Kaplan-Meier estimation method for Nonparametric Distribution Analysis (Right Censoring)

Characteristics of Variable – Kaplan-Meier estimation method

The 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: Temp80

Variable: Temp80

Censoring Censoring Information Count Uncensored value 37 Right censored value 13 Censoring value: Cens80 = 0

Nonparametric Estimates

Characteristics of Variable Standard 95.0% Normal CI Mean(MTTF) Error Lower Upper Q1 Median Q3 IQR 63.7123 3.83453 56.1968 71.2279 48 55 * *

Interpretation

The characteristics of the variable are shown for the engine windings that are tested at 80° C.

The MTTF (63.7123) is a sensitive statistic because outliers and the tails in a skewed distribution significantly affect its values.

The median (55) and the IQR are resistant statistics because the tails in a skewed distribution and outliers do not significantly affect their values.
Note

In this example, due to censoring, there is not sufficient failure data to calculate where 75% fail or 25% survive (Q3). Therefore, Minitab displays a missing value * for Q3 and IQR.

Kaplan-Meier Estimates – Kaplan-Meier estimation method

The survival probabilities indicate the probability that the product survives until a particular time. Use these values to determine whether your product meets reliability requirements or to compare the reliability of two or more designs of a product.

Nonparametric estimates do not depend on any particular distribution and therefore are good to use when no distribution adequately fits the data.

Example output

Distribution Analysis: Temp80

Variable: Temp80

Censoring Censoring Information Count Uncensored value 37 Right censored value 13 Censoring value: Cens80 = 0

Nonparametric Estimates

Characteristics of Variable Standard 95.0% Normal CI Mean(MTTF) Error Lower Upper Q1 Median Q3 IQR 63.7123 3.83453 56.1968 71.2279 48 55 * *
Kaplan-Meier Estimates Number Number Survival Standard 95.0% Normal CI Time at Risk Failed Probability Error Lower Upper 23 50 1 0.980000 0.0197990 0.941195 1.00000 24 49 1 0.960000 0.0277128 0.905684 1.00000 27 48 2 0.920000 0.0383667 0.844803 0.99520 31 46 1 0.900000 0.0424264 0.816846 0.98315 34 45 1 0.880000 0.0459565 0.789927 0.97007 35 44 1 0.860000 0.0490714 0.763822 0.95618 37 43 1 0.840000 0.0518459 0.738384 0.94162 40 42 1 0.820000 0.0543323 0.713511 0.92649 41 41 1 0.800000 0.0565685 0.689128 0.91087 45 40 1 0.780000 0.0585833 0.665179 0.89482 46 39 1 0.760000 0.0603987 0.641621 0.87838 48 38 3 0.700000 0.0648074 0.572980 0.82702 49 35 1 0.680000 0.0659697 0.550702 0.80930 50 34 1 0.660000 0.0669925 0.528697 0.79130 51 33 4 0.580000 0.0697997 0.443195 0.71680 52 29 1 0.560000 0.0701997 0.422411 0.69759 53 28 1 0.540000 0.0704840 0.401854 0.67815 54 27 1 0.520000 0.0706541 0.381521 0.65848 55 26 1 0.500000 0.0707107 0.361410 0.63859 56 25 1 0.480000 0.0706541 0.341521 0.61848 58 24 2 0.440000 0.0701997 0.302411 0.57759 59 22 1 0.420000 0.0697997 0.283195 0.55680 60 21 1 0.400000 0.0692820 0.264210 0.53579 61 20 1 0.380000 0.0686440 0.245460 0.51454 62 19 1 0.360000 0.0678823 0.226953 0.49305 64 18 1 0.340000 0.0669925 0.208697 0.47130 66 17 1 0.320000 0.0659697 0.190702 0.44930 67 16 2 0.280000 0.0634980 0.155546 0.40445 74 13 1 0.258462 0.0621592 0.136632 0.38029
Empirical Hazard Function Hazard Time Estimates 23 0.0200000 24 0.0204082 27 0.0212766 31 0.0217391 34 0.0222222 35 0.0227273 37 0.0232558 40 0.0238095 41 0.0243902 45 0.0250000 46 0.0256410 48 0.0277778 49 0.0285714 50 0.0294118 51 0.0333333 52 0.0344828 53 0.0357143 54 0.0370370 55 0.0384615 56 0.0400000 58 0.0434783 59 0.0454545 60 0.0476190 61 0.0500000 62 0.0526316 64 0.0555556 66 0.0588235 67 0.0666667 74 0.0769231

Interpretation

For the engine windings tested at 80° C, 0.42, or 42%, of the windings survived for at least 60.0 hours.

