Bertrand Thirion

Functional neuroimaging group studies

Multi-subject statistical analysis is an essential step of neuroimaging studies, as it makes it possible to draw conclusions that hold with a prescribed confidence level for the population under study. The use of the linear assumption to model activation signals in brain images and their modulation by various factors has opened the possibility to rely on relatively simple estimation and statistical testing procedures. Specifically, the analysis of functional neuroimaging signals is typically carried out on a per-voxel basis, in the so-called mass univariate framework. However, the lack of power in neuroimaging studies has incited neuroscientists to develop new procedures to improve this framework: various solutions have been set up to take into account the spatial context in statistical inference or to deal with violations of distributional assumptions of the data. In this chapter, we review the general framework for group inference, the ensuing mixed-effects model design and its simplifications, together with the various solutions that have been considered to improve the standard massunivariate testing framework.

Introduction

Most neuroimaging statistical problems deal with the comparison of sets of images that embed some features of brain structure or function across a group of subjects, with the purpose to detect some common effects across individuals or some differences across sub-populations. The most standard framework consists in comparing images on a voxel-by-voxel basis (or a vertex-by-vertex basis if the data is sampled on a mesh), after resampling in a common spatial referential, such as the referential defined by the Montreal NeuroImaging Institute (MNI) template [18], or a coordinate system on the cortical surface [7]. This resampling is assumed to correct for pose and shape differences across individuals, so that possible remaining differences are related to the anatomical or functional feature of interest and not to a mere residual of between-subject registration. Note that this is an assumption, and that the limitations of the brain image coregistration procedures used for this purpose are key to understanding the difficulty of statistical inference in the context of neuroimaging group studies.

The simplest statistical inference procedure for image-based data, a.k.a. massunivariate inference, relies on the computation of a statistic in each voxel. This statistic is then compared to a reference distribution, that represents its likely values when the hypothetical effect is absent. If the actual statistic value is extreme with respect to this distribution -its p-value is low-one can conclude that it would not likely be observed under the null hypothesis, hence it is more likely explained by some alternative hypothesis: the null hypothesis is rejected. This inference scheme 1 Figure 1: Illustration of the deviation from normality of functional neuroimaging datasets: the map shows the p-value of a test [START_REF] Pearson | Testing for departures from normality[END_REF] rejecting the normality of the distribution of the z-transformed activation statistics across subjects using the dataset presented in section 7 (n = 573). Note that the Gaussian hypothesis is significantly rejected in all cortical regions. is known as classical statistics. In practice, the question of how low the p-value should be to support the conclusion is dealt with arbitrarily, and the corresponding choice (p < .05 is commonly accepted in the literature) cannot be justified within the framework of classical statistics [23].

The choice of the decision statistic is very important. As many studies involve comparing the mean value of a given image-derived feature across populations, parametric tests such as Student's t-test or Fisher's F-test are natural choices. However, they are limited in two regards:

• In order to yield accurate p-values, they involve a normality hypothesis which is most often violated (see e.g. Fig. 1). Depending on how whether standard parametric statistics are robust enough to these deviations or other nonparametric statistics should be used instead is an important question. In practice, however, the price to pay in terms of computation cost or loss of sensitivity when using non-parametric inference is most often too high, while accurate pvalues for t or F tests can be obtained under weak assumptions by permutation testing [2,[START_REF] Nichols | Nonparametric permutation tests for functional neuroimaging: a primer with examples[END_REF].

• The individual values that are compared in the test are not directly observed; instead they are estimated with some level of uncertainty from the acquired data. Taking into account the uncertainty in the statistical evaluation leads formally to a mixed-effects model [START_REF] Mclean | A Unified Approach to Mixed Linear Models[END_REF]. Such a model is more accurate, but also arguably more expensive than simple random effects model. We review the mixed-effects formulation and its simplifications in detail in section 4.

Then comes an issue that is standard in statistics, yet particularly problematic neuroimaging, namely that of multiple comparisons: since many tests are performed simultaneously, the risk of observing low p-values by chance, hence of making false detections, is high. Depending on the statistical guarantees required for the analysis, a suitable correction can be implemented, such as family-wise error control (of which the Bonferroni correction, that corrects the significance level by the number of tests performed, is the simplest example) or false discovery rate control. Again, the most reliable procedure is given by permutation tests (see section 6).

One of the major issues with image-based statistical works is that those are often carried out on small samples of subjects; for instance, most cognitive neuroimaging findings are based on cohorts of no more than 20 subjects, due to the cost of data acquisition and processing. This situation leads to a degradation of the ability to separate signal from noise; in practice, enforcing a strict control on type I errors jeopardizes the analyst's ability to detect the actual signal. As a result, neuroimaging studies typically suffer from a lack of power and in a low reproducibility of the findings [3].

For the sake of sensitivity, the mass-univariate setting can be enhanced by taking into account the image context in statistical inference. Smoothing is a relatively standard image analysis procedure, but it strongly biases the shape of the signal of interest and thus can only be used sparingly. However, on can also consider the continuous structure of the signals of interest embedded in the images by focusing on the size of the connected components of supra-threshold areas for a given detection threshold. Assuming that the distribution of such sizes under the global null hypothesis (that there is no effect present in any region of the image) is known, observing larger region sizes typically indicates the presence of an effect in these regions. This type of cluster-level inference has become a standard in neuroimaging [26]; it has also been extended to general procedures that avoid the prior selection of a cluster-forming threshold [START_REF] Stephen | Threshold-free cluster enhancement: addressing problems of smoothing, threshold dependence and localisation in cluster inference[END_REF]. This and other related procedures are discussed in section 5. We give a brief account of permutation testing approaches in Section 6 and conclude in Section 7 with illustrating examples.

Variability of brain shape and function

Variability of brain organization and brain imaging

Neuroimaging group analyses test the effect of some external variables of interest on the image signal, i.e. they compare the amount of signal explained by the variables of interest to the residual signal after fitting a linear model. They do thus measure the fraction of between-subject signal variance that is correlated with the target variable. However, this variability has a complex nature, as it encodes functional and structural features unique to each individual, together with limitations or various signal corruptions in the imaging process; in any case, it cannot be simply conceptualized as an additive random noise in the observations. Building suitable imaging features that achieve some robustness against the observed between-subject variability is thus in important challenge. It requires some efforts to understand and capture the information of interest in the presence of distortions and structured noise.

Variability of brain shape

The variability of brain size and shape is mostly observed through anatomical imaging and various computational geometric procedures that measure the thickness, the regularity of the cortical surface, or some of its singularities (sulcal pitts, sulci fundi etc.). The variability of such features readily poses a challenge for the comparison of brains from different subjects: what makes each brain location unique from an anatomical perspective ? Or put differently, how to warp each individual brain such that the localized individual features can be considered as corresponding to each other ? The best approaches so far consist of first compensating for differences in image pose and brain size through linear transformations, then using high-dimensional diffeomorphic registration to align individual gray matter outline approximately [START_REF] Ashburner | Diffeomorphic registration using geodesic shooting and gauss-newton optimisation[END_REF], and then to perform statistical analysis, yet in the absence of further guarantee on the identity of the tested structures. This framework results in an uncertainty of about 10mm on the actual voxel correspondences, which can be taken as a blur on the results of any group analysis [START_REF] Hellier | Retrospective evaluation of intersubject brain registration[END_REF][START_REF] Stiers | Mapping multiple visual areas in the human brain with a short fmri sequence[END_REF]. Current attempts to improve upon this situation rely on surface-based mapping [8] -yet without any formal guarantee of accurately aligning cyto-architectural areas nor functional areas-or using functional localizer experiments to define individual regions of interest [START_REF] Nieto-Castañón | Subject-specific functional localizers increase sensitivity and functional resolution of multi-subject analyses[END_REF].

A point relevant for all neuroimaging studies is that these differences are not Figure 2: Illustration of the relative magnitude of within-and between-subject variability, through the average between/within variance ratio of the dataset presented in section 7. While the ratio is close to 1 in many regions, it is larger in regions that display a non-zero mean effect across the population (compare with Fig. 5).

modeled, because current imaging contrasts are not sufficient to disambiguate the nature and organization of all brain areas; it is also clear that human cortical folding cannot be mapped diffeomorphically across individuals [START_REF] Rivière | Automatic recognition of cortical sulci of the human brain using a congregation of neural networks[END_REF]. The corresponding variability is thus inaccurately handled as unstructured noise.

Variability of brain function

Besides the variability of brain shape and anatomical organization, different subjects may display different brain activation patterns, which can be interpreted as differences in functional organization, cognitive strategies, attention, or more simply signal-to-noise ratio of the imaging data related e.g. to presence of motion or various acquisition artifacts. In the absence of external data, these sources of variability cannot be identified easily, nor can they be removed. They are often considered as an additional additive noise. Note that functional MRI is not a quantitative modality, in the sense that the corresponding measurements of the BOLD (blood oxygen-level dependent) signal are not expressed in absolute physical units. Practitioners have found that expressing the signal fluctuations as a percentage of the baseline level was a practical measure, yet it is unclear how invariant the resulting quantification is to parameters of no interest in statistical analysis, such as MR sequence parameters (see an illustration in Fig.

5(b)).

A rough measure of the between-subject variability can be given by comparing the amount of between-subject variance to average amount of within-subject variance (related to observation noise). An example of such a ratio is displayed in Fig. 2.

Mixed-effects and fixed-effects analyses

Fig. 2 illustrates a situation that arises frequently in functional neuroimaging, namely the presence of two or more heterogeneous sources of variance in the data.

