Fabien Llobell

Davide Giacalone

Amaury Labenne

Mostafa El

Qannari

El Mostafa Qannari

Assessment of the agreement and cluster analysis of the respondents in a CATA experiment

Keywords: Agreement, Segmentation, Cluster analysis, Atypical subjects, CATA experiment, Permutation test. I

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Check all that apply (CATA) experiment is among those so-called rapid methods of sensory evaluation that have gained ground these last two decades or so [START_REF] Varela | Novel techniques in sensory characterization and consumer profiling[END_REF]Meyners, Castura, Carr, 2013). In this experiment, respondents are instructed to associate attributes from a predefined list to each one of the products to be evaluated. For more details regarding this experiment, we refer to the paper by [START_REF] Meyer | Existing and new approaches for the analysis of CATA data[END_REF].

Besides setting up graphical displays that depict the similarity and the dissimilarity of the products, practitioners have been interested in assessing the agreement among the respondents. The importance of assessing the agreement of respondents in a CATA experiment was stressed by Worch and Piqueras-Fizman (2015). For that purpose, these authors proposed to use strategies based on multiple factor analysis on contingency tables and McNemar tests. It is worth noting that [START_REF] Meyners | Statistical evaluation of panel repeatability in Check-All-That-Apply questions[END_REF] remarked that although McNemar's test provides insight into products differences, it is not appropriate to assess the repeatability in a CATA experiment. The discrimination of the products by the various attributes (e.g. Cochran's test, Meyners, Castura, Carr, 2013) are also important topics.

We propose a unified framework to address some of these issues. In this regard, the data from each respondent are plunged in a Euclidean space. The cosine between two matrices associated with two respondents stands as a similarity index between these two

respondents. An overall agreement index is derived from the first eigenvalue of the matrix that contains the pairwise similarity indices between the respondents. The associated eigenvector yields individual agreement indices that reflect how each respondent agrees with the general point of view of the panel. It is clear that this analysis bears high similarities to the STATIS method where the RV coefficient is used as a similarity index between the datasets [START_REF] Lavit | The act (static method)[END_REF]. A permutation test is performed in order to assess the significance of the overall agreement index. Following a similar strategy, we investigate the agreement of the respondents for each attribute taken separately. If the agreement index associated with a given attribute turns out to be non-significant, this means that this attribute has been misunderstood by the subjects or interpreted differently by them. Therefore, it is very likely that this inconsistency will result in a non-discrimination of the products. Drawing on selected case studies, we will compare the outcomes of the permutation tests regarding the consistency of the attributes with those of Cochran's test, which is commonly used to assess the discrimination of the attributes (Meyners, Castura, Carr, 2013).

In a subsequent section, we tackle the issue of clustering the subjects participating to a CATA experiment. We discuss a refinement of a clustering strategy called CLUSCATA [START_REF] Llobell | A new approach for the analysis of data and the clustering of subjects in a CATA experiment[END_REF]. The idea is to use an additional cluster, called "noise cluster" or "K+1" cluster, to the standard CLUSCATA procedure, with the aim of catching those respondents who are atypical in that sense that they do not fit to the pattern of any main cluster. The concept of noise cluster was originally introduced by [START_REF] Dave | Characterization and detection of noise in clustering[END_REF], and has previously been adapted to various situations in sensometrics [START_REF] Vigneau | Segmentation of consumers in preference studies while setting aside atypical or irrelevant consumers[END_REF][START_REF] Berget | Segmentation in projective mapping[END_REF]Llobell, Vigneau, Qannari, 2019).

The paper is organized as follows. In the section "Theory", we start by outlining a brief reminder of the similarity among pairs of respondents. Then, we discuss how the overall agreement of the respondents or, equivalently, the homogeneity of the panel can be assessed and how a permutation test can be set up to evaluate the significance of this overall index. We also tackle the issue of assessing the agreement of the respondents for each attribute. Subsequently, we discuss how the strategy of clustering the respondents called CLUSCATA can be adapted to include a noise cluster. In section 3, the general strategy of analysis is illustrated of the basis of selected case studies. We end the paper by a conclusion (section 4).

