Adjusting effective multiplicity ( M eff ) for family-wise error rate in functional near-infrared spectroscopy data with a small sample size.

Significance: The advancement of multichannel functional near-infrared spectroscopy (fNIRS) has enabled measurements across a wide range of brain regions. This increase in multiplicity necessitates the control of family-wise errors in statistical hypothesis testing. To address this issue, the effective multiplicity ( $M_{\text{eff}}$ ) method designed for channel-wise analysis, which considers the correlation between fNIRS channels, was developed. However, this method loses reliability when the sample size is smaller than the number of channels, leading to a rank deficiency in the eigenvalues of the correlation matrix and hindering the accuracy of $M_{\text{eff}}$ calculations.

Aim: We aimed to reevaluate the effectiveness of the $M_{\text{eff}}$ method for fNIRS data with a small sample size.

Approach: In experiment 1, we used resampling simulations to explore the relationship between sample size and $M_{\text{eff}}$ values. Based on these results, experiment 2 employed a typical exponential model to investigate whether valid $M_{\text{eff}}$ could be predicted from a small sample size.

Results: Experiment 1 revealed that the $M_{\text{eff}}$ values were underestimated when the sample size was smaller than the number of channels. However, an exponential pattern was observed. Subsequently, in experiment 2, we found that valid $M_{\text{eff}}$ values can be derived from sample sizes of 30 to 40 in datasets with 44 and 52 channels using a typical exponential model.

Conclusions: The findings from these two experiments indicate the potential for the effective application of $M_{\text{eff}}$ correction in fNIRS studies with sample sizes smaller than the number of channels.