Multiple failure modes analysis (Kaplan-Meier estimation method) for Nonparametric Distribution Analysis (Right Censoring)

Characteristics of variable – Multiple failure mode (Kaplan-Meier estimation method)

The mean (MTTF, mean time to failure) and the median are measures of the center of the distribution. The IQR is a measure of the spread of the distribution.

Example output

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Break
Censoring Censoring Information Count Uncensored value 25 Right censored value 40

Estimation Method: Maximum Likelihood

Distribution: Weibull

Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.97672 0.276587 1.50260 2.60044 Scale 891.929 90.8270 730.552 1088.96

Log-Likelihood = -188.140

Goodness-of-Fit Anderson-Darling (Adjusted) 1.373
Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 790.635 80.2800 647.957 964.730 Standard Deviation 417.662 64.7460 308.228 565.951 Median 740.979 79.9464 599.746 915.471 First Quartile(Q1) 474.898 67.4629 359.485 627.365 Third Quartile(Q3) 1052.19 107.155 861.801 1284.64 Interquartile Range(IQR) 577.290 83.4174 434.907 766.287
Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 87.0276 30.6339 43.6548 173.493 2 123.896 37.7877 68.1466 225.252 3 152.497 42.3555 88.4796 262.833 4 176.847 45.7243 106.541 293.548 5 198.502 48.3870 123.105 320.077 6 218.260 50.5811 138.583 343.746 7 236.594 52.4406 153.227 365.317 8 253.812 54.0493 167.205 385.279 9 270.130 55.4632 180.636 403.963 10 285.703 56.7217 193.608 421.606 20 417.625 64.8194 308.086 566.111 30 529.457 69.7943 408.905 685.548 40 634.964 74.3928 504.686 798.871 50 740.979 79.9464 599.746 915.471 60 853.343 87.6525 697.736 1043.65 70 979.746 99.1411 803.489 1194.67 80 1134.71 117.529 926.234 1390.11 90 1360.10 152.029 1092.51 1693.23 91 1391.24 157.433 1114.50 1736.69 92 1425.26 163.497 1138.28 1784.59 93 1462.89 170.393 1164.31 1838.05 94 1505.19 178.371 1193.22 1898.73 95 1553.77 187.816 1226.02 1969.15 96 1611.28 199.369 1264.30 2053.50 97 1682.59 214.223 1311.01 2159.50 98 1778.36 235.032 1372.53 2304.18 99 1931.34 270.138 1468.25 2540.49

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Obstruction
Censoring Censoring Information Count Uncensored value 40 Right censored value 25

Estimation Method: Maximum Likelihood

Distribution: Lognormal

Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 5.75328 0.271171 5.22179 6.28476 Scale 1.95933 0.238720 1.54311 2.48780

Log-Likelihood = -284.213

Goodness-of-Fit Anderson-Darling (Adjusted) 98.132
Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 2149.00 1303.69 654.415 7056.97 Standard Deviation 14492.1 15344.4 1819.17 115449 Median 315.222 85.4790 185.266 536.337 First Quartile(Q1) 84.0770 22.7774 49.4399 142.981 Third Quartile(Q3) 1181.83 418.719 590.174 2366.65 Interquartile Range(IQR) 1097.76 407.901 529.932 2274.01
Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 3.30424 1.78563 1.14571 9.52940 2 5.63679 2.72980 2.18177 14.5631 3 7.91050 3.55915 3.27511 19.1066 4 10.2074 4.33709 4.43857 23.4741 5 12.5595 5.08849 5.67682 27.7867 6 14.9838 5.82646 6.99250 32.1079 7 17.4916 6.55916 8.38765 36.4772 8 20.0913 7.29230 9.86408 40.9221 9 22.7896 8.03022 11.4236 45.4641 10 25.5926 8.77646 13.0681 50.1206 20 60.5984 17.2863 34.6455 105.993 30 112.822 29.6226 67.4371 188.749 40 191.884 49.8160 115.359 319.171 50 315.222 85.4790 185.266 536.337 60 517.841 152.725 290.505 923.079 70 880.729 291.401 460.480 1684.51 80 1639.73 627.451 774.563 3471.28 90 3882.58 1807.19 1559.26 9667.69 91 4360.12 2080.97 1710.97 11111.0 92 4945.69 2424.60 1892.04 12927.8 93 5680.72 2866.84 2112.69 15274.7 94 6631.50 3454.60 2388.85 18409.2 95 7911.58 4269.92 2747.04 22785.7 96 9734.61 5470.91 3235.47 29288.6 97 12561.2 7407.95 3953.98 39904.9 98 17628.0 11054.7 5157.08 60256.0 99 30072.1 20656.8 7824.62 115575

