A Bayesian approach for sensitivity analysis of incomplete multivariate longitudinal data with potential nonrandom dropout

Abstract

Experiments involving repeated observations of multivariate outcomes are common in biomedical and public health researches which lead to multivariate longitudinal data. These kinds of data have a unique property in the sense that they allow the researcher to study the joint evolution of the multiple outcomes over the time. Recently, there has been a considerable amount of interest on using Bayesian modelling of longitudinal data, data which commonly suffer from incomplete observations. Those Bayesian models for longitudinal data that rely on the ignorability assumption of the dropout mechanism might give misleading inferences. Hence, there is a need to further study the impact of departures from the ignorability assumption on the Bayesian estimates of the model parameters. Current methodology for Bayesian sensitivity analysis mostly involves single response variable in both cross-sectional and longitudinal studies. In this paper, we propose a multivariate extension of the Bayesian index of sensitivity to non-ignorability for the general case of multivariate longitudinal studies with the possibility of having mixed correlated outcomes and a vector of multiple non-ignorability parameters in the missing mechanism. To simultaneously model the mixed responses over the time, we use a random effect latent variable approach. We illustrate the method conducting some simulation studies and analyzing a real data set from a longitudinal study for the comparison of two oral treatments for toenail dermatophyte onychomycosis.

Appendix: Bayesian ISNI derivation

Assume that there is a $s=(m+n)$ dimensional vector of non-ignorability parameters $\Gamma _1=(\gamma _{11},\ldots , \gamma _{1s})$ in the missing mechanism and that $\eta =(\Theta , \Gamma _0)$ is the vector of model parameters. We define the Bayesian ISNI in the case of multiple non-ignorability parameters as the partial first derivatives of the posterior mean of $\eta$ with respect to $\Gamma _1$, i.e.,

$$\begin{aligned}&ISNI[\tilde{\eta }(\Gamma _1)]=\frac{\partial E(\eta |w^{obs},R,\Gamma _1)}{\partial \Gamma _1^T}|_{\Gamma _1=0}\\&\quad \!=\! \left( \! \frac{\partial E(\eta |w^{obs},R,\Gamma _1)}{\partial \gamma _{11}}|_{\Gamma _1=0}, \frac{\partial E(\eta |w^{obs},R,\Gamma _1)}{\partial \gamma _{12}}|_{\Gamma _1=0},\ldots , \frac{\partial E(\eta |w^{obs},R,\Gamma _1)}{\partial \gamma _{1s}}|_{\Gamma _1=0} \!\right) \end{aligned}$$

where the elements of the above vector could be calculated as follows:

$$\begin{aligned} \frac{\partial E(\eta |w^{obs},R,\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0}= & {} \frac{\partial }{\partial \gamma _{1j}} \int \eta \frac{f_{\eta }(w^{obs},R|\Gamma _1)\pi (\eta ) }{\int f_{\eta }(w^{obs},R|\Gamma _1)\pi (\eta ) d \eta } d \eta \vert _{\Gamma _1=0}\\= & {} \int \eta \frac{\partial f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}} \frac{\pi (\eta )}{\int f_{\eta }(w^{obs},R|\Gamma _1)\pi (\eta ) d \eta }d \eta |_{\Gamma _1=0}\\&- \left[ \frac{\int \eta f_{\eta }(w^{obs},R|\Gamma _1)\pi (\eta ) d \eta }{(\int f_{\eta }(w^{obs},R|\Gamma _1)\pi (\eta ) d \eta )^2}\times \int \frac{\partial f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}} \pi (\eta ) d \eta \right] _{\Gamma _1=0}\\= & {} \int \left\{ \eta \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0} \frac{f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) }{\int f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) d \eta } \right\} d \eta \\&- \int \eta \frac{f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) }{\int f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) d \eta } d\eta \\&\times \int \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0} \frac{f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) }{\int f_{\eta }(w^{obs},R|\Gamma _1=0)\pi (\eta ) d \eta } d \eta \\= & {} E_I \left( \eta \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0} \right) \\&- E_I (\eta ) E_I \left( \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0}\right) \\= & {} COV_I(\eta , \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}|_{\Gamma _1=0}), ~~\forall j=1,\ldots , s \end{aligned}$$

where $E_I(.)$ and $COV_I(.)$ denote the posterior mean and covariance functions for the ignorable model. Hence the vector of Bayesian sensitivity index could be denoted as:

$$\begin{aligned} ISNI[\tilde{\eta }(\Gamma _1)]= & {} COV_{I}\left( \eta ,\frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \Gamma _1^T}\vert _{\Gamma _1=0}\right) \nonumber \\= & {} \left[ COV_I(\eta ,\frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{11}}\vert _{\Gamma _1=0}),\right. \nonumber \\&\left. \ldots ,COV_I\left( \eta ,\frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1s}}\vert _{\Gamma _1=0}\right) \right] \end{aligned}$$

