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Tools for assessing and mitigating confounding effects in Machine-Learning based studies of MRI data
Elisa Ferrari1,2,3, Giovanna Spera1, Letizia Palumbo1, and Alessandra Retico1

1INFN, sez. Pisa, Pisa, Italy, 2Scuola Normale Superiore, Pisa, Italy, 3University of Pisa, Pisa, Italy

Synopsis

Using Machine Learning (ML) techniques on neuroanatomical data obtained with magnetic resonance imaging (MRI) is becoming increasingly popular in the study of Psychiatric Disorders (PD). However, this kind of analyses can be affected by overfitting and thus be sensitive to biases in the dataset, producing hardly reproducible results. It is therefore important to identify and correct possible bias sources in the sample. We present two tools aimed at addressing this matter: a methodology to assess the confounding power of a variable in a specific classification task, and a cost function to use during classifier training on highly biased data.

INTRODUCTION

In the latest MRI-based ML studies1, the awareness of potential sources of uncontrolled variation (especially those related to the MRI acquisition sites2) and of the difficulties in normalizing these data is growing. This motivates researchers to limit their analyses to a few sites and homogeneous groups of subjects, sacrificing sample size. However, it is not always clear which variables must be considered during dataset stratification to avoid confounding biases. Furthermore, even when all the confounding factors are known, it is difficult to isolate a group of subjects that are adequately homogeneous with respect to all of them. To tackle this problem, ML approaches have been proposed that aim at minimizing a saddle-shaped cost function during training3. This function represents the compromise between optimizing classification performances while avoiding to learn from bias-generated differences. However, when the value of a confounding variable and the class label are highly correlated within subjects, this tug of war risks hampering the classifier ability to actually learn anything. In this study we present a cost function that improves the probability of learning the correct classification pattern in unbalanced datasets overcoming these difficulties. Furthermore, a practical method for assessing the confounding effect of a variable is introduced.

CONFOUNDING POWER INDEX

To study the confounding power CPx of a categorical variable $X\in\{0,1\}$ we propose to measure the effect that a strong bias (related to X) in the dataset has in a two-group classification. To this purpose, a classifier must be trained on three different dataset configurations (see figure 1). From the relationship in the white box, an index to quantify the confounding power can be derived as:

$$CP_x=(AUC_2-AUC_3)/AUC_1$$

If X is not confounding, the AUCs will not depend on the partitioning of the dataset with respect to the X variable and will be very similar to each other and thus $CP_x\approx0$ . If X is confounding, CPx is expected to be sensibly greater than 0. An extension of CPx in the case of a continuously distributed variable X is described in figure 2.

A key aspect of CPX is that it measures the impact of the bias during the classification task under study. Thus, its value represents the relative impact of X in the specific analysis being carried on. For an example of this, see figure 3, in which the value of CPFIQ for the variable Full Intelligence Quotient (FIQ) is reported for a male/female classification task (orange) and for an Autism Spectrum Disorder (ASD) subjects / Healthy Controls (HC) classification task (blue).

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COST FUNCTION FOR HIGHLY BIASED DATASETS

An ML algorithm is usually trained to minimize a Mean Squared Loss Function (MSLF), representing the error on the subject class prediction. When the dataset is affected by a bias related to a variable X, a term can be added to MSLF, constraining the training to also maximize the classification error on the subject X label. However, when the bias is strong, this can be counterproductive. To avoid this effect, we developed a cost function that penalizes the errors deriving from a wrong learning pattern related to X. This cost function is: $$MSLF_c=a(output_i,input_i,X_i)∙(output_i-input_i)^2$$

Where inputi ∈{0,1} is the class of the i-th subject, outputi ∈[0,1] represents the classifier estimated probability that the subject belongs to class 1 and a ∈ {1,k} with k>1. Considering the case described in fig. 4, which has a highly biased dataset, the condition for which a=k (as opposed to a=1) is: $$IF (|output_i-input_i|>=0.5\wedge(input_i - X_i)>0)\rightarrow a=k$$

This weights more the errors made when a classification pattern based on the value of X rather than on the class is used. This function has been tested on datasets organized as described in fig. 4.

In fig. 5 the results on a male/female classification task are shown as a function of the value of k used in the cost function. The MRI acquisition site was used as confounding variable. Finally, comparing the performances obtained by two networks trained with a traditional MSLF and with MSLFc respectively, it can be observed that, for the first one, 83% of the misclassified subjects belongs to the underrepresented acquisition site, while this accounted for only 54% of the errors of the second network.

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CONCLUSION

The proposed index CPx represents a valid instrument for finding confounding variables, while the proposed cost function MSLFc has proved to be successful in minimizing their effects. Despite this study validates this tool in a relatively simple situation, the satisfactory results obtained suggest that extensions of this technique may generalize well.

Acknowledgements

No acknowledgement found.

References

1. Abraham, Alexandre, et al. "Deriving reproducible biomarkers from multi-site resting-state data: An Autism-based example." NeuroImage 147 (2017): 736-745.

2. Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432–441, 2008.

3. Han Zhao, Shanghang Zhang, Guanhang Wu, João P Costeira, José MF Moura, and Geoffrey J Gordon. Multiplesource domain adaptation with adversarial training of neural networks. arXiv preprint arXiv:1705.09684, 2017.

4. Di Martino, Adriana, et al. "The autism brain imaging data exchange: towards a large-scale evaluation of the intrinsic brain architecture in autism." Molecular psychiatry 19.6 (2014): 659.

5. Di Martino, Adriana, et al. "Enhancing studies of the connectome in autism using the autism brain imaging data exchange II." Scientific data 4 (2017): 170010.

6. Fischl, Bruce. "FreeSurfer." Neuroimage 62.2 (2012): 774-781.

Figures

Figure 1: Classifiers necessary to determine whether a categorical variable X is confounding for a binary classification. AUC1 obtained with the first classifier shows classification performances in unbiased configuration. If AUC2 ≥ AUC1 the features depend on X and thus the classifier learnt additional differences. However, this dependency could be insufficient to confound the classification. To define X as confounding AUC3 ≤ AUC1 should also hold.

Figure 2: Schematic of the analysis on the confounding power of a continuously distributed variable X. In order to compute the value of CPx for such a variable, it is necessary to discretize its values. This involves two steps:

• Selecting a range of values that represents the discretization step.
• Assessing the confounding power of X, carefully choosing the subjects in the training and validation sets, so that they come from two different discrete intervals. These intervals are characterized by: (1) The distance they have from each other, (2) The starting point of the first unit

Figure 3: Example of the differences of the CPx value calculated for two different classification tasks. Given that the confounding features under examination (FIQ) is continuously distributed, CPFIQ has been evaluated as a function of the parameter d (see fig. 2). In these analyses, the features representing the subjects, the dimension of the sample and the kind of classifier used are the same. Thus, the differences in CPX are caused by the different impact that FIQ has on the different classification tasks. Data were taken from ABIDE database4,5. The features used have been extracted from structural MRI with Freesurfer6.

Figure 4: Schematic of a severely biased configuration used for the identification of a cost function able to not be confounded by the X variable.

Figure 5: Performance increment in the male/female classification based on a biased dataset due to the use of MLSFc. Every data point was calculated averaging the results of 50 different analyses on different sub-datasets. In each of these analyses the obtained AUCs come from networks trained and validated on the same subjects. AUCMLSFc achieves a maximum of 0.95 +/- 0.08, while AUCMLSF on an unbiased dataset reaches 0.86 +/- 0.09. This shows the potential of this approach: the correction is able to perform better than the standard approach on an unbiased dataset, thanks to the higher number of subjects available.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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