Estimating the correlation between rates of change in longitudinal studies in presence of dropout

Technical Summary: Estimating Correlation Between Rates of Change in Longitudinal Studies with Dropout

Problem Statement
In longitudinal neuroimaging and biomarker studies, a common objective is to estimate the correlation between subject-specific rates of change (slopes) of two different signals (e.g., brain atrophy and cognitive decline). The standard approach, termed the "slopes-first" strategy, involves fitting an ordinary least squares (OLS) regression for each subject to obtain individual slope estimates, followed by calculating the Pearson correlation of these estimates.

While algebraically equivalent to a contrast-based estimator under complete data, the authors demonstrate that the slopes-first approach becomes biased in the presence of missing data, specifically independent dropout. The core issue is that the slopes-first method implicitly applies heterogeneous weights to the cross-covariance between the two signals based on the specific missing-data patterns of the subjects. If the cross-covariance between the signals is not constant over time, these data-dependent weights introduce bias into the correlation estimate.

Methodology
The paper formalizes the relationship between the slopes-first estimator and the joint variance-covariance matrix of the signals. It proposes an alternative "contrast-based" strategy that estimates the joint covariance matrix before applying the contrast to derive the correlation.

The authors define two primary strategies for estimation:

1. Strategy 1 (Slopes-First): Estimate subject-specific slopes via OLS for each individual using their available data points, then compute the correlation of these slopes.
2. Strategy 2 (Joint Covariance): Estimate the joint variance-covariance matrix ($\Sigma$) of the signals across all timepoints, then compute the correlation using pre-defined weights derived solely from the measurement times.
   • Strategy 2a: Estimates $\Sigma$ using only subjects with complete data (complete-case analysis) or pairwise complete observations.
   • Strategy 2b: Estimates $\Sigma$ using a linear mixed model (LMM) with Restricted Maximum Likelihood (REML), incorporating all available data, including incomplete trajectories.

The paper further extends these methods to estimate partial correlations (adjusting for covariates such as baseline values) by deriving conditional covariance matrices.

Key Results
Through simulation studies and a real-world case study using Alzheimer's Disease Neuroimaging Initiative (ADNI) data, the authors present the following findings:

• Bias in Slopes-First: Under conditions where the cross-covariance between signals varies over time, Strategy 1 produces biased estimates of the correlation. The bias is exacerbated under Missing At Random (MAR) dropout mechanisms. In simulations with non-constant cross-covariance, Strategy 1 showed biases ranging from 0.059 to 0.165, whereas the true correlation was often significantly different.
• Unbiasedness of Joint Covariance: Strategy 2b (LMM-based) remains unbiased under both Missing Completely At Random (MCAR) and MAR conditions, provided the model assumptions hold. Strategy 2a (pairwise/complete-case) is generally unbiased but suffers from higher variance (lower efficiency) compared to Strategy 2b, particularly in realistic covariance structures.
• Efficiency: Strategy 2b utilizes information from all subjects, including those with incomplete follow-up, resulting in lower empirical standard errors compared to Strategy 2a.
• Real-World Impact: In the case study analyzing the correlation between brain-predicted age change and grey matter volume change, the different strategies yielded different point estimates and confidence intervals. The authors note that these differences were sufficient to potentially lead to different scientific conclusions regarding the strength and significance of the association.

Significance and Claims
The paper claims that the choice of estimator is critical in longitudinal studies with dropout. The authors argue that researchers should not rely on the standard slopes-first approach when data are missing and the cross-covariance structure is complex or time-varying.

Instead, they recommend:

1. Inspecting the missing data pattern.
2. Utilizing the joint covariance approach (specifically Strategy 2b via linear mixed models) to ensure unbiased estimation of the correlation between rates of change.
3. Recognizing that the slopes-first approach may lead to erroneous scientific conclusions in the presence of non-constant cross-covariance and dropout.

The work provides a formal mathematical justification for the bias in common practices and offers a computationally feasible, unbiased alternative for biostatisticians and neuroimaging researchers.