(15)

Now if we want to calculate the Bayesian ISNI for the multivariate mixed longitudinal model of Sect. 2, we have to compute the first order derivations of the corresponding log likelihood function (logarithm of the quantity in Eq. (4)).

Assume that $y_{i}^{obs}=(y_{i1},\ldots ,y_{iM_i})$, $z_{i}^{obs}=(z_{i1},\ldots ,z_{iM_i})$, where for $t=1,\ldots , M_i$, $y_{it}=\{y_{ijt}: 1\le j \le m\}$ and $z_{it}=\{z_{ikt}: 1\le k \le n \}$, and also assume that the non-ignorability parameters in Eq. (3) are denoted by, $\Gamma _1=(\Gamma _{11}, \Gamma _{12})$, where $\Gamma _{11}=(\xi _{1},\ldots ,\xi _{m})$ and $\Gamma _{12}=(\eta _{1},\ldots ,\eta _{n})$. The log likelihood function for the multivariate mixed longitudinal model of Sect. 2 could be decomposed as the sum of the following three terms;

$$\begin{aligned} l_1= & {} \sum _{i=1}^{N} log~\left[ \int \prod _{t=1}^{M_i}f(w_{it}|B_i)\phi (B_i)dB_i\right] \\ l_2= & {} \sum _{i=1}^{N}\sum _{t=2}^{M_i} log~P(R_{it}=1|w_{it},w_{it-1})\\ l_3= & {} \sum _{i; M_i<T_i} log~ \left\{ \int P(R_{i,M_i+1}=0|w_{i,M_i+1}^{mis}, w_{iM_i}) f(w_{i,M_i+1}^{mis}|w_{iM_i})dw_{i,M_i+1}^{mis}\right\} \end{aligned}$$

According to the above decomposition, $\frac{\partial l_1}{\Gamma _1^T}=0$ since $l_1$ is not a function of $\Gamma _1$. Also the derivatives of $l_2$ and $l_3$ are as follows;

$$\begin{aligned} \frac{\partial l_2}{\partial \Gamma _{1}^T}|_{\Gamma _1=0}= & {} \sum _{i=1}^{N}\sum _{t=2}^{M_i} \frac{g_{it}'}{g_{it}}|_{\Gamma _1=0}w_{it},\\ \frac{\partial l_3}{\partial \Gamma _{11}^T}|_{\Gamma _1=0}= & {} \sum _{i; M_i<T_i}-\frac{\partial g_{i,M_i+1}/\partial \Gamma _{11}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0} \\&\times \left[ \sum _{mis}\int _{mis}y_{i,M_i+1}^{mis} f(w_{i,M_i+1}^{mis}|w_{iM_i})d z_{i,M_i+1}^{mis}\right] _{\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i} \frac{\partial g_{i,M_i+1}/\partial \Gamma _{11}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0}\left[ \int _{B_i}\int _{mis}\sum _{mis}y_{i,M_i+1}^{mis}\right. \\&\times \left. f(w_{i,M_i+1}^{mis}|B_i)\phi (B_i|w_{iM_i})d z_{i,M_i+1}^{mis}dB_i\right] {\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i}\frac{\partial g_{i,M_i+1}/\partial \Gamma _{11}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0}\\&\times \left[ \int _{B_i}\sum _{mis}y_{i,M_i+1}^{mis} f(y_{i,M_i+1}^{mis}|B_i)\phi (B_i|w_{iM_i})dB_i\right] _{\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i}\frac{\partial g_{i,M_i+1}/\partial \Gamma _{11}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0} E[\mu _{i,M_i+1}^{Y}(\Theta ,B_i)|w_{iM_i}]\}_{\Gamma _1=0}, \end{aligned}$$

