Example of a parametric distribution analysis with arbitrarily censored data
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Suppose you work for a company that manufactures tires. You are interested in finding out how many miles it takes for various proportions of the tires to "fail," or wear down to 2/32 of an inch of tread. You are especially interested in knowing how many of the tires last past 45,000 miles.

You inspect each tire at regular intervals (every 10,000 miles) to see if the tire has failed, then enter the data into the Minitab worksheet.

1    Open the worksheet TIREWEAR.MTW.

2    Choose Stat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Parametric Distribution Analysis

3    In Start variables, enter Start. In End variables, enter End.

4    In Frequency columns, enter Freq.

5    From Assumed distribution, choose Smallest extreme value.

6    Click Graphs. Check Survival plot, then click OK.

7    Click Estimate. In Estimate probabilities for these times (values), enter 45000. Click OK in each dialog box.

Session window output

Distribution Analysis, Start = Start and End = End

 

 

Variable Start: Start  End: End

Frequency: Freq

 

Censoring Information    Count

Right censored value        71

Interval censored value    694

Left censored value          8

 

Estimation Method: Maximum Likelihood

 

Distribution:   Smallest Extreme Value

 

 

Parameter Estimates

 

                     Standard   95.0% Normal CI

Parameter  Estimate     Error    Lower    Upper

Location    77538.0   547.040  76465.8  78610.2

Scale       13972.0   445.019  13126.5  14872.1

 

Log-Likelihood = -1465.913

 

Goodness-of-Fit

Anderson-Darling (adjusted) = 2.426

 

 

Characteristics of Distribution

 

                                    Standard   95.0% Normal CI

                          Estimate     Error    Lower    Upper

Mean(MTTF)                 69473.1   646.639  68205.7  70740.5

Standard Deviation         17919.8   570.759  16835.4  19074.1

Median                     72417.0   599.541  71242.0  73592.1

First Quartile(Q1)         60130.2   849.036  58466.2  61794.3

Third Quartile(Q3)         82101.7   538.928  81045.4  83158.0

Interquartile Range(IQR)   21971.5   699.808  20641.8  23386.8

 

 

Table of Percentiles

 

                     Standard   95.0% Normal CI

Percent  Percentile     Error    Lower    Upper

      1     13264.5   2216.24  8920.79  17608.3

      2     23020.0   1916.28  19264.1  26775.8

      3     28756.5   1741.64  25342.9  32170.0

      4     32848.0   1618.18  29676.4  36019.5

      5     36038.3   1522.71  33053.9  39022.8

      6     38658.9   1444.90  35827.0  41490.9

      7     40886.6   1379.29  38183.3  43590.0

      8     42826.9   1322.59  40234.6  45419.1

      9     44547.8   1272.70  42053.3  47042.2

     10     46095.8   1228.18  43688.6  48503.0

     20     56580.8   939.304  54739.8  58421.8

     30     63133.8   777.321  61610.3  64657.3

     40     68152.6   670.956  66837.5  69467.6

     50     72417.0   599.541  71242.0  73592.1

     60     76316.5   555.679  75227.4  77405.6

     70     80131.6   537.646  79077.8  81185.3

     80     84187.1   548.165  83112.7  85261.4

     90     89191.1   600.473  88014.2  90368.0

     91     89816.2   609.626  88621.4  91011.1

     92     90483.5   619.952  89268.4  91698.6

     93     91203.3   631.705  89965.2  92441.4

     94     91990.6   645.247  90725.9  93255.3

     95     92867.9   661.132  91572.1  94163.7

     96     93871.7   680.261  92538.4  95205.0

     97     95067.8   704.270  93687.4  96448.1

     98     96596.6   736.691  95152.7  98040.5

     99     98875.8   788.141  97331.0   100421

 

 

Table of Survival Probabilities

 

                      95.0% Normal CI

 Time  Probability     Lower     Upper

45000     0.907181  0.890261  0.921608

Graph window output

 

Interpreting the results

As shown in the Characteristics of Distribution table, the mean and median miles until the tires fail are 69,473.1 and 72,417.0 miles, respectively.

To see the times at which various percentages or proportions of the tires fail, look at the Table of Percentiles. For example, 5% of the tires fail by 36,038.3 miles and 50% fail by 72,417.0 miles.

In the Table of Survival Probabilities, you can see that 90.72% of the tires last past 45,000 miles.