Crow Extended - Continuous Evaluation

The Crow Extended model is designed for a single test phase. However, in most cases, testing for a system will be conducted in multiple phases. The Crow Extended - Continuous Evaluation model is designed for analyzing data across multiple test phases, while considering the data for all phases as one data set. To perform the analysis in the RGA software, you would use the Multi-Phase data sheet.

The Crow Extended - Continuous Evaluation (3-parameter) model is an extension of the Crow Extended model, and is designed for practical testing situations where we need the flexibility to apply corrective actions at the time of failure or at a later time during the test, at the end of the current test or during a subsequent test phase. This three-parameter model is free of the constraint where the test must be stopped and all BD modes must be corrected at the end of the test, as in the Crow Extended model. The failure modes can be corrected during the test or when the testing is stopped for the incorporation of the fixes, or even not corrected at all at the end of the test phase. Based on this flexibility, the end time of the test is also not predefined, and it can be continuously updated with new test data. This is the reason behind the name, "continuous evaluation."

Classifications
Under the Crow Extended - Continuous Evaluation model, corrective actions can be fixed at the time of failure or delayed to a later time (before time $$T\,\!$$, at time $$T\,\!$$ or after time $$T\,\!$$, where $$T\,\!$$ indicates the test's end time). The definition of delayed is expanded to include all type B failure modes corrected after the time of failure. This will include most, if not all, design-related failure modes requiring a root cause failure analysis. Failure modes that are corrected at the time of failure are typically related to human factors, such as manufacturing, operator error, etc.

For the Crow Extended - Continuous Evaluation model, the classifications are defined as follows:


 * A indicates that a corrective action will not be performed (management chooses not to address these modes for technical, financial or other reasons).
 * BC is defined as a failure mode that receives a corrective action at the time of failure and before the testing resumes. Typically, a type BC failure mode does not require extensive root cause failure analysis, and therefore can be corrected quickly. Type BC modes are generally easy to fix, and are usually related to issues such as quality, manufacturing, operator, etc.
 * BD is defined as a failure mode that receives a corrective action at a test time after the first occurrence of the failure mode. Therefore, a fix is considered delayed if it is not implemented at the time of failure. A delayed fix can then occur at a later time during the test, at the end of the test (before the next phase) or during another test phase. Type BD failure modes typically require failure analysis and time to fabricate the corrective action. During the period $$(0,T)\,\!$$ into the test, there may be BD modes with corrective actions incorporated into the systems and other BD modes that have been seen, but not yet fixed.

The following table shows a comparison between the definitions of the classifications in the Crow Extended and Crow Extended - Continuous Evaluation models:

Reliability growth is achieved by decreasing the failure intensity. The failure intensity for the A failure modes will not change. Therefore, reliability growth can be achieved only by decreasing the BC and BD mode failure intensities. In general, the only part of the BD mode failure intensity that can be decreased is the part that has been seen during testing. The BC failure modes and fixed BD modes (delayed fixes implemented during the test) are corrected during testing, and their failure intensities will not change any more at the end of test.

Event Codes
A Multi-Phase data sheet that is analyzed with the Crow Extended - Continuous Evaluation model has an Event column that allows you to indicate the types of events that occurred during testing. The possible event codes that can be used in the analysis are:


 * F: indicates a failure time.
 * I: denotes that a fix has been implemented for a BD failure mode at the specific time. BD modes that have not received a corrective action by time $$T\,\!$$ would not have an associated I event in the data set.
 * Q: indicates that the failure was due to a quality issue. An example of this might be a failure caused by a bolt not being tightened down properly. You have the option to decide whether or not to include quality issues in the analysis. This option can be specified by checking or clearing the Include Q Events check box under Continuous Options on the Analysis tab.
 * P: indicates that the failure was due to a performance issue. You can determine whether or not to include performance issues in the analysis. This option can be specified by checking or clearing the Include P Events check box under Continuous Options on the Analysis tab.
 * AP: indicates an analysis point. Analysis points can be shown in a multi-phase plot to track overall project progress and can be compared to an idealized growth curve.
 * PH: indicates the end of a test phase. Test phases can be shown in a multi-phase plot to track overall project progress and can be compared to planned growth phases. The RGA software allows for a maximum of seven test phases to be defined.
 * X: indicates that the data point will be excluded from the analysis. An X can be placed in front of any existing event code or entered by itself. The row of data with the the X will not be included in the analysis.

