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| ===Confidence Limits for the MCF===
| | #REDIRECT [[Non-Parametric Recurrent Event Data Analysis]] |
| Upper and lower conifidence limits for <math>M({{t}_{i}})</math> are:
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| ::<math>\begin{align}
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| & {{M}_{U}}({{t}_{i}})= {{M}^{*}}({{t}_{i}}).{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{*}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}} \\
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| & {{M}_{L}}({{t}_{i}})= \frac{{{M}^{*}}({{t}_{i}})}{{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{*}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}}}
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| \end{align}</math>
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| where <math>\alpha </math> ( <math>50%<\alpha <100%</math> ) is confidence level, <math>{{K}_{\alpha }}</math> is the <math>\alpha </math> standard normal percentile and <math>Var[{{M}^{*}}({{t}_{i}})]</math> is the variance of the MCF estimate at recurrence age <math>{{t}_{i}}</math> . The variance is calculated as follows:
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| ::<math>Var[{{M}^{*}}({{t}_{i}})]=Var[{{M}^{*}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]</math>
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| where <math>r</math> is defined in Eqn.(R), <math>{{R}_{i}}</math> is the set of the units that have not been suspended by <math>i</math> and <math>{{d}_{ji}}</math> is defined as follows:
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| ::<math>\begin{align}
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| & {{d}_{ji}}= 1\text{ if the }{{j}^{\text{th }}}\text{unit had an event recurrence at age }{{t}_{i}} \\
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| & {{d}_{ji}}= 0\text{ if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}}
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| \end{align}</math>
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| '''Example 2:'''
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| {{Example: Recurrent Events Data Non-parameteric MCF Bound Example}}
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| '''Example 3:'''
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| {{Example: Recurrent Events Data Non-parameteric Transmission Example}}
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