A sensitivity approach to modeling longitudinal bivariate ordered data subject to informative dropouts

Abstract

Incomplete data abound in epidemiological and clinical studies. When the missing data process is not properly investigated, inferences may be misleading. An increasing number of models that incorporate nonrandom incomplete data have become available. At the same time, however, serious doubts have arisen about the validity of these models, known to rely on strong and unverifiable assumptions. A common conclusion emerging from the current literature is the clear need for a sensitivity analysis. We propose in this paper a detailed sensitivity analysis using graphical and analytical techniques to understand the impact of missing-data assumptions on inferences. Specifically, we explore the influence of perturbing a missing at random model locally in the direction of non-random dropout models. Data from a psychiatric trial are used to illustrate the methodology.

Appendices

Appendix 1

Numerical integration of the profile log-likelihood

In consideration of a bounded continuous mixing distribution, the Gauss–Legendre quadrature is chosen to approximate the integration. Reasonable lower and upper bounds of the integration can be obtained from the estimates of the standard deviation of a fully parametric normal model. Specifically, for each dimension of the random effects, the range of integration is chosen to be $(-5 SD_{\ell u}, 5 SD_{\ell u})$, where $SD_{\ell u}$ is the estimated standard deviation of the uth normal random effects of the $\ell$th outcome. The corresponding product space approximately supports all probability mass of the mixing distribution. In confining the support of the random effects to a finite space, the starting density of the mixing distribution is chosen to be uniform given by

$$f^{0}(b)=\prod_{\ell=1,2}\prod_{u=1}^{r_{\ell}}\frac{1}{10 SD_{\ell u}}, \quad \forall b,$$

where $r_{\ell}$, for $\ell=1,2$ is dimension of the random effects design vector $z_{\ell}$. When the random effects dimension $r_{1}+r_{2}$ is large, say greater than 4, a Monte Carlo simulation is used to perform the integration. It is widely known that for higher dimensional integrals, the number of support points in a quadrature approach increases exponentially compared to that of a Monte Carlo approach, which increases only linearly.

Appendix 2

Optimization of the profile log-likelihood for fixed λ

The marginal non-ignorable log-likelihood for a sample of K independent subjects is given by

$$L(\alpha, \beta_{3}, \lambda)=\sum_{k=1}^{K} \log \mathcal{L}(y_{k,1:(t_{k}-1)}^{\rm obs}, t_k).$$

Although this non-ignorable log-likelihood is a continuous function of $(\alpha, \beta_3)$ for fixed λ, it involves recursive functions. Specifically, the smoothing parameter ν of the mixing distribution is imbedded in recursive functions, which do not allow explicit derivatives. Thus, maximizing the log-likelihood with respect to (α, β₃) and the smoothing parameter ν using Newton–Raphson approach with analytical derivatives is proved to be a difficult task. We use the Davidon–Fletcher–Powell (DFP) method, which only requires evaluations of the objective function to search for an optimum point. This is a quasi-Newton method that builds up an approximation of the inverse Hessian and is often regarded as the most sophisticated approach for solving unconstrained problems. Earliest schemes for constructing the inverse Hessian was originally proposed by Davidon (1959) and later developed by Fletcher and Powell (1963). It has the interesting property that, for a quadratic objective, it simultaneously generates the directions of the conjugate gradient method while constructing the inverse Hessian. The method is also referred to as the variable metric method. The DFP method has theoretical properties that guarantee superlinear (fast) convergence rate and global convergence under certain conditions. It should be noted however, that this method could fail for general non-linear problems. Specifically, DFP is highly sensitive to inaccuracies in line searches and can get stuck on a saddle-point or may obtain only a local maximum. To address this issue, we used different starting values to examine whether the algorithm converges to the same point.

Finally, all the computations are performed assuming the bivariate probit model, but similar calculations can be done for the bivariate logistics model.