Confidence Levels_Failure Rates_Confidence Intervals

MTBF = 2T/X2( , 2r+2) MTBF = mean time between failures T = Operating time = 1 confidence level r = # failures, if 0, 2r + 2 = 2, which is the degrees of freedom MTBF95% conf = 2T/ X2(0.05, 2) MTBF50% conf = 2T/ X2 (0.5, 2)

2T = (MTBF50% conf)(X2

Substituting: MTBF95% conf = (MTBF50% conf)(X Therefore (X2(0.5, 2)/ X2(0.05, 2)) = 0.23137 And, MTBF95% conf = MTBF50% conf (0.23137) R = exp[-t/MTBF]

(0.5, 2)) 2 (0.5, 2))/ X (0.05, 2)

2

= MTBF50% conf (X2(0.5, 2)/ X2(0.05, 2))

From table look-up[1], X2(0.05, 2) = 5.99146; X2(0.5, 2) = 1.38629

[1] Probability & Statistics for Engineers & Scientists 7th Ed., Walpole, Myer, Myer, and Ye, Prentice Hall Publishers, NJ, 2002

Reliability @ 50% Time: Failure Rate: MTBF result: R: 1 hr 3.30E-06 units 303030.303 same units 0.99999670000544500000 0.999994733 Change 1-hr to 30 Min. Failure Rate: 60 MIN: 30 Min.: R: 5.23E-06 8.71119E-08 2.61336E-06 0.999997387

Change the 50% to 95% CL Reliability @ 95% Time: 1 hr

MTBF result: R:

70112.12121 0.999985737

Confidence IntervalsConfidence intervals for fraction nonconforming -- normal distribution. Z=x- W Where: x = any value of interest, such as a specification limit. W = the population standard deviation Z = normal distribution value for a given confidence level Example: The upper specification limit for a process is 90. The process average is 88.5 and the standard deviation is 1.23. Both of these values were estimated from a relatively large sample size (n = 150). Using the above equation, calculate: Z = 90 - 88.5 = 1.22 1.23 Using Excel: x = 90 = 88.5 W =1.23 Z = 1.22

ESTIMATING SAMPLE-SIZE FOR MAGNITUDE-BASED INFERENCES Sample sizes via statistical significance appear further down on this sheet. Author: Will G Hopkins, AUT University, NZ. Email: [email protected] The method is based on keeping chances of what I call Type 1 and Type 2 clinical errors acceptably low. A Type 1 clinical error occurs when, on the basis of your study, you decide to use an effect that is actually harmful. I have set the default chances to Type 1 = 0.5% and Type 2 = 25%. You need this kind of disparity if you want a reasonably low chance of an You may change cells with numbers in blue. Sample size and other useful stats are in red and plum. Do not change the cells in red. You may change cells in plum to see the effect on The smallest change, difference, correlation, relative risk or odds ratio is the smallest value of the statistic that has a substantial effect on the health, wealth, The larger the true effect, the smaller the sample size for a clear outcome. To estimate the sample size for a suspected true effect, replace the Ignore the error-variance factor, which is used to calculate the sample size and other stats. Ignore the grey cells on the far right. Sample sizes Type 2 chance of benefit

-0.50

0.50

0.50most unlikely

25unlikely

15unlikely

0most unlikely

91likely

9unlikely

2.3very unlikely

0.2most unlikely

15

"" 0.50

Choose a harmful value

Choose a beneficial value

Chances the true effect is: harmful beneficial

Decide to use effect clinically

Chances of these outcomes when the true effect is null Observe the following magnitudes unclear trivial any non-trivial likely non-trivial very likely nontrivial

Chances of observin true effe >Type 2 chance of benefit

-0.50

0.50

0.50most unlikely

25unlikely

17unlikely

1very unlikely

89likely

10unlikely

2.2very unlikely

0.1most unlikely

17

"" 0.20







0.20

-0.20

0.5most unlikely

25unlikely

17unlikely

1very unlikely

88likely

10unlikely

2.2very unlikely

0.1most unlikely

17

"" approx







0.10

-0.10

0.10

0.5most unlikely

25unlikely

17unlikely

1very unlikely

89likely

10unlikely

2.1very unlikely

0.1most unlikely

17

"vfz" 1.10







1.10

0.91

0.5most unlikely

25unlikely

17unlikely

2very unlikely

88likely

10unlikely

2.1very unlikely

0.1most unlikely

17

"vfz" 1.10







1.10

0.91

0.5most unlikely

25unlikely

17unlikely

2very unlikely

88likely

10unlikely

2.1very unlikely

0.1most unlikely

17

"" 0.35

"" 0.35

"" 0.14

"" approx

0.07

"vfz" 1.07

"vfz" 1.07

6 trials:

100*LOG-Transformed Data (Clear any #NUM! Restore formula for new data.) 1 2 3 4 0 0Alex -35.7 Ariel 104.0 Ashley -34.2 Bernie 269.5 Casey -123.8 Chris 119.7 Corey 56.5 Courtney 212.9 Devon 130.0 Drew 65.2 Dylan -63.5 Frances -281.3 Gene 101.2 Jaimie 54.8 Jean 77.5 Jesse 1.0 Jo 0.0 Jody 55.4 Jordan 313.9 Kade 39.9 Mean 53.15 SD 132.66 N 20 Include which trials? 1 DF 19 1.7 Back-transformed mean SD as a / factor 3.77 SD as a CV (%) 276.8 156.2 31.5 119.1 76.5 -49.4 72.8 148.4 219.8 173.5 131.9 -75.5 -177.2 100.8 217.5 85.9 138.6 32.9 70.3 292.1 214.0 98.99 110.82 20 1 19 2.7 3.03 202.9 202.6 34.4 150.2 269.7 -6.2 81.1 107.8 244.5 239.0 148.2 51.9 -77.7 195.0 107.5 0.0 150.9 10.4 91.2 328.3 149.5 123.91 104.41 20 1 19 3.5 2.84 184.1 104.0 135.1 #NUM! #NUM! #NUM! #NUM! #NUM! 265.9 205.5 190.4 87.1 -61.6 156.7 198.1 47.0 242.4 30.7 62.1 330.0 218.0 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

Do not modify

Mean

92.0 116.6 0 0 0 0 0 0 0 0 0

0

0

Mean 2.5 3.21 220.9

Measure4-3 -4 Mean Clear any #NUM! from the log-transformed data to stop them plotting as zeros. Restore the log-transformation formula to those cells if you enter new data.400 300 250 200 150 100 50 0 -50 0 -100 -150 -200 -250 Trial 1

Factors Facto

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2.88 2.40 3.73 1.25 1.01 30

200 Trial 2 100 0 -500 -100 0 -200 -300 Trial 1 500 Trial 2-1

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0.52 0.44 0.68 1.25 1.01

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350 300 250 200 150 100 50 0 -200 -50 0 -100 Trial 2 350

250 200 150 Trial 3-2 100 50 0 -400 200 400 -200-50 0 -100 -150 Trial 2 150 200 400

Conf. limits Typical

Trial 3

Conf. limits a Bias appr Standardi

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0.75 0.57 0.87 2.6 8.5 19 30 4.3 17.7

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Trial 4

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1 0.8 Trial 5 0.6 0.4 0.2 0 -200 0 Trial 4 200 400 -200 Trial 5-4

1 0.8 0.6 0.4 0.2 0 0 200 Trial 4 400

B Intra

Measures

1 0.8 Trial 6 0.6 0.4 0.2 0 0 0.5 Trial 5 1

1 0.8 Trial 6-5 0.6 0.4 0.2 0 0 0.5 Trial 5 1

Bia

Stdized

Std

Bia

Ef

Hover cursor for >6 trials:

Change scores of Log-transformed data 2-1 3-2 4-3 -4191.9021 -72.5466 153.3378 -192.983 74.3578 -46.94 91.85609 6.89136 43.49975 66.67604 -12.0144 104.1454 -0.3643 162.663 8.393445 137.6344 32.93037 14.92124 -21.8391 174.129 Mean 45.8 SD 95.1 N 20 46.31669 2.877896 31.09651 193.1184 43.24209 8.338161 -40.54651 24.66173 65.44907 16.25189 127.3816 99.54281 94.22288 -109.9749 -85.86616 12.22176 -22.49437 20.91852 36.17804 -64.49174 24.9 71.7 20 -98.5236 100.7077

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Do not modify this col.