One is related to the observation process and can be viewed as noise, while the other is related to the difference between individuals, as discussed in Section 2. We refer to the corresponding variance estimates as first-level and second-level variance respectively. This sources of variability are jointly embedded in the observations, yet the terms are separated in the so-called mixed effects modeling framework [START_REF] Mclean | A Unified Approach to Mixed Linear Models[END_REF], which we will discuss in detail in section 4. It should be noted that one has the possibility to neglect the second-level variance, which leads to a fixed-effects model: such a model typically assumes that a given effect has been observed in all subjects as if it were a repeated measurement on the same individual, and thus aims at deriving the mean effect and variance of this common effect without considering cross-subjects fluctuations. Such an inference cannot be used as a population-level inference, given that it ignores the variability that stems from the subject effect [START_REF] Friston | How many subjects constitute a study?[END_REF]. The difference between fixed-and mixed-effects inference is illustrated in Fig. 3.

Group analysis for functional neuroimaging

In this section, we first review the two-level linear model for functional neuroimaging, then discuss the estimation of the mixed-effects model and ensuing statistical tests.

Problem setting and notations

As a preliminary note, it is assumed here that some functional data are observed at a given set of brain locations across multiple subjects, be these locations cortical surface nodes or voxels (see e.g. [START_REF] Tucholka | An empirical comparison of surface-based and volume-based group studies in neuroimaging[END_REF]).

To clarify the notations, we use bold capital letters for matrices (e.g. A), bold letters for vectors (e.g. a) and small letters for scalars (e.g. a). We denote N (µ, Σ) the multivariate Gaussian distribution with mean µ and covariance matrix Σ. Given n scalars (σ 2 1 , .., σ 2 n ), we denote by diag(σ 2 1 , .., σ 2 n ) the (n × n) diagonal matrix with these scalars on the diagonal. We use the following conventions: n subjects denotes the number of subjects, p the number of voxels and m(s) the number of scans of a given dataset in a given subject s.

Within-subject model

We consider a study performed over n subjects subjects. For any subject s ∈ [n subjects ], let Y s denote a set of observations (fMRI scans) obtained in this subject. To simplify the statistical formalism, each fMRI scan is flattened to a vector representation, where each coordinate of the vector represents the fMRI activity in a given voxel within a suitable brain mask. Let us denote m(s) the length of the time series in subject s. Y s is thus a matrix of shape (m(s) × p) and the value Y s (i, j) for 1 ≤ i ≤ m(s) and 1 ≤ j ≤ p represents the fMRI signal at voxel j acquired at time i. Let (X s ) s∈[n subjects ] represent the design matrices that model experimental and nuisance effects that are likely to be reflected in brain signals. The first level general linear model (GLM) takes the form

∀s ∈ [n subjects ], Y s = X s B s + E s , (1)
where (B s ) for each s ∈ [n subjects ] is a matrix of shape (n reg × p) that yields the coefficients associated with the columns of the design matrix and (E s ) is the unmodeled signal, considered as observation noise.

In practice, only a certain combination of the parameters is of interest and will be the subject of further inference. For instance, the effects associated with motion parameters may not be considered in population-level analyses, while the activation signal associated with a combination of experimental conditions quantifies a cognitive response of interest that one wishes to compare with individual characteristics. To keep the setting clear yet comprehensive, we thus rewrite equation 1 with split design and parameter matrices to introduce a distinction between the coefficients of interest and those modeling other effects. Note that singling out a parameter of interest amounts to defining a contrast, i.e. a linear combination of the effects B s , that represents it in the data. Here we assume that the design matrix is written in a form such that the contrast of interest corresponds to the first column x i s of X s , i.e. X s = [x i s , X r s ] ; correspondingly, the effects of interest and residual effects are written written b s and B r s , i.e.

X s B s = x i s b s + X r s B r s , thus yielding ∀s ∈ [n subjects ], Y s = x i s b s + X r s B r s + E s (2)

Group-subject model

B = Zβ + E, (3)
where β is a matrix of shape (n f actors × n voxels ) that represents the population-level effects. Without loss of generality, Eq. 1-3 can thus be rewritten

Y s = x i s b s + X r s B r s + E s , ∀s ∈ [n subjects ] (4) B = [b 1 , .., b n subjects ] T = Zβ + E (5)
Finally, the question of interest is where in the brain a certain combination of the factors of interest yields a positive effect on average in the population, i.e., whether c T β > 0, where c is a suitable vector of contrast on the population-level effects: if Z contains three variates related to the sex, the age of the subjects and an intercept, setting c = (0, 1, 0) will display the effect of age on the observed BOLD signal. Different types of contrasts correspond to different statistical questions, see appendix A for an overview. The problem consists thus in estimating with which confidence one can reject the null hypothesis c T β = 0. Since the model defined in Eqs. 4-5 can be handled at each brain location independently, we proceed with voxel-level analysis (estimation and statistical analysis). We keep focusing on voxellevel probabilistic assessment till the end of this section; taking into account the joint signal distribution over voxels is deferred to section 5: such an approach is typical of a mass-univariate modeling frameworks.

Interpretation as a mixed effects model

It is straightforward to observe that Eqs. 1-3 can be concatenated into one equation:

∀s ∈ [n subjects ], Y s = x i s Z| s β + x i s E| s + X r s B r s + E s . ( 6
)
where Z| s and E| s denote the restrictions of matrix Z and E to their row s. This means that the observed data Y s are actually composed of four different effects:

• the effect of interest, to be tested:

x i s Z| s β • A random subject effect x i
s E| s • Some within-subject effects of no interest or nuisance effects, handled as fixed effects X r s B r s

• Some observation noise E s .

The model thus comprises both fixed and random effects, hence the reference to a mixed effects model.

Estimation

It is then generally assumed that the two random components are Gaussian distributed, with unknown variance: let (Λ s ) be the (m × m) variance-covariance matrix of the noise E s in a subject s ∈ [1..n subjects ] and ∆ be the variance-covariance matrix (n subjects × n subjects ) of the random effects. While Λ s are often taken as the covariance matrix of a (temporal) auto-regressive process scaled by an unknown variance, ∆ can be assumed to be diagonal, as (E| 1 , .., E| n subjects ) have been sampled independently. In the absence of additional information on the population structure, ∆ = γ 2 I n subjects , where I n is the n × n identity matrix. The estimation of the parameters of the mixed effects model, is carried out, generally following the maximum likelihood principle. The parameters to estimate are

Θ = (Λ s , B r s ), β, γ 2 , ( 7
)
given (Y s , X s ) s=1..n and Z:

Θ = argmax Θ N (B; Zβ, γ 2 I n subjects ) n subjects s=1 N (Y s ; x i s β s + X r s B r s , Λ s )
For the sake of simplicity and computation efficiency, it is frequent to use a twostep method: first, the estimation of the effects of interest and their covariance, independently in each subject, and then the estimation of the population parameters. Standard solutions include: the EM algorithm [START_REF] Worsley | A general statistical analysis for fmri data[END_REF][START_REF] Roche | Mixedeffect statistics for group analysis in fmri: a nonparametric maximum likelihood approach[END_REF][START_REF] Tucholka | An empirical comparison of surface-based and volume-based group studies in neuroimaging[END_REF], Bayesian methods [39] or the Gauss-Newton method or some variant thereof [START_REF] Thirion | Analysis of a large fmri cohort: Statistical and methodological issues for group analyses[END_REF]. Here we follow the EM approach. First level model fit (for notational simplicity, we assume that Λ s is given, while it is voxel-specific and data-dependent in practice). The first-level estimation procedure yields estimates ( bs , σ2 s ) of the individual effects and associated variance. These estimates have a large number of degrees of freedom m(s) -n reg . The group model (Eq. 4-5) then boils down to bs

= b s + e s , ∀s ∈ [n subjects ] B = [b 1 , .., b n subjects ] T = Zβ + E,
where for each subject s ∈ [n subjects ], σ 2 s = var(e s ) is estimated with a large number of degrees of freedom, hence we assume that it is exact, while γ 2 = var(E) is unknown, and has to be estimated. (β, γ 2 ) can then be estimated so that they maximize the likelihood

L( B; β, γ 2 ) = N B; Zβ, γ 2 I n subjects + diag(σ 2 1 , .., σ 2 n subjects ) (8)
using an EM algorithm: If we denote N ( bs , s2 s ) the variational distribution of b s , and B = [ b1 , .., bn subjects ], the EM algorithm consists in iterating the two steps:

E-step: ∀s ∈ [1..n subjects ], s2 s = 1 γ 2 + 1 σ 2 s -1
, bs = s2

s bs γ 2 + β σ 2 s (9) M-step: β = (Z T Z) -1 Z T B, γ 2 = 1 n subjects n subjects s=1 s2 s + B -Zβ 2 (10)
It converges toward a local maximum of the likelihood function. Although there is no guarantee that the reached maximum is global, it can be observed in many cases that the likelihood function only has one maximum. See e.g. in Fig. 4 the likelihood as a function of the parameters (β, v) obtained for the toy data used in Fig. 3. It should be noted that the above algorithm may not yield the optimal estimators for the variance parameters; for instance, more accurate estimates of the variance may be obtained by using Restricted Maximum likelihood approaches [START_REF] Harville | Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems[END_REF].

Statistical inference

The main question remains to test whether the effect of interest is different from zero, i.e. c T β > 0 or c T β = 0. It is not obvious what test should be used. A Wald statistic can be computed as

t = c T β c T (Z T W -1 Z) -1 c , ( 11
)
where W (γ) = γ 2 I n subjects + diag(σ 2 1 , .., σ 2 n subjects ). However, it does not conform exactly to a student distribution under the null hypothesis [39,4]. This means that non-parametric methods have to be used to obtain an estimate of its null distribution. Given this requirement, the likelihood ratio statistic -that is known as the most powerful test for the model-is the best possible test. This is defined as follows

Λ = 2 sup β,γ log L( B; β, W (γ)) -sup β:c T β=0,γ log L( B; β, W (γ)) ( 12
)
where L is defined as in equation 8. Λ is readily computed by running the EM algorithm twice, once in a constrained mode (c T β = 0), once in an unconstrained mode.