II. Theory

Similarity measure between respondents

Consider a CATA experiment involving subjects who are instructed to evaluate products on the basis of attributes. The data from each respondent can be coded as a matrix ( × ) whose rows refer to the products and columns to the attributes. The entry of this matrix is equal to 1 if the respondent has ticked the attribute in the corresponding column for the product in the corresponding row. Otherwise, the entry is equal to 0 (Meyners, Castura, Carr, 2013). Thus, the data of each respondent are plunged in a Euclidean space, namely the space of matrices of dimensions × . We can show that the cosine between two matrices and associated with two respondents is given by , =

,

| | | | = √ √
, where is the total number of times that both respondents checked the same attributes for the same products, and (resp. ) is the total number of checks across attributes and products of respondent (resp. ). Indeed, since and contains only 0 and 1, it is easy to check that < , >= ∑ ∑ = ,

| | = ∑ ∑ = and similarly | | =
. This index , stands as a similarity coefficient between and and is referred to the Ochiai coefficient [START_REF] Ochiai | Zoogeographic studies on the soleoid fishes found in Japan and its neighbouring regions[END_REF]. We can show that , = 0 if and only if the two respondents at hand have checked completely different attributes for each product. We can also show that , = 1 if and only if =

, which means that the two respondents perfectly agree. More details regarding this similarity index can be found in (Llobell et al., 2019).

It is worth mentioning that, within the context of CATA experiments, [START_REF] Carr | Multidimensional Alignment (MDA): A simple numerical tool for assessing the degree of association between products and attributes on perceptual maps[END_REF] and Meyners, Castura and Carr (2013) proposed an approach of analysis called Multidimensional Alignment (MDA) based on the cosine between pairs of vectors. The aim of MDA is to investigate the relationships between products and attributes. The aim of the approach adopted herein is different since we seek to assess the similarity between the respondents participating to the CATA experiment.

Overall and individual agreement indices

Let us denote by ! , … , # the 0-1 matrices associated with the repondents.

We also denote by $ = % , & the Ochiai coefficient between % and ' . Thus, the matrix ( = $ ,$)*,...,, of dimension × is formed by positive entries. As a consequence, its largest eigenvalue, which we denote by -, is positive and its corresponding eigenvector . = . * , … , . , can be chosen in such a way that all its components are positive [START_REF] Meyer | Existing and new approaches for the analysis of CATA data[END_REF]. As an overall index of agreement among the respondents or homogeneity of the panel, we propose / = 0 , . We can show that this index is comprised between This permutation procedure is repeated a large number of times (say, 1000 times) and the distribution of the values / * thus obtained will stand for the distribution of the test statistic under 4 5 . This means that we can compute a p-value as the proportion of simulated values / * that are larger or equal than the observed value, /. As a consequence, the hypothesis 4 5 will be rejected if this p-value is smaller than a significance level chosen by the practitioner (e.g., 5%).

The agreement of respondent 1 1 = 1, … , can be assessed by . which is, as stated above, the 1 :; component of the eigenvector of matrix < associated with the largest eigenvalue, -. The coefficients . are comprised between 0 and 1 but unlike the overall agreement index, these coefficients should not be compared to 1 but to each other. This means that they enable us to rank the respondents according to their agreement with the general point of view of the panel. Obviously, their interpretation depends on the value of the overall index of agreement discussed above.

Attributes consistency

We consider the degree of agreement of the respondents regarding the evaluation of the products with respect to each attribute. More specifically, we are interested in the hypothesis test that evaluates whether there is no agreement at all among the respondents when evaluating the products for the attribute under consideration against the alternative hypothesis that stipulates that there is some degree of consistency among the respondents.

In case the null hypothesis is not rejected, the attribute in consideration is very unlikely to discriminate the products. However, the opposite implication is not necessarily true since the respondents may perfectly agree on the assessment of an attribute that is not intrinsically discriminant. To assess the discrimination of the attributes, it is recommended to perform Cochran's test [START_REF] Cochran | The comparison of percentages in matched samples[END_REF]Meyners, Castura, Carr, 2013), which is devoted to investigate the differences among several treatments (i.e., products in the case of CATA)

when the data at hand are binary. The consistency test proposed herein may complement the Cochran's test in the sense that it checks whether the non-discrimination of the products by the attribute under consideration can be explained by the disagreement among the respondents.

The hypothesis test for the respondents' consistency in evaluating the products by means of a given attribute follows exactly the same process as for the evaluation of the overall agreement (see previous section). Since we are interested in one attribute at a time, the data can be stored in a table where the rows refer to the products and the columns refer to the respondents. The entries of this table are 1 or 0 indicating whether each respondent (column) has checked the attribute under consideration for each product (row) or not. We can note in passing that this is precisely the setting for performing the Cochran's test.

We will not give further details regarding the consistency test since it follows point by point the test regarding overall agreement. It proceeds as if we had a CATA experiment with one attribute, that is the attribute for which we investigate the consistency.

CLUSCATA while setting aside atypical respondents

In a previous paper (Llobell et al., 2019), a cluster analysis specially tailored for the CATA data -called CLUSCATA -was introduced. The rationale behind this method of analysis is to determine homogeneous clusters of respondents and, for each cluster, a group average dataset which is as close as possible to the datasets associated with the respondents in this cluster.

