Metabolic ratios are used extensively magnetic resonance spectroscopy (MRS) literature due to their simplicity and ability to account for hardware imperfections and cerebrospinal fluid content. Different metabolites often shift in opposing directions in many pathologies, such as multiple sclerosis (MS) in which N-acetyl-aspartate (NAA) levels decrease while creatine (Cr), choline (Cho) and myo-inositol (mI) increase. On the one hand, it seems that the ratio of two metabolites displaying opposing changes might increase their statistical favorability; on the other hand, metabolic ratios display increased coefficients of variation¹ (CVs) compared to absolute quantification due to the combined variances of both numerator and denominator. To adequately decide whether ratios are statistically superior to absolute quantification, CVs alone are insufficient and one must examine the expected change and the CVs simultaneously. Herein we examine metabolite ratios effect on sample size and statistical significance.Assuming both numerator and denominator are normally distributed, X₁~N(μ₁,σ₁²), X₂~N(μ₂,σ₂²), their ratio X₁/X₂ has been previously derived². Assuming a coefficient of variation CV₂≡σ₂/μ₂CV<0.25 for the denominator, one can perform a series expansion and obtain a normal distribution X₁/X₂~N(μ_R,σ_R²) with μ_R=μ₁/μ₂ and σ_R²=μ_R²·(CV₁²-2·ρ·CV₁·CV₂+CV₂²). ρ is the correlation coefficient. Given knowledge of a normally distributed variable in control X_c~N(μ_c,σ_c²) and patient X_p~N(μ_p,σ_p²) populations, the sample size required to observe a one-sided change in the mean Δμ=μ_p-μ_c with statistical significance α and power β is N_s=(z_1‑ασ_p‑z_βσ_c)²/Δμ² (with z~N(0,1)).

We computed N_s for both NAA, Cr and NAA/Cr, using literature values³ for white matter (WM) NAA (7.7±0.6 mM) and Cr (4.9±0.5 mM) in healthy controls, assuming the standard deviations of both remain unchanged in patient populations. The patient WM NAA and Cr values were varied by up to 30% (NAA decrease, Cr increase) typical of many pathologies. The sample sizes N_NAA, N_Cr and N_ratio required to observe the single-sided change between controls and patients for NAA, Cr, and NAA/Cr with a statistical significance α=5% and power β=80% were calculated.

Global WM changes in NAA, Cr and NAA/Cr were examined using data from two previously acquired datasets^4,5: a longitudinal study of relapsing-remitting multiple sclerosis (MS) (18 patients, ages 21-45, scanned every 6 months for 3 years), and a cross-sectional mild traumatic brain injury (mTBI) study (26 patients, ages 33±11). A Shapiro-Wilks test with a significance level of α=0.05 was used to validate the normality of numerator, denominator and ratio. Pearson correlation coefficients were computed to estimate whether one can assume ρ=0. To examine whether metabolite ratios yield any improvements to statistical significance, an unpaired two-sided t-test was conducted among patient and control groups for NAA, Cr and NAA/Cr.

The $$$\left( \mu_{NAA}^{(patient)}, \mu_{Cr}^{(patient)} \right)$$$ plane (Fig. 1) divides in three mutually exclusive regions: N_ratio<min(N_NAA,N_Cr) (black), N_ratio<min(N_NAA,N_Cr) (gray) and N_ratio=min(N_NAA,N_Cr) (white), where all sample sizes were rounded up to the nearest integer. Clearly, a non-trivial relationship exists between numerator, denominator and ratio distributions. For example, when NAA declines by 5% and Cr rises by 5%, N_Cr=26, N_NAA=14, N_ratio=10, rendering ratios superior, while a decline of 5% in NAA and rise of 2.5% in Cr yields NCr=104, NNAA=14, Nratio=19, making absolute quantification preferable.

The statistical tests confirmed normality and lack of correlations between metabolites in our cohorts' metaboilte datasets. Reduction in WM NAA and increase in WM Cho, Cr and mI concentrations were observed in several - but not all - time points in MS patients compared with the pooled mean of controls. In contrast, the ratio NAA/Cr differs significantly at all time points between patients and controls, as determined by the t-test (Fig. 2); all time points remain significant even as the significance level is lowered to a strict α=0.01 level. For the mTBI cohort, only NAA and NAA/mI were statistically different between patients and controls (α=0.05) (Fig. 3). Sample size estimations based on the estimated population variance and mean yielded the smallest sample size for NAA, with a slightly higher sample size required for NAA/mI (16 vs. 17).

It is often implicitly assumed that whenever two metabolites’ means shift in an opposing manner, their ratio must improve detection. Herein we have shown that, depending on the means and variances of both numerator and denominator, metabolic ratios can either enhance or diminish statistical significance and sample size requirements in a non-trivial manner. This impacts study design considerations and choice of diagnostic biomarkers. Our conclusions can be considered an extension of previous studies, which have already noted the increased CVs exhibited by ratios albeit without examining their effect on statistical tests.