The random effects t-test

A crude approximation of the above model consists in neglecting the first-level variance; this amounts to assuming that the individual variances are identical (σ

2 1 = • • • = σ 2
n subjects ) and clearly simplifies the ensuing statistical inference. The assumption yields a simple random effects model:

B = Zβ + E , ( 13
)
where E ∼ N (0, g 2 I n subjects ), for which the decision statistic is simply

t RF X = c T β g 2 c T (Z T Z) -1 c , ( 14
)
where β and g 2 are easily estimated with a least-squares fit. Note that it is the most widely used model for population level inference in neuroimaging. Similarly, an F statistic can be defined when one is interested in unsigned or multi-dimensional contrasts In that case, and under the Gaussian hypothesis, the log-likelihood ratio defined in equation 12 is a monotonous function of the Fisher statistic, or, if one is interested in signed effects, of the t statistic. We provide an empirical comparison of the t statistic with the full mixed-effects model in section 7. We also provide a table of the most frequently used statistical models (one-sample t-test, two-sample t-test, paired-t-test, two way ANOVA) in table 1. See [START_REF] Poldrack | Handbook of functional MRI data analysis[END_REF] for a more complete discussion on this topic.

Taking into account the spatial context in statistical inference

Multiple comparisons in the mass univariate framework So far, we have considered each voxel independently, which is a requirement at the estimation stage, given the computation cost of the procedure, but comes at the price of low sensitivity when performing statistical inference. In the mass-univariate setting, one statistical test is performed per voxel, and the significance is then simply corrected for the complete set of tests: Corrected p-values can be obtained by family-wise error-rate (FWER) control, i.e. the probability of performing one false detection, or the generally more lenient false discovery rate (FDR) control, that controls for the proportion of false discoveries among the detections. We discuss the practical computation of FWER-corrected thresholds in section 6. For false discovery rate control, a simple reference procedure is presented in the neuroimaging context in [START_REF] Christopher R Genovese | Thresholding of statistical maps in functional neuroimaging using the false discovery rate[END_REF] and is most often used due to its simplicity, but non-parametric alternatives exist too [START_REF] Ge | Resampling-based multiple testing for microarray data analysis[END_REF]. This framework is usually adopted for its convenience and relatively simple interpretability. However, it is limited by its inability to take into account the image structure of the data: the observed cross-subject variability in brain organization limits the relevance of a voxel-by-voxel description of the data, and yields a reduced sensitivity to detect the true effects.

Spatial regularization

The first way to take into account the image structure in statistical modeling is to regularize spatially the data: such a regularization is most often implemented with data smoothing in the volume or on the surface, with a 6 to 10mm full width at half maximum (fWHM) kernel. Choosing the adapted kernel directly leads to a bias-variance trade-off: larger kernels reduce the variance yet yield a large bias on the activation peaks position. Smoothing can instead be replaced by Markov Random field modeling [6] or anisotropic smoothing [32], but such procedures are not widely used by practitioners due to the lack of efficient implementations.

Inference on the regions size

A second possibility to take into account the image structure in the inference procedure is to consider the size of the connected components of the set of supra-threshold voxels [START_REF] Friston | Assessing the significance of focal activations using their spatial extent[END_REF], assuming that the chosen threshold (e.g. t>3 for a t-test) separates correctly the peaks of the image from the background noise. The size statistic has to be compared with the reference distribution of the size of such structures obtained when no effect is present, i.e. under the null hypothesis. Such a distribution is in general not known a priori, but it can be approximated under the hypothesis that the data follows a known parametric distribution [26], or more simply by permutation [15] -see section 6. The latter approach is typically recommended. The underlying intuition is that, especially when looking at population-level statistics maps, narrow regions are unlikely to represent truly active brain structures. There are however two major drawbacks with such an approach: First, it depends on an arbitrary cluster-forming threshold, so that the result does not outline intrinsic characteristics of the data. Second, varying this threshold potentially yields abrupt changes in the detected areas, thus leading to a fundamental instability of the results.

It is however relatively popular in the neuroimaging community due to its enhanced sensitivity, which is however attributable to the weakness of the test: it rejects the global null hypothesis for the whole cluster, but does not make it possible to localize the activation within the supra-threshold cluster. Hence, it should only be used with high cluster-forming thresholds [START_REF] Anderson M Winkler | Permutation inference for the general linear model[END_REF].

Cluster-mass testing and Threshold-free cluster enhancement (TFCE)

An improvement upon these procedures consists in combining the size and height of peak regions in a common statistic [START_REF] Poline | Combining spatial extent and peak intensity to test for activations in functional imaging[END_REF][START_REF] Hayasaka | Combining voxel intensity and cluster extent with permutation test framework[END_REF]. However, more general combinations of extent and height can be used alternatively, resulting in the so-called thresholdfree cluster enhancement (TFCE) procedure [START_REF] Stephen | Threshold-free cluster enhancement: addressing problems of smoothing, threshold dependence and localisation in cluster inference[END_REF]. Let v → φ(v) be the statistical map obtained from the univariate inference step; the TFCE statistic is computed as follows:

TFCE(v) = φ(v) 0 z H a v (z) E dz, ( 15
)
where a(z) is the size of the connected component of the map thresholded at level z that contains voxel v, H and E are two non-negative numbers. In general H = 2 and E = .5, as in the reference TFCE implementation in [START_REF] Stephen | Threshold-free cluster enhancement: addressing problems of smoothing, threshold dependence and localisation in cluster inference[END_REF]. Note that H = 1 and E = 0 yields a cluster mass statistic, while H = 0 and E = 1 yields a generalized cluster size statistic integrated over thresholds. The setting H = 2, E = .5 was designed to represent optimally paraboloid-shaped activations.

Randomized Parcellation-Based Inference (RPBI)

A recent alternative to TFCE has been proposed in [START_REF] Benoit Da Mota | Randomized parcellation based inference[END_REF], and consists in using multiple parcellations of the brain volume. A parcellation is a partition of the brain volume into connected regions (parcels), for instance 1000 or 2000 parcels; taking parcel-based averages of the input signal can be understood as a dimension reduction procedure well suited for smooth images. The parcel-based signal estimation can be carried in a group of subjects in parallel, and thus used in a population model, where standard statistics (e.g. Eq. 11) are computed, resulting in a parcel-based statistical map. While the signal compression brought by parcel-based analysis is certainly beneficial to sensitivity, the resulting map is heavily biased by the initial parcellation specification: to avoid such biases, one has to marginalize out the parcellation choice. The RPBI statistic is thus defined as the voxel-based sum of binary variables testing whether the decision statistic in the parcel including the voxel was above a given threshold or not, computed over a large enough number of initial parcellations. Let P be a set of parcellations, and V be the set of voxels under consideration. Given a voxel v and a parcellation P , the parcel-based thresholding function θ t is defined as:

θ t (v, P ) = 1 if F (Φ P (v)) > t 0 otherwise (16)
where Φ P : V → P is the mapping function that associates each voxel with a parcel from the parcellation P . For a predefined test, F returns the F -statistic associated with the average signal of a given parcel (a t or other statistic is also possible). Finally, the aggregating statistic at a voxel v is given by the counting function C t :

C t (v, P) = P ∈P θ t (v, P ). ( 17
)
C t (v, P) represents the number of times the voxel v was part of a parcel associated with a statistical value larger than t across the folds of the analysis conducted on the set of parcellations P. The parameter t should be set to ensure a Bonferronicorrected control at p < 0.1 in each of the parcel-level analyzes. In practice, the results are weakly sensitive to mild variations of t.

Type I error control with permutation testing

Permutation testing is the reference approach for statistical inference, as it provides valid and accurate p-values under a typically restricted set of hypotheses.

In practice, many statistical procedures described previously (cluster size, TFCE, RPBI) dot not have a well-defined distribution under the null hypothesis. Even the family-wise error control in the presence of correlations does not have a perfectly known distribution and is correctly approximated only in some peculiar settings [START_REF] Worsley | An improved theoretical p value for spms based on discrete local maxima[END_REF]. While some reference distributions could be estimated or even simulated under the assumption of Gaussian noise with pre-defined covariance -a framework popularized as Gaussian Random Field Theory [START_REF] Friston | Assessing the significance of focal activations using their spatial extent[END_REF]-the hypotheses involved in such a procedure are relatively strong, hence easily proved wrong. In practice, the smoothness of the signal can vary across regions of an image, challenging the stationarity of the random field [START_REF] Salimi-Khorshidi | Adjusting the neuroimaging statistical inferences for nonstationarity[END_REF]. In general, permutation-based inference is thus the reference solution to obtain accurate control on type I errors.

As the statistics in ( 11) and ( 12) are pivotal, the advised procedure is to use Westfall-Young kind of correction for multiple comparisons [START_REF] Westfall | Resampling-based multiple testinging: Examples and methods for p-value adjustment[END_REF]: family-wise error rate (FWER) corrected p-values are obtained by sampling the null distribution of the maximal statistic through a permutation scheme: let us consider a statistical map v → φ(v) in which the values follow the same distribution under the null hypothesis (this is called the pivotality hypothesis). We assume that the data are exchangeable under the null hypothesis: for instance, we compare the mean effect across two populations of subjects, and do not have any other covariate in the data model. Then for any permutation π of the data (i.e. permutation of the rows of the secondlevel design matrix Z), one can compute the corresponding map v → φ π (v), by applying the usual estimation procedure. Assuming that J such permutation-based estimations are carried out the critical threshold for the family-wise-error-corrected p-value α is given by

φ c (α) = Q 1-α (max v∈[1..p] φ πj (v)) j∈[1..J] , (18)
Where Q 1-α stands for the (1 -α) quantile of the values (in this case, across permu-

tations (π 1 , • • • , π J ))
. The significance at level α of the non-permuted statistics φ(v) is thus assessed by thresholding of these values against φ c α . Slightly more complex strategies can be used to control the False discovery rate instead [START_REF] Ge | Resampling-based multiple testing for microarray data analysis[END_REF].

