Hazard and density estimates for Nonparametric Distribution Analysis (Right Censoring)

Hazard estimates – actuarial estimation method

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

Although the nonparametric hazard function is not dependent on any specific distribution, you can use it to help determine which distribution might be appropriate for modeling the data if you decide to use parametric estimation methods. Select a distribution that has a hazard function that resembles the nonparametric hazard function.

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 Median Error Lower Upper 56.1905 3.36718 49.5909 62.7900
Additional Time from Time T until 50% of Running Units Fail Proportion of Running Additional Standard 95.0% Normal CI Time T Units Time Error Lower Upper 20 1.00 36.1905 3.36718 29.5909 42.7900 40 0.84 20.0000 3.08607 13.9514 26.0486
Actuarial Table Conditional Interval Number Number Number Probability Standard Lower Upper Entering Failed Censored of Failure Error 0 20 50 0 0 0.000000 0.000000 20 40 50 8 0 0.160000 0.051846 40 60 42 21 0 0.500000 0.077152 60 80 21 8 4 0.421053 0.113269 80 100 9 0 6 0.000000 0.000000 100 120 3 0 3 0.000000 0.000000
Table of Survival Probabilities Survival Standard 95.0% Normal CI Time Probability Error Lower Upper 20 1.00000 0.0000000 1.00000 1.00000 40 0.84000 0.0518459 0.73838 0.94162 60 0.42000 0.0697997 0.28320 0.55680 80 0.24316 0.0624194 0.12082 0.36550 100 0.24316 0.0624194 0.12082 0.36550 120 0.24316 0.0624194 0.12082 0.36550
Hazards and Densities Hazard Standard Density Standard Time Estimates Error Estimates Error 10 0.0000000 * 0.0000000 * 30 0.0086957 0.0030627 0.0080000 0.0025923 50 0.0333333 0.0068579 0.0210000 0.0034900 70 0.0266667 0.0090867 0.0088421 0.0027959 90 0.0000000 * 0.0000000 * 110 0.0000000 * 0.0000000 *

Interpretation

For engine windings that run at 80° C, the likelihood of failure is approximately 3.07 (0.0266667/0.0086957) times greater after 70 hours than after 30 hours.

Density estimates – actuarial estimation method

The density estimates describe the distribution of failure times and provide a measure of the likelihood that a product fails at particular times.

Although the nonparametric density function is not dependent on any specific distribution, you can use it to help determine which distribution might be appropriate for modeling the data if you decide to use a parametric estimation methods. Select a distribution that has a density function that resembles the nonparametric density function.

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 Median Error Lower Upper 56.1905 3.36718 49.5909 62.7900
Additional Time from Time T until 50% of Running Units Fail Proportion of Running Additional Standard 95.0% Normal CI Time T Units Time Error Lower Upper 20 1.00 36.1905 3.36718 29.5909 42.7900 40 0.84 20.0000 3.08607 13.9514 26.0486
Actuarial Table Conditional Interval Number Number Number Probability Standard Lower Upper Entering Failed Censored of Failure Error 0 20 50 0 0 0.000000 0.000000 20 40 50 8 0 0.160000 0.051846 40 60 42 21 0 0.500000 0.077152 60 80 21 8 4 0.421053 0.113269 80 100 9 0 6 0.000000 0.000000 100 120 3 0 3 0.000000 0.000000
Table of Survival Probabilities Survival Standard 95.0% Normal CI Time Probability Error Lower Upper 20 1.00000 0.0000000 1.00000 1.00000 40 0.84000 0.0518459 0.73838 0.94162 60 0.42000 0.0697997 0.28320 0.55680 80 0.24316 0.0624194 0.12082 0.36550 100 0.24316 0.0624194 0.12082 0.36550 120 0.24316 0.0624194 0.12082 0.36550
Hazards and Densities Hazard Standard Density Standard Time Estimates Error Estimates Error 10 0.0000000 * 0.0000000 * 30 0.0086957 0.0030627 0.0080000 0.0025923 50 0.0333333 0.0068579 0.0210000 0.0034900 70 0.0266667 0.0090867 0.0088421 0.0027959 90 0.0000000 * 0.0000000 * 110 0.0000000 * 0.0000000 *

Interpretation

For engine windings running at 80° C, the likelihood of failure is greater at 50 hours (0.021000) than at 70 hours (0.0088421).

Comparison of survival curves – actuarial 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 different. 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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