Bias reduction using surrogate endpoints as auxiliary variables

Abstract

Recently, it is becoming more active to apply appropriate statistical methods dealing with missing data in clinical trials. Under not missing at random missingness, MLE based on direct-likelihood, or observed likelihood, possibly has a serious bias. A solution to the bias problem is to add auxiliary variables such as surrogate endpoints to the model for the purpose of reducing the bias. We theoretically studied the impact of an auxiliary variable on MLE and evaluated the bias reduction or inflation in the case of several typical correlation structures.

Appendices

Appendix A: MLE based on direct-likelihood with $Y_a$

We shall obtain MLE based on direct-likelihood with $Y_a$ according to Anderson (1957) in which MLE is more easily derived by using characteristics of normal distribution and reparametrization.

Assuming that $(X,Y,Y_a)$ have a normal distribution in (5), the conditional distribution of Y given $(X,Y_a)$ follows a normal distribution with mean $\beta _0+\beta _{x} X + \beta _{y_a} Y_a$ and variance $\sigma ^2_e$, where:

$$\begin{aligned} \beta _0&= \mu _y - \beta _{x}\mu _x - \beta _{y_a} \mu _{y_a}, \quad \beta _{x} = \sigma _{xx \cdot y_a}^{-1} \sigma _{xy \cdot y_a}, \quad \beta _{y_a} = \sigma _{y_ay_a \cdot x}^{-1} \sigma _{y_ay \cdot x}, \nonumber \\ \sigma ^2_e&=\sigma _{yy \cdot xy_a}= \sigma _{yy \cdot x} - \sigma _{yy_a \cdot x}^2 \sigma _{y_ay_a \cdot x}^{-1}. \end{aligned}$$

(31)

Hence, the direct-likelihood $DL_+$ can be rewritten by the reparametrization as follows:

$$\begin{aligned} DL_+ =&\prod _{i=1}^m f_{Y|XY_a}(Y_i|X_i,Y_{a,i};\beta _0,\beta _x,\beta _{y_a},\sigma ^2_{e}) \nonumber \\&\times \prod _{i=1}^n f_{XY_a}(X_i,Y_{a,i}|\mu _x,\mu _{y_a}, \sigma _{xx},\sigma _{y_ay_a}, \sigma _{xy_a}). \end{aligned}$$

(32)

MLEs of $\mu _x, \mu _{y_a}, \sigma _{xx}, \sigma _{xy_a}, \sigma _{y_ay_a}$ are easily obtained from the second factor of the $DL_+$ as follows:

$$\begin{aligned} \hat{\mu }_x&=\bar{X}_{(n)}=\frac{1}{n} \sum _{i=1}^n X_i, \quad \hat{\mu }_{y_a}=\bar{Y}_{a_{(n)}}=\frac{1}{n} \sum _{i=1}^n Y_{a,i}, \nonumber \\ \hat{\sigma }_{xx}&=S_{xx_{(n)}} = \frac{1}{n} \sum _{i=1}^n \left( X_i-\bar{X}_{{(n)}}\right) ^2, \quad \hat{\sigma }_{y_ay_a} =S_{y_ay_{a_{(n)}}}= \frac{1}{n} \sum _{i=1}^n \left( Y_{a,i}-\bar{Y}_{a_{(n)}}\right) ^2, \nonumber \\ \hat{\sigma }_{xy_a}&=S_{xy_{a_{(n)}}}= \frac{1}{n} \sum _{i=1}^n (X_i-\bar{X}_{(n)})(Y_{a,i}-\bar{Y}_{a_{(n)}}). \end{aligned}$$

The MLE of $\theta _{a1}$, $\theta _{a2}$, and $\theta _{a3}$ will easily be found from the results.

MLEs of the remaining parameters of $\beta _0, \beta _x,\beta _{y_a}, \sigma ^2_{e}$ are obtained from the first factor of $DL_+$ from the standard results on the linear regression as follows:

$$\begin{aligned} \hat{\beta }_{x}&=S_{xx \cdot y_{a_{(m)}}}^{-1} S_{xy \cdot y_{a_{(m)}}}, \quad \hat{\beta }_{y_a} =S_{y_ay_a \cdot x_{(m)}}^{-1} S_{y_ay \cdot x_{(m)}}, \nonumber \\ \hat{\beta }_0&= \bar{Y}_{(m)} - \hat{\beta }_{x} \bar{X}_{(m)} - \hat{\beta }_{y_a} \bar{Y}_{a_{(m)}}, \nonumber \\ \hat{\sigma }^2_{e}&=\hat{\sigma }_{yy \cdot xy_a} =S_{yy \cdot xy_{a_{(m)}}} =S_{yy \cdot x_{(m)}} - S_{yy_a \cdot x_{(m)}}^2S_{y_ay_a \cdot x_{(m)}}^{-1}, \end{aligned}$$