A particular case is when there is no regressor in the model, i.e. the test only consists in the intercept, meaning that the inference is about the mean effect being larger than zero. In that case, permuting the data does not change the statistic. What is done instead is to further assume that the distribution under the null hypothesis is symmetric, and then swap the sign of the observations: for n observations, this generates up to 2 n resamplings.

Limitations of permutation testing.

There are however some cases where the use of permutation tests requires some care, because simplistic implementations lead to inaccurate results. One such case is when models include some covariates, on top of the effects of interest: due to the possible correlations between the contrasts of interest and other covariates, the permutation test needs to be handled with specific procedures [9,[START_REF] Anderson M Winkler | Permutation inference for the general linear model[END_REF].

Another obvious limitation of permutation testing is its cost: to guarantee a stable estimate of φ c α in Eq. 18, it is necessary to use J 1 α 2 permutations.

Illustration of various inference strategies on an example

The difference between the output of a mixed-effects and a fixed-effects model is usually large in neuroimaging settings, where inter-subject variability is large with respect to intra-subject variability (see e.g. Fig. 3). As neuroimaging studies are concerned with population-level validity of the findings, fixed-effects inference is ruled out. By contrast, the difference between the mixed-effects statistic and the random effects statistic (i.e. the test in Eq. 14 as opposed to the test in Eq. 12) is more subtle and deserves some consideration. Neglecting first-level variance is actually equivalent to assuming that it is equal in all subjects, i.e. that all individual observations are equally reliable (see section 4.4). Hence any difference between truly mixed-effects are random-effects model clearly outlines the impact of the difference in reliability across samples in the population.

Two examples are described next.

• The first example in Fig. 5(a) shows the results of permutation-controlled α = .05, J = 10 4 mixed-effects and random-effects tests to detect the mean effect of a computation task (from which a simple reading effect has been subtracted [START_REF] Pinel | Fast reproducible identification and large-scale databasing of individual functional cognitive networks[END_REF]), based on a sample of n = 30 subjects.

• The second example in Fig. 5 In both cases, random-and mixed-effects maps are presented, after thresholding at the p < .05 level, corrected for multiple comparisons by a permutation procedure. While the two kinds of maps are clearly similar, one can observe more sensitivity in the case of mixed-effects inference in the two settings:

• In the one-sample test, more active voxels are found in the ventral striatum, the anterior cingular cortex, the insula and the right intraparietal sulcus, that are regions of the dorsal attentional network. Moreover, the detected regions display more significant p-values.

• In the two-sample test, the effects observed in the mixed-effects analysis are wider and more significant in all regions, although this is more visible in the ventral striatum and the thalamus.

Note that the superiority of the mixed model in terms of sensitivity has already been reported in the literature [39,[START_REF] Roche | Mixedeffect statistics for group analysis in fmri: a nonparametric maximum likelihood approach[END_REF][START_REF] Tucholka | An empirical comparison of surface-based and volume-based group studies in neuroimaging[END_REF]4]. Next, to assess the impact of spatially-aware statistics, we perform a qualitative comparison of the maps obtained through the different spatial models, coupled with the mixed-effects statistic: peak significance (no spatial context), cluster size significance, TFCE and RPBI. The results are shown in Fig. 6, and are based on the one-sample test example.

They clearly shows that for a given significance level (p < 0.05, corrected for multiple comparison through an max-control permutation-based procedure), spatially informed approaches yield more active voxels than voxel-level inference, a gain that can be interpreted as a higher sensitivity. For instance, some temporal or frontal activation foci are not found in the voxel-based approach, but are detected with the other approaches. It should be reminded however that this increase in sensitivity is mitigated by two important drawbacks: • Only the voxel-level inference provides a guarantee on the active status of each detected voxel, as the other approaches only provide a rejection of the null hypothesis for some (unknown) voxel in the detected areas.

• Non-voxel-based methods require additional parameters. For instance, in our procedure, we relied on a cluster-forming threshold of p < 10 -4 , the E and H parameters of the TFCE (we kept the default E = .5, H = 2), the internal threshold of the RPBI approach as well as the number of atlases and their resolution. See section 5 for a description of these parameters.

Importantly, it has been reported that the increased sensitivity afforded by spatially-aware method comes with higher reproducibility, i.e. a higher chance of reproducing the results on another sample of images (see e.g. [START_REF] Benoit Da Mota | Randomized parcellation based inference[END_REF]), which is acknowledged as a criterion for model selection in the neuroimaging and other communities [START_REF] Chen | Fmri group analysis combining effect estimates and their variances[END_REF]. This potentially means that in the low-sample/low-power regime common to most neuroimaging studies, the increase in sensitivity afforded by these methods offsets their drawbacks (see also [START_REF] Thirion | Analysis of a large fmri cohort: Statistical and methodological issues for group analyses[END_REF]).

Conclusion

The flexibility afforded by the linear model has made it possible for neuroimagers to make inference on brain functional organization and its variability across groups of subjects in a relatively intuitive way. The central concept is arguably that of contrast, that corresponds to stating a precise question to identify the effect of variables on brain activation signals. Current gains in computational efficiency has led to the development of permutation-based approaches, that are typically more accurate and make it possible to draw inference from quantities that do not have any known or simple distribution under the null hypothesis.

In practice, neuroscientists often rely on the solution offered by the software that they use most often, such as SPM, FSL or AFNI, which make different choices: SPM relies mostly on parametric statistics, while FSL gives access to the popular TFCE statistic and relies more on permutation tools; AFNI yields a wider choice of statistics. The development and diffusion of methods in open-source software and more user-friendly environments will condition the adoption of the more advanced tools by the community.

Methodological research still has to address some hard challenges, such as the design of more efficient methods to take into account the spatial structure in the images while performing probabilistic inference. Also, the same level of statistical rigor needs to be achieved in more complex statistical analyses that consider the signals from multiple regions, such as multivariate pattern analysis and functional connectivity analysis.

1 0 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 0 -1 0 1 -1 1 0 -1 0 1 -1 0 1 0 -1 1 -1 0 1 0 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1                     β mean β Y1 β Y'1 β Y2 β Y1 Y'1 β Y1 Y'2      F-

Figure 3 :

 3 Figure 3: Illustration of the difference between fixed-and mixed-effects inference: Given n = 10 observations associated with a given level of uncertainty (left), one can perform a fixed-effects inference that ignores cross-subject variability of the observations and thus leads to population effects with tight uncertainty (middle), or consider this variance and then obtain wider uncertainty estimates (right). Only the mixed effects model yields a valid inference on the population from which the observations were sampled.

Figure 4 :

 4 Figure 4: Likelihood of the observations displayed in Fig. 3 as a function of the parameters (β, γ 2 ) (which are 2 scalars in that case). It can be seen that the likelihood function has a unique maximum.

Figure 5 :Figure 6 :

 56 Figure 5: Difference between a mixed-effects model and a random effects model for statistical analysis. (a) One-sample test that aims at detecting the mean effect on a computation task (from which a simple reading effect has been subtracted [24]), based on a sample of n subjects = 30 subjects; top: mixed-effects model; bottom: random-effects model. Both maps are thresholded at the p < 0.05 level, corrected for multiple comparisons, using permutation testing. (b) Illustration of the difference on a two-sample test, where the differential effect of two MRI scanners on the activation obtained in the same task as (a). The mixed-(top) and random-(bottom) effects maps, both thresholded at p < 0.05, FWER-corrected by permutation, show the same effects, but again the mixed-effects inference is more sensitive.

tests for all contrasts c T β = 0 H

 0

  (b) is a two-sample test that targets the differential effect of two MRI scanners on the activation obtained after completing the same task as the first example. More precisely, it aims at detecting difference between a Siemens Trio 3T scanner (data acquired in 2007-2008[START_REF] Tucholka | An empirical comparison of surface-based and volume-based group studies in neuroimaging[END_REF]) and a 3T Bruker scanner (data acquired in 2003-2006[START_REF] Pinel | Fast reproducible identification and large-scale databasing of individual functional cognitive networks[END_REF]). The acquisition parameters of the Bruker dataset are: TR= 2400ms, TE= 60ms, the matrix size is 64 × 64, FOV = 24cm × 24cm. Each volume consisted of n a 3-mm-or 4mm-thick axial slices without gap, where n a varied from 26 to 40 according to the session. A session comprised 130 scans. The acquisition parameters of the Trio dataset are: TR= 2400ms, TE= 30ms, the matrix size is 64 × 64, FOV = 19.2cm × 19.2cm and the number of slices is 40, while the session duration is unchanged. Note that the two sets of images have been preprocessed in the same way. The cohort studied comprises n 1 = 72 and n 2 = 78 subjects from the Bruker and Siemens scanners respectively.

Table 1 :

 1 Example of Statistical models used in neuroimaging. The first column describes the model, the second column describes how the data are ordered in the outcome vector, the third column shows the design matrix, and the last column illustrates the hypothesis tests and corresponding contrasts. Note, in the ANOVA example F -tests are used for all contrasts, whereas t-tests are used for the other examples. We use the notations from Eq. 13. This table is adapted from[START_REF] Poldrack | Handbook of functional MRI data analysis[END_REF].

Acknowledgment

We are very thankful to Philippe Pinel who provided the data used in the comparison. This work was also possible thanks to the software developments carried out by Benoit da Mota, Virgile Fritsch and Gaël Varoquaux. Many thanks also to Ana Luisa Grilo Pinho who suggested the addition of the statistical tests table. We acknowledge funding from the Human Brain Project, the Digiteo project iConnectom and the Microsoft Research (MediLearn project).