The next figure shows an example of a Crow Extended - Continuous Evaluation folio with all of the possible event codes. As you can see, each failure is indicated with A, BC or BD in the Classification column. In addition, any text can be used to specify the mode. In this figure, failure codes were used in the Mode column for simplicity, but you could just as easily use Seal Leak, or whatever designation you deem appropriate for the failure mode.



$$p\,\!$$ Ratio
In the Crow Extended - Continuous evaluation, there is a certain probability of not incorporating a corrective action by time $$T\,\!$$. This is the additional (third parameter), as compared to the Crow Extended model. We define $$p\,\!$$ as:


 * $$p=\frac{\text{All modes such that, if seen, will be corrected at time }T\text{, or later}}{\text{All failure modes such that, if seen, the corrective action will be delayed}}\,\!$$

It is assumed that the ratio $$p\,\!$$ remains fixed over $$(0,T)\,\!$$. This implies that each time a distinct failure mode is seen, the probability that the corrective action is delayed until time $$T\,\!$$ or later is $$p\,\!$$. In other words, each time a new BD mode is seen over $$(0,T)\,\!$$ there is a probability $$p\,\!$$ that the corrective action for that mode will not have been incorporated into the system by time $$T\,\!$$. Under the Crow Extended two-parameter model, this probability is always equal to 1 at time $$T.\,\!$$

Effectiveness Factors
It is very important to note that failure modes are rarely totally eliminated by a corrective action. After failure modes have been found and fixed, a certain percentage of the failure intensity will be removed, but a certain percentage of the failure intensity will also remain. For each BD mode, an effectiveness factor (EF) is required to estimate how effective the corrective action will be in eliminating the failure intensity due to the failure mode. The EF is the fractional decrease in a mode's failure intensity after a corrective action has been made, and it must be a value between 0 and 1. A study on EFs showed that an average EF, $$d,\,\!$$ is about 0.7. Therefore, about $$30%\,\!$$, (i.e., $$100(1-d)%\,\!$$) of the BD mode failure intensity will typically remain in the system after all of the corrective actions have been implemented. However, individual EFs for the failure modes may be larger or smaller than the average.

Similar to the Crow Extended model, each BD mode has an effectiveness factor that represents the decrease in failure intensity for that mode once the corrective action has been incorporated into the system. In addition, a delayed fix can be incorporated any time after the first occurrence of the failure mode. Therefore, delayed fixes can be incorporated before the end of the test phase, at the end of the test phase (just like the Crow Extended delayed fixes) or not incorporated at all at the end of the current test phase but postponed for a subsequent test phase. For calculation purposes, any delayed fixes that are incorporated during the test (those with an I event code) do not need to have an effectiveness factor specified, since the fix is already incorporated in the system. The next figure shows how effectiveness factors are defined in the Crow Extended - Continuous Evaluation model in the RGA software. The figure also shows that you can specify whether the delayed fix was actually implemented at the end of the current test phase, at a later phase or not implemented at all.



For a Type BD failure mode that is not yet corrected but still deferred, the Actual Effectiveness Factor for that mode is zero. The Actual Effectiveness Factor for a deferred Type BD failure mode will stay at zero until the point when the corrective action will be incorporated is reached. At that time, the Actual Effectiveness Factor is changed to equal the Assigned Effectiveness Factor for that mode. At this point, the Nominal Effectiveness Factor (the effectiveness factor, assuming that fixes are implemented at the end of the specific phase) and the Actual Effectiveness Factor are the same. In other words, if a fix is not incorporated for a BD mode, its actual effectiveness for reducing failure intensity is zero, and the assigned effectiveness factor will be used only for projecting the MTBF (or failure intensity). This topic will be discussed in the Growth Potential and Projections section.

The next figure shows an event report in the RGA software. At the end of the test phase, depending on whether the BD mode was specified as fixed or not fixed, the actual EF is zero (e.g., mode BD3000) or equal to the nominal EF (e.g., mode BD1500).



