Bayesian second-order sensitivity of longitudinal inferences to non-ignorability: an application to antidepressant clinical trial data

Here, the details of statistical inference and Bayesian sensitivity analysis of Bivariate longitudinal simulation study are given. Based on the simulation set up in subsection 4.1, the logarithm of the joint probability function of the observed response variables and the missing indicator needed for non-ignorable model estimation is

where the data is assumed to be sorted by the missing pattern, so that the first N ₁(≤N) cases are completely observed. In addition, f μ 2.1 , σ 2.1 2 y i 2 miss | y i 1 is the normal pdf with mean μ 2.1 = μ 2 + σ 12 σ 1 2 ( y i 1 − μ 1 ) and variance σ 2.1 2 = σ 2 2 − σ 12 2 σ 1 2 . Consequently, the integral part of the above formula can be rewritten as follows:

The above expectation could be calculated based on the following lemma [39]:

Implementing lemma 1, the explicit form of the expectation in (13) would be:

Hence, the logarithm of the non-ignorable joint pdf needed for the global sensitivity analysis plotted in Figures 1 and 2 can be rewritten as follows:

To calculate Bayesian ISNI and ISSNI, only the ignorable model assuming ψ ₁ = 0 in (16) needs to be fitted. Ignorable posterior inference for the vector of transformed parameters ϕ = μ 1 , σ 1 2 , β 20.1 , β 21.1 , σ 2.1 2 T , where

would be performed based on the following ignorable hierarchical posterior distributions:

where

and

We implemented R to get posterior samples of the ignorable model parameters ϕ , using conditional posterior distributions (17)–(21). For this model, sampling was continued until 12,000 draws were available in total and to have plausible results, each scenario of occurrence of missingness is repeated 100 times. In addition to have posterior samples of ψ ₀₀ and ψ ₀₁, based on the probit model (7) and prior (8), OpenBUGS software is used. The number of simulations and repetitions were the same as previous sampling procedure. All posterior samples were checked to be uncorrelated and have plausible mc-errors.

It should be noticed that ϕ is a one-to-one function of the vector of interesting parameters θ = μ 1 , σ 1 2 , μ 2 , σ 2 2 , ρ T , where ρ = σ ₁₂/σ ₁ σ ₂. Posterior samples of θ are extracted based on the inverse transformation of posterior sample draws of ϕ . The ISNI is calculated as follows:

where Φ i ′ = Φ ′ ( ψ 00 + ψ 01 y i 1 ) = ϕ ( ψ 00 + ψ 01 y i 1 ) , the pdf of standard normal distribution and E ( Y i 2 | y i 1 ) = μ 2.1 = μ 2 + ρ σ 2 σ 1 ( y i 1 − μ 1 ) . The ISSNI based on (5) would be:

where Φ i ′ ′ = Φ ′ ′ ( ψ 00 + ψ 01 y i 1 ) = ( ψ 00 + ψ 01 y i 1 ) ϕ ( ψ 00 + ψ 01 y i 1 ) .

In the three-time longitudinal simulation setting of Section 4.2, let θ = ( β 0 , β 1 , σ , σ u 0 ) T and the data be sorted by the missing pattern. Actually, it is assumed that there are three monotone patterns of missingness denoted by the vector of missing indicators M _i  = (M _i2, M _i3) (All subjects are assumed to be observed at the baseline):

1. Completers: M _i  = (0, 0) for i = 1, …, N ₁

2. Subjects who drop put at the third visit: M _i  = (0, 1) for i = N ₁ + 1, …, N ₂.

3. Subjects who drop put at the second visit: M _i  = (1, 1) for i = N ₂ + 1, …, N.

Based on these assumption and the mentioned drop-out mechanism, the logarithm of the non-ignorable joint pdf of the observed response variables and the missing indicators can be decomposed as follows,

where

and

The integral parts of (24) have a logistic-normal form [39], which could be rewritten as

and

To have explicit forms for the above expectations, one-probit approximation approach [39] is applied:

where according to the minimax criterion m = 1.7 is chosen. Therefore, implementing the one-probit approximation along with lemma 1, the explicit form of the expectations would be:

and

where a = ψ ₀₀ + 2ψ ₀₁ + ψ ₀₂ y _i1 and b = ψ ₀₀ + 3ψ ₀₁ + ψ ₀₂ y _i2. Substituting the above formulas in (24), the logarithm of the non-ignorable joint pdf can be summarized and rewritten as follows:

As mentioned before, to calculate Bayesian ISNI and ISSNI ignorable posterior samples of the parameters β ₁, σ (square root of σ ²) and σ u 0 (square root of σ u 0 2 ) are just needed. We have implemented brms package in R for sampling model parameters and OpenBUGS software for drop-out mechanism parameters. For this model, sampling was continued until 800,000 draws were available in total, excluding the initial 400,000 burn-in samples and to have plausible results, each scenario of occurrence of missingness is repeated 50 times. It is absolutely essential that ignorable posterior samples converge, since true Bayesian indices must be calculated based on the efficient ignorable samples and not realizing that, can lead to misleading results. Therefore, all posterior samples were checked to be uncorrelated and have plausible mc-errors.

According to the ignorable posterior samples, ISNI and ISSNI are calculated as,

and

where h(.) = expit(.), h ′ ( . ) = expit ( . ) 1 + exp ( . ) and h ′ ′ ( . ) = expit ( . ) ( 1 − exp ( . ) ) ( 1 + exp ( . ) ) 2 .