From now on, the datasets % 1 = 1, … , associated with the various respondents are assumed to be scaled in order to have their norm equal to one. This is achieved by dividing the original binary datasets by their respective norms. This standardization makes it possible to account for the total number of checked attributes which may significantly differ from one respondent to another. For what follows, we also need to extend the similarity index s to measure the similarity between two datasets which are not necessarily binary.

This is given for two matrices = and > (say) by =, > = < =, > > = :?@AB => C D:?@AB == E :?@AB >> C .

Obviously, this is the cosine between = and > considered as elements of the space of matrices of dimension × . For more details, we refer to the paper by Llobell et al. (2019).

In that paper, we proposed an optimisation criterion that clearly highlights the rationale behind CLUSCATA. A new formulation of CLUSCATA which is more conducive to the 'noise cluster' idea is to seek to maximize the quantity:

4 = ∑ ∑ 8 % , F & 9 ∈H I J $)* 1 where F & = ∑ . % ∈H I
, with ∑ .

∈H I = 1. F & is the group average dataset in cluster K $ , and the weight . in cluster K $ reflects the degree of agreement of respondent 1 with F & .

Clearly, this criterion reflects the idea that we are seeking clusters of respondents so that in each cluster, K $ , the datasets % are as much close (i.e., high similarity) as possible to the group average dataset, F & associated with K $ . As a matter of fact, we can show the following equality:

∑ || % , )* || = ∑ LF & L J $)* + ∑ ∑ || % - ∈H I J $)* . F & ||². Since
∑ LF & L J $)* ∑ || % , )* || P
, which is equal to

∑ Q R S T SUR V
, (i.e., the overall homogeneity) also reflects how the W X are far removed from each others. The lower bound of this quantity, which is equal to Y/ , means there is no grounds for clustering (i.e., between clusters variation negligible) and the upper bound, which is equal to 1, entails that we have perfectly separate clusters and within each cluster the datasets are similar (i.e., within clusters variation is equal to 0).

In order to solve this maximization problem, a hierarchical algorithm followed by a partitioning algorithm was proposed (Llobell et al., 2019). We can also show that 4

= ∑ -$ J $)*
where -$ is the largest eigenvalue of matrix S, which contains the pairwise similarities as measured by s between the respondents in group 3. As previously, we propose to assess the homogeneity in cluster 3 by / $ = 0 I

, I where $ is the number of respondents in cluster 3.

We will focus hereinafter on the partitioning algorithm, which is akin to the 3-means algorithm since the datasets are iteratively moved in and out of the clusters depending on their similarity with the average datasets of these clusters. In parallel, the average datasets are updated after each change (i.e., removal and inclusion of datasets in the clusters). The partitioning algorithm requires to fix the number of cluster and an initial partition of the respondents as a starting point. This can be achieved by examining the outcomes from the hierarchical strategy of clustering [START_REF] Everitt | Cluster Analysis, 5th Edition[END_REF]Llobell et al., 2019).

More precisely, the evolution of the aggregation criterion in the course of the clustering process is reflected by the lengths of branches of the dendrogram or hierarchical tree.

Therefore, we can choose the number of clusters that corresponds to a significant jump of the aggregation criterion when passing from a partition with K clusters to a partition with K-1 clusters. The solution thus obtained is further improved by running the partitioning algorithm associated with CLUSCATA (Llobell et al., 2019).

We will discuss how the partitioning algorithm can be extended in order to identify and set aside the respondents who are atypical in that sense that they do not fit to the pattern of any cluster. Following Dave's idea [START_REF] Dave | Characterization and detection of noise in clustering[END_REF], we introduce, besides the main Y clusters, an additional cluster called "noise cluster" or "K+1 cluster", which will contain the respondents who are deemed to be atypical. This strategy of analysis requires selecting a threshold value, [, between 0 and 1, below which the similarity between a dataset associated with a respondent and the average dataset associated with a given cluster is considered as weak. Naturally, the partitioning algorithm seeks to assign a respondent to the cluster for which this similarity is the largest but, if this similarity is smaller than the threshold value [, the respondent is assigned to the "K+1 cluster". Formally, we seek to maximize the following criterion:

4 \ = ∑ ∑ 8] $ 8 % , F & 9 + ] J^* [ 9 , )* J $)* 2
where ] $ is the Kronecker symbol which is equal to 1 if the dataset % belongs to cluster 3 and 0 otherwise. We have the constraint ∑ ] $ = 1 J^* $)*

, which stipulates that a subject belongs to one and only one cluster, including the noise cluster. To solve this maximization criterion, we run the following algorithm:

Step 0 (initial partition). Choose a partition into Y clusters of datasets (i.e., subjects) as a starting point. This can be done by a random assignment of the datasets at hand to Y clusters or by cutting the hierarchical tree at the level corresponding to Y clusters.