where

$$\begin{aligned} \bar{X}_{(m)}&=\frac{1}{m} \sum _{i=1}^m X_i, \quad \bar{Y}_{a_{(m)}}= \frac{1}{m} \sum _{i=1}^m Y_{a,i}, \quad \bar{Y}_{(m)}= \frac{1}{m} \sum _{i=1}^m Y_i, \nonumber \\ S_{xx_{(m)}}&=\frac{1}{m} \sum _{k=1}^m (X_i - \bar{X}_{(m)})^2, \quad S_{y_ay_{a_{(m)}}}=\frac{1}{m} \sum _{k=1}^m \left( Y_{a,i} - \bar{Y}_{a_{(m)}}\right) ^2, \quad \nonumber \\ S_{xy_{a_{(m)}}}&=\frac{1}{m} \sum _{i=1}^m (X_i - \bar{X}_{(m)}) (Y_{a,i} - \bar{Y}_{a_{(m)}}), \nonumber \\ S_{uv \cdot w_{{(m)}}}&=S_{uv_{{(m)}}} - S_{uw_{{(m)}}}S_{ww_{{(m)}}}^{-1}S_{wv_{{(m)}}}. \end{aligned}$$

It follows from (31) and the relationship in parameters between (5) and (6) that

$$\begin{aligned} \theta _1&=\mu _y-\sigma _{yx}\sigma _{xx}^{-1}\mu _x = (\beta _0 + \beta _x \mu _x + \beta _{y_a} \mu _{y_a}) - \theta _2 \mu _x,\\ \theta _2&=\sigma _{xx}^{-1} \sigma _{xy} = \sigma _{xx}^{-1} (\sigma _{xx}\beta _x + \sigma _{y_ax}\beta _{y_a}) = \beta _x + \sigma _{xx}^{-1} \sigma _{xy_a} \beta _{y_a},\\ \theta _3&=\sigma _{yy \cdot x} = \sigma ^2_e + \beta _{y_a}^2 \sigma _{y_ay_a \cdot x},\\ \theta _{a4}&=\sigma _{y_ay \cdot x} = \sigma _{y_ay_a \cdot x} \beta _{y_a}. \end{aligned}$$

Hence, we obtain MLE of these parameters as follows, which means the results in Proposition 2:

$$\begin{aligned} \hat{\theta }_1&= (\hat{\beta }_0 + \hat{\beta }_x \hat{\mu }_x + \hat{\beta }_{y_a} \hat{\mu }_{y_a}) - \hat{\theta }_2 \hat{\mu }_x,\\ \hat{\theta }_2&= \hat{\beta }_x+ \hat{\sigma }_{xx}^{-1} \hat{\sigma }_{xy_a} \hat{\beta }_{y_a}, \nonumber \\ \hat{\theta }_3&=\hat{\sigma }^2_{e} + \hat{\beta }_{y_a}^2 \hat{\sigma }_{y_ay_a \cdot x},\\ \hat{\theta }_{a4}&= \hat{\sigma }_{y_ay_a \cdot x} \hat{\beta }_{y_a}. \end{aligned}$$

Appendix B: Limit of statistics using complete cases only

Here, we shall show the following convergences for limits of $\bar{X}_{(m)}$ and $S_{xx_{(m)}}$ as n tends to infinity. The limits of the other statistics also have the same properties.

$$\begin{aligned} \bar{X}_{(m)}&{\mathop {\rightarrow }\limits ^{P}} \mu _x + \frac{\sigma _{xz}}{\sigma _{zz}}E[Z-\mu _z|R_Y=1], \end{aligned}$$

(33)

$$\begin{aligned} S_{xx_{(m)}}&{\mathop {\rightarrow }\limits ^{P}} \sigma _{xx} + \frac{\sigma _{xz}^2}{\sigma _{zz}^2} (Var[Z|R_Y=1] - \sigma _{zz}). \end{aligned}$$

(34)

Assuming that $(X,Y,Y_a,Z)$ have a normal distribution in addition to (5), where the mean and variance of Z are $\mu _z$ and $\sigma _{zz}$, respectively, and covariance between Z and $(X,Y,Y_a)$ is $(\sigma _{zx}, \sigma _{zy}, \sigma _{zy_a})$.