The Average Nominal EF is:


 * $${{d}_{N}}=\frac{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{Ni}}}{M}\,\!$$

where $$M\,\!$$ is the total number of open and distinct BD modes at time $${{T}_{J}}\,\!$$, and $${{d}_}\,\!$$ is the Nominal Effectiveness Factor as specified for each of the BD mode.

The Average Actual EF is:


 * $${{d}_{A}}=\frac{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{Ai}}}{M}\,\!$$

where $$M\,\!$$ is the total number of open and distinct BD modes at time $${{T}_{J}}\,\!$$, and $${{d}_}\,\!$$ is the Actual Effectiveness Factor at time $${{T}_{j}}\,\!$$ for each BD mode.

Growth Potential and Projections
The failure intensity that remains in the system will depend on the management strategy that determines the classification of the A, BC and BD failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, the failure intensity depends on $$h(t)\,\!$$, which is the rate at which unique BD failure modes are being discovered during testing. The rate of discovery drives the opportunity to take corrective actions based on the seen failure modes and it is an important factor in the overall reliability growth rate. The reliability growth potential is the limiting value of the failure intensity as time $$T\,\!$$ increases. This limit is the maximum MTBF that can be attained with the current management strategy. The maximum MTBF will be attained when all BD modes have been observed and fixed.

If all seen BD modes are corrected by time $$T\,\!$$, that is, no deferred corrective actions at time $$T\,\!$$, then the growth potential is the maximum attainable based on the type BD designation of the failure modes, the corresponding assigned effectiveness factors and the remaining A modes in the system. This is called the nominal growth potential.

If some seen BD modes are not corrected at the end of the current test phase, then the prevailing growth potential is below the maximum attainable with the type BD designation of the failure modes and the corresponding assigned effectiveness factors.

The Crow-AMSAA (NHPP) model is used to estimate the current demonstrated MTBF or $$MTB{{F}_{D}}.\,\!$$ The demonstrated MTBF does not take into account any type of projected improvements. Refer to the Crow-AMSAA (NHPP) chapter for more details.

The corresponding current demonstrated failure intensity is:


 * $$\begin{align}

{{\lambda }_{D}}=\lambda \beta {{T}^{\beta -1}} \end{align}\,\!$$

or:


 * $${{\lambda }_{D}}=\frac{1}{MTB{{F}_{D}}}\,\!$$

The nominal growth potential factor is:


 * $${{\lambda }_{NGPFactor}}=\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{Ni}} \right)\frac{T}\,\!$$

where:


 * $$M\,\!$$ is the total number of distinct unfixed BD modes at time $${{T}_{j}}\,\!$$.
 * $${{d}_{Ni}}\,\!$$ is the assigned (nominal) EF for the $${{i}^{th}}\,\!$$ unfixed BD mode at time $${{T}_{j}}.\,\!$$
 * $${{N}_{i}}\,\!$$ is the total number of failures over (0, $${{T}_{j}}\,\!$$ ) for the distinct unfixed BD mode $$i\,\!$$.

The nominal growth potential factor signifies the failure intensity of the $$M\,\!$$ modes after corrective actions have been implemented for them, using the nominal values for the effectiveness factors.

Similarly, the actual growth potential factor is:


 * $${{\lambda }_{AGPFactor}}=\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{Ai}} \right)\frac{T}\,\!$$

where $${{d}_{Ai}}\,\!$$ is the actual EF for the $${{i}^{th}}\,\!$$ unfixed BD mode at time $${{T}_{j}}.\,\!$$

The actual growth potential factor signifies the failure intensity of the $$M\,\!$$ modes after corrective actions have been implemented for them, using the actual values for the effectiveness factors.

Based on the definition of BD modes for the Crow Extended - Continuous Evaluation model, the estimate of $$p\,\!$$ at time $${{T}_{j}}\,\!$$ is calculated as follows:


 * $$p=\frac{\text{Total number of distinct unfixed BD modes at time }{{T}_{j}}}{\text{Total number of distinct BD modes at time }{{T}_{j}}\text{ (both fixed and unfixed)}}\,\!$$

The unfixed BD mode failure intensity at time $${{T}_{j}}\,\!$$ is:


 * $$\lambda_{BDunfixed} = \frac{\text{Total number of unfixed BD failures at time} T_j}{T_j}\,\!$$

Similar to the Crow Extended model, the discovery function at time $$T\,\!$$ for the Crow Extended - Continuous Evaluation model is calculated using all the first occurrences of the all the BD modes, both fixed and unfixed. $$h(t)\,\!$$ is the unseen BD mode failure intensity and is also the rate at which new unique BD modes are being discovered.