Step 1 (group average datasets). In each cluster K $ , the group average F & is determined as a weighted average of the datasets in this cluster, where the weights are derived from the first eigenvector of the matrix which contains the pairwise similarities between the respondents in cluster K $ .

Step 2 (changing clusters). New clusters of subjects are formed by moving each dataset, % , to the cluster K $ for which the quantity % , F & is the largest providing that this quantity is larger than [ otherwise, % is assigned to the cluster "K+1".

Steps 1 and 2 are iterated until the datasets stop changing clusters.

As a matter of fact, CLUSCATA is an adaptation of a clustering approach called CLUSTATIS to the particular case of CATA data (Llobell et al., 2019). Within the framework of CLUSTATIS, a strategy to select an appropriate threshold value was proposed and backed up by several considerations (Llobell, Vigneau, Qannari, 2019). The same considerations could be applied to CLUSCATA. This leads us to select the following quantity:

[ = ∑ `a % , F & % ^ a % , F 8& % b9 c d eUR , 3
where F & % is the weighted average dataset of the cluster to which the 1 :; dataset belongs, and F & % g is the weighted average dataset of the second nearest cluster to the 1 :; dataset, that is the cluster for which % , F & % g is the largest after % , F & % The rationale behind the selection of this parameter is to set up an intermediary value that delineates the boundary between the clusters, so much so that those subjects who are straddling two or more clusters are assigned to the noise cluster.

III. Illustrations

Beers dataset

The data used to illustrate the general strategy of analysis discussed herein pertain to a CATA experiment where 9 beer images were evaluated by 76 consumers using 15 attributes. The consumers were instructed to check the relevant usage contexts to drink each beer. The attributes (or contexts) are shown in Table 1. The beer images and more details regarding these data are given in [START_REF] Giacalone | Situational appropriateness of beer is influenced by product familiarity[END_REF].

Attribute Abbreviation Attribute Abbreviation

As a gift for someone Table 1: CATA attributes and product names (beer brands) in the beer dataset.

Respondents' agreement

As indicated above, the data from respondent 1 1 = 1, … , 76 were converted into a binary matrix % . These matrices are scaled by dividing them by their respective norm.

From the matrix, (, containing the pairwise similarity coefficients, we computed the largest eigenvalue, which, once divided by = 76 yields a homogeneity index I=46.8%. This highlights a rather fair agreement among the respondents. By performing the permutation test, this value turned out to be significant (p-value< 0.001). Figure 1 shows the agreement coefficient, . , associated to the individual respondents. As indicated above, these weights are the components of the eigenvector of matrix ( associated with the largest eigenvalue.

We can see that some subjects (e.g., subjects 46 and 50) have very small weights indicating a very poor agreement with the panel. Following the strategy of analysis called CATATIS (Llobell et al., 2019), the coefficients

. are used to compute a weighted average F = ∑ . % , )*

, which stands as a compromise configuration. Thereafter, F is submitted to Correspondence Analysis (CA; [START_REF] Greenacre | Correspondence analysis in practice[END_REF].

Figure 2 gives the biplot depicting the beers and the attributes. The first axis, which explains up to 85% of the total variation, shows an opposition of contexts that call for a celebration (e.g., fine dining, achievement, treat) to casual contexts (e.g., fishing, rugby). Accordingly, the beers are positioned along the first axis depending on their association with the contexts reflected by the first axis. All in all, the findings from this analysis agree to a large extent to those obtained by [START_REF] Giacalone | Situational appropriateness of beer is influenced by product familiarity[END_REF] and we refer to this paper for a more comprehensive discussion of these findings.

Figure 2: CA on the weighted average of the beer data

Attributes consistency

Each attribute was considered in turn and submitted to a permutation test to assess whether the respondents were (in)consistent in their evaluation. All the permutation tests were significant, implying that for none of the attributes, the panel of respondents were completely inconsistent. In parallel, we performed Cochran's test on the data associated to each attribute to assess whether these attributes were discriminant. All the attributes turned out to be discriminant.

CLUSCATA

The hierarchical algorithm of CLUSCATA suggests to consider four clusters (Figure 3).

Indeed, we can observe a significant jump mnopq of the criterion 4 (Eq. ( 1) ) when passing from a partition in four clusters to a partition with three clusters. This suggests that, at this step, heterogeneous clusters are being merged. By way of comparing methods, we performed, in a first stage, the CLUSCATA partitioning algorithm without the option commending the detection of atypical subjects.