Test Description

Order of Data

Zβ

Hypothesis test

One-sample t-test, 5 observations


Version française

A l'origine, l'étude du non-recours porte expressément sur les prestations sociales financières.

C'est pourquoi la définition initiale renvoie le non-recours à toute personne éligible à une prestation sociale [financière], qui -en tout état de cause -ne la perçoit pas. Au Royaume-Uni où l'on parle de non take-up of social benefits, la question est apparue dans les années 1930.

Son émergence a clairement correspondu au besoin politique d'évaluer la bonne affectation de ces prestations, en particulier celles ciblées sur des populations particulières (means-testing benefits). Même si l'introduction du thème du non-recours a des origines différentes dans d'autres pays, partout la question du non-recours a servi à rendre plus explicite et davantage opérationnelle la préoccupation gestionnaire de l'effectivité de l'offre de prestations financières (impacts prévus/impacts réels). D'une façon générale, la prise en compte de cette question est liée au besoin récurrent de savoir si l'offre atteint bien les populations à qui elle est destinée.

Cette histoire délimite fortement l'objet du non-recours. Pourtant, il est possible et nécessaire de procéder à son élargissement pour découvrir les enjeux d'un phénomène qui interroge la pertinence des politiques publiques.

Un phénomène étendu

Deux raisons principales militent pour élargir l'objet du non-recours au-delà du domaine des prestations sociales financières. La première est liée au besoin de dépasser la définition orthodoxe du non-recours, qui permet d'en parler uniquement lorsque l'on peut définir précisément une population éligible, comme dans le cas -pour le coup bien particulier -des prestations sociales. La seconde raison est que dans d'autres domaines les acteurs sociaux s'inquiètent de situations de non-recours.

L'éligibilité à des prestations sociales est le critère à partir duquel la notion de non-recours a été construite au tout début. Des règles de droit décident de qui est éligible et de qui ne l'est pas.

Ces règles formelles portent sur une diversité de paramètres, comme la composition familiale, l'âge, le genre, les ressources, etc., mais aussi les statuts officiellement reconnus et contrôlés qui ouvrent l'accès à des droits, parfois successifs dès qu'un premier est rendu actif. Elles définissent les populations éligibles et les conditions de l'accès à l'offre publique. Il est alors possible de calculer des taux de non-recours à partir du différentiel entre population potentiellement éligible (Ne) et population éligible et effectivement bénéficiaire (NeR).

Selon cette approche orthodoxe, on ne peut parler de non-recours et le mesurer que lorsqu'une population potentiellement éligible (Ne) est identifiée. De ce point de vue, le non-recours ne peut être objectivé que pour une partie très limitée des droits et services qui constituent l'offre publique. Il est en général difficile d'estimer, et a fortiori d'identifier, des publics cibles. Les prestations sociales étant définies pour des groupes ou des populations suivant des critères donnés, elles se prêtent à cette approche. Mais pour le reste cela ne va de soi : la notion de public potentiel est bien vague, et elle n'est jamais assortie d'estimations et surtout de comptages précis. C'est tout le problème de l'étude du non-recours comme l'a signalé l'économiste Antoine Math, qui fût parmi les tout premiers à introduire la question en France.

Pour cette raison, il ne serait donc pas possible de parler de non-recours aux transports collectifs urbains, aux crèches municipales, aux dispositifs d'aide aux devoirs initiés par les établissements scolaires, à des dispositifs d'aide ou d'accompagnement à l'insertion ou de prévention de risques sanitaires, etc. Toutefois, il s'agit là d'un postulat et celui-ci ne résiste pas, pour trois raisons. Laissons de côté cette discussion sur les limites de l'approche orthodoxe du non-recours pour constater maintenant que bien d'autres domaines que celui des prestations sociales sont concernés par le non-recours [START_REF] Odenore | L'envers de la fraude sociale. Le scandale du non-recours[END_REF]. Dans celui de la santé, par exemple, des questions sont posées en termes de retards et d'abandons de soins, d'inobservance de prescriptions, etc. Des domaines aussi différents que ceux de la justice, des transports ou de l'énergie, peuvent aussi être concernés par le non-recours à des droits subjectifs ayant un caractère obligatoire ou facultatif pour les institutions prestataires : la demande d'aide juridictionnelle, le recours juridictionnel, la tarification sociale, etc. De plus, il existe dans ces domaines comme dans d'autres des connexités fortes avec les prestations sociales : l'ouverture de certains minima sociaux commande normalement l'accès à des aides sociales en matière de transport ou à l'aide juridictionnelle ; l'accès au juge ou au service de probation peut être nécessaire pour l'obtention des prestations sociales ; etc. Les situations de non-recours peuvent donc exister bien ailleurs que dans le domaine des prestations sociales, indépendamment de celui-ci ou parfois de façon liée dès lors que les offres s'emboîtent du fait des règles de l'éligibilité. Par conséquent, on ne peut pas s'en tenir à un domaine particulier du fait de la fréquente connexité des dispositifs. En même temps, l'enchaînement des problématiques de l'exclusion du travail (Alain Touraine), des précarités sociales (Robert Castel) et des sécessions urbaines (Jacques Donzelot), comme celles de la mobilité/accessibilité de l'offre publique et des discriminations, oblige à tenir compte de l'accumulation de situations de non-recours au-delà du seul domaine des prestations sociales.

La question de la délimitation des domaines concernés par le non-recours ne peut pas se régler du seul point de vue orthodoxe, par un calcul de différentiel entre population potentiellement éligible et population éligible et effectivement bénéficiaire. Il convient d'intégrer par conséquent une diversité d'offres publiques relatives à l'aide sociale, à l'action sociale et collective (dispositifs d'accompagnement et de médiation en particulier), mais aussi aux services publics et aux institutions régaliennes, indépendamment de leur caractère obligatoire, alternatif ou contingenté. C'est pour cela que nous parlons de « Non-recours aux droits et services ». Il faut également tenir compte des populations éligibles, mais aussi de celles qui perdent ce statut, pour les conséquences différées -en termes de non-recours -que ce changement peut induire.

Par conséquent, la définition initiale du non-recours se trouve modifiée : ainsi, le non-recours renvoie à toute personne qui -en tout état de cause -ne bénéficie pas d'une offre publique, de droits et de services, à laquelle elle pourrait prétendre.

L'élargissement proposé de l'objet du non-recours conduit cependant à revoir les typologies conçues initialement pour les prestations sociales à partir de l'approche orthodoxe [START_REF] Warin | Le non-recours aux politiques sociales[END_REF]. En ouvrant le regard, d'autres formes de non-recours apparaissent, et avec elles la complexité du phénomène.

Pour essayer de dépasser les limites de typologies ou de modèles d'analyse largement inspirés d'une théorie du choix rationnel, l'Observatoire des non-recours aux droits et services a présenté une typologie explicative qui comprend aujourd'hui quatre principales formes de nonrecours :

 La non connaissance, lorsque l'offre n'est pas connue,  La non demande, quand elle est connue mais pas demandée,  La non réception, lorsqu'elle est connue, demandée mais pas obtenue.  La non proposition, lorsque l'offre n'est pas activée par les agents prestataires malgré l'éligibilité du demandeur, que celui-ci connaisse ou pas l'offre.

Pour chacune des quatre formes, une série d'explications typiques (qui est censée évoluer au fur et à mesure de l'étude du phénomène) est présentée, de façon à construire un modèle d'analyse dynamique fondé sur la combinaison des explications qui peuvent être observées.

Cette typologie propose une grille d'analyse générale, applicable à l'ensemble des situations de non-recours. Outre son but explicatif, son principal apport est de souligner la possibilité d'un non-recours volontaire. Dans ce cas, le non-recours n'apparaît pas comme un incident plus ou moins fortuit, notamment dans le processus de demande (par non connaissance de l'offre ou problème dans la mise en oeuvre), mais comme un rapport social à l'offre publique et aux institutions qui la servent. Ce non-recours caractérise en particulier les personnes qui ne demandent pas l'offre qui est proposée. Plus avant, cette grille distingue aussi un non-recours contraint et un non-recours choisi. Elle permet du coup de questionner l'effectivité de l'offre mais également sa pertinence, c'est-à-dire son sens pour les (non) usagers. Ainsi, le nonrecours est non seulement un enjeu gestionnaire -comme il fût pensé au départ -mais aussi une question politique. La dimension politique du non-recours s'exprime aussi bien quand il signale un désintérêt ou un désaccord avec ce qui est proposé, que lorsqu'il est l'effet d'impossibilités, aussi variées soient-elles. Dans ces différents cas, la question du non-recours permet d'interroger la possibilité de « ruptures de citoyenneté » dans la mise en oeuvre de l'offre publique. Aussi pour finir, choisissons-nous d'illustrer, même rapidement, certains enjeux de la non demande parce que cette forme exprime au mieux la double dimension du volontaire/choisi et du volontaire/contraint qui permet d'introduire cette dimension politique.

Aperçu des enjeux du non-recours

La non demande ne peut pas être assimilée au modèle de la « défection » chère aux théories du consommateur informé et rationnel qui décide de ne pas prendre (ou acheter) et de reporter/retarder sa demande. D'autres explications entrent en jeu. En particulier, les situations de non demande découlent souvent de conflits de normes et de pratiques. Ces conflits renvoient aux désaccords entre, d'une part, les règles qui déterminent le contenu de l'offre et ses conditions d 'accès, et, d'autre part, les valeurs, représentations, expériences, situations et attentes à travers lesquelles les destinataires les perçoivent.