Empirical hazard function – Kaplan-Meier estimation method

The hazard function provides a measure of the likelihood of failure as a function of how long a unit has survived (the instantaneous failure rate at a particular time, t).

The empirical hazard function always results in an increasing function; therefore, the likelihood of failure is assumed to increase as a function of age.

Example output

Distribution Analysis: Temp80

Variable: Temp80

Censoring Censoring Information Count Uncensored value 37 Right censored value 13 Censoring value: Cens80 = 0

Nonparametric Estimates

Characteristics of Variable Standard 95.0% Normal CI Mean(MTTF) Error Lower Upper Q1 Median Q3 IQR 63.7123 3.83453 56.1968 71.2279 48 55 * *
Kaplan-Meier Estimates Number Number Survival Standard 95.0% Normal CI Time at Risk Failed Probability Error Lower Upper 23 50 1 0.980000 0.0197990 0.941195 1.00000 24 49 1 0.960000 0.0277128 0.905684 1.00000 27 48 2 0.920000 0.0383667 0.844803 0.99520 31 46 1 0.900000 0.0424264 0.816846 0.98315 34 45 1 0.880000 0.0459565 0.789927 0.97007 35 44 1 0.860000 0.0490714 0.763822 0.95618 37 43 1 0.840000 0.0518459 0.738384 0.94162 40 42 1 0.820000 0.0543323 0.713511 0.92649 41 41 1 0.800000 0.0565685 0.689128 0.91087 45 40 1 0.780000 0.0585833 0.665179 0.89482 46 39 1 0.760000 0.0603987 0.641621 0.87838 48 38 3 0.700000 0.0648074 0.572980 0.82702 49 35 1 0.680000 0.0659697 0.550702 0.80930 50 34 1 0.660000 0.0669925 0.528697 0.79130 51 33 4 0.580000 0.0697997 0.443195 0.71680 52 29 1 0.560000 0.0701997 0.422411 0.69759 53 28 1 0.540000 0.0704840 0.401854 0.67815 54 27 1 0.520000 0.0706541 0.381521 0.65848 55 26 1 0.500000 0.0707107 0.361410 0.63859 56 25 1 0.480000 0.0706541 0.341521 0.61848 58 24 2 0.440000 0.0701997 0.302411 0.57759 59 22 1 0.420000 0.0697997 0.283195 0.55680 60 21 1 0.400000 0.0692820 0.264210 0.53579 61 20 1 0.380000 0.0686440 0.245460 0.51454 62 19 1 0.360000 0.0678823 0.226953 0.49305 64 18 1 0.340000 0.0669925 0.208697 0.47130 66 17 1 0.320000 0.0659697 0.190702 0.44930 67 16 2 0.280000 0.0634980 0.155546 0.40445 74 13 1 0.258462 0.0621592 0.136632 0.38029
Empirical Hazard Function Hazard Time Estimates 23 0.0200000 24 0.0204082 27 0.0212766 31 0.0217391 34 0.0222222 35 0.0227273 37 0.0232558 40 0.0238095 41 0.0243902 45 0.0250000 46 0.0256410 48 0.0277778 49 0.0285714 50 0.0294118 51 0.0333333 52 0.0344828 53 0.0357143 54 0.0370370 55 0.0384615 56 0.0400000 58 0.0434783 59 0.0454545 60 0.0476190 61 0.0500000 62 0.0526316 64 0.0555556 66 0.0588235 67 0.0666667 74 0.0769231

Interpretation

For the engine windings tested at 80° C, the likelihood of failure is 2 (0.0500000/0.0250000) times greater after the windings run for 61 hours than after the windings run for 45 hours.

Comparison of survival curves – Kaplan-Meier estimation method

Use the log-rank and Wilcoxon tests to compare the survival curves of two or more data sets. Each test detects different types of differences between the survival curves. Therefore, use both tests to determine whether the survival curves are equal.

The log-rank test compares the actual and expected number of failures between the survival curves at each failure time.

The Wilcoxon test is a log-rank test that is weighted by the number of items that still survive at each point in time. Therefore, the Wilcoxon test weights early failure times more heavily.

Example output

Test Statistics Method Chi-Square DF P-Value Log-Rank 7.7152 1 0.005 Wilcoxon 13.1326 1 0.000

Interpretation

For the engine windings data, the test is to determine whether the survival curves for the engine windings running at 80° C and 100° C are the same. Because the p-value for both tests is less than an α-value of 0.05, the engineer concludes that a significant difference exists between the survival curves.

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