Then, in a second stage, we run CLUSCATA including the "K+1" strategy. We computed the threshold parameter [ from equation ( 4) and found 0.68. This resulted in 18 consumers being set in the noise cluster (Table 2) with an almost even repartition in the four clusters.

By removing these respondents, the overall homogeneity increased from 54.0% to 61.7%. Figure 4 shows the graphical representation of the compromise of cluster 1 without identification of atypical subjects (left) and after the removal of atypical subjects (right). We can see that the removal of the atypical subjects resulted in some changes in the positions of the beers and the attributes. For instance, the beer LR appears as extreme along the second axis in the configuration on the right figure (i.e., without the atypical respondents). The association of this beer with the attribute "Rugby" is more emphasized than in the case when the atypical subjects were not removed.

Figure 4: The beers data: Representation of the products on the first two components of Cluster 1 with the atypical respondents included (left) and without these respondents (right).

Other case studies

By way of illustrating the strategy of analysis on a broader basis, we considered seven additional case studies from previously published CATA experiments to which we applied the general approach discussed in this paper. The objective is also to give orders of magnitude regarding the homogeneity indices, the number of attributes for which the respondents were not consistent, the gain in terms of homogeneity when we perform CLUSCATA, the number of respondents who are assigned to the "K+1" cluster, etc. The characteristics of the case studies and the bibliographical references are shown in Table 3. We can see that the number of products varies from 4 to 12 and the number of respondents ranges between 73 and 154. In Table 4, we show the main results regarding the various steps of the strategy of analysis.

We can see that the homogeneity indices ranges between 24.8% for case study D1 to 70.7% for case study D4. However, if we exclude these two extreme case studies, the homogeneity of the other case studies is average to poor. The permutation tests applied to each case study in turn to assess the significance of the homogeneity or agreement indices lead to the conclusion that the null hypothesis stipulating that the respondents are in complete disagreement should be rejected.

As expected, the overall homogeneity increased after performing CLUSCATA and this increase was even more noticeable after the removal of those respondents who are deemed to be atypical. On average, one third of the respondents were set in the noise cluster.

The permutation tests regarding the consistency of the respondents for each attribute taken separately highlighted that in case studies D3, D4, D5 and D7, very few attributes revealed a consistency problem. However, more than 10 attributes were involved in a consistency problem for the remaining case studies (up to 40% of the evaluated attributes). For these case studies, the original number of attributes was relatively large and the agreement among the respondents as assessed by the homogeneity indices was poor. In parallel to the permutation test applied to the various attributes, we also run Cochran's test to assess their discrimination ability. The conclusions drawn from both tests (i.e., the Cochran's and permutation tests) were the same for almost all 151 attributes across all the case studies. There were three exceptions: one attribute was assessed as non-discriminant but consistent, and two attributes were assessed as inconsistent but discriminant. These marginal differences can be explained by the fact the two hypothesis tests do not have exactly the same aim or can be attributed to the chance of making errors associated with a hypothesis test.

Case

Conclusion

A strategy of analysis for the statistical treatment of CATA data was proposed. This strategy encompasses the evaluation of the agreement among the respondents at a global level and for each attribute considered separately, the clustering of the respondents while setting aside atypical respondents. The relevance of this analytical strategy was demonstrated on the basis of several case studies. The outputs from these investigations showed cases with poor agreements among the respondents and cases with a relatively high degree of agreement. In all the cases, the clustering of the respondents yields an improvement of the overall agreement, which was even more noticeable when a noise cluster was added to the main clusters. The introduction of a noise cluster entails the selection of a noise parameter, [. We have proposed an analytical expression for this parameter which, on the basis of the presented case studies, seems appropriate. It is clear that this parameter depends on the number of clusters, on the structure of the data (e.g., within and between variation associated with the clusters), on the number of datasets in each cluster, etc. Further developments regarding these aspects are needed.

This analytical strategy highlights an important issue related to the CATA experiment, that is, the number of attributes that are checked by each respondent for the various products. We proposed a standardization which consists in dividing each binary dataset by the square root of the total number of checked attributes for all the products.

Notwithstanding, it still remains that those respondents who have a tendency to check more attributes will have a higher contribution to the analyses than those who check attributes more parsimoniously. A way to counteract this problem may be to instruct the respondents to check for each product the most salient attributes up to a pre-specified number of attributes (see e.g. [START_REF] Campo | Aroma properties of young Spanish monovarietal white wines: a study using sorting task, list of terms and frequency of citation[END_REF]).

Yet another issue concerns the number of attributes which can cause fatigue to the respondents when it is large. This problem is even more acute when the number of products to be assessed is itself large. Indeed, we can remark from the case studies discussed above that as a general rule the agreement among the respondents is rather poor when the number of attributes is high.