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Break, Obstruction
Censoring Censoring Information Count Uncensored value 65

Estimation Method: Maximum Likelihood

Distribution: Weibull, Lognormal

Table of Percentiles 95.0% Normal CI Percent Percentile Lower Upper 1 3.30048 1.24014 10.5054 2 5.62678 2.33098 15.6865 3 7.89119 3.47003 20.2810 4 10.1752 4.67345 24.6448 5 12.5102 5.94720 28.9128 6 14.9126 7.29455 33.1540 7 17.3928 8.71781 37.4099 8 19.9581 10.2189 41.7082 9 22.6145 11.7997 46.0691 10 25.3668 13.4619 50.5080 20 59.0237 34.9419 101.298 30 106.275 66.5640 168.017 40 170.417 110.481 254.467 50 254.292 169.625 361.847 60 360.567 248.567 490.105 70 493.871 354.763 642.021 80 666.812 501.548 831.290 90 923.218 721.881 1112.21 91 958.607 751.307 1151.99 92 997.221 782.953 1195.79 93 1039.86 817.281 1244.69 94 1087.69 854.949 1300.22 95 1142.48 896.936 1364.74 96 1207.13 944.817 1442.15 97 1286.96 1001.40 1539.63 98 1393.62 1072.59 1672.94 99 1562.71 1175.48 1890.85

Probability Plot for Cycles

Probability Plot for Cycles

Interpretation

For the dishwasher data, the estimated median failure times are:
  • 729.3 cycles for breaks
  • 257.47 cycles for obstructions
  • 195.94 cycles for either breaks or obstructions

To maximize the overall product reliability, the engineers should focus improvement efforts on reducing spray arm obstructions.

Kaplan-Meier estimates – multiple failure mode analysis (Kaplan-Meier estimation method)

The survival probabilities are the probability that the product survives until a particular time. Use these values to determine whether your product meets reliability requirements or to determine which failure modes impact the overall reliability.

Example output

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Break
Censoring Censoring Information Count Uncensored value 25 Right censored value 40

Estimation Method: Maximum Likelihood

Distribution: Weibull

Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.97672 0.276587 1.50260 2.60044 Scale 891.929 90.8270 730.552 1088.96

Log-Likelihood = -188.140

Goodness-of-Fit Anderson-Darling (Adjusted) 1.373
Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 790.635 80.2800 647.957 964.730 Standard Deviation 417.662 64.7460 308.228 565.951 Median 740.979 79.9464 599.746 915.471 First Quartile(Q1) 474.898 67.4629 359.485 627.365 Third Quartile(Q3) 1052.19 107.155 861.801 1284.64 Interquartile Range(IQR) 577.290 83.4174 434.907 766.287
Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 87.0276 30.6339 43.6548 173.493 2 123.896 37.7877 68.1466 225.252 3 152.497 42.3555 88.4796 262.833 4 176.847 45.7243 106.541 293.548 5 198.502 48.3870 123.105 320.077 6 218.260 50.5811 138.583 343.746 7 236.594 52.4406 153.227 365.317 8 253.812 54.0493 167.205 385.279 9 270.130 55.4632 180.636 403.963 10 285.703 56.7217 193.608 421.606 20 417.625 64.8194 308.086 566.111 30 529.457 69.7943 408.905 685.548 40 634.964 74.3928 504.686 798.871 50 740.979 79.9464 599.746 915.471 60 853.343 87.6525 697.736 1043.65 70 979.746 99.1411 803.489 1194.67 80 1134.71 117.529 926.234 1390.11 90 1360.10 152.029 1092.51 1693.23 91 1391.24 157.433 1114.50 1736.69 92 1425.26 163.497 1138.28 1784.59 93 1462.89 170.393 1164.31 1838.05 94 1505.19 178.371 1193.22 1898.73 95 1553.77 187.816 1226.02 1969.15 96 1611.28 199.369 1264.30 2053.50 97 1682.59 214.223 1311.01 2159.50 98 1778.36 235.032 1372.53 2304.18 99 1931.34 270.138 1468.25 2540.49

