Matched pairs:
two samples when each observation in one sample pairs with an observation in the other.
Such matched pairs data commonly occur in studies with repeated measurement of subjects, such as longitudinal studies.
Let \(Y_{it}=\)response at time \(t\) for subject \(i\)(1=success, 0=failure).
Then conditional model for the matched pair is \[ \text{logit}\{P(Y_{it}=1)\}=\alpha_i+\beta x_{it},\mbox{ }\mbox{ }t=1,2, \] with \(x_{it}=0\) for \(t=1\); \(x_{it}=1\) for \(t=2\).
Traditional approach estimates \(\beta\) conditional on sufficient statistics for nuisance parameters \(\{\alpha_i\}\).
Assume \((Y_{i1},Y_{i2})\) are independent, given \(\alpha_i,\beta\), and independent of different matched pairs. Joint probability function of \(\{(y_{11}, y_{12}),\ldots,(y_{n1},y_{n2})\}\) is \[ \prod_{i=1}^n \left(\frac{e^{\alpha_i}}{1+e^{\alpha_i}}\right)^{y_{i1}}\left(\frac{1}{1+e^{\alpha_i}}\right)^{1-y_{i1}} \left(\frac{e^{\alpha_i+\beta}}{1+e^{\alpha_i+\beta}}\right)^{y_{i2}}\left(\frac{1}{1+e^{\alpha_i+\beta}}\right)^{1-y_{i2}}\\ \propto\exp\left( \sum_i \alpha_i(y_{i1}+y_{i2})+\beta\left(\sum_iy_{i2} \right)\right). \] Then, sufficient statistics for \(\{\alpha_i\}\) are \(\{S_i= y_{i1}+y_{i2}\}\).
Thus, eliminate \(\{\alpha_i\}\) by conditioning on \(\{S_i= y_{i1}+y_{i2}\}\).
Given \(S_i=0\), \(P(Y_{i1}=Y_{i2}= 0|S_i=0)=1\).
Given \(S_i=2\), \(P(Y_{i1}=Y_{i2}= 1|S_i=2)=1\)
Given \(S_i=1\), \(P(Y_{i1}= 1 \mbox{ or } Y_{i2}= 1|S_i=1)\) has two cases: \((Y_{i1}= 1, \mbox{ } Y_{i2}= 0)\), \((Y_{i1}= 0, \mbox{ } Y_{i2}= 1)\).
\(\implies\)Conditional distribution of \((Y_{i1}, Y_{i2})\) depends on \(\beta\) only when \(S_i=1\).
Given \(Y_{i1}+Y_{i2}= 1\), \[\begin{eqnarray*} P(Y_{i1}=y_{i1}, Y_{i2}= y_{i2}|S_i=1)&=& \frac{P(Y_{i1}=y_{i1}, Y_{i2}= y_{i2})}{P(Y_{i1}=1, Y_{i2}=0)+ P(Y_{i1}=0, Y_{i2}=1)} \\ &=& \frac{\left(\frac{e^{\alpha_i}}{1+e^{\alpha_i}}\right)^{y_{i1}}\left(\frac{1}{1+e^{\alpha_i}}\right)^{1-y_{i1}}\left(\frac{e^{\alpha_i+\beta}}{1+e^{\alpha_i+\beta}}\right)^{y_{i2}}\left(\frac{1}{1+e^{\alpha_i+\beta}}\right)^{1-y_{i2}}}{{\left(\frac{e^{\alpha_i}}{1+e^{\alpha_i}}\right)\left(\frac{1}{1+e^{\alpha_i+\beta}}\right)+ \left(\frac{1}{1+e^{\alpha_i}}\right) \left(\frac{e^{\alpha_i+\beta}}{1+e^{\alpha_i+\beta}}\right)}}\\ &=&\begin{cases} \frac{e^\beta}{1+e^\beta}& y_{i1}=0, y_{i2}=1, \\ \frac{1}{1+e^\beta} & y_{i1}=1, y_{i2}=0.\end{cases} \end{eqnarray*}\] Thus, conditional on \(S_i=1\), joint distribution is \[ \prod_{S_i=1}\left(\frac{e^\beta}{1+e^\beta}\right)^{y_{i2}}\left(\frac{1}{1+e^\beta}\right)^{y_{i1}}=\frac{(e^\beta)^{\sum y_{i2}}}{(1-e^\beta)^{\sum y_{i1}+y_{i2}}}. \] Conditional log likelihood is \[ l(\beta)=\beta\sum_i y_{i2}-\log(1-e^\beta)\left(\sum_i y_{i1}+y_{i2}\right)\\ \implies\frac{d\mbox{ }l(\beta)}{d\beta}= 0 \\ \implies \hat\beta=\log\frac{\sum_i y_{i2}}{\sum_i y_{i1}}=\log\frac{n_{21}}{n_{12}}. \]
Conditional model의 관심은 nuisance parameter를 제외하고 관심이 있는 parameter만을 추정하는 것이다.
그러기 위해서 full likelihood에서 Nuisance parameter의 sufficient statistic을 찾아낸 다음, conditional likelihood를 구해낸다.