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

In This Topic

Characteristics of Variable – Kaplan-Meier estimation method
Kaplan-Meier Estimates – Kaplan-Meier estimation method
Empirical hazard function – Kaplan-Meier estimation method
Comparison of survival curves – Kaplan-Meier estimation method

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

Variable: Temp80

Censoring

Censoring Information	Count
Uncensored value	37
Right censored value	13

Nonparametric Estimates

Characteristics of Variable

	Standard Error	95.0% Normal CI
Mean(MTTF)	Standard 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

Variable: Temp80

Censoring

Censoring Information	Count
Uncensored value	37
Right censored value	13

Nonparametric Estimates

Characteristics of Variable

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

Kaplan-Meier Estimates

	Number at Risk	Number Failed	Survival Probability	Standard Error	95.0% Normal CI
Time	Number at Risk	Number Failed	Survival Probability	Standard 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

Time	Hazard 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, 0.40, or 40%, 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

Variable: Temp80

Censoring

Censoring Information	Count
Uncensored value	37
Right censored value	13

Nonparametric Estimates

Characteristics of Variable

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

Kaplan-Meier Estimates

	Number at Risk	Number Failed	Survival Probability	Standard Error	95.0% Normal CI
Time	Number at Risk	Number Failed	Survival Probability	Standard 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

Time	Hazard 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.