Some cautions regarding the number of respondents involved in a CATA experiment should be taken when segmenting the subjects. For instance, take the scenario of the first case study where 76 respondents were involved in a CATA experiment. The segmentation of these respondents led to four clusters. It would be unreasonable to draw strong conclusions regarding the clusters with relatively small sample sizes. The findings from the cluster analysis should be taken for what they are worth that is, indications about the presence of segments and some sources of variation between these segments. In any case, further actions such as increasing the number of respondents are needed.

In the future, more investigations will be dedicated to the tricky problem of determining the appropriate number of clusters when performing CLUSCATA. So far, the decision has been made on the basis of a visual examination of the dendrogram but a more formal choice based, for instance, on a hypothesis testing framework would be more indicated.

The analyses discussed in this paper with the exception of the permutation tests are available in the R package ClustBlock (Llobell, Vigneau, Cariou, Qannari, 2019).

  || % || = 1 (standardization), it follows that ∑ || % , )* || = . This quantity stands as the total variation in the original datasets. The quantity ∑ ||² stands as the within clusters variation and ∑ LF & L J $)* stands as the between clusters variation. It follows that the ratio

  abbreviations between brackets: Steilager Classic (SC), Gold Medal Ale (GMA), Lion Red (LR), Mac's Hop Rocker (MHR), Monteith's Black Beer (MBB), Stoke Gold (STG), Hopwired Ipa (HPW), Stonecutter Scotch Ale (SSA) and Pot Kettle Black (PKB).

Figure 1 :

 1 Figure 1: Weight associated with each respondent.

Figure 3 :

 3 Figure 3: Dendrogram given by the CLUSCATA hierarchical clustering analysis and variation of the criterion 4.

Table 2 :

 2 Size, homogeneity and overall homogeneity indices using CLUSCATA without and with the "K+1" strategy.

Table 3 :

 3 CATA case studies

	Overall	homogeneity with	"K+1" cluster		30.5%	48.2%	60.1%	79.9%	62.8%	40.7%	44.1%
	N of respondents	in "K+1" cluster (%)			53 (34.4%)	20 (27.4%)	33 (34.0%)	22 (29.9%)	39 (34.8%)	22 (26.5%)	43 (32.1%)
	Threshold	s parameter			0.45	0.61	0.67	0.83	0.66	0.55	0.55
	Overall	homogeneity	without "K+1"	cluster	24.8%	42.7%	51.8%	74.7%	51.1%	35.8%	37.3%
	N of	clusters			3	3	4	3	2	3	3
	N of	inconsistent	attributes	(%)	10 (37%)	11 (28.9%)	0 (0%)	1 (10%)	1 (5.9%)	12 (40%)	3 (21.4%)
	Homogeneity	index			21.3%	38.2%	45.5%	70.7%	48.5%	31.8%	31.7%
	Case	study			D1	D2	D3	D4	D5	D6	D7

Table 4 :

 4 Main results of the various strategies of analysis

	IV.