Anysubject? #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0

21.39076 -33.4275 42.19944 35.24996 16.03427 -38.3656 90.5999 47.00036 91.54054 20.31247 -29.1706 1.748231 68.52687 22.4 55.7 15

Total no. subjects 0 0

Trials

2-1

3-2

4-3

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-

Mean

1.58 1.28 Lower conf. limit 1.09 0.97 Upper conf. limit 2.28 1.69 1.44 1.32 Factor typical error 1.96 1.66 Lower conf. limit 1.71 1.50 Upper conf. limit 2.51 2.00 1.21 1.16 ias correction factor 1.01 1.01 Degrees of freedom 19 19

0

0

0

hange in mean (%) 58.1 28.3 Lower conf. limit 9.5 -2.8 Upper conf. limit 128.5 69.3 59.5 36.0 96.0 66.1 Lower conf. limit 70.6 49.6 Upper conf. limit 151.4 100.4 1.46 1.42 1.02 1.02 Change in mean 0.37 0.23 Lower conf. limit 0.07 -0.03 Upper conf. limit 0.68 0.49

onf. limits as value 0.30 0.26 Typical error 0.55 0.47 Lower conf. limit 0.44 0.37 Upper conf. limit 0.75 0.65 1.31 1.31 Bias correction factor 1.01 1.01

earson correlation 0.71 0.78 Lower conf. limit 0.45 0.57 Upper conf. limit 0.86 0.89 Bias correction factor 1.01 1.01 raclass correlation 0.72 0.80 Lower conf. limit 0.48 0.60 Upper conf. limit 0.86 0.90

2-1

3-2

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-4

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Mean

Change in mean 45.83 24.92 Lower conf. limit 9.05 -2.81 Upper conf. limit 82.62 52.66 Typical error 67.27 50.72 Lower conf. limit 53.41 40.27 Upper conf. limit 92.19 69.51 ias correction factor 1.01 1.01 DF 19 19

#DIV/0! #DIV/0! #DIV/0! 0 0 0 #DIV/0!

0.37 0.23 Lower conf. limit 0.07 -0.03 Upper conf. limit 0.68 0.49 tdized typical error 0.55 0.47 Lower conf. limit 0.44 0.37 Upper conf. limit 0.75 0.65

Pearson r 0.71 0.78 Lower conf. limit 0.45 0.57 Upper conf. limit 0.86 0.89 ias correction factor 1.01 1.01 Intraclass r 0.72 0.80 Lower conf. limit 0.48 0.60 Upper conf. limit 0.86 0.90 Effective no. of trials 2.0 2.0 F 6.1 8.8 Numerator DF 19 19 Denominator DF 19 19 F lower 2.8 4.1 F upper 13.2 19.1

0

0

0

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

MIL-HDBK-213F, Table 4-1: R Major Concerns: MODELS: * Are all functional elements included in the reliability block diagram /model? * Are all modes of operation considered in the math mode?. * Does the math model results show that the design achieves the reliability requirement? ALLOCATION: * Are system reliability requirements allocated (subdivided) to useful levels? * Does the allocation process consider complexity, design flexibility, and safety margins? PREDICTION: * Does the sum of the parts equal the value of the module or unit? - Are environmental conditions and part quality representative of the requirements? * Are the circuit and part temperatures defined and do they represent the design? * Are equipment, assembly, subassembly and part reliability drivers identified? * Are alternate (Non MIL-HDBK-217) failure rates highlighted along with the rationale for their use? * Is the level of detail for the part failure rate models sufficient to reconstruct the result? - Are critical components such as VHSIC, Monolithic Microwave Integrated Circuits (MMIC), Application Specific Integrated Circuits (ASIC) or Hybrids highlighted?

Comments: * System design drawing/diagrams must be reviewed to be sure that the reliability model/diagram agrees with the hardware. * Duty cycles, alternate paths, degraded conditions and redundant units must be defined and modeled * Unit failure rates and redundancy equations are used from the detailed part predictions in the system math model (See MIL-HDBK-338 Revision B Electronic Reliability Design Handbook) * Useful Ievels are defined as: equipment for subcontractors, assemblies for sub-subcontractors, circuit boards for designers. * Conservative values are needed to prevent reallocation at every design change. * Many prediction neglect to include all the parts producing optimistic results (check for solder connections, connectors, circuit boards). * Optimistic quality levels and favorable environmental conditions are often assumed causing optimistic results * Temperature is the biggest driver of part failure rates; low temperature assumptions will cause optimistic results. * Identification is needed so that corrective actions for reliability improvement can be considered. * Use of alternate failure rates, if deemed necessary; require submission of backup data to provide credence in the values. * Each component type should be sampled and failure rates completely reconstructed for accuracy. * Prediction methods for advanced technology parts should be carefully evaluated for impact on the module and system.

