What is the paper about and what does it try to contribute:
The paper provides a detailed review of the statistical analysis methods used in functional near-infrared spectroscopy (fNIRS) research. It aims to present these methods comprehensively and coherently, enabling fNIRS practitioners to access necessary information easily without extensive searching through multiple sources. The paper emphasizes a unified approach to statistical analysis, demonstrating that various methods like t-tests, ANOVA, and GLM can be derived from a general mixed model with restricted maximum likelihood (ReML) estimation, thus helping students and researchers to understand and develop their statistical approaches for specific problems.

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Introduction (we already learn most of this):
Functional near-infrared spectroscopy (fNIRS) measures brain activity non-invasively by detecting changes in light absorption, with distinct spectra for oxy-hemoglobin (HbO) and deoxy-hemoglobin (HbR). It offers high temporal resolution, robustness to motion artifacts, and can be used during various activities, making it advantageous over other neuroimaging methods like fMRI. However, fNIRS lacks anatomical precision and is affected by noise and artifacts, necessitating careful statistical analysis. Initially, heuristic methods were used, but rigorous statistical approaches like t-tests, ANOVA, and GLM have since been developed to improve accuracy.
*the important part is that we need robust statistical method in order to use the fNIRS data.

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Imaging Physics (a technical explanation of how fNIRS works):
The modified Beer–Lambert law is utilized in fNIRS to quantify changes in oxy-hemoglobin (HbO) and deoxy-hemoglobin (HbR) concentrations by measuring optical density changes. It accounts for the absorption of near-infrared light in a highly scattering medium, allowing determination of HbO and HbR changes using their extinction coefficients. Tomographic mapping, a more rigorous approach, derives optical density changes from the photon diffusion equation, addressing the nonlinear inverse problem with various regularization methods for improved topographic mapping accuracy.
*In the full paper you can see the exact rules and function used.

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Signal Processing of fNIRS Signals (for Motion Artifact Correction):
fNIRS measurements capture changes in hemoglobin oxygenation in the brain's gray matter, but also include systemic confounds from the superficial layers of the head. Advanced signal processing techniques are crucial to remove these confounds before statistical analysis.

Two main strategies are employed for motion artifact correction:
Use of Additional Sensors: Some methods involve using accelerometers to detect motion and apply filters like recursive least squares (RLS) to remove artifacts. This category relies on external measures to improve accuracy.

Signal Processing Techniques: This approach does not require additional sensors and includes methods like wavelet-based filtering, Kalman filtering, and Independent Component Analysis (ICA). ICA, for instance, identifies and removes motion-specific components by assuming statistical independence among the source signals. Techniques like wavelet filtering are effective in isolating rapid signal changes caused by motion, providing a clear separation of hemodynamic signals from artifacts.

Effectiveness of Methods:
Studies comparing various methods found that approaches like multiple-channel regression and ICA significantly enhance the signal-to-noise ratio. Spline interpolation, in particular, has shown a substantial reduction in mean-squared error by accurately interpolating motion-related signal segments, thereby preserving the integrity of hemodynamic response data.
*processing of other fNIRS signals can be viewed in the full paper.

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Statistical Inferences for fNIRS Signals:
This section details the evolution of statistical methods in fNIRS studies from basic visual inspections to sophisticated analyses. Initially, simple visual methods assessed hemoglobin changes, but were prone to errors from noise. As a result, statistical techniques like t-tests, ANOVA, and correlation analyses were introduced to improve accuracy. These methods evolved to include Fourier analysis for periodic tasks and General Linear Models (GLM) to handle complex data variations. Advanced software tools such as HomER and fOSA integrated these statistical methods, enhancing data analysis capabilities. Newer tools like NIRS-SPM and NinPy further advanced the field by offering more precise interpolation and statistical analysis suited for the unique challenges of fNIRS data.

let take a look at and example of a statistical technique:
Generalized linear mixed model (GLM, brief summary):
The Generalized Linear Mixed Model (GLMM) is an extension of the General Linear Model (GLM) that incorporates both fixed and random effects, allowing for a more flexible and comprehensive analysis of data with complex variance structures. It integrates fixed effect parameters and random effects through a covariance matrix, accommodating correlations within and across groups in the data.

* more detail and more techniques are available in the paper.

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Functional connectivity analysis:
Compared to fMRI, the temporal resolution of fNIRS is significantly higher. This gives us an opportunity to investigate the functional con-nectivity of the brain by exploiting the temporal correlations between multiple channels. In calculating the connectivity maps, two different approaches exist: seed-based analysis and independent component analysis. This area of research is still evolving, so we briefly review some of the representative works.

Seed-Based Approaches:
Seed-based approaches in functional connectivity analysis involve selecting a specific channel (the seed) and calculating the correlation between its time series and those of other channels to understand brain network interactions. This method has been applied in various studies using both fNIRS and fMRI technologies, showing that fNIRS can effectively capture spontaneous hemodynamic fluctuations similar to those identified in BOLD signals from fMRI. Additionally, variations in connectivity related to frequency bands have been observed, indicating that different cognitive functions might be linked to specific frequency ranges in hemodynamic signals.

ICA-Based Approach:
In research by H. Zhang et al. (2010), Independent Component Analysis (ICA) was utilized to analyze resting-state brain connectivities from fNIRS data, which is well-suited due to its large number of temporal samples and fewer channels. The process included detrending the data to remove trends, using PCA to reduce data dimensionality, and applying ICA to identify sensorimotor and visual components. The results were then statistically analyzed using a two-tailed one-sample t-test on z-maps derived from ICA outputs. This ICA approach was found to be more effective than traditional seed-based methods, especially in scenarios with high noise levels, offering greater sensitivity and specificity in detecting functional connectivity.

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Developmental changes of functional connectivities from neonates to 6 months old infants, characterized by spontaneous fluctuations of HbO signals.
(a) Probe settings of the 94-channel fNIRS system.
(b) Representative time courses of HbO measured from a 6 month old infant. Correlation between HbO signals from P3 and P4 was higher than that from P3 and Fp

Conclusion and outlooks
In this paper, we have provided an extensive review of statistical analyses of fNIRS data. Applications of classical statistical analysis such as t- and F-statistics as well as more advanced SPM-type ap-proaches have been discussed. For a unified understanding of various classical statistical analyses, we provided a linear mixed model with ReML covariance estimation and showed that most of the classical analyses can be explained with-in this framework. We also provided a complete derivation of group analysis using a general linear mixed model with ReML framework, which is novel and was not available before.
 

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