Une non demande surgit notamment lorsque l'offre impose des conditions de comportements qui paraissent irréalisables ou inacceptables. Une conditionnalité qui demande aux destinataires de démontrer leur autonomie et responsabilité, véhicule des modèles de « l'accomplissement de soi ». Ces modèles sont difficilement accessibles pour certains, tant pour des raisons sociales, économiques et psychologiques, que pour des raisons morales ou politiques. En particulier, le principe de l'activation, avec ce qu'il suppose comme engagements à respecter, peut susciter : une non demande par dénigrement de ses propres capacités, une non demande par découragement devant la complexité de l'accès, ou encore une non demande par non adhésion aux principes de l'offre. Dans ces différents cas, les situations de non-recours paraissent contraintes et renvoient aux inégalités sociales, c'est-à-dire ici aux handicaps liés à l'appartenance sociale, au manque de capacités, au statut imposé que l'offre avive. Elles ne peuvent être comprises indépendamment des positions sociales, des psychologies et des valeurs individuelles, elles-mêmes inscrites dans les histoires de vie. On peut se demander alors quels sont les effets des normes imposées par l'offre publique, notamment lorsque la précarité met à mal l'estime de soi. Le recours/non-recours dépend pour partie de la confiance en soi (des individus en eux-mêmes), de la confiance dans le contenu de l'offre et le prestataire, et de la confiance dans la tournure des événements (dans l'échange lui-même). L'offre pouvant être « passive » ou « active », c'est-à-dire s'appliquer suivant des règles et procédures préétablies ou bien être mise en oeuvre en fonction de relations d'obligations entre acteurs (destinataires/prestataires), il faut considérer que le recours ou le non-recours dépend aussi de la façon dont l'offre sollicite différemment ces régimes de confiance. Mais il s'agit aussi de citoyenneté. En effet, l'Etat qui assurait jusque-là l'autonomie des individus par une « égalité de protection », maintenant la leur renvoie. A eux de montrer qu'ils sont autonomes et responsables : c'est la nouvelle condition pour être aidé, la nouvelle logique de la solidarité. La première édition du WP 1 date de juin 2010. Elle a été publiée dans laviedesidees.fr, le 1er juin 2010

© laviedesidees.fr

English version

Originally, non take-up was a term applied specifically within the context of financial social benefits, which is why the original definition for non-take up was all persons or households entitled to receive financial social benefits who are unaware of their entitlement. In the United Kingdom, in which the term non take-up of social benefits originated, this phenomenon was first identified in the 1930s, and clearly coincided with a political preoccupation with ensuring that social security, particularly when targeted at specific sections of the population (meanstesting benefits), was being correctly allocated. Even if the debate surrounding non take-up has different origins across different countries, it has always been a sign of political concern over the issue of effectiveness of social spending, the difference between planned versus actual impact, and most importantly, that the benefits on offer actually reach those they are intended for.

The history of non take-up has lead to it being interpreted and applied in a rather limited way.

It is therefore necessary to widen the scope of analysis somewhat to look more broadly at other aspects of a phenomenon which has the potential to call into question traditional public policy.

A widespread phenomenon

There are two main reasons to expand the definition of non take-up of benefits beyond purely financial terms. Firstly, because the traditional definition of non take-up can only be applied if, as is the case with financial social benefits, a specific entitled population is identified, and secondly, because the problem of non take-up affects many different public services.

The very notion of non take-up initially emerged based on the criteria of benefit entitlement.

Entitlement is determined both by the law, which takes into account a diverse range of factors such as family status, age, gender and income, and also by other official regulations which uphold certain social security entitlements, some of which automatically establish successive entitlements. These laws essentially define eligible sections of the population and the conditions under which social security benefits can be accessed. The rate of non take-up can therefore be measured as the difference between the entitled population and the entitled population actually claiming and receiving their entitlement.

Using this traditional method to calculate rates of non take-up means that non take-up can only be measured if a population entitled to claim is first identified, and thus can only be applied to a very limited section of the social security system as a whole as it is usually quite difficult to estimate, much less identify, a target population of benefit recipients. While financial social benefits, which are provided to meet the needs of specific groups within the population according to very specific criteria, lend themselves well to this method, for other parts of the social security system the idea of a potential target population can be vague, even impossible to identify. The French economist Antoine Math, who was amongst the first to introduce the topic for discussion in France, explains that this difficulty in identifying entitled populations remains the main obstacle to any study of the phenomenon, which is precisely why any study of non take-up rates of other services such as public transport, childcare, after-school services and social inclusion and public health programmes, would be so problematic. It must be said, however, that this argument has its weaknesses.

Firstly, the fact that an entitled population cannot be identified does not mean that non take-up of benefits is less relevant or does not take place. The issue does not disappear simply because of a lack of figures; it is the difficulty in obtaining the figures in the first place which results in less coverage and discussion of the issue. One might take nursery places as an example. As a result of the limited number of places, a significant number of parents might underestimate their chances of success and decide not to claim places for their children, despite being entitled to do so. This is non take-up. However, how do we calculate how many entitled parents failed to claim, and what the total population of entitled parents is? These important questions, which can be so difficult to answer accurately, must still be acknowledged and accounted for.

Another point to consider is that the traditional method of calculating non take-up rates fails to take into account the often frequent changes which are made to the eligibility criteria which claimants must meet in order to access many social security benefits and services. Persons or households once eligible may, under new guidelines, lose their entitlement to claim and receive benefits. Since the original definition of non take-up refers only to entitled sections of the population and not to those previously entitled, it would, strictly speaking, be misleading to apply it here. Despite this fact however, many argue that the frequent changes made to the eligibility criteria within the social security system is a direct cause of non take-up, an argument also very often included within the growing debate on social security systems becoming less generous, which emerged first in the United Kingdom and United States and later in the Netherlands, Germany and in France. A further point to consider is increasing socio-economic vulnerability and the possible effects that this might have in further complicating entitlement conditions, particularly in terms of unemployment and low-wage incomes. This question has been examined in the context of 'double destabilisation' and reduced social security protection.

Studies of non take-up must therefore also consider populations who are either suddenly no longer eligible or who face increasingly complex and demanding entitlement conditions, both immediately and over a period of time. It is also important to consider how changes to entitlement conditions can affect the opinions and behaviour of potential claimants, such as pushing some to stop claiming altogether. Loss of entitlement and the effect that this loss of entitlement can have on take-up rates is thus an important part of the debate.

A final point to make is that this method ignores the obvious fact that non take-up inevitably exists in every situation, since no entitled population will ever be forced to claim benefits or access services. Regardless of the need to identify a potentially entitled population, the possibility for non take-up to occur exists in all situations in which there is social security available, and there are two reasons for this. Firstly, the entitled potential recipient is never under any obligation to claim; the only obligation lies with the service provider, which is why the provision of benefits can be described as a subjective right. Potential recipients of social benefits and of all public services in general, are never under any obligation to accept the benefits and services which the service providers are legally required to provide. In other words, no-one is obliged to take-up. As is becoming increasingly clear, fewer and fewer people are claiming, even amongst those who are in need of financial or other assistance, for there are almost always other alternatives available to entitled potential recipients, which in turn limits their dependence upon the social security system. Even in the case of financial benefits, for which there is very little equivalent, many still have the option of financial assistance from family and friends. Indeed, a whole range of alternative sources of assistance exists, ranging from semi-governmental and private programmes to other forms of private support, all of which is likely to increase during periods when the state social security system and its entitlement conditions become overly restrictive. This is a far cry from the image of a dependent population dutifully paying into insurance schemes, which one might mistakenly associate with the phenomenon of non take-up.

Having acknowledged that the traditional method of measuring non take-up of benefits has certain limitations, let us now examine the prevalence of non take-up in other areas [START_REF] Odenore | L'envers de la fraude sociale. Le scandale du non-recours[END_REF]. In the context of the health service, examples of non take-up can include delays or premature discontinuation of treatment and non-adherence to medication. The non take-up of benefits and services which service providers are nevertheless legally obliged to provide has been recorded in areas as diverse as the legal system and the transport and energy sectors.

Relevant examples include legal aid and legal representation, and social tariffs. A further factor to consider is the way in which many of these sectors are tied to the larger social security system in terms of benefit allocation. For example, in order to access certain benefits such as travel discounts within the transport sector or legal aid within the justice system, you must first be entitled to receive the minimum social security benefits. Similarly, in order to obtain certain other benefits, a requirement might be that you first access court or probation services. Non take-up occurs across a whole range of sectors, both entirely separately from the social security system and at times also linked through combined entitlement conditions, which is why we cannot simply focus on one area alone. What's more, the combined problems of workforce exclusion, social vulnerability and urban alienation, as well as discrimination and increasingly restricted access to benefits, further highlight the apparent increase in non take-up outside the social security system.

In order to identify where non take-up is occurring, it is therefore insufficient to rely solely on the traditional method of calculating the difference between the entitled population and the entitled population actually claiming and receiving their entitlement. The whole range of public services which exists as part of the wider social security system, including all social work and community outreach programmes, in particular guidance and mediation schemes, as well as government services and ruling institutions, regardless of their binding, optional or conditional nature, should be considered together. This is why we refer to the non take-up of entitlements and public services. We must also remember to consider both entitled populations and those that have lost their entitlements and the possible effects that this could have in terms of non take-up. The original definition of non take-up should thus be modified to refer to all persons or households that are not claiming the benefits and services to which they are entitled.

Interpreting non take-up more broadly also challenges the original classifications of non take-up which were formed based on the traditional measurement method referred to above [START_REF] Warin | Le non-recours aux politiques sociales[END_REF]. By broadening our scope, other forms of non take-up become apparent, and the complexity of the phenomenon grows.