Joy Khayat

Stéphane Champely

Ahmad Diab

Ahmad Rifai Sarraj

Patrick Fargier

Effect of mental calculus on the performance of complex movements

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Several everyday activities require the mental computation of arithmetical calculations. For example, a subtraction may be useful for determining the exact amount of money remaining after a purchase or the remaining time before the beginning of a scheduled activity. Mental calculation is included in school curricula and as a consequence it is also a topic of interest for educational research (e.g. [START_REF] Fuson | More complexities in subtraction[END_REF]. Several authors have also aimed at understanding the neurofunctional mechanisms of mental arithmetic (e.g. [START_REF] Arsalidou | Is 2 + 2 = 4? Meta-analyses of brain areas needed for numbers and calculations[END_REF][START_REF] Arsalidou | Brain areas associated with numbers and calculations in children: Meta-analyses of fMRI studies[END_REF] and the neural bases of dyscalculia (e.g. [START_REF] Butterworth | Foundational numerical capacities and the origins of dyscalculia[END_REF][START_REF] Butterworth | Dyscalculia: From brain to education[END_REF]. These studies have suggested the possible involvement of the same cerebral mechanisms during both mental calculation and movement preparation. A possibility that is consistent with the hypothesis that any conceptual representation is grounded to sensorimotor experience [START_REF] Barsalou | Grounded cognition[END_REF][START_REF] Barsalou | Grounded cognition: Past, present, and future[END_REF][START_REF] Wiemers | Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect[END_REF]. This might explain the use of fingers to learn simple arithmetic operations, as it has been observed among children of different cultures (e.g. [START_REF] Previtali | Nature or nurture in finger counting: A review on the determinants of the direction of number-finger mapping[END_REF]. This might also explain the positive effect of training with an abacus on the performance of mental calculation [START_REF] Hanakawa | The role of rostral Brodmann area 6 in mental-operation tasks: An integrative neuroimaging approach[END_REF]. These mechanisms learned from infancy may remain influential during calculation in adults (e.g. [START_REF] Moeller | Effects of finger counting on numerical development -the opposing views of neurocognition and mathematics education[END_REF]. This is supported by behavioral studies in the case of fingers where it was shown that finger gnosia (the ability to represent, differentiate and name the fingers, e.g. [START_REF] Ardila | Angular gyrus syndrome revisited: Acalculia, finger agnosia, right-left disorientation and semantic aphasia[END_REF] may predict the performance of arithmetical tasks in both children and adults (e.g. [START_REF] Newman | Does finger sense predict addition performance?[END_REF][START_REF] Noël | Finger gnosia: A predictor of numerical abilities in children?[END_REF][START_REF] Penner-Wilger | The foundations of numeracy: Subitizing, finger gnosia, and fine motor ability[END_REF][START_REF] Penner-Wilger | Subitizing and finger gnosis predict calculation fluency in adults[END_REF]. This is consistent with neuroimaging studies that showed, in the brain cortex, a significant functional activation of finger somatosensory areas during both mental subtractions and multiplications performed without explicit movement by children [START_REF] Berteletti | Perceiving fingers in single-digit arithmetic problems[END_REF] and adults (Andres, Michaux, & Pesenti, 2012). [START_REF] Berteletti | Perceiving fingers in single-digit arithmetic problems[END_REF] also found significant functional activations of finger motor areas in children during mental subtraction. Similarly [START_REF] Hanakawa | The role of rostral Brodmann area 6 in mental-operation tasks: An integrative neuroimaging approach[END_REF] found a significant activation of the dorsolateral Premotor Cortex in adults performing arithmetical additions (usually overlapping the Primary Motor area active during actual finger movements).

Spatial representations involved during mental calculation and during control of movement direction might also require similar cerebral attentional mechanisms. During mental calculation such representation of number space may lead to right-or upward and left-or downward attentional shifts (e.g. [START_REF] Knops | Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect[END_REF]. This might interfere with the attentional control of movement direction, as it is supported by the study by [START_REF] Wiemers | Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect[END_REF] showing the effect of the direction of arm motion on the performance of mental calculation. These authors found that the performance of subtraction tasks was impaired by right-and upward arm movements, while mental additions were impaired by left-and downward movements. Such effects were confirmed with up-and downward passive motions of the whole body [START_REF] Lugli | Counting is easier while experiencing a congruent motion[END_REF] and with right-and leftward walk [START_REF] Anelli | Walking boosts your performance in making additions and subtractions[END_REF], again during mental additions and subtractions.

Overall while both neuroimaging and behavioral studies have suggested a possible link between mental calculation and movement these studies did not consider the case of highintensity movements and the influence of mental calculation upon the performance of a subsequent movement remained to be examined. To our knowledge only [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF] have produced results regarding this possible influence in the case of a complex movement (see [START_REF] Wulf | Principles derived from the study of simple skills do not generalize to complex skill learning[END_REF] of squat vertical jump (SVJ) with maximal intensity. [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF] examined the possible effect of action verbs upon jump height in adult subjects and the results showed for example that SVJ height was higher after reading the verb « jump » than after watching a black screen (baseline).

Different words were used as controls and mental subtraction as a control of attentional task.

It was interestingly found that mental subtraction increased SVJ height almost as much as the verb "jump", a result that required both confirmation and further examination.

The present study thus aimed at examining the respective influence, upon SVJ performance, of number processing (e.g. number viewing) and the application of operation rules [START_REF] Arsalidou | Brain areas associated with numbers and calculations in children: Meta-analyses of fMRI studies[END_REF] as both are involved during mental subtraction. The present study also examined whether the effect of mental subtraction on SVJ performance is affected by the numerical format (i.e. numbers written in Arabic digits or numbers written as words) as [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF] considered the case of Arabic digits only. The possible influence of the numerical format was tested in the present study as studies on numerical cognition (e.g. [START_REF] Dehaene | Varieties of numerical abilities[END_REF] have suggested that cerebral mechanisms involved in number processing and/or calculation may partly differ depending on the used numerical format.

In addition the investigation was extended to a simpler movement than SVJ, i.e. an upper limb motion (manual-pointing task, MPT), in accordance with previous studies suggesting a link between cognitive stimuli and motor performance (e.g. [START_REF] Boulenger | Cross-talk between language processes and overt motor behavior in the first 200 msec of processing[END_REF][START_REF] Boulenger | Subliminal display of action words interferes with motor planning: A combined EEG and kinematic study[END_REF]Nazir et al., 2008).