 * $$\begin{align}

\widehat{h}(T|BD)&= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ & = & \frac{M{{\widehat{\beta }}_{BD}}}{T} \end{align}\,\!$$

where:


 * $${{\widehat{\beta }}_{BD}}\,\!$$ is the unbiased estimated of $$\beta \,\!$$ for the Crow-AMSAA (NHPP) model based on the first occurrence of $$M\,\!$$ distinct BD modes.
 * $${{\widehat{\lambda }}_{BD}}\,\!$$ is the unbiased estimated of $$\lambda \,\!$$ for the Crow-AMSAA (NHPP) model based on the first occurrence of $$M\,\!$$ distinct BD modes.

The parameters $${{\widehat{\beta }}_{BD}}\,\!$$ and $${{\widehat{\lambda }}_{BD}}\,\!$$ are also known as the Rate of Discovery Parameters.

The nominal growth potential failure intensity is:


 * $$\lambda_{NGP} = \lambda_D - \lambda_{BD\text{unfixed}} + \lambda_{NGP\text{Factor}} + d_N\cdot p\cdot h(T) - d_N h(T) \,\!$$

and the nominal growth potential MTBF is:


 * $$MTB{{F}_{NGP}}=\frac{1}\,\!$$

The nominal projected failure intensity at time $$T\,\!$$ is:


 * $$\begin{align}

{{\lambda }_{NP}}={{\lambda }_{NGP}}+{{d}_{N}}h(T) \end{align}\,\!$$

and the nominal projected MTBF at time $$T\,\!$$ is:


 * $$MTB{{F}_{NP}}=\frac{1}\,\!$$

The actual growth potential failure intensity is:


 * $$\lambda{AGP} = \lambda_D - \lambda_{BD\text{unfixed}} + \lambda{AGP\text{Factor} + d_A\cdot p\cdot h(T) - d_A h(T)} \,\!$$

and the actual growth potential MTBF is:


 * $$MTB{{F}_{AGP}}=\frac{1}\,\!$$

The actual projected failure intensity at time $$T\,\!$$ is:


 * $${{\lambda }_{AP}}={{\lambda }_{AGP}}+{{d}_{A}}\cdot h\left( T \right)\,\!$$

and the actual projected MTBF at time $$T\,\!$$ is:


 * $$MTB{{F}_{AP}}=\frac{1}\,\!$$

In terms of confidence intervals and goodness-of-fit tests, the calculations are the same as for the Crow Extended model.

=Example with One Phase=

=Example with Six Phases=

=Discrete Data= The Crow Extended - Continuous Evaluation model also can be applied for discrete data from one-shot (success/failure) testing. In the RGA software, you can create a multi-phase data sheet that is configured for mixed data to accommodate data from tests where a single unit is tested for each successive configuration (individual trial-by-trial), where multiple units are tested for each successive configuration (configurations in groups) or a combination of both.

Corrective actions cannot take place at the time of failure for discrete data. With that in mind, the mixed data type does not allow for BC modes. The delayed corrective actions over time can be either fixed or unfixed, based on a subsequent implementation (I) event. So, for discrete data there are only unfixed BD modes or fixed BD modes. As a practical application, think of a multi-phase program for missile systems. Because the systems are one-shot items, the fixes for the failure modes are delayed until at least the next trial.

Note that for calculation purposes, it is required to have at least three failures in the first interval. If that is not the case, then the data set needs to be grouped until this requirement is met before calculating. The RGA software performs this operation in the background.

Multi-Phase Discrete Data Example
A one-shot system underwent reliability growth development for a total of 20 trials. The test was performed as a combination of configuration in groups and individual trial-by-trial. The following table shows the obtained data set. The Failures in Interval column specifies the number of failures that occurred in each interval, and the Cumulative Trials column specifies the cumulative number of trials at the end of that interval.

