Template:Example: Kaplan-Meier Example: Difference between revisions

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'''Kaplan-Meier Example'''
#REDIRECT [[Non-Parametric_Life_Data_Analysis]]
 
A group of 20 units are put on a life test with the following results.
 
<center><math>\begin{matrix}
  Number & State & State  \\
  in State & (F or S) & End Time  \\
  3 & F & 9  \\
  1 & S & 9  \\
  1 & F & 11  \\
  1 & S & 12  \\
  1 & F & 13  \\
  1 & S & 13  \\
  1 & S & 15  \\
  1 & F & 17  \\
  1 & F & 21  \\
  1 & S & 22  \\
  1 & S & 24  \\
  1 & S & 26  \\
  1 & F & 28  \\
  1 & F & 30  \\
  1 & S & 32  \\
  2 & S & 35  \\
  1 & S & 39  \\
  1 & S & 41  \\
\end{matrix}</math></center>
 
Use the Kaplan-Meier estimator to determine the reliability estimates for each failure time.
 
 
'''Solution'''
 
Using the data and the reliability equation of the Kaplan-Meier estimator , the following table can be constructed:
 
<center><math>\begin{matrix}
  State & Number of & Number of & Available  & {} & {}  \\
  End Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & \tfrac{{{n}_{i}}-{{r}_{i}}}{{{n}_{i}}} & \mathop{}_{}^{}\prod\tfrac{{{n}_{i}}-{{r}_{i}}}{{{n}_{i}}}  \\
  9 & 3 & 1 & 20 & 0.850 & 0.850  \\
  11 & 1 & 0 & 16 & 0.938 & 0.797  \\
  12 & 0 & 1 & 15 & 1.000 & 0.797  \\
  13 & 1 & 1 & 14 & 0.929 & 0.740  \\
  15 & 0 & 1 & 12 & 1.000 & 0.740  \\
  17 & 1 & 0 & 11 & 0.909 & 0.673  \\
  21 & 1 & 0 & 10 & 0.900 & 0.605  \\
  22 & 0 & 1 & 9 & 1.000 & 0.605  \\
  24 & 0 & 1 & 8 & 1.000 & 0.605  \\
  26 & 0 & 1 & 7 & 1.000 & 0.605  \\
  28 & 1 & 0 & 6 & 0.833 & 0.505  \\
  30 & 1 & 0 & 5 & 0.800 & 0.404  \\
  32 & 0 & 1 & 4 & 1.000 & 0.404  \\
  35 & 0 & 1 & 3 & 1.000 & 0.404  \\
  39 & 0 & 1 & 2 & 1.000 & 0.404  \\
  41 & 0 & 1 & 1 & 1.000 & 0.404  \\
\end{matrix}</math></center>
 
As can be determined from the preceding table, the reliability estimates for the failure times are:
 
<center><math>\begin{matrix}
  Failure Time & Reliability Est.  \\
  9 & 85.0%  \\
  11 & 79.7%  \\
  13 & 74.0%  \\
  17 & 67.3%  \\
  21 & 60.5%  \\
  28 & 50.5%  \\
  30 & 40.4%  \\
\end{matrix}</math></center>

Latest revision as of 08:10, 10 August 2012

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