Fault Tree Analysis (FTA)DEFINITIO N Fault Tree Analysis ( FTA ) is a top-down approach to failure mode analysis. An FTA identifies failures and strives to eliminate the cause of the failure.

Reliability Block DiagramsDEFINITION Reliability Block Diagrams ( RBD 's) establish system reliability on a modular/block basis rather than a component basis using a block diagram approach.

SITUATION While troubleshooting a failure or trying to identify possible causes to a specific failure effect, an FTA can be a very useful tool. SITUATION For complex systems, RBD's make the reliability of a system much easier to understand, expose weaknesses much quicker, and make what-if analyses much easier.

OBJECTIV E FTA is a systematic, deductive method for defining a single specific undesirable event and determining all possible failures that could VALUE TO YOUR ORGANIZATION Although an FTA can be very useful in the initial product design phase as an evaluation tool, it is probably more powerful as a troubleshooting tool after an event (or proposed event) has taken

OBJECTIVES To develop a reliability model using blocks or modules to make the model easier to understand and change.

VALUE TO YOUR ORGANIZATION Used early in a design cycle, it allocates reliability among blocks, guiding architecture and design decisions to achieve an overall system reliability requirement. Used on established designs, it sorts every component into blocks according to thei function to expose the distribution of failure rates.

RELIABILITY INTEGRATION An example of Reliability Integration during Fault Tree Analysis is as follows:

RELIABILITY INTEGRATION An example of Reliability Integration during Reliability Block Diagramming (RBD) is as follows: Information from Reliability Block Diagrams feed directly into design decisions If the block diagram concludes that the only way to meet the design goal is to add redundancy, then this becomes input to the design early on so that cost and schedule impacts are kept to a minimum. METHODOLOGY We start out with a high levelBOM or functional block diagram and from that, we establish redundancy paths and assign an MTBF to each block. We make this assignment from vendor data, past products, benchmarking, and a variety of other means. Then we calculate the total system reliability based on each block.

Using FTA's during HALT planning: When a FMECA identifies a critical effect, an FTA is often deployed to evaluate all possible failure modes that can also cause the same critical effect. This is especially helpful when planning a HALT so that the appropriate stresses can be applied and so that the failure can easily be troubleshot if the critical effect is exposed during

METHODOLOGY When we perform an FTA, we start with an undesired event. The undesired FMECA's are very similar in this a fault tree diagram. FTA's and event constitutes the top event inregard but the goal is We much different. Whereas a FMECA is trying to identifying all How we decide whether to use an FTA or a FMECA

CASE STUDIES/OPTIONS The following case studies and options provide example approaches. We shall tailor our approach to meet your specific situation.

FTA is preferred over FMECA when: A small number of top events can be identified Product functionality is highly complex The product is not repairable once initiated FMECA is preferred over FTA when:

1) RDB's Make Understanding Reliability of Complex Systems Easier For a Networking company with a highly complex core switch, we developed a reliability block diagram to make the reliability of the system easier to understand.

2) Using RDB's to Help Drive Design Decisions A Medical Device company had a system comprised of many blocks. We developed a Reliability Block diagram to help them understand the driving forces of reliability early in a design effort and created a functional block diagram to determine where to concentrate design efforts.

The events cannot be limited to a small number Multiple successful functional profiles are possible Identification of "all possible" failure modes is important

3) Using RDB's to Drive Redundancy Decisions A Military contractor had a specific reliability goal in mind and had a choice of using redundancies but needed to make this decision very early in the design process. We developed a Reliability Block Diagram for them and showed them where they could take advantage of redundancies to meet their goals.

CASE STUDIES/OPTIONS The following case studies and options provide example approaches. We shall tailor our approach to meet your specific 1) Using FTA's to Identify Safety-Critical Failures A Computer manufacturer had a safety-critical failure in their product and they wanted to identify if there were any other failure 2) Using FTA's During Failure Analysis For a Medical Device company, we showed them how to use FTA

Confidence Levels_Failure Rates_Confidence Intervals

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