Statistical analysis of longitudinal neuroimage data with Linear Mixed Effects models

Abstract

Longitudinal neuroimaging (LNI) studies are rapidly becoming more prevalent and growing in size. Today, no standardized computational tools exist for the analysis of LNI data and widely used methods are sub-optimal for the types of data encountered in real-life studies. Linear Mixed Effects (LME) modeling, a mature approach well known in the statistics community, offers a powerful and versatile framework for analyzing real-life LNI data. This article presents the theory behind LME models, contrasts it with other popular approaches in the context of LNI, and is accompanied with an array of computational tools that will be made freely available through FreeSurfer - a popular Magnetic Resonance Image (MRI) analysis software package. Our core contribution is to provide a quantitative empirical evaluation of the performance of LME and competing alternatives popularly used in prior longitudinal structural MRI studies, namely repeated measures ANOVA and the analysis of annualized longitudinal change measures (e.g. atrophy rate). In our experiments, we analyzed MRI-derived longitudinal hippocampal volume and entorhinal cortex thickness measurements from a public dataset consisting of Alzheimer's patients, subjects with mild cognitive impairment and healthy controls. Our results suggest that the LME approach offers superior statistical power in detecting longitudinal group differences.

Keywords: Linear Mixed Effects models; Longitudinal studies; Statistical analysis.

Figure 1

Locally weighted smoothed mean measurement trajectory (lowess plot) for each of the four clinical groups. This method produces a smooth curve by centering a window of fixed size at each time-point and fitting a straight line to the data within that window. The lowess estimate of the mean at a time-point is simply the predicted values at that time-point from the fitted regression line. In this plot, the fraction of the total number of data points included in the sliding window was set to 0.7. HC: healthy control; sMCI: stable MCI; cMCI: converter MCI; AD: Alzheimer patients. (A) Hippocampal volume (HV). (B) Entorhinal cortex thickness (ECT).

Figure 1

Locally weighted smoothed mean measurement trajectory (lowess plot) for each of the four clinical groups. This method produces a smooth curve by centering a window of fixed size at each time-point and fitting a straight line to the data within that window. The lowess estimate of the mean at a time-point is simply the predicted values at that time-point from the fitted regression line. In this plot, the fraction of the total number of data points included in the sliding window was set to 0.7. HC: healthy control; sMCI: stable MCI; cMCI: converter MCI; AD: Alzheimer patients. (A) Hippocampal volume (HV). (B) Entorhinal cortex thickness (ECT).

Figure 2

Statistical power versus alpha (false positive rate) to discriminate the atrophy rates of stable and converter MCIs. HV: hippocampal volume. ECT: entorhinal cortex thickness.

Figure 3

Locally weighted smoothed mean measurement trajectory (lowess plot) for two groups. This method produces a smooth curve by centering a window of fixed size at each time-point and fitting a straight line to the data within that window. The lowess estimate of the mean at a time-point is simply the predicted values at that time-point from the fitted regression line. In this plot, the fraction of the total number of data points included in the sliding window was set to 0.7. HC: healthy controls who remained so throughout the study; and cHC: converter HCs, who were healthy at baseline but progressed to MCI or AD during follow-up. Mean time to progression was 2.6 years from baseline. (A) Hippocampal volume (HV). (B) Entorhinal cortex thickness (ECT).

Figure 4

Statistical power versus alpha (false positive rate) to discriminate the atrophy rates of stable and converter healthy controls (HC). HV: hippocampal volume. ECT: entorhinal cortex thickness.

Figure 5

The mean absolute difference between non-parametric and parametric p-values for three statistical methods in comparing hippocampal volume loss rates between healthy controls (HC) and Alzheimer patients (AD) (Experiment 3) as a function of total sample size. LME: Linear Mixed Effects model with random intercept and slope. Rm-ANOVA: random effects ANOVA. X-Slope: GLM-based cross-sectional analysis of annualized rate of atrophy (slope).

Figure 6

Detection rate (the frequency of true positives) in differentiating hippocampal volume loss rates between healthy controls and AD patients (Experiment 3), as a function of alpha (p-value threshold) with 2N=20 subjects. LME: Linear Mixed Effects model with random intercept and slope. Rm-ANOVA: random effects ANOVA. X-Slope: GLM-based cross-sectional analysis of annualized rate of atrophy (slope).

Figure 7

Repeatability (the frequency at which a method differentiates hippocampal volume loss rates between healthy controls and AD patients in two independent samples of 2N= 20) versus alpha (p-value threshold) (Experiment 3). LME: Linear Mixed Effects model with random intercept and slope. Rm-ANOVA: random effects ANOVA. X-Slope: GLM-based cross-sectional analysis of annualized rate of atrophy (slope).

Figure 8

The influence of including subjects with a single time-point on LME-based inference results. MRI-derived total hippocampal volume was the dependent variable. The full sample contained 50 HC and 50 AD subjects, all with 4 visits (scans). We had 1000 random simulations, in which a reduced dataset was generated, by treating 20 random AD subjects as dropouts and discarding their last three scans. The y-axis shows the average difference between the coefficient estimates obtained on the reduced sample by including (black bars) or discarding (white bars) the 20 dropout AD patients, and the coefficients from the full sample. The error bars show the standard deviations across 1000 random simulations. These results suggest that including the subjects with a single time-point increases the accuracy of the model fit and introduces minimal bias.