Using the response indicator $R_Y$, $\bar{X}_{(m)}$ is expressed in the form:

$$\begin{aligned} \bar{X}_{(m)}&= \frac{1}{\sum _{i=1}^n R_{Y_i}} \sum _{i=1}^n R_{Y_i} X_i. \end{aligned}$$

By the weak law of large numbers, we obtain

$$\begin{aligned} \bar{X}_{(m)}&{\mathop {\rightarrow }\limits ^{P}} \frac{E[R_YX]}{E[R_Y]} = \frac{E[X|R_Y=1] P(R_Y=1)}{P(R_Y=1)} = E[X|R_Y=1]. \end{aligned}$$

By using the condition $X \mathop {\perp \!\!\!\!\perp }R_Y |Z$, we obtain (33) shown as follows:

$$\begin{aligned} E[X|R_Y=1] = E[E[X|Z]|R_Y=1] = \mu _x+ \frac{\sigma _{xz}}{\sigma _{zz}} E[Z-\mu _z|R_Y=1]. \end{aligned}$$

(35)

For $S_{xx_{(m)}}$, we can rewrite using response indicator $R_Y$ as follows:

$$\begin{aligned} S_{xx_{(m)}}&= \frac{1}{\sum _{i=1}^n R_{Y_i}} \sum _{i=1}^n R_{Y_i} \left( X_i - \bar{X}_{(m)} \right) ^2. \end{aligned}$$

By applying the weak law of large numbers,

$$\begin{aligned} S_{xx_{(m)}} =&\frac{1}{\sum _{i=1}^n R_{Y_i}} \sum _{i=1}^n R_{Y_i}X_i^2 - \left( \bar{X}_{(m)} \right) ^2 \nonumber \\ {\mathop {\rightarrow }\limits ^{P}}&\frac{E \left[ R_YX^2 \right] }{E[R_Y]} - \left( E \left[ X|R_Y=1 \right] \right) ^2 \nonumber \\ =&E \left[ X^2|R_Y=1 \right] - \left( E \left[ X|R_Y=1 \right] \right) ^2 \nonumber \\ =&E \left[ \left\{ X - E[X|R_Y=1] \right\} ^2 | R_Y=1 \right] = Var[X|R_Y=1] \nonumber \\ =&E \left[ \left\{ (X - E[X|Z]) + (E[X|Z]- E[X|R_Y=1]) \right\} ^2 | R_Y=1 \right] \nonumber \\ =&E \left[ \left\{ X - E[X|Z] \right\} ^2 | R_Y=1 \right] \nonumber \\&+ E \left[ \left\{ E[X|Z]- E[X|R_Y=1] \right\} ^2 | R_Y=1 \right] \nonumber \\&+ 2E \left[ \left\{ X - E[X|Z] \right\} \left\{ E[X|Z]- E[X|R_Y=1] \right\} | R_Y=1 \right] . \end{aligned}$$

(36)

By noting that $X \mathop {\perp \!\!\!\!\perp }R_Y|Z$, we can evaluate the third term as follows:

$$\begin{aligned}&2E \left[ \left\{ X - E[X|Z] \right\} \left\{ E[X|Z]- E[X|R_Y=1] \right\} | R_Y=1 \right] \nonumber \\&\quad = 2E \left[ E \left[ \left\{ X - E[X|Z] \right\} \left\{ E[X|Z]- E[X|R_Y=1] \right\} |Z,R_Y=1 \right] | R_Y=1 \right] \nonumber \\&\quad = 2E \left[ E \left[ X - E[X|Z]|Z,R_Y=1 \right] \left\{ E[X|Z]- E[X|R_Y=1] \right\} | R_Y=1 \right] \nonumber \\&\quad = 2E \left[ E \left[ X - E[X|Z]|Z \right] \left\{ E[X|Z]- E[X|R_Y=1] \right\} | R_Y=1 \right] \nonumber \\&\quad = 0. \end{aligned}$$

The first term is written as follows:

$$\begin{aligned} E \left[ \left\{ X - E[X|Z] \right\} ^2 | R_Y=1 \right]&= E \left[ E \left[ \left\{ X - E[X|Z] \right\} ^2 |Z \right] | R_Y=1 \right] \nonumber \\&= \sigma _{xx \cdot z}. \end{aligned}$$

(37)

The second term is written by using (35) as follows:

$$\begin{aligned}&E \left[ \left\{ E[X|Z]- E[X|R_Y=1] \right\} ^2 | R_Y=1 \right] \nonumber \\&\quad = E \left[ \frac{\sigma _{xz}^2}{\sigma _{zz}^2} (Z-E[Z|R_Y=1])^2|R_Y=1 \right] \nonumber \\&\quad = \frac{\sigma _{xz}^2}{\sigma _{zz}^2} Var[Z|R_Y=1]. \end{aligned}$$

(38)

Hence, we finally obtain:

$$\begin{aligned} S_{xx_{(m)}} {\mathop {\rightarrow }\limits ^{P}} \sigma _{xx \cdot z} + \frac{\sigma _{xz}^2}{\sigma _{zz}^2} Var[Z|R_Y=1] = \sigma _{xx} + \frac{\sigma _{xz}^2}{\sigma _{zz}^2} (Var[Z|R_Y=1] - \sigma _{zz}). \end{aligned}$$

Similar derivations have been used in Kano (2015).