Quantile Regression in the Presence of Monotone Missingness with Sensitivity Analysis

A missing data pattern is monotone if, for each individual, there exists a measurement occasion \(j\) such that \(R_1 = ··· = R_{j−1} = 1\) and \(R_j = R_{j+1} = · · · = R_J = 0\); that is, all responses are observed through time \(j − 1\), and no responses are observed thereafter. \(S\) is called follow-up time.

• Assumption: \(Y_{i1}\) is always observed.
• Interested: \(\tau\)-th marginal quantile regression coefficients \(\mathbf \gamma_j = (\gamma_{j1}, \gamma_{j2}, \ldots, \gamma_{jp})^T\), \[ Pr (Y_{ij} \leq \mathbf x_i^{T} \mathbf \gamma_j ) = \tau, \mbox{ for } j = 1, \ldots, J, \] \[ p_k(Y) = p(Y | S = k), \quad p_{\geq k} (Y) = p(Y | S \geq k) \]

• 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, \[ \tau = Pr (Y_{ij} \leq \mathbf x_{i}^T \mathbf \gamma_j ) = \sum_{k=1}^J \phi_kPr_k (Y_{ij} \leq \mathbf x_{i}^T \mathbf \gamma_j ) \mbox{ for } j = 1, \] and \[ \begin{align} \tau &= Pr (Y_{ij} \leq \mathbf x_{i}^{T} \mathbf \gamma_j ) = \sum_{k=1}^J \phi_kPr_k (Y_{ij} \leq \mathbf x_{i}^{T} \mathbf \gamma_j ) \\ & = \sum_{k=1}^J \phi_k \int\cdots \int Pr_k (Y_{ij} \leq \mathbf x_{i}^{T} \mathbf \gamma_j | \mathbf y_{ij^{-}} ) p_k (y_{i(j-1)}| \mathbf y_{i(j-1)^{-}}) \nonumber \\ & \quad \cdots p_k (y_{i2}| y_{i1}) p_k(y_{i1}) dy_{i(j-1)}\cdots dy_{i1}. \mbox{ for } j = 2, \ldots, J .\nonumber \end{align} \]

• 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 \(k < j\): \[ p_k(y_j|y_1, \ldots, y_{j-1}) = p_{\geq j}(y_j|y_1, \ldots, y_{j-1}). \]

\[ Y_{ij}|\mathbf Y_{ij^{-}}, S_i = k \sim \begin{cases} \textrm{N} \big (\Delta_{ij} + \mathbf y_{ij^{-}}^T \mathbf \beta_{Y,j-1}, \sigma_j \big), & k \geq j ; \\ \textrm{N} \big (\Delta_{ij} + \mathbf y_{ij^{-}}^T \mathbf \beta_{y,j-1} + h_{0}^{(k)}, \sigma_j \big), & k < j ; \\ \end{cases}, \mbox{ for } 2 \leq j \leq J, \]

• 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 \[ \begin{align} L_i(\mathbf \xi| \mathbf y_i, S_{i} = k) & = \phi_kp_k (y_k | y_1, \ldots, y_{k-1}) p_k (y_{k-1}|y_1, \ldots, y_{k-2}) \cdots p_{k} (y_1), \\ & = \phi_k p_{\geq k} (y_k | y_1, \ldots, y_{k-1}) p_{\geq k-1} (y_{k-1}|y_1, \ldots, y_{k-2}) \cdots p_{k} (y_1), \nonumber \end{align} \]

• 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.