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

  • Elahe Momeni Roochi and Samaneh Eftekhari Mahabadi ORCID logo EMAIL logo
Published/Copyright: November 27, 2023
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 20 Issue 2

Abstract

Incomplete data is a prevalent complication in longitudinal studies due to individuals’ drop-out before intended completion time. Currently available methods via commercial software for analyzing incomplete longitudinal data at best rely on the ignorability of the drop-outs. If the underlying missing mechanism was non-ignorable, potential bias arises in the statistical inferences. To remove the bias when the drop-out is non-ignorable, joint complete-data and drop-out models have been proposed which involve computational difficulties and untestable assumptions. Since the critical ignorability assumption is unverifiable based on the observed part of the sample, some local sensitivity indices have been proposed in the literature. Specifically, Eftekhari Mahabadi (Second-order local sensitivity to non-ignorability in Bayesian inferences. Stat Med 2018;59:55–95) proposed a second-order local sensitivity tool for Bayesian analysis of cross-sectional studies and show its better performance for handling bias compared with the first-order ones. In this paper, we aim to extend this index for the Bayesian sensitivity analysis of normal longitudinal studies with drop-outs. The index is driven based on a selection model for the drop-out mechanism and a Bayesian linear mixed-effect complete-data model. The presented formulas are calculated using the posterior estimation and draws from the simpler ignorable model. The method is illustrated via some simulation studies and sensitivity analysis of a real antidepressant clinical trial data. Overall, the numerical analysis showed that when repeated outcomes are subject to missingness, regression coefficient estimates are nearly approximated well by a linear function in the neighbourhood of MAR model, but there are a considerable amount of second-order sensitivity for the error term and random effect variances in Bayesian linear mixed-effect model framework.

Keywords: second-order local sensitivity; normal longitudinal data; informative drop-out; Bayesian approach; linear mixed-effect model
Corresponding author: Samaneh Eftekhari Mahabadi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, E-mail:

  1. Research ethics: Not applicable.

  2. Code availability: The simulation and data analysis codes for this study are available on Code Ocean.

  3. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Competing interests: The authors state no conflict of interest.

  5. Research funding: None declared.

  6. Data availability: The raw data can be obtained on request from the corresponding author.

Appendix A

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

ln f μ , R , ψ 0 y obs , m | ψ 1 = ∑ i = 1 N ln f μ , R y i 1 , y i 2 obs + ∑ i = 1 N 1 ln ( 1 − Φ ( ψ 00 + ψ 01 y i 1 + ψ 1 y i 2 ) ) + ∑ i = N 1 + 1 N ln ∫ Φ ψ 00 + ψ 01 y i 1 + ψ 1 y i 2 miss f μ 2.1 , σ 2.1 2 y i 2 miss | y i 1 d y i 2 miss ,

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:

(13) ∫ Φ ψ 00 + ψ 01 y i 1 + ψ 1 y i 2 miss f μ 2.1 , σ 2.1 2 y i 2 miss | y i 1 d y i 2 miss = E ( Φ ψ 00 + ψ 01 Y i 1 + ψ 1 Y i 2 miss | Y i 1 = y i 1 ) .

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

Lemma 1

Let U ∼ N(0, σ 2), then for all s ∈ R

(14) E ( Φ ( s + U ) ) = Φ s 1 + σ 2

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

(15) E ( Φ ψ 00 + ψ 01 Y i 1 + ψ 1 Y i 2 miss | Y i 1 = y i 1 ) = Φ ( ( ψ 00 + ψ 01 y i 1 + ψ 1 μ 2.1 ) / 1 + ψ 1 2 σ 2.1 2 ) .

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:

(16) ln f μ , R , ψ 0 y obs , m | ψ 1 = ∑ i = 1 N ln f μ , R y i 1 , y i 2 obs + ∑ i = 1 N 1 ln ( 1 − Φ ( ψ 00 + ψ 01 y i 1 + ψ 1 y i 2 ) ) + ∑ i = N 1 + 1 N ln Φ ψ 00 + ψ 01 y i 1 + ψ 1 μ 2.1 1 + ψ 1 2 σ 2.1 2 .

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

β 21.1 = σ 12 σ 1 2 ,   β 20.1 = μ 2 − μ 1 β 21.1

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

(17) N σ ̂ 1 2 σ 1 2 | y obs ∼ χ ( N ) 2

(18) μ 1 | σ 1 2 , y obs ∼ N μ 1 ̂ , σ 1 2 N

(19) N 1 σ ̂ 2.1 2 σ 2.1 2 | y obs ∼ χ ( N 1 − 1 ) 2

(20) β 21.1 | σ 2.1 2 , y obs ∼ N β ̂ 21.1 , σ 2.1 2 N 1 S 11

(21) β 20.1 | β 21.1 , σ 2.1 2 , y obs ∼ N y ̄ 2 − β 21.1 y ̄ 1 , σ 2.1 2 N 1 ,

where

σ ̂ 1 2 = 1 N ∑ i = 1 N ( y i 1 − μ 1 ̂ ) 2 , μ 1 ̂ = 1 N ∑ i = 1 N y i 1 , σ ̂ 2.1 2 = S 2.1 2 , β ̂ 21.1 = S 12 S 11

and

y ̄ j = 1 N 1 ∑ i = 1 N 1 y i j , j = 1,2 , S j k = 1 N 1 ∑ i = 1 N 1 ( y i j − y ̄ j ) ( y i k − y ̄ k ) , j , k = 1,2 S 2.1 2 = S 22 − S 12 2 S 11 .