Materials and Methods

The study was approved by the ethical committee of the AZM Center for Research in Biotechnology and its Application (Tripoli, Lebanon). The total number of subjects was of 161 undergraduate male students, from the Faculty of Public Health of the Lebanese University (Beyrouth, Lebanon). They further gave their written and informed consent. All subjects were healthy, had normal or corrected-to-normal vision and were English-speaking.

Experimental design

The possible effect of mental subtraction upon motor performance was studied in two separate series of experiments respectively concerning the performance of SVJ and MPT (named Series SVJ and Series MPT). Before each experiment the subjects were informed of the scope of the study and of the experimental tasks, but they were left unaware of possible effects of any stimulus used in the experimental protocol. In each experiment the subjects realized some warm-up trials and particularly of the movement performed during the experiment to obtain a correct performance of this movement. All of them were systematically asked to achieve the highest possible SVJ and to realize the MPT reaching movement as fast as possible.

After warm-up each subject performed three series of either six SVJs (Series SVJ), or six MPTs (Series MPT), each series of six being thereafter called a block (Fig. 1). Between two consecutive blocks there was a 3 mn rest. In each block the first three movements were performed without any previous cognitive task (subjects in front of a black screen during 10 sec). The measured performance of the movements constituted a baseline. The three ensuing movements were performed after a cognitive task, i.e. each of them was performed after a mental subtraction or after reading loudly a number (during 10 sec) or an action verb (specific to the movement realized in the experiment, i.e. "jump" in Series SVJ and "reach" in Series MPT; performed during 10 sec). The latter was taken as control (see [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF]. The blocks and the numerical stimuli inside a block were randomly assigned.

Insert Fig. 1. Here

Series SVJ

The 101 subjects participating to this series of experiments were randomly distributed into three groups of 21, 40 and 40, forming respectively a control group and experimental groups 1 and 2. The mean age (years ± SD) of the participants was respectively of 20.1 ± 1.7, 20.1 ± 1.5 and 20.3 ± 1.5 and the mean body-mass index (BMI ± SD in kg/m 2 ) was respectively 24 ± 1.0, 24 ± 1.2 and 23.5 ± 1.0 (considered to be the normal BMI; e.g. [START_REF] Garrouste-Orgeas | Body mass index. An additional prognostic factor in ICU patients[END_REF][START_REF] Zhao | Is body mass index associated with outcomes of mechanically ventilated adult patients in intensive critical units? A systematic review and meta-analysis[END_REF].

Experimental procedure

The control experiment was done to check a possible effect of movement repetition on SVJ performance. This was performed according to the experimental protocol described in Fig. 1 ( § 2.1.) with the exception that no cognitive stimulus was given (subjects simply stood in front of a black screen). Experiments 1 and 2 were done also as described in Fig. 1 ( § 2.1.). In experiment 1, each number was written as word. The numbers to be read and the subtractions were respectively: (a) "four", "seven" and "seventeen" and (b) "seventeen -eight", "sixteennine" and "twelve -four" (only number notation differed between experiments 1 and 2 to verify the possible influence of the numerical format on motor performance). In experiment 2, the same numbers and mental subtractions as in experiment 1 were written in Arabic digits.

Each stimulus was written in white (Times New Roman, font size 96) and it was projected on a black wall. The image was 1.3 m in diagonal and the subjects stood at 3.5 m from the wall, in accordance with the specifications of the manufacturer (Optoma © /ThemeSceneH projector, 92100 Boulogne-Billancourt, France).

Squat vertical jump

The jumps realized in Series SVJ were performed as described and analyzed by [START_REF] Fargier | Fast regulation of vertical squat jump during push-off in skilled jumpers[END_REF]. Jump height was determined by measuring the time of flight by an Optojump Next ® apparatus (Microgate France, 38330 St-Ismier) connected to a laptop.

Series MPT

The participants (a total number of 60 other subjects participated to these experiments of the MPT Series) were randomly distributed in three groups of 20 subjects, respectively forming a control group and experimental groups 3 and 4 similarly to Series SVJ.

The mean age (years ± SD) of the participants to the control experiment and to the experiments 3 and 4 was respectively 20.1 ± 2.0, 20.8 ± 1.8 and 20.2 ± 1.8. The subjects were right handed as it was assessed by using the Edinburgh Inventory [START_REF] Oldfield | The assessment and analysis of handedness: The Edinburgh inventory[END_REF]. The mean scores obtained with the Edinburgh Inventory (number of points ± SD) of the participants to the control experiment and to experiments