Normative modeling of statistical parameters

1928 

Abstract Submission 

Richard Dinga¹, Thomas Wolfers², Mostafa Kia¹, Marijn van Wingerden¹

¹Tilburg University, Tilburg, Netherlands, ²Laboratory for Mental Health Mapping, University of Tübingen, Tübingen, BW

Richard Dinga  
Tilburg University
Tilburg, Netherlands

Thomas Wolfers  
Laboratory for Mental Health Mapping, University of Tübingen
Tübingen, BW

Mostafa Kia  
Tilburg University
Tilburg, Netherlands

Marijn van Wingerden  
Tilburg University
Tilburg, Netherlands

Normative modeling, inspired by pediatric growth charting, is a method in neuroimaging research to model the distribution of image-derived phenotypes in a reference population[1]. It quantifies individual atypicality by comparing measurements to reference centiles. A complication that hinders the development of normative models in neuroimaging is that, in contrast to body measurements in pediatrics, many variables derived from fMRI images, such as the effects of a stimulus or functional connectivity between regions, are only estimated, which means that they are measured with non-negligible error. This measurement error is not accounted for in standard normative modeling, resulting in the overestimated variance of the normative distribution. In this work, we propose a method to account for the known measurement error in normative modeling, leading to a more accurate estimation of normative models.

Our goal is to model an estimated neuro-imaging variable, such as the strength of an effect at a specific location or functional connectivity between regions, as a function of one or more variables, such as age and sex, and accurately estimate centiles of the distribution of these variables in a population. For this purpose, we propose a meta-regression random-effects model [2] for the kth subject:

y_k = θ + β_1x_1k + .. .+ β_nx_nk + ϵ_k+ ζ_k

Where θ is an intercept, β_1, … , β_n are coefficients of independent variables x_1, … x_n (e.g., age and sex). ζ_k are the between subjects random-effects assumed to be normally distributed with N(0, v), and ϵ_k is a normally distributed residual error term. Estimation of the normative distribution variance equals the estimation of the v from the random-effects model. Centiles are then calculated based on the centiles of the corresponding normal distribution.

Experimental validation:
We have randomly simulated data for 500 subjects with a uniformly distributed age variable between 20 and 80. The measurement-error-free outcome variable was simulated as 100 - age*0.1 - (age - 50)^2*0.005 + e where e was normally distributed with mean 0 and variance 1. Then, we simulated measurement error by adding a random noise with a mean of 0 and variance of 1.25 to the measurement error-free outcome (Figure 1A). We compared our proposed model with an ordinary least-squares model that did not account for the measurement error.

Results can be seen in Figures 1B and 1C for the proposed method and standard method, respectively. The estimated centiles of the underlying normative distribution using the proposed method closely follow the true centiles, while using the standard method, they are too far apart. The proposed method estimated a variance of 0.99, and the standard method estimated a variance of 1.59. This is because the standard method does not differentiate between the measurement error and the variance of the underlying normative distribution and the estimated variance is estimated based on the mixture of the two distributions.

The proposed method provides a crucial step for valid normative models of task activation data or resting state data, which are measured with noise. The limitation of the method is that it assumes a normal distribution of variables, which might not be the case in neuro-imaging data [3][4], and that the variance of the measurement error needs to be known or estimated beforehand and needs to be reasonably accurate since an underestimation of the measurement error will lead to an overestimation of the normative distribution and vice versa.


[1] A. F. Marquand, I. Rezek, J. Buitelaar, C. F. Beckmann, Biol. Psychiatry 2016, 80, 552.
[2] W. Viechtbauer, J. Stat. Softw. 2010, 36, 1.
[3] C. J. Fraza, R. Dinga, C. F. Beckmann, A. F. Marquand, Neuroimage 2021, 245, 118715.
[4] R. Dinga, C. J. Fraza, J. M. M. Bayer, S. M. Kia, C. F. Beckmann, A. F. Marquand, bioRxiv 2021, 2021.06.14.448106.

Aging

Lifespan Development Other ²

Methods Development ¹

Univariate Modeling

Aging

Development

Meta- Analysis

NORMAL HUMAN

Statistical Methods

Other - normative modeling