The table also gives the classifications of the failure modes. There are 5 BD modes. Of these 5 modes, 2 are corrected during the test (BD3 and BD4) and 3 have not been corrected by time $$T=20\,\!$$ (BD1, BD2 and BD5). Do the following:


 * 1)	Calculate the parameters of the Crow Extended - Continuous Evaluation model.
 * 2)	Calculated the demonstrated reliability at the end of the test.
 * 3)	Calculate parameter $$p\,\!$$.
 * 4)	Calculate the unfixed BD mode failure probability.
 * 5)	Calculate the nominal growth potential factor.
 * 6)	Calculate the nominal average effectiveness factor.
 * 7)	Calculate the discovery failure intensity function at the end of the test.
 * 8)	Calculate the nominal projected reliability at the end of the test.
 * 9)	Calculate the nominal growth potential reliability at the end of the test.

Solution


 * 1)	The next figure shows the data entered in the RGA software.

The parameters $$\beta \,\!$$ and $$\lambda \,\!$$ are calculated as follows (the calculations for these parameters are presented in detail in the Crow-AMSAA (NHPP)chapter):


 * $$\widehat{\beta }=0.8572\,\!$$


 * and:


 * $$\widehat{\lambda }=0.4602\,\!$$


 * 2)	The corresponding demonstrated unreliability is calculated as:


 * $$\begin{align}

{{f}_{D}}=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0 \end{align}\,\!$$


 * Using the parameter estimates given above, the instantaneous unreliability at the end of the test (or $$T=20\,\!$$) is equal to:


 * $${{f}_{D}}(20)=0.8572\cdot 0.4602\cdot {{20}^{0.8572-1}}=0.2572\,\!$$


 * The demonstrated reliability is:


 * $$\begin{align}

{{R}_{D}}= & 1-{{f}_{D}} \\ = & 1-0.2572=0.7428 \end{align}\,\!$$


 * Note that in discrete data, we calculate the reliability and not MTBF because we are dealing with the number of trials and not test time.


 * 3)	Assume that the following effectiveness factors are assigned to the unfixed BD modes:


 * The parameter $$p\,\!$$ is the total number of distinct unfixed BD modes at time $$T\,\!$$ divided by the total number of distinct BD (fixed and unfixed) modes.


 * In this example:


 * $$p=\frac{3}{5}=0.6\,\!$$


 * 4)	The unfixed BD mode failure probability at time $$T\,\!$$ is the total number of unfixed BD failures (classified at time $$T\,\!$$ ) divided by the total trials. Based on the table at the beginning of the example, we have:


 * $${{f}_{BD, unfixed}}=\frac{4}{20}=0.2\,\!$$


 * 5)	The nominal growth potential factor is:


 * $$\begin{align}

{{\lambda }_{NGPFactor}}= & \underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{i}} \right)\frac{T} \\ = & \left( 1-{{d}_{1}} \right)\frac{T}+\left( 1-{{d}_{2}} \right)\frac{T}+\left( 1-{{d}_{5}} \right)\frac{T} \\ = & \left( 1-0.65 \right)\frac{1}{20}+\left( 1-0.70 \right)\frac{2}{20}+\left( 1-0.75 \right)\frac{1}{20} \\ = & 0.06 \end{align}\,\!$$


 * 6)	The nominal average effectiveness factor is:


 * $$\begin{align}

{{d}_{N}}= & \frac{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{Ni}}}{M} \\ = & \frac{0.65+0.70+0.75}{3} \\ = & 0.70 	\end{align}\,\!$$
 * 7)	The discovery function at time $$T\,\!$$ is calculated using all the first occurrences of all the BD modes, both fixed and unfixed. In our example, we calculate $${{\widehat{\beta }}_{BD}}\,\!$$ and $${{\widehat{\lambda }}_{BD}}\,\!$$ using only the unfixed BD modes and excluding the second occurrence of BD2, which results in the following:


 * $${{\widehat{\beta }}_{BD}}=0.6602\,\!$$


 * and:


 * $${{\widehat{\lambda }}_{BD}}=0.6920\,\!$$


 * So the discovery failure intensity function at time $$T=20\,\!$$ is:


 * $$\begin{align}

\widehat{h}(T|BD)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ = & 0.6920\cdot 0.6602\cdot {{20}^{0.6602-1}} \\ = & 0.16507 \end{align}\,\!$$


 * The next figure shows the plot for the discovery failure intensity function.