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Obstruction
Censoring Censoring Information Count Uncensored value 40 Right censored value 25

Estimation Method: Maximum Likelihood

Distribution: Lognormal

Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 5.75328 0.271171 5.22179 6.28476 Scale 1.95933 0.238720 1.54311 2.48780

Log-Likelihood = -284.213

Goodness-of-Fit Anderson-Darling (Adjusted) 98.132
Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 2149.00 1303.69 654.415 7056.97 Standard Deviation 14492.1 15344.4 1819.17 115449 Median 315.222 85.4790 185.266 536.337 First Quartile(Q1) 84.0770 22.7774 49.4399 142.981 Third Quartile(Q3) 1181.83 418.719 590.174 2366.65 Interquartile Range(IQR) 1097.76 407.901 529.932 2274.01
Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 3.30424 1.78563 1.14571 9.52940 2 5.63679 2.72980 2.18177 14.5631 3 7.91050 3.55915 3.27511 19.1066 4 10.2074 4.33709 4.43857 23.4741 5 12.5595 5.08849 5.67682 27.7867 6 14.9838 5.82646 6.99250 32.1079 7 17.4916 6.55916 8.38765 36.4772 8 20.0913 7.29230 9.86408 40.9221 9 22.7896 8.03022 11.4236 45.4641 10 25.5926 8.77646 13.0681 50.1206 20 60.5984 17.2863 34.6455 105.993 30 112.822 29.6226 67.4371 188.749 40 191.884 49.8160 115.359 319.171 50 315.222 85.4790 185.266 536.337 60 517.841 152.725 290.505 923.079 70 880.729 291.401 460.480 1684.51 80 1639.73 627.451 774.563 3471.28 90 3882.58 1807.19 1559.26 9667.69 91 4360.12 2080.97 1710.97 11111.0 92 4945.69 2424.60 1892.04 12927.8 93 5680.72 2866.84 2112.69 15274.7 94 6631.50 3454.60 2388.85 18409.2 95 7911.58 4269.92 2747.04 22785.7 96 9734.61 5470.91 3235.47 29288.6 97 12561.2 7407.95 3953.98 39904.9 98 17628.0 11054.7 5157.08 60256.0 99 30072.1 20656.8 7824.62 115575

Distribution Analysis: Cycles

Variable: Cycles Failure Mode: Failure = Break, Obstruction
Censoring Censoring Information Count Uncensored value 65

Estimation Method: Maximum Likelihood

Distribution: Weibull, Lognormal

Table of Percentiles 95.0% Normal CI Percent Percentile Lower Upper 1 3.30048 1.24014 10.5054 2 5.62678 2.33098 15.6865 3 7.89119 3.47003 20.2810 4 10.1752 4.67345 24.6448 5 12.5102 5.94720 28.9128 6 14.9126 7.29455 33.1540 7 17.3928 8.71781 37.4099 8 19.9581 10.2189 41.7082 9 22.6145 11.7997 46.0691 10 25.3668 13.4619 50.5080 20 59.0237 34.9419 101.298 30 106.275 66.5640 168.017 40 170.417 110.481 254.467 50 254.292 169.625 361.847 60 360.567 248.567 490.105 70 493.871 354.763 642.021 80 666.812 501.548 831.290 90 923.218 721.881 1112.21 91 958.607 751.307 1151.99 92 997.221 782.953 1195.79 93 1039.86 817.281 1244.69 94 1087.69 854.949 1300.22 95 1142.48 896.936 1364.74 96 1207.13 944.817 1442.15 97 1286.96 1001.40 1539.63 98 1393.62 1072.59 1672.94 99 1562.71 1175.48 1890.85

Probability Plot for Cycles

Probability Plot for Cycles

Interpretation

For the dishwasher data, the survival probabilities are as follows:
  • 95% (or 0.951351) of the spray arms survived breaks for at least 141.90 cycles
  • 95% (or 0.953846) of the spray arms survived obstructions for at least 10.02 cycles
  • 95% (or 0.953846) of the spray arms survived both types of failures for at least 10.02 cycles

To have the greatest impact on improving the reliability of the dishwashers, the engineers should focus on the spray arm obstructions.

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