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:

(22) ISNI ( θ ̂ B ( ψ 1 ) ) = ∑ i = N 1 + 1 N E post Φ i ′ Φ i Cov post ( θ , E ( Y i 2 | y i 1 ) ) ,

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:

(23) ISSNI ( θ ̂ B ( ψ 1 ) ) = ∑ i = N 1 + 1 N E post Φ i ′ ′ Φ i Cov post ( θ , E Y i 2 2 | y i 1 ) − ∑ i = N 1 + 1 N E post Φ i ′ Φ i 2 C o v post ( θ , E 2 ( Y i 2 | y i 1 ) ) + Cov post θ , ∑ i = N 1 + 1 N Φ i ′ Φ i 2 − 2 ∑ i = 1 N 1 y i 2 E post Φ i ′ 1 − Φ i ∑ i = N 1 + 1 N E post Φ i ′ Φ i × Cov post ( θ , E ( Y i 2 | y i 1 ) ) − 2 ∑ i = N 1 + 1 N E post Φ i ′ Φ i E ( Y i 2 | y i 1 ) − ∑ i = 1 N 1 y i 2 E post Φ i ′ 1 − Φ i × ISNI ( θ ̂ B ( ψ 1 ) ) ,

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 ) .

Appendix B

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,

(24) ln f θ , ψ 0 y obs , m | ψ 1 = ∑ i = 1 N ln f θ y i 1 , y i 2 obs , y i 3 obs + ∑ i = 1 N 1 ∑ j = 2 3 ln 1 − expit ( ψ 00 + ψ 01 j + ψ 02 y i j − 1 + ψ 1 y i j ) + ∑ i = N 1 + 1 N 2 ln 1 − expit ( ψ 00 + 2 ψ 01 + ψ 02 y i 1 + ψ 1 y i 2 ) + ∑ i = N 2 + 1 N ln ∫ expit ( ψ 00 + 2 ψ 01 + ψ 02 y i 1 + ψ 1 y i 2 miss ) × f μ 2.1 , σ 2.1 2 y i 2 miss | y i 1 d y i 2 miss + ∑ i = N 1 + 1 N 2 ln ∫ expit ( ψ 00 + 3 ψ 01 + ψ 02 y i 2 + ψ 1 y i 3 miss ) × f μ 3.12 , σ 3.12 2 y i 3 miss | y i 1 , y i 2 obs d y i 3 miss ,

where

Y i 2 miss | Y i 1 = y i 1 ∼ N μ 2.1 , σ 2.1 2 , μ 2.1 = β 0 + β 1 + σ u 0 2 σ 2 + σ u 0 2 ( y i 1 − β 0 ) , σ 2.1 2 = 2 σ 2 σ u 0 2 + ( σ 2 ) 2 σ 2 + σ u 0 2 ,

and

Y i 3 miss | Y i 1 = y i 1 , Y i 2 obs = y i 2 obs ∼ N μ 3.12 , σ 3.12 2 , μ 3.12 = β 0 + 2 β 1 + σ u 0 2 y i 1 + y i 2 obs − 2 β 0 − β 1 2 σ u 0 2 + σ 2 , σ 3.12 2 = σ 2 3 σ u 0 2 + σ 2 σ 2 + 2 σ u 0 2 .

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

∫ expit ( ψ 00 + 2 ψ 01 + ψ 02 y i 1 + ψ 1 y i 2 miss ) f μ 2.1 , σ 2.1 2 y i 2 miss | y i 1 d y i 2 miss = E expit ( ψ 00 + 2 ψ 01 + ψ 02 Y i 1 + ψ 1 Y i 2 miss ) | Y i 1 = y i 1

and

∫ expit ( ψ 00 + 3 ψ 01 + ψ 02 y i 2 + ψ 1 y i 3 miss ) f μ 3.12 , σ 3.12 2 y i 3 miss | y i 1 , y i 2 obs d y i 3 miss = E expit ( ψ 00 + 3 ψ 01 + ψ 02 Y i 2 + ψ 1 Y i 3 miss ) | Y i 1 = y i 1 , Y i 2 obs = y i 2 obs .