In dynamic analysis model based on these explanations, so that a general framework for analysis can then be applied to all situations in which the various types of non take-up might be observed. Aside from its explanatory function, the framework's main contribution is that it highlights the possibility of voluntary non take-up, which is less the result of coincidental factors such as lack of awareness or administrative problems with claiming, and more about a deliberate social relationship between the potential recipient and the public service and the institutions which offer it. This form of non take-up in particular can be observed especially often amongst those entitled populations who are offered benefits without having explicitly requested them themselves. The framework also differentiates between what it identifies as compelled non take-up and chosen non take-up, which calls into question not only the effectiveness of the benefits system but also its relevance and significance for potential recipients. Non take-up is thus not only, as was originally claimed, an administrative issue but also a political one. The political dimension of non take-up is present just as much in situations where it occurs through indifference or through some deliberate decision not to claim what is being offered by the social security system as when it is due to factors beyond the control of the potential recipient. In all these cases, non take-up represents a potential break between the individual and the state. This political dimension will be examined briefly in the final section of the text through an examination of the decision not to claim, since it is this type of non take-up in particular which best illustrates the voluntary/chosen and voluntary/compelled binary classifications which have potential political implications.

An overview of non take-up

The customer defection model, in which the consumer makes informed and rational decisions not to purchase a product or to delay their purchase, and which is so intrinsic to theories of consumer behaviour, should not be applied to non take-up. We must consider other factors, in particular the fact that non take-up often occurs as a result of incompatible standards and practices in which the benefits and entitlement conditions clash with the experiences, personal circumstances and expectations of the potential recipients.

The decision by entitled populations not to claim is usually the result of perceived unrealistic or unacceptable requirements imposed upon them in order to receive the benefits, such as the requirement that they demonstrate 'self-fulfillment', autonomy and responsibility. This can be difficult for some, for socio-economic and psychological reasons as much as for moral or political ones. The expectation that entitled populations take on certain commitments can increase non take-up rates if those populations either underestimate their own capacities to meet those requirements, become discouraged when faced with complicated eligibility conditions or if they refuse altogether to adhere to the entitlement conditions. These examples provide a good illustration not only of the degree to which a potential recipient can be compelled by circumstances to engage in non take-up, but also the link between non take-up and social inequality, and the disadvantages that can stem from social background, low achievement and the lower status which claiming welfare benefits can draw attention to. Non take-up cannot be properly understood without taking into account the social status, psychologies and personal values which characterise each individual story. Considering the extent to which increased social and financial vulnerability can damage self-esteem, an interesting point to consider might therefore be what effect the social security requirements have when they are imposed on recipients in this way. Take-up depends to some degree on the self-esteem of individuals, on the confidence which they have in the service and the service provider, and on their confidence in how the situation will pan out (in terms of the actual exchange of services or benefits between claimant and provider). The fact that different benefits and services may elicit different levels of trust and public confidence from potential claimants, especially given that some benefits and services are classified as being 'passive' or 'active', that is to say either based on pre-established rules and procedures or based on more personalized shared mutual obligations between recipients and service providers, should be included in the debate on non take-up rates. It is also a question of the relationship between citizen and state. The State, whose obligation it is to safeguard the autonomy of the individual through the principle of equal protection for all, now demands of the individual that this autonomy and responsibility be proven in order to receive assistance. This is the new logic of social solidarity. Faced with such a proposition, the refusal to take-up can be interpreted both as a rejection and a challenge, as a refusal to accept this model of autonomy, but also as a sign of distress and isolation, in which the individual feels compelled to reject take-up. This forms the basis of what Robert Castel refers to as the individualism of disengagement.

Non take-up can also result from indifference to what the social security system has to offer.

Research looking at the optional social security benefits allocated showed rates of non take-up due to this particular reason as being as high. The conditions to access these benefits were so complicated, and the benefits themselves so modest and at times unpredictable, that they were not deemed worth applying for. Generally speaking, and bearing in mind that entitled populations are under no obligation to claim, the nature of the benefit in material terms can be an important factor to consider. Since chosen non take-up is essentially a form of consumer choice, it can be used to measure the usefulness of public services, which is exactly what the British government did to push through budget cuts, particularly between 1979 and 1990 under

Thatcher. The argument of rational choice has been used since the debate on non take-up first emerged in the United Kingdom, with the aim of identifying the usefulness and effectiveness of means-tested benefits. This went on to form the object of a heated political debate between the Labour Party and the Conservatives after the end of the Second World War, which questioned what type of social security model the country most needed. This political interest in whether or not the benefits system is reaching those it is aimed at has been at the heart of the debate ever since.

Indifference towards the benefits system can also spread between individuals if, for example, they see that friends and family have failed to benefit from the system. This is precisely the reason given most often by the younger recipients of what was previously the French equivalent of the Jobseeker's allowance to explain why they had refused to sign professional insertion contracts; they had already seen their peers fail to find employment. Non take-up can thus be a symptom of both social and psychological frustration which extends far beyond any resentment the individual might feel towards the benefits system and their own material living conditions. Once the individual rejects the entitlements and public services, the latter and the conditions by which to claim them become unattainable, which is precisely why social service providers often emphasize the need to work extensively with individuals on the receiving end of social security to promote social recognition and improve self-esteem and confidence, before any attempt is made to tackle non take-up. Social workers are also increasingly highlighting the growing difficulty which some sections of the population, not necessarily the economically worst off, are having in expressing their needs and making demands. For example, local mother and child welfare services (Protection Maternelle Infantile) have described how parents are finding it increasingly difficult to communicate the needs of their young children. Another possible form of non take-up could thus be described as difficulty in expressing needs, which overlaps somewhat with the issues of illiteracy and low levels of educational attainment. These issues were signaled by the organization ATD Quart Monde and the Oheix report on poverty in France in the early 1980s, as becoming particular problems amongst poorer sections of the population.

The personal concerns of potential claimants are also important to consider, for the individual may believe, rightly or wrongly, that there are possible risks associated with claiming benefits.

In research, the first of its kind, carried out by the economist Anne Reinstadler for the Caisse Nationale d'Allocations Familiales (the body in France which deals with family income benefits), it was found that parents identified as being vulnerable were choosing not to take-up benefits targeted at them from fear that claiming would lead to them losing custody of their children.

Another example looked at the non take-up of home assistance services aimed at elderly people claiming the APA (Aide Personnalisée d'Autonomie), and found that a significant number of them are under-consuming either through rejection of the proposed services or by special arrangement with the service providers, most likely where there is a shortage in the services available or because they already have family members caring for them. Non reception, as in the previous example, and the decision not to claim, can also occur due to a desire to reduce dependence on entitlements and public services. This could be a personal strategy to protect oneself from the perceived negative consequences of claiming, but could also represent a form of moderate individualism, stemming from a personal belief in the need to reduce unnecessary consumption.

* * *

Non take-up is thus not merely a question of the non-allocation of financial social security benefits but has wider implications for the entire social security system as a whole, as well as other areas of public intervention. What's more, due to the many possible explanations for non take-up which have slowly emerged, it would be equally misleading to associate the phenomenon solely with the negative connotations of deficiency, or to depict benefits and public services as the cause and consequence of individual and family problems. Non take-up exists amongst all social groups, and it is not just about deficiency, passivity, disadvantage or domination. It is also about the freedom of individuals to express their indifference, their disagreement and their rejection of the system, which politicians would do well to take into account when defining social need. By looking at the private decisions which people make and identifying different forms of voluntary non take-up (compelled vs. chosen) we may question the very relevance and purpose of social security, instead of merely focusing on whether or not it is being effectively allocated (in which case lack of awareness and non reception would be the most relevant forms to examine). The fact that voluntary non take-up can be affected by such a wide range of factors, from self-esteem, confidence in the system and individual concerns to social recognition, citizenship and personal moral beliefs, is indicative of the extent to which non take-up is capable of promoting wider debate which looks collectively at both the social and political consequences of public decisions and policy change. Non take-up has the potential to generate an evaluation of public policy which would place the interests of service users, particularly those on the margins of society who are most at risk of missing out, firmly at its centre. Above all, it allows the least visible and the least vocal in our society the opportunity to express their opinions on public policy, and this is its greatest gift.

