in Position 3
1	2
F1	F1
S1	S1
F2	F2
S2	F3
F3	S2

It can be seen that F2 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 (MON), given by:

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.

F3 in Position 3		or	F3 in Position 4			or	F3 in Position 5
1	2		1	2	3		1	2	3
F1	F1		F1	F1	F1		F1	F1	F1
F2	F2		S1	F2	F2		S1	F2	F2
F3	F3		F2	S1	S2		F2	S1	S2
S1	S2		F3	F3	F3		S2	S2	S1
S2	S1		S2	S2	S1		F3	F3	F3

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: F1	1	13%
2: F2	2.25	36%
3: F3	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
Item number	*State, "F" or "S"**	Life of item, hr
1	F1	1,000
2	S1	1,100
3	S2	1,200
4	S3	1,300
5	F2	10,000
* F - Failed, S - Suspended.

Case 2
Item number	*State, "F" or "S"**	Life of item, hr
1	F1	1,000
2	S1	9,700
3	S2	9,800
4	S3	9,900
5	F2	10,000
* F - Failed, S - Suspended.

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.

The estimated parameters using the method just described are the same for both cases (1 and 2):

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 η, 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 x is a continuous random variable with pdf and cdf:

where θ1, θ2,..., θk are the k unknown parameters which need to be estimated from R observed failures at T1, T2 ... TR, and M observed suspensions at S1, S2 ... SM then the likelihood function is formulated as follows: [Note: Note that a binomial constant term may also be included in this likelihood function for completeness. In the interest of clarity and consistency, this term is omitted since it drops out in the subsequent formulation (i.e. when taking partial derivatives of the log-likelihood function, the derivative of a constant is zero).]

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 A to time B (or from time 0 to time B if left censored), where A < B. (Note: In Weibull++, left censored data are entered the same way as interval but by entering 0 (zero) for the Last Inspected time entry in the Data Sheet.)

In the case of interval data, and given P interval observations, the likelihood function is modified by multiplying the likelihood function with an additional term as follows:

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.

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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,

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