Synopsis

Keywords: Data Analysis, Diffusion/other diffusion imaging techniques, Quantitative Medical Imaging

We propose a novel quantity to correctly assess the extent individuals deviate from the median of a heavy-tailed distribution. We compute a percentile-based score, we call ‘P-score’, which normalizes the deviation of an individual from the sample median by incorporating the individual’s position in the left/right tail of the sample distribution and the corresponding length between the sample median and the 5th/95th percentile edge values of the sample distribution, respectively. We demonstrate the skewness present in diffusion MRI data and the bias introduced when Z-scores are used and further show the control of this bias using the proposed ‘P-scores’ approach.

Introduction

Quantitative MRI metrics hold the promise of improving individual assessment by comparing measurements with a normative distribution. Typically, Z-scores are used, especially, when comparing quantities at different scales, such as, those often derived from clinical tests and neuroimaging data. However, the interpretation can be biased¹ for data that do not follow a Gaussian distribution, and some MRI metrics may be heavily skewed. A method is required to accurately assess individual deviations from the central tendencies of a heavily skewed distribution. We address this issue by computing a novel quantity, we call ‘P-score’, which accurately estimates an individual's deviation from a normative distribution.

Methods

We acquired diffusion imaging data from 48 control subjects using a 3T Siemens Prisma scanner with a 32-channel head coil. The data was preprocessed using the TORTOISE pipeline^2-3. Various diffusion metrics were computed and average values within ROIs⁴ were extracted. Here, we highlight fractional anisotropy (FA), propagator anisotropy (PA) and mean diffusivity (MD), which were age-corrected using quantile regression^5-6 of the median age. To compute ‘P-scores’, first, we obtain subject-wise percentiles for each ROI and use the 5^th and 95^th boundary values from the sample distribution in the second step; where, we normalize the difference between a subject’s diffusion metric and the sample median, by the corresponding difference between the sample median and the 5^th/95^th percentile edge values, depending on the subject’s location in the left/right tail, respectively. Equation (1) was used to compute percentiles –
$$p_{rank_{ij}} = \frac{n_{C\leq{x_{ij}}}}{n_{C}} \times100;\space\space i = 1,2,3 ...n_{C}, \space j = 1,2,3 ...M\space\space\space(1)$$
where $$$x_{ij}$$$ represents diffusion metric of the $$$i^{th}$$$ individual from the control sample $$$n_{C}$$$ for the $$$j^{th}$$$ ROI, with $$$i$$$ and $$$j$$$ representing $$$1...n_{C}$$$ individuals and $$$1...M$$$ ROIs, respectively. $$$n_{C\leq{x_{ij}}}$$$ represents the number of subjects in $$$n_{C}$$$ with values equal or less than $$$x_{ij}$$$. The ‘P-scores’ are computed using equations (2) and (3) –
$$P_{5_{ij}} = -1.645\frac{d_{x_{ij}}}{d_{5}};\space\space \begin{cases} -d_{x_{ij}} = x_{ij} - m_{C_{j}} & x_{ij} < m_{C_{ij}} \\d_{5} = m_{C_{j}} - x_{C_{j_{5}}} & x_{ij} < m_{C_{ij}}\end{cases}\space\space\space(2)$$
$$P_{95_{ij}} = +1.645\frac{d_{x_{ij}}}{d_{95}};\space\space \begin{cases} +d_{x_{ij}} = x_{ij} - m_{C_{j}} & x_{ij} > m_{C_{ij}} \\d_{95} = x_{C_{j_{95}}} - m_{C_{j}} & x_{ij} > m_{C_{ij}}\end{cases}\space\space\space(3)$$
where $$$x_{ij}$$$, $$$i$$$ and $$$j$$$ are the same as in (1). $$$d_{x_{ij}}$$$ is the difference between $$$x_{ij}$$$ and the median, $$$m_{C_{j}}$$$ of the sample group for the j^th ROI. $$$-d_{x_{ij}}$$$ means the current subject is located below $$$m_{C_{j}}$$$ on the left-hand tail, between $$$m_{C_{j}}$$$ and the 5^th boundary value, $$$x_{C_{j_{5}}}$$$ in the sample distribution, giving $$$d_{5}$$$ as the difference between $$$m_{C_{j}}$$$ and $$$x_{C_{j_{5}}}$$$. Contrarily, $$$+d_{x_{ij}}$$$ indicates the current subject is located above $$$m_{C_{j}}$$$ on the right-hand tail, between $$$m_{C_{j}}$$$ and the 95^th boundary value, $$$x_{C_{j_{95}}}$$$, giving $$$d_{95}$$$ as the difference between $$$x_{C_{j_{95}}}$$$ and $$$m_{C_{j}}$$$. $$$P_{5_{ij}}$$$ and $$$P_{95_{ij}}$$$ takes the polarity of $$$d_{x_{ij}}$$$, producing negative and positive ‘P-scores’, respectively. Please note, owing to $$$n_{C} = 48$$$, the smallest resolution of percentiles is ~2.08%. Therefore, suitable boundaries of ~5^th and ~95^th percentiles were chosen, which gives $$$P_{score} = 1$$$, for subjects at the 5^th/95^th boundaries. Thus, if ‘P-scores’ are multiplied with a scalar “1.645” (Z-score magnitude of 5^th/95^th percentiles from a normal distribution), they attain the same scale of Z-scores. This helps comparing the two scores and assessing which one attains a closer fit to the normal distribution.

Results and Discussion

Figures 1(a), 2(a) and 3(a) show heatmaps of Z-scores (left) and ‘P-scores’ (right) for FA, PA and MD, respectively. Overall, we observe a negative bias in Z-scores from FA, more strongly from PA, and a positive bias from MD. The correction for imbalance in positive and negative extreme values can be visually inspected from the 'P-score' heatmaps. Figures 1(b), 2(b) and 3(b) show the histograms of Z-scores (top) and ‘P-scores’ (bottom) from all ROIs for FA, PA and MD, respectively. We fitted a normal distribution curve in each histogram to assess which score follows it closely. The Z-score distributions (top) are negatively skewed for FA and PA, and positively skewed for MD, while the P-score distributions (bottom) for all three are controlled for this skewness, bringing it closer to a normal distribution. The imbalance in the % of these extreme values from Z-scores is quantified in Table1, e.g., for PA, “%neg < -1.65” is ~6 times of “%pos > 1.65” and we can appreciate that the ‘P-scores’ correct this imbalance having a 1:1 ratio between “%neg < -1.65” and “%pos > 1.65”. Similarly, FA and MD can be evaluated. A limitation of the method is that we used a set pair of boundaries (5^th/95^th) to compute ‘P-scores’. However, given a sufficient sample size, a normative distribution with incremental percentile boundaries can be obtained and individual deviations can be assessed correctly even if the data originates from a heavy-tailed distribution.

Conclusion

We proposed using a novel quantity called ‘P-score’ to correct for skewness in the data and provide a more accurate position of an individual from the median of a normative distribution. We demonstrated the presence of skewness in three diffusion MRI metrics and how ‘P-scores’ correct the overestimation and underestimation of extreme values when Z-scores are used in heavily skewed data. Although applied in a diffusion imaging setting, the 'P-score' method has the advantage to be universally applied in any imaging and clinical data.

Acknowledgements

No acknowledgement found.