Template:Confidence limits for the MCF: Difference between revisions

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===Confidence Limits for the MCF===
#REDIRECT [[Non-Parametric Recurrent Event Data Analysis]]
Upper and lower confidence limits for  <math>M({{t}_{i}})</math>  are:
 
::<math>\begin{align}
  & {{M}_{U}}({{t}_{i}})= {{M}^{*}}({{t}_{i}}).{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{*}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}} \\
& {{M}_{L}}({{t}_{i}})=  \frac{{{M}^{*}}({{t}_{i}})}{{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{*}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}}} 
\end{align}</math>
 
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:
 
::<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>
 
where  <math>{r}_{i}</math>  is defined in the equation of the survivals,  <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:
 
::<math>\begin{align}
  & {{d}_{ji}}= 1\text{  if the }{{j}^{\text{th }}}\text{unit had an event recurrence at age }{{t}_{i}} \\
& {{d}_{ji}}=  0\text{  if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}} 
\end{align}</math>
 
 
'''Example 2:'''
{{Example: Recurrent Events Data Non-parameteric MCF Bound Example}}
 
 
'''Example 3:'''
{{Example: Recurrent Events Data Non-parameteric Transmission Example}}

Latest revision as of 02:11, 16 August 2012

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