 * 8)	The nominal projected failure probability at time $$T=20\,\!$$ is:


 * $$\begin{align}

{{f}_{NP}}= & {{f}_{NGP}}+{{d}_{N}}h(T) \\ = & 0.0701+0.7\cdot 0.16507 \\ = & 0.1865 \end{align}\,\!$$


 * Therefore, the nominal projected reliability is:


 * $$\begin{align}

{{R}_{P}}= & 1-0.1856= \\ = & 0.8135 \end{align}\,\!$$


 * 9)	The nominal growth potential unreliability is:


 * $${{f}_{NGP}} = {{f}_{D}} - {{f}_{BD unfixed}} + {{\lambda}_{NGPFactor}} + {{d}_{N}}\cdot p \cdot h(T) - {{d}_{N}}h(T)\,\!$$


 * Based on the previous calculation for this example, we have:


 * $$\begin{align}

{{f}_{NGP}}= & 0.2572-0.2+0.06+0.7\cdot 0.6\cdot 0.16507-0.7\cdot 0.16507 \\ = & 0.0709 \end{align}\,\!$$


 * So the nominal growth potential reliability is:


 * $$\begin{align}

{{R}_{NGP}}= & 1-0.0709 \\ = & 0.9291 \end{align}\,\!$$

The next figure shows the plot for the instantaneous demonstrated, projected and growth potential reliability.



=Statistical Tests for Effectiveness of Corrective Actions= The purpose of the statistical tests is to explore the effectiveness of corrective actions during or at the end of a phase. Say we have two phases, phase 1 and phase 2. Suppose that corrective actions are incorporated during or at the end of phase 1. The system is then operated during phase 2. The general question is whether or not the corrective actions have been effective.

There are two questions that can be addressed regarding the effectiveness of the corrective actions:

Question 1. Is the average failure intensity for phase 2 statistically less than the average failure intensity for phase1?

Question 2. Is the average failure intensity for phase 2 statistically less than the Crow-AMSAA (NHPP) instantaneous failure intensity at the end of phase 1?

Average Failure Intensities Test
The purpose of this test is to compare the average failure intensity during phase 2 with the average failure intensity during phase 1.

The average failure intensity for phase 1 is:


 * $${{\overline{r}}_{1}}=\frac\,\!$$

where $${{T}_{1}}\,\!$$ is the phase 1 test time and $${{N}_{1}}\,\!$$ is the number of failures during phase 1.

Similarly, the average failure intensity for phase 2 is:


 * $${{\overline{r}}_{2}}=\frac\,\!$$

where $${{T}_{2}}\,\!$$ is the phase 2 test time and $${{N}_{2}}\,\!$$ is the number of failures during phase 2.

The overall test time, $$T\,\!$$, is:


 * $$\begin{align}

T={{T}_{1}}+{{T}_{2}} \end{align}\,\!$$

The overall number of failures, $$N\,\!$$, is:


 * $$\begin{align}

N={{N}_{1}}+{{N}_{2}} \end{align}\,\!$$

Define $$P\,\!$$ as:


 * $$P=\frac{T}\,\!$$

If the cumulative binomial probability $$B(k;P,N)\,\!$$ of up to $${{N}_{2}}\,\!$$ failures is less than or equal to the statistical significance $$\alpha \,\!$$, then the average failure intensity for phase 2 is statistically less than the average failure intensity for phase 1 at the specific significance level.

The cumulative binomial distribution probability is given by:


 * $$B(k;P,N)=\underset{f=0}{\overset{k}{\mathop \sum }}\,\left( \begin{matrix}

N \\ f \\ \end{matrix} \right){{P}^{f}}{{\left( 1-P \right)}^{N-f}}\,\!$$

which gives the probability that the test failures, $$f\,\!$$, are less than or equal to the number of allowable failures, or acceptance number $$k\,\!$$ in $$N\,\!$$ trials, when each trial has a probability of succeeding of $$P\,\!$$.