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

expit ( x ) ≃ Φ x m , m > 1 ,

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:

E expit ( ψ 00 + 2 ψ 01 + ψ 02 Y i 1 + ψ 1 Y i 2 miss ) | Y i 1 = y i 1 = Φ a + ψ 1 μ 2.1 1 . 7 2 + ψ 1 2 σ 2.1 2

and

E expit ( ψ 00 + 3 ψ 01 + ψ 02 Y i 2 + ψ 1 Y i 3 miss ) | Y i 1 = y i 1 , Y i 2 obs = y i 2 obs = Φ b + ψ 1 μ 3.12 1 . 7 2 + ψ 1 2 σ 3.12 2 ,

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:

(25) ln f θ , ψ 0 y obs , m | ψ 1 = ∑ i = 1 N ln f θ y i 1 , y i 2 obs , y i 3 obs + ∑ i = 1 N 2 ∑ j = 2 3 ln 1 − expit ( ψ 00 + ψ 01 j + ψ 02 y i j − 1 + ψ 1 y i j ) + ∑ i = N 2 + 1 N 1 ln 1 − expit ( ψ 00 + 2 ψ 01 + ψ 02 y i 1 + ψ 1 y i 2 ) + ∑ i = N 1 + 1 N ln Φ a + ψ 1 μ 2.1 1 . 7 2 + ψ 1 2 σ 2.1 2 + ∑ i = N 2 + 1 N 1 ln Φ b + ψ 1 μ 3.12 1 . 7 2 + ψ 1 2 σ 3.12 2 .

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,

ISNI ( θ ̂ B ( ψ 1 ) ) = ∑ i = N 2 + 1 N 1 E post h ′ ( b ) h ( b ) Cov post ( θ , μ 3.12 ) + ∑ i = N 1 + 1 N E post h ′ ( a ) h ( a ) Cov post ( θ , μ 2.1 )

and

ISSNI ( θ ̂ B ( ψ 1 ) ) = ∑ i = N 2 + 1 N 1 E post h ′ ′ ( b ) h ( b ) Cov post θ , σ 3.12 2 + μ 3.12 2 − E post h ′ 2 ( b ) h 2 ( b ) Cov post θ , μ 3.12 2 + ∑ i = N 1 + 1 N E post h ′ ′ ( a ) h ( a ) Cov post θ , σ 2.1 2 + μ 2.1 2 − E post h ′ 2 ( a ) h 2 ( a ) Cov post θ , μ 2.1 2 + Cov post θ , ∑ i = N 2 + 1 N 1 h ′ ( b ) h ( b ) μ 3.12 + ∑ i = N 1 + 1 N h ′ ( a ) h ( a ) μ 2.1 2 − 2 ∑ i = 1 N 2 ∑ j = 2 3 y i j h ′ ( ψ 00 + ψ 01 j + ψ 02 y i j − 1 ) 1 − h ( ψ 00 + ψ 01 j + ψ 02 Y i j − 1 ) + ∑ i = N 2 + 1 N 1 y i 2 h ′ ( b ) 1 − h ( b ) × ISNI ( θ ̂ B ( ψ 1 ) ) − 2 ∑ i = N 2 + 1 N 1 E post h ′ ( b ) h ( b ) μ 3.12 + ∑ i = N 1 + 1 N E post h ′ ( a ) h ( a ) μ 2.1   − ∑ i = 1 N 2 ∑ j = 2 3 y i j E post h ′ ( ψ 00 + ψ 01 j + ψ 02 y i j − 1 ) 1 − h ( ψ 00 + ψ 01 j + ψ 02 y i j − 1 ) − ∑ i = N 2 + 1 N 1 y i 2 E post h ′ ( a ) 1 − h ( a ) ,

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

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Received: 2022-01-26
Accepted: 2023-10-19
Published Online: 2023-11-27

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2022-0014
Keywords for this article
second-order local sensitivity; normal longitudinal data; informative drop-out; Bayesian approach; linear mixed-effect model

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Random forests for survival data: which methods work best and under what conditions?
  4. Flexible variable selection in the presence of missing data
  5. An interpretable cluster-based logistic regression model, with application to the characterization of response to therapy in severe eosinophilic asthma
  6. MBPCA-OS: an exploratory multiblock method for variables of different measurement levels. Application to study the immune response to SARS-CoV-2 infection and vaccination
  7. Detecting differentially expressed genes from RNA-seq data using fuzzy clustering
  8. Hypothesis testing for detecting outlier evaluators
  9. Response to comments on ‘sensitivity of estimands in clinical trials with imperfect compliance’
  10. Commentary
  11. Comments on “sensitivity of estimands in clinical trials with imperfect compliance” by Chen and Heitjan
  12. Research Articles
  13. Optimizing personalized treatments for targeted patient populations across multiple domains
  14. Statistical models for assessing agreement for quantitative data with heterogeneous random raters and replicate measurements
  15. History-restricted marginal structural model and latent class growth analysis of treatment trajectories for a time-dependent outcome
  16. Revisiting incidence rates comparison under right censorship
  17. Ensemble learning methods of inference for spatially stratified infectious disease systems
  18. The survival function NPMLE for combined right-censored and length-biased right-censored failure time data: properties and applications
  19. Hybrid classical-Bayesian approach to sample size determination for two-arm superiority clinical trials
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