  Tout d'abord, ce n'est pas parce qu'une population potentiellement éligible ne peut être précisément identifiée qu'il n'existe pas de non-recours. Ce n'est pas parce qu'un chiffrage n'est pas possible que la question disparaît. C'est cette difficulté qui fait que cette question reste autant invisible. Mais prenons un exemple : du fait d'un nombre limité de places dans les crèches municipales, « bon nombre » de parents (combien ? sur un total possible de combien ? on ne peut le savoir) ne cherchent pas à demander l'accueil de leurs enfants. Ils dénigrent leurs chances et s'abstiennent de toute demande, alors qu'ils pourraient être bénéficiaires. C'est du non-recours. Il faut donc accepter cette difficulté et tout faire méthodologiquement pour la résorber, car on ne peut tronquer une question aussi cruciale. Ensuite, l'approche orthodoxe ignore les populations qui perdent leur éligibilité. Du fait de l'évolutivité souvent rapide des conditions d'accès à diverses prestations, aides, ou dispositifs d'accompagnement, des populations qui étaient éligibles ne le sont plus subitement et ne peuvent plus bénéficier de l'offre en question. Dans ce cas, il ne s'agit pas à proprement parler de non-recours selon la définition initiale qui ne s'intéresse qu'aux populations éligibles. Néanmoins, beaucoup d'auteurs ont saisi ces changements pour soutenir que le non-recours est une conséquence de changements intervenus dans les règles d'attribution. Ainsi, le transfert du thème du non take-up du Royaume-Uni et des Etats-Unis vers les Pays-bas, l'Allemagne puis la France a-t-il été étroitement lié au débat sur la réduction de la générosité des systèmes de protection sociale. Par ailleurs, le constat récurrent d'une déstabilisation des statuts sociaux engendrée par les phénomènes de précarisation socio-économique conduit à considérer également la déstabilisation des situations d'éligibilité. Notamment, les dynamiques d'entrée et de sortie du chômage et les situations de sous-emploi tendent à complexifier les critères d'éligibilité à différentes prestations sociales. Ce phénomène a été étudié au travers de la notion de « double déstablisation » et de ses conséquences en termes de perte des protections sociales. L'étude du non-recours doit donc tenir compte aussi des populations qui subitement ne sont plus éligibles ou qui font face -on va le voir -à la multiplication des conditionnalités de l'offre (simultanée et dans le temps). Ceci est d'autant plus important que les changements de critères d'éligibilité produisent des effets sur les perceptions, représentations et comportements des demandeurs potentiels. Notamment, une perte d'éligibilité pousse certains à ne plus rien demander ; c'est en partie la question de la rupture de droits et de ses effets en termes d'abandons. Enfin, cette approche orthodoxe ne tient pas du tout compte du fait que le non-recours existe de partout, simplement puisqu'aucune offre n'a de public contraint et captif. Autrement dit, indépendamment de la question méthodologique de l'identification d'une population potentiellement éligible, il y a une possibilité de non-recours dès lors qu'une offre s'adresse à un public, et ce pour deux raisons. D'une part, l'offre publique, quelle qu'elle soit, n'est jamais obligatoire pour son créancier : elle n'a jamais d'usager contraint. Elle ne peut être obligatoire que pour son débiteur ; ce qui renvoie à la définition d'un droit subjectif. Les offres de prestations sociales, comme l'offre publique en générale, n'ont jamais de destinataires contraints, alors même qu'elles constituent une obligation pour les prestataires. Autrement dit, personne n'est obligé de recourir. Comme on le voit de plus en plus, beaucoup ne demandent plus rien, même s'ils ont besoin d'une aide financière ou d'autre chose. D'autre part, des alternatives existent la plupart du temps, si bien qu'il n'y a pas non plus d'usager captif. En effet, l'offre publique n'a rien de monopolistique. C'est pareil avec les prestations financières : certes elles n'ont pas d'équivalent, mais cela n'empêche pas des alternatives, comme le soutien financier venant de proches. Des possibilités d'alternatives à l'offre publique existent donc presqu'à chaque fois ; elles reposent notamment sur la mobilisation d'offres parapubliques ou privées, d'autres ressources ou bien des soutiens personnels. Il est même probable que le recours aux alternatives soit d'autant plus important lorsque l'offre publique est contingentée et soumise à des conditions d'accès restrictives ou contraignantes. Par conséquent, le phénomène de non-recours ne peut être lié à la représentation idéalisée d'un public contraint et captif, inspirée de la population des cotisants et assurés sociaux.

  Face à une offre qui change sur le fond, ne pas recourir peut être à la fois un signe de désaccord, la non demande choisie exprime alors une contestation, un refus (ne pas accepter ce régime d'autonomie), mais aussi un signe de perdition, et dans ce cas la non demande contrainte fait le lit de l'individualisme de déliaison dont parlait Robert Castel.Les non demandes peuvent aussi résulter d'un faible intérêt pour l'offre publique. L'exemple des aides sociales facultatives montre que pour cette raison le non recours peut atteindre des taux très élevés. L'accès à ces aides est si compliqué, le résultat si modeste et bien souvent aléatoire, que le jeu n'en vaut pas la chandelle. D'une façon générale, le contenu de l'offre est un élément actif du non-recours ; les destinataires n'étant ni contraints, ni captifs, répétons-le.Dans ce cas, la non demande choisie, qui s'apparente ici au libre choix du consommateur, peut alors devenir l'étalon de mesure de l'utilité de l'offre publique. C'est ainsi que les Britanniques en firent par moments usage dans le but de procéder à des coupes budgétaires, notamment sous le gouvernement de Margaret Thatcher de 1979 à 1990. L'explication en termes de choix rationnels fût d'ailleurs présente dès le départ dans le débat sur le non-recours. C'est effectivement en termes d'utilité ou de non utilité des prestations ciblées pour les destinataires potentiels que la question du non take-up (dont la traduction littérale est celle de non-usage, non participation) a été posée au Royaume-Uni dans un débat politique majeur entre Travaillistes et Conservateurs sur : quel modèle de protection sociale, à la sortie de la seconde guerre mondiale. Cela orienta durablement le thème du non-recours dans un questionnement politiquement centré sur l'intérêt des usagers pour l'offre publique.Parfois, la non demande par manque d'intérêt pour l'offre apparaît comme une forme héritée. C'est le cas lorsque les personnes considèrent que l'offre ne leur sera pas utile, parce qu'elle ne l'a pas été pour des proches. Nos observations ont montré que parmi les allocataires de l'ex-RMI, les plus jeunes expliquent principalement la non signature d'un contrat d'insertion pour cette raison, ayant constaté chez des aînés l'impasse de l'offre d'insertion. Dans ces situations, comme dans bien d'autres, le non-recours renvoie à un état de frustration sociale mais aussi psychique, qui va au-delà du ressentiment à l'égard de l'offre proposée lié aux conditions matérielles d'existence. L'offre publique de droits ou de services devenant dans ce cas une « non idée » pour les personnes, son contenu et ses règles d'accès deviennent imprescriptibles. C'est pour cela que des intervenants sociaux attirent l'attention -à commencer celle de leur hiérarchie et financeurs -pour que, avant de parler d'accès aux droits et d'action contre le nonrecours, ils puissent tout d'abord procéder pleinement à un travail de reconnaissance sociale avec les individus, dans le but de leur redonner une image positive d'eux-mêmes. Par ailleurs, des travailleurs sociaux signalent la difficulté croissante d'une partie de leur public, qui n'est pas forcément précaire sur le plan économique, à formuler des besoins et par là même à présenter des demandes. C'est le cas, par exemple, de services locaux de Protection maternelle infantile (PMI) qui observent que des parents ont de plus en plus de difficultés à mettre en mots les besoins de leurs jeunes enfants. Cela indique la possibilité d'une non demande par difficulté à exprimer des besoins. Cette observation repose d'une certaine façon la question de l'illettrisme que le rapport Oheix sur la pauvreté en France ou ATD Quart Monde avaient signalé au début des années 1980 pour les populations pauvres, pour parler du manque de capacités requises, liées en particulier au déficit d'apprentissages ou de transmissions de connaissances. A côté de l'intérêt de l'offre, l'intérêt des individus entre aussi en ligne de compte dans l'explication de la non demande. C'est par exemple le cas lorsque les conséquences (réelles ou supposées) induites par l'offre paraissent risquées. Parmi les travaux précurseurs initiés par la CNAF, ceux de l'économiste Anne Reinstadler ont signalé un non-recours à l'Aide aux parents isolés (API) par crainte que leur demande, induisant une enquête de situation, aboutisse à des décisions de placement des enfants. Autre cas de figure, nos travaux sur la non utilisation des heures de services d'aide à domicile, préconisées à des personnes âgées bénéficiaires de l'Aide personnalisée d'autonomie (APA), indiquent que bon nombre d'entre elles sont « sous consommatrices » par non adhésion aux propositions d'aides à domicile, ou par arrangement avec le prestataire, notamment à cause du contingentement de l'offre ou du souci de préserver les aidants familiaux. En même temps, la non réception, comme dans cet exemple, ou la non demande peut correspondre à une volonté de moins consommer de droits ou de services. Cela peut s'expliquer par des raisons tactiques individuelles (se préserver, par crainte d'effets induits), mais aussi par des normes personnelles (par exemple, par souci de juste consommation, qui peut être significatif d'un individualisme modéré, tempéré par des valeurs ou des règles morales. * * * La question du non-recours ne porte donc pas sur les seules prestations sociales financières, elle touche la protection sociale dans son ensemble et au-delà bien d'autres domaines de l'intervention publique. En même temps, la diversité des explications mises peu à peu à jour empêchent de dégager une représentation systématiquement déficitaire du phénomène, selon laquelle le non-recours à l'offre publique serait à la fois la conséquence et la cause de multiples difficultés individuelles ou familiales. Faut-il rappeler que le non-recours concerne a priori toutes les catégories sociales. Le non-recours n'est pas que manque, passivité, incapacité, domination ; il rend compte également du libre choix des individus et exprime des désintérêts, des désaccords ou des impossibilités, que les acteurs des politiques pourraient prendre en compte pour définir la « demande sociale ». La représentation contrastée du non-recours volontaire (choisi/contraint), entraperçue ici à travers quelques remarques sur la non demande, permet de s'interroger sur la pertinence de l'offre publique et non seulement sur son effectivité (telle qu'elle apparaît au travers du non-recours par non connaissance ou par non réception). La diversité des enjeux liés à cette forme particulière (enjeux de confiance, d'intérêt, de reconnaissance sociale, de citoyenneté sociale, de valeurs morales) montre que le non-recours est une question intermédiaire permettant d'approcher les conséquences sociales et politiques des choix publics et des changements de logiques dans les politiques. Dit autrement, la question du non-recours peut probablement aider une évaluation des politiques publiques qui serait résolument tournée vers les destinataires, et en particulier vers ceux qui manquent à l'appel. Elle ne sera probablement jamais plus utile qu'en donnant la possibilité de frotter les choix publics au sens critique de ceux que l'on n'entend pas ou ne voit plus. Références Odenore. 2013. L'envers de la fraude sociale. Le scandale du non-recours. Paris : La Découverte. Warin P. 2016. Le non-recours aux politiques sociales. Grenoble : Presses Universitaires de Grenoble.

  order to go beyond classifications and models based almost exclusively on a theory of rational choice, the Observatory of Non Take-up of Social Rights and Public Services in France identified what it suggests are the four main types of non take-up:  Lack of awareness, in which the potential claimant is unaware of the benefits available or of his entitlement to claim  Decision not to claim, in which the potential claimant is aware of his entitlement but chooses not to claim  Non reception, in which the claimant is aware of his entitlement and decides to claim but does not obtain his entitlement  Non proposition, in which the provider does not propose a benefit to the potential claimant For each of these four types, various typical explanations are suggested, which are expected to change and develop over the course of the Observatory's research. The aim is to create a