Quantile Regression in the Presence of Monotone Missingness with Sensitivity Analysis

• Mixture models factor the joint distribution of response and missingness as $$p (\mathbf Y, \mathbf S |\mathbf x, \mathbf \omega) = p (\mathbf Y|\mathbf S, \mathbf x, \mathbf \omega) p (\mathbf S | \mathbf x, \mathbf \omega).$$
• The full-data response distribution is given by $$p (\mathbf Y | \mathbf x, \mathbf \omega) = \sum_{S \in \mathcal{S}} p(\mathbf Y| \mathbf S, \mathbf x, \mathbf \theta) p (\mathbf S | \mathbf x, \mathbf \phi),$$ where $\mathcal{S}$ is the sample space for follow-upch time $S$ and the parameter vector $\mathbf \omega$ is partitioned as $(\mathbf \theta, \mathbf \phi)$.
• The conditional distribution of response within patterns can be decomposed as $$P (Y_{obs}, Y_{mis} | \mathbf S, \mathbf \theta) = P (Y_{mis}|Y_{obs}, \mathbf S, \mathbf \theta_E) Pr (Y_{obs} | \mathbf S, \mathbf \theta_{y, O}),$$
• $\mathbf \theta_E$: extrapolation distribution
• $\mathbf \theta_{y, O}$ : distribution of observed responses

• Multivariate normal distributions within each pattern $$\begin{align} Y_{i1}|S_i = k & \sim N (\Delta_{i1} + \mathbf \beta_1^{(k)}, \sigma_1^{(k)} ), k = 1, \ldots, J,\\ Y_{ij}|\mathbf Y_{ij^{-}}, S_i = k & \sim \begin{cases} \textrm{N} \big (\Delta_{ij} + \mathbf y_{ij^{-}}^T \mathbf \beta_{Y,j-1}, \sigma_j \big), & k \geq j ; \\ \textrm{N} \big ( \chi(\mathbf x_{i}, \mathbf Y_{ij^{-}}), \sigma_j \big), & k < j ; \\ \end{cases}, \mbox{ for } 2 \leq j \leq J, \\ S_{i} = k|\: \mathbf x_{i} & \sim \textrm{Multinomial}(1, \mathbf \phi). \end{align}$$

• The marginal quantile regression models: $$Pr (Y_{ij} \leq \mathbf x_{i}^T \mathbf \gamma_j ) = \tau,$$
• $\chi(\mathbf x_{i}, \mathbf y_{ij^{-}})$ is the mean of the unobserved data distribution and allows sensitivity analysis by varying assumptions on $\chi$; for computational reasons, we assume that $\chi$ is linear in $y_{ij^{-}}$.
• Here we specify $$\chi(\mathbf x_{i}, \mathbf y_{ij^{-}}) = \Delta_{ij} + \mathbf y_{ij^{-}}^T \mathbf \beta_{y,j-1} + h_{0}^{(k)}.$$

$\Delta_{ij}$ are subject/time specific intercepts determined by the parameters in the model and are determined by the marginal quantile regressions,

• Embed the marginal quantile regressions directly in the model through constraints in the likelihood of pattern mixture models
• The mixture model allows the marginal quantile regression coefficients to differ by quantiles. Otherwise, the quantile lines would be parallel to each other.
• The mixture model also allows sensitivity analysis.

• Mixture models are not identified due to insufficient information provided by observed data.
• Specific forms of missingness are needed to induce constraints to identify the distributions for incomplete patterns, in particular, the extrapolation distribution
• In mixture models , MAR holds (Molenberghs et al. 1998; Wang & Daniels, 2011) if and only if, for each $j \geq 2$ and $k < j$: $$p_k(y_j|y_1, \ldots, y_{j-1}) = p_{\geq j}(y_j|y_1, \ldots, y_{j-1}).$$

• When $2 \leq j \leq J$ and $k < j$, $Y_j$ is not observed, thus $h_0^{(k)}$ can not be identified from the observed data.
• $\mathbf \xi_s = (h_0^{(k)})$ is a set of sensitivity parameters (Daniels & Hogan 2008), where $k =1, ..., J-1$.
• $\mathbf \xi_s = \mathbf \xi_{s0} = \mathbf 0$, MAR holds.
• $\mathbf \xi_s$ is fixed at $\mathbf \xi_s \neq \mathbf \xi_{s0}$, MNAR.
• We can vary $\mathbf \xi_s$ around $\mathbf 0$ to examine the impact of different MNAR mechanisms.

The observed data likelihood for an individual $i$ with follow-up time $S_i = k$ is

• To facilitate computation of the $\Delta$'s and the likelihood, we propose a tricky way to obtain analytic forms for the required integrals.
• Use the bootstrap to construct confidence interval and make inferences.

• Weights were recorded at baseline ($Y_0$), 6 months ($Y_1$) and 18 months ($Y_2$).
• We are interested in how the distributions of weights at six months and eighteen months change with covariates.
• The regressors of interest include AGE, RACE (black and white) and weight at baseline ($Y_0$).
• Weights at the six months ($Y_1$) were always observed and 13 out of 224 observations (6%) were missing at 18 months ($Y_2$).
• The AGE covariate was scaled to 0 to 5 with every increment representing 5 years.
• We fitted regression models for bivariate responses $\mathbf Y_i = (Y_{i1}, Y_{i2})$ for quantiles (10%, 30%, 50%, 70%, 90%).
• We ran 1000 bootstrap samples to obtain 95% confidence intervals.

We also did a sensitivity analysis based on an assumption of MNAR.

• Based on previous studies of pattern of weight regain after lifestyle treatment (Wadden et al. 2001; Perri et al. 2008) we assume that $$E(Y_2 - Y_1| R=0) = 3.6 \mbox{kg},$$ which corresponds to 0.3kg regain per month after finishing the initial 6-month program.

• Specify $\chi(\mathbf x_{i}, Y_{i1})$ as

  $$\chi(\mathbf x_{i}, y_{i1}) = 3.6 + y_{i1},$$

• There are no large differences for estimates for $Y_2$ under MNAR vs MAR.

• This is partly due to the low proportion of missing data in this study.

• Developed a marginal quantile regression model for data with monotone missingness.
• Used a pattern mixture model to jointly model the full data response and missingness.
• Estimate marginal quantile regression coefficients instead of conditional on random effects.
• Allows for sensitivity analysis which is essential for the analysis of missing data (NAS 2010).
• Allows the missingness to be non-ignorable.
• Recursive integration simplifies computation and can be implemented in high dimensions.

Future Work

• Simulation results showed that the mis-specification of the error term did have an impact on the extreme quantile regression inferences.
• Working on replacing it with a non-parametric model, for example, a Dirichlet process with mixture of normals.