Example
Suppose a test is being conducted and is divided into two phases. The test time for the first phase is $${{T}_{1}}=27\,\!$$ days, and the test time for the second phase is $${{T}_{2}}=18\,\!$$ days. The number of failures during phase 1 is $${{N}_{1}}=11\,\!$$, and the number of failures during phase 2 is $${{N}_{2}}=2.\,\!$$

The average failure intensity for phase 1 is:


 * $${{\overline{r}}_{1}}=\frac=\frac{11}{27}=0.4074.\,\!$$

Similarly, the average failure intensity for phase 2 is:


 * $${{\overline{r}}_{2}}=\frac{T2}=\frac{2}{18}=0.1111\,\!$$

Although the average failure intensities are different, we want to see, if at the 10% statistical significance level, the average failure intensity for phase 2 is statistically less than the average failure intensity for phase 1.

Solution

Concerning the total test time, we have:


 * $$\begin{align}

T={{T}_{1}}+{{T}_{2}}=27+18=45 \end{align}\,\!$$

The total number of failures is equal to:


 * $$\begin{align}

N={{N}_{1}}+{{N}_{2}}=11+2=13 \end{align}\,\!$$

Then, we calculate $$P\,\!$$ as:


 * $$P=\frac{T}=\frac{18}{45}=0.4\,\!$$

Using the cumulative binomial distribution probability equation, we have:


 * $$\begin{align}

B(k;P,N)= & B({{N}_{2}};P,N) \\ = & \underset{f=0}{\overset{\mathop \sum }}\,\left( \begin{matrix}  N  \\   {{N}_{2}}  \\ \end{matrix} \right){{P}^{f}}{{\left( 1-P \right)}^{N-f}} \\ = & \underset{f=0}{\overset{2}{\mathop \sum }}\,\left( \begin{matrix}  13  \\   f  \\ \end{matrix} \right){{0.4}^{f}}{{\left( 1-0.4 \right)}^{13-f}} \\ = & 0.058 \end{align}\,\!$$

Because 0.058 is lower than 0.10, the conclusion is that at the 10% significance level of the the average failure intensity for phase 2 is statistically less than the average failure intensity for phase 1. The conclusion would be different for a different significance level. For example, at the 5% significance level, since 0.058 is not lower than 0.05, we fail to reject the null hypothesis. In other words, we cannot statistically prove any significant difference between the average failure intensities at the 5% level.

Average vs. Demonstrated Failure Intensities Test
The purpose of this test is to compare the average failure intensity during phase 2 with the Crow-AMSAA (NHPP) instantaneous (demonstrated) failure intensity at the end of phase 1.

Once again, the average failure intensity for phase 2 is given by:


 * $$\overline{r_{2}}=\frac{N_{2}}{T_{2}}$$

where $$T_{2}\,\!$$ is the phase 2 test time and $$N_{2}\,\!$$ is the number of failures during phase 2.

The Crow-AMSAA (NHPP) model estimate of failure intensity at time is $$\hat{r}(T_{1})$$. Dr. Larry H. Crow [16] showed that the Crow-AMSAA (NHPP) estimate is approximately distributed as a random variable with standard deviation $$\textstyle \sqrt{\frac{N_{1}}{2}}$$.

We therefore treat $$\hat{r}(T_{1})$$ as an approximate Poisson random variable with number of failures:


 * $$N_{1}^{*}=\frac{N_{1}}{2}$$

We also set:


 * $$T_{1}^{*}=\frac{N_{1}^{*}}{\hat{r}(T_{1})}$$

Then we define:


 * $$T=T_{1}^{*} + T_{2}$$

and:


 * $$N=N_{1}^{*} + N_{2}$$

Let:


 * $$P=\frac{T_{2}}{T}$$

If the cumulative binomial probability $$B(k;P,N)\,\!$$ of up to $$N_{2}\,\!$$ failures is less than or equal to the statistical significance $$\alpha_{1}\,\!$$, then the average failure intensity for phase 2 is statistically less than the Crow-AMSAA (NHPP) instantaneous failure intensity at the end of phase 1, at the specific significance level. The cumulative binomial distribution probability is again given by:


 * $$B(k;P,N)=\underset{f=0}{\overset{k}{\mathop \sum }}\,\left( \begin{matrix}

N \\ f \\ \end{matrix} \right){{P}^{f}}{{\left( 1-P \right)}^{N-f}}\,\!$$

Example
Suppose a test is being conducted and is divided into two phases. The test time for the first phase is $$T_{1}=21\,\!$$ days, and the test time for the second phase is $$T_{2} = 18\,\!$$ days. The number of failures during phase 1 is $$N_{1}=11\,\!$$ and the number of failures during phase 2 is $$N_{2}=2\,\!$$. The Crow-AMSAA (NHPP) parameters for the first phase are $$\beta = 0.7189\,\!$$ and $$\lambda = 1.0288\,\!$$.

