To which extent the Cramér-Rao bound (CRB) is a reliable benchmark in quantitative MRS?
Guilhem Pagès1 and Jean-Marie Bonny1

1AgroResonance - UR370 QuaPA, INRA, Saint Genès Champanelle, France


Following MRS data fitting, absolute or relative Cramér-Rao bounds (CRB and rCRB, respectively) are often computed as indices of parameter uncertainties. However, the unknown true values of parameters are required to compute CRB. Here, we studied the effect of substituting the true values by the noisy estimates on the bounds (noted CRB* and rCRB*). We showed by simulations that both CRB* and rCRB* are particularly sensitive to noise. Also, the mode of rCRB* distribution is left-shifted, which leads to a significant false positive risk when rCRB* is thresholded. A threshold not exceeding 20% is recommended to limit this risk.


Quantification of complex mixtures from noisy NMR spectra is generally based on the fit of a parametric model to the data. A popular indicator of quantification reliability is the Cramér-Rao bound (CRB) and its relative expression rCRB. CRB provides the minimal variances obtained by the best estimator. However, as parameter true values are required to compute CRB, only an estimate of CRB, i.e. CRB* (and its relative counterpart rCRB*), can be obtained from noisy data. Here, we analyzed by computer simulations the consequences of substituting parameter true values by estimates on CRB* and rCRB* of peak amplitudes.


FID signals mimicking spectra of various complexities were computed. For each condition, 105 noisy FIDs were generated by adding zero-mean complex Gaussian noise to the FID. Parameter estimates were then fitted in the least squares sense using the exact model (i.e. sum of exponentially damped sinusoids). CRB* and rCRB* were computed from each of the 105 sets of estimates. These Monte-Carlo simulations allowed obtaining smooth probability density functions (pdfs) for (r)CRB* at different noise levels, i.e. which correspond to different level of true rCRB. Because rCRB* is currently compared to a threshold for accepting or rejecting an estimate of metabolite concentration, the probabilities to wrongly accept or reject a fitted spectrum were also calculated.


Figure 1 represents the evolution of the CRB* and rCRB* pdfs for the peak amplitude parameter as function of rCRB values, i.e. several noise levels. Because of noise propagation, the width of CRB* and rCRB* pdfs rapidly increases with rCRB. rCRB* distribution has the specificity of showing a left asymmetry. It thus tends to indicate a lower value than the ground truth. The insets in Fig. 1 display the pdfs for the peak amplitude estimates. Besides, both false positive and negative risks were calculated for several thresholds η as shown on Fig. 2. It can be noted that the curves are steeper for lower η. It indicates that, by keeping η small, incorrect rejection and false acceptance errors are likely to occur only when rCRB is close to the threshold.


Recently, several papers warned on the use of rCRB* as fitting quality indicator.1,2 We show here that, beyond a noise level corresponding to an rCRB of around 10%, the uncertainties on both CRB* and rCRB* cannot be neglected. In other words, substituting the true values of parameters by noisy estimates lead to CRB* which are likely to be far from the minimal variances. As expected, how CRB* deviates from its true value increases with noise level (Fig. 1). Besides, Kreis recently suggested to use the CRB* instead of the rCRB*.2 Our data show that the modes of the CRB* distributions well match to the CRBs. However, the width of the CRB* pdf is larger than the one of rCRB*. rCRB* is currently used as an acceptance/rejection criterion when compare to a threshold. To limit the false positive and negative risks, the threshold should be set with caution. For example, for a true rCRB of 80%, almost one time out of two noisy rCRB* were below 50% (purple curve in Fig. 2a). Reducing the threshold to 20% limits this risk to nearly zero (Fig. 2a, green curve).


Both CRB* and rCRB* of peak amplitudes are spoiled by noise. In accordance with the well-known compromise between precision and accuracy, CRB* provides more accurate indications than rCRB*, but less precise. Notably, rCRB* is prone to the underestimation of the variance. Consequently, when an amplitude estimate is accepted or rejected according to its rCRB* in MRS, a conservative threshold cutoff level of 20% should not be exceeded to limit the risk of false positives.



1. Tisell A, Leinhard OD, Warntjes JBM, Lundberg P. Procedure for quantitative 1H magnetic resonance spectroscopy and tissue characterization of human brain tissue based on the use of quantitative magnetic resonance imaging. Magn Reson Med. 2013;70(4):905-915.

2. Kreis R. The trouble with quality filtering based on relative Cramér-Rao lower bounds. Magn Reson Med. 2016;75(1):15-18.


Figure 1: Normalized pdfs of the CRB* (blue) and rCRB* (red) represented in % for an rCRB value (black dashed line) set to (a) 10, (b) 20, (c) 30 and (d) 50%. The insets represent the normalized pdfs of the amplitude estimates (Ampl*). The true amplitude was 1 and is represented by the black dashed line.

Figure 2: Probabilities to obtain (a) false positive (i.e. false acceptance or type II error) and (b) false negative (i.e. incorrect rejection or type I error) as function of the threshold level η.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)