$$\begin{aligned} \frac{\partial l_3}{\partial \Gamma _{12}^T}|_{\Gamma _1=0}= & {} -\sum _{i; M_i<T_i}\frac{\partial g_{i,M_i+1}/\partial \Gamma _{12}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0}\\&\times \left[ \sum _{mis}\int _{mis}z_{i,M_i+1}^{mis}f(w_{i,M_i+1}^{mis}|w_{iM_i})d z_{i,M_i+1}^{mis}\right] _{\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i} \frac{\partial g_{i,M_i+1}/\partial \Gamma _{12}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0}\\&\times \left[ \int _{B_i}\int _{mis}\sum _{mis}z_{i,M_i+1}^{mis}f(w_{i,M_i+1}^{mis}|B_i)\phi (B_i|w_{iM_i})d z_{i,M_i+1}^{mis}dB_i\right] _{\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i}\frac{\partial g_{i,M_i+1}/\partial \Gamma _{12}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0}\\&\times \left[ \int _{B_i}\int _{mis}z_{i,M_i+1}^{mis} f(z_{i,M_i+1}^{mis}|B_i)\phi (B_i|w_{iM_i})dB_i\right] _{\Gamma _1=0}\\= & {} -\sum _{i; M_i<T_i}\frac{\partial g_{i,M_i+1}/\partial \Gamma _{12}^T}{1-g_{i,M_i+1}}\vert _{\Gamma _1=0} E[\mu _{i,M_i+1}^{z}(\Theta ,B_i)|w_{iM_i}]_{\Gamma _1=0} \end{aligned}$$

where,

$$\begin{aligned} \mu _{i,M_i+1}^{Y}(\Theta ;B_i)= & {} E[Y_{i,M_i+1}|B_i]=\sum _{mis}y_{i,M_i+1}^{mis}f(y_{i,M_i+1}^{mis}|B_i),\\ \mu _{i,M_i+1}^{Z}(\Theta ;B_i)= & {} E[Z_{i,M_i+1}|B_i]=\int _{mis}z_{i,M_i+1}^{mis}f(z_{i,M_i+1}^{mis}|B_i)d z_{i,M_i+1}^{mis}, \end{aligned}$$

and $g_{i,M_i+1}=P(R_{i,M_i+1}=1|W_{i,M_i+1},W_{i,M_i},X,R_{i,M_i}=1)$ which has been modeled in Eq. (3). Also, the posterior means are calculated with respect to the posterior distribution of $B_i$ given $W_i^{obs}=(y_{i}^{obs}, z_{i}^{obs})$ for the ignorable model (i.e., $\phi (B_i|W_{i,M_i})$).

According to Eq. (15), we need the posterior covariance of the above quantities with the vector of parameters $\eta$ where we have, $COV_{I}(\eta ,\frac{\partial l_2}{\partial \Gamma _1^T}\vert _{\Gamma _1=0})=0$. Hence, the vector of Bayesian ISNI with multiple non-ignorability parameters for the multivariate longitudinal data would be reduced to:

$$\begin{aligned} ISNI[\tilde{\eta }(\Gamma _1)]= & {} -\left( COV_I \left( \eta , \sum _{i; M_i<T_i}\frac{g_{i}'|_{\Gamma _1=0}}{1-g_{i}|_{\Gamma _1=0}} E[\mu _{i,M_i+1}^{Y}(\Theta ,B_i)\left| w_{iM_i}\right] \right) ,\right. \\&\times \left. COV_I \left( \eta , \sum _{i; M_i<T_i}\frac{g_{i}'|_{\Gamma _1=0}}{1-g_{i}|_{\Gamma _1=0}} E[\mu _{i,M_i+1}^{Z}(\Theta ,B_i)\left| w_{iM_i}\right] \right) \right) . \end{aligned}$$

On the other hand, one can approximate the above vector of covariances using the Delta method which leads to the following approximation:

$$\begin{aligned} COV_I\left( \eta , \frac{\partial log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \gamma _{1j}}_{\Gamma _1=0}\right) \approx VAR_I(\eta )\times \frac{\partial ^2 log~ f_{\eta }(w^{obs},R|\Gamma _1)}{\partial \eta \partial \Gamma _1^T}|_{\Gamma _1=0} \end{aligned}$$

where $VAR_I(\eta )$ is the covariance matrix of the posterior distribution of $\eta$ evaluated at the ignorable model.