The demonstrated failure intensity at the end of phase 1 is calculated as follows:


 * $$\begin{align}

\hat{r}(T_{1}) &= \lambda \beta T^{\beta - 1} \\ &= 1.0288 \cdot 0.7189 \cdot 27^{0.7189-1} \\ &= 0.2929 \end{align}$$

The average failure intensity for phase 2 is:


 * $$\overline{r}_{2} = \frac{N_{2}}{T_{2}} = \frac{2}{8} = 0.1111$$

Determine if the average failure intensity for phase 2 is statistically less than the demonstrated failure intensity at the end of phase 1 at the 10% significance level.

Solution

We calculate the number of failures $$N_{1}^{*}$$ as:


 * $$N_{1}^{*}=\frac{N_{1}}{2}= \frac{11}{2}=5.5$$

Then:


 * $$T_{1}^{*}= \frac{N_{1}^{*}}{\hat{r}(T_{1})}= \frac{5.5}{0.2929}= 18.78$$

Concerning the total test time, we have:


 * $$T = T_{1}^{*} + T_{2} = 18.78 + 18 = 36.78$$

Concerning the total number of failures, we have:


 * $$N = N_{1}^{*} + N_{2} = 5.5 + 2 = 7.5$$

Then we calculate $$P$$ as:


 * $$P = \frac{T_{2}}{T} = \frac{18}{36.78} = 0.4894$$

Since the number of failures $$N = 7.5\,\!$$ is not an integer, we are going to calculate the cumulative binomial probabilities for $$N = 7\,\!$$ and $$N = 8\,\!$$, and then interpolate to $$N = 7.5\,\!$$.

For $$N = 7\,\!$$, we have:


 * $$\begin{align}

B(k;P,N) &= B(N_{2};P,N)\\ &= B(2;0.4894,7) \\ &= \sum_{f=0}^{N_{2}}\begin{pmatrix}N\\ N_{2}\end{pmatrix}p^{f}(1-P)^{N-f} \\ &= \sum_{f=0}^{2}\begin{pmatrix}7\\ f\end{pmatrix}0.4894^{f}(1-0.4849)^{7-f} \\ &= 0.244 \end{align}$$

And for $$N=8\,\!$$, we have:


 * $$\begin{align}

B(k;P,N) &= B(N_{2};P,N)\\ &= B(2;0.4894,8) \\ &= \sum_{f=0}^{N_{2}}\begin{pmatrix}N\\ N_{2}\end{pmatrix}p^{f}(1-P)^{N-f} \\ &= \sum_{f=0}^{2}\begin{pmatrix}8\\ f\end{pmatrix}0.4894^{f}(1-0.4849)^{8-f} \\ &= 0.158 \end{align}$$

A linear interpolation between two data points $$(x_{a},y_{a})\,\!$$ and $$(x_{b},y_{b})\,\!$$ at the $$(x,y)\,\!$$ interpolant is given by:


 * $$y = y_{a} + (x - x_{a})\frac{(y_{b}-y_{a})}{(x_{b}-x_{a})}\,\!$$

So for $$N = 7.5\,\!$$ we would have that:


 * $$\begin{align}

B(N_{2};P,7.5) &= B(N_{2};P,7) + (7.5 - 7)\frac{B(N_{2};P,8)-B(N_{2};P,7)}{8-7} \\ &= 0.244 + (7.5 - 7) \frac{0.158 - 0.244}{8 - 7} \\ &= 0.201 \end{align}$$

Since 0.201 is greater than 0.10, the conclusion is that at the 10% significance level the average failure intensity for phase 2 is not statistically different compared to the demonstrated failure intensity at the end of phase 1.

Using the example with the six phases that was given earlier in this chapter, the following figure shows the application of the two statistical tests in RGA.