This note is for the paper “Sensitivity Analysis for Clinical Trials with Missing Continuous Outcome Data Using Controlled Multiple Imputation: A Practical Guide.” The key concept in this paper is the use of multiple conditional probabilities. While this may seem complex initially, it is essentially an application of Bayes’ theorem: \(P(A|B)=\frac{P(B\cap A)}{P(B)}=\frac{P(B|A)P(A)}{P(B)}\). Another important point to remember is that \(P(A,B)=P(A\cap B)\) and \(P(A,B,C)=P(A\cap B \cap C)\)…
We first prove the equation \((1)\) in the paper”
\[f(R_i|Y_i,X_i,\phi)f(Y_i|X_i,\theta)=f(Y_i,R_i|X_i,\theta,\phi)=f(Y_i|R_i,X_i,\theta)f(R_i|X_i,\phi)\tag{1}\]
Prove \(f(R_i|Y_i,X_I,\phi)f(Y_i|X_i,\theta)=f(Y_i,R_i|X_i,\theta,\phi)\)
Since \(\theta\) and \(\phi\) are assumed to be separate/distinct, we first just ignore them and try to prove
\[f(R_i|Y_i,X_i)f(Y_i|X_i)=f(Y_i,R_i|X_i)\tag{2}\]
\[f(Y_i,R_i|X_i)=\frac{f(X_i,R_i,Y_i)}{f(X_i)}=\frac{f(R_i|X_i,Y_i)f(X_i,Y_i)}{f(X_i)}=\frac{f(R_i|X_i,Y_i)f(Y_i|X_i)f(X_i)}{f(X_i)}=f(R_i|X_i,Y_i)f(Y_i|X_i)\tag{3}\] This proved the equation \((2)\),similarly, we ignore \(\theta\) and \(\phi\) we can prove \(f(R_i,Y_i|X_i)=f(Y_i|R_i,X_i)f(R_i|X_i)\tag{4}\)
\[f(R_i,Y_i|X_i)=\frac{f(X_i,R_i,Y_i)}{f(X_i)}=\frac{f(Y_i|X_i,R_i)f(X_i,R_i)}{f(X_i)}=\frac{f(Y_i|X_i,R_i)f(R_i|X_i)f(X_i)}{f(X_i)}=f(Y_i|X_i,R_i)f(R_i|X_i)\tag{5}\] combine \((3)\) and \((5)\) and plug in \(\theta\) and \(\phi\) as condition we proved \((1)\)
We can prove the equation directly,
\[\begin{align*} f(Y_i,R_i|X_i,\theta,\phi)&=\frac{f(Y_i,R_i,X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=\frac{f(R_i|Y_i,X_i,\theta,\phi)f(Y_i,X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=\frac{f(R_i|Y_i,X_i,\theta,\phi)f(Y_i|X_i,\theta,\phi)f(X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=f(R_i|Y_i,X_i,\theta,\phi)f(Y_i|X_i,\theta,\phi)\\&=f(R_i|Y_i,X_i,\phi)f(Y_i|X_i,\theta) \end{align*}\] Note, the last step is due to, \(R_i\) dose not depend on \(\theta\) and \(Y_i\) dose not depend on \(\phi\)
\[\begin{align*} f(Y_i,R_i|X_i,\theta,\phi)&=\frac{f(Y_i,R_i,X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=\frac{f(Y_i|R_i,X_i,\theta,\phi)f(R_i,X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=\frac{f(Y_i|R_i,X_i,\theta,\phi)f(R_i|X_i,\theta,\phi)f(X_i,\theta,\phi)}{f(X_i,\theta,\phi)}\\&=f(Y_i|R_i,X_i,\theta,\phi)f(R_i|X_i,\theta,\phi)\\&=f(Y_i|R_i,X_i,\phi)f(R_i|X_i,\theta) \end{align*}\] Still, the last step is due to, \(R_i\) dose not depend on \(\theta\) and \(Y_i\) dose not depend on \(\phi\)
Here, I demonstrate the derivation of equation (1), which serves as the theoretical foundation for the selection model and mixture pattern model used in sensitivity analysis of missing values.
The next equation in the paper is:
\[f(Y_i|R_i,X_i,\theta,\phi)f(R_i|X_i,\theta,\phi)=f(Y_{iM}|Y_{iO},R_i,X_i,\theta)f(Y_{iO}|R_i,X_i,\theta)f(R_i|X_i,\phi)\] The prove of this equation is similar as the prove of equation \((1)\), since there is \(f(R_i|X_i,\theta,\phi)\) on both sides, we just ignore this part.
We can show that \[\begin{align*} f(Y_i|R_i,X_i,\theta)&=f(Y_{iO},Y_{iM}|R_i,X_i,\theta)\\&=\frac{f(Y_{iO},Y_{iM},R_i,X_i,\theta)}{f(R_i,X_i,\theta)}\\&=\frac{f(Y_{iM}|Y_{iO},R_i,X_i,\theta)f(Y_{iO},R_i,X_i,\theta)}{f(R_i,X_i,\theta)}\\&=\frac{f(Y_{iM}|Y_{iO},R_i,X_i,\theta)f(Y_{iO}|R_i,X_i,\theta)f(R_i,X_i,\theta)}{f(R_i,X_i,\theta)}\\&=f(Y_{iM}|Y_{iO},R_i,X_i,\theta)f(Y_{iO}|R_i,X_i,\theta) \end{align*}\] This proves the second equation of the paper.
In another interesting paper on the pattern-mixture model (Fiero, Mallorie H., Chiu‐Hsieh Hsu, and Melanie L. Bell. “A pattern‐mixture model approach for handling missing continuous outcome data in longitudinal cluster randomized trials.” Statistics in medicine 36.26 (2017): 4094-4105.), the authors written the pattern-mixture model as:
\[p(Y_{obs},Y_{miss},R|X)=p(R|X)p(Y_{obs},Y_{miss}|R,X)\]
The proof of this equation is similar as in Cro’s paper.
\[p(Y_{obs},Y_{miss},R|X)=\frac{p(Y_{obs},Y_{miss},R,X)}{p(X)}=\frac{p(Y_{obs},Y_{miss}|R,X)P(R,X)}{p(X)}=\frac{p(Y_{obs},Y_{miss}|R,X)P(R|X)p(X)}{p(X)}=p(Y_{obs},Y_{miss}|R,X)P(R|X)\] Q.E.D
Cro, Suzie, et al. “Sensitivity analysis for clinical trials with missing continuous outcome data using controlled multiple imputation: a practical guide.” Statistics in medicine 39.21 (2020): 2815-2842.
Fiero, Mallorie H., Chiu‐Hsieh Hsu, and Melanie L. Bell. “A pattern‐mixture model approach for handling missing continuous outcome data in longitudinal cluster randomized trials.” Statistics in medicine 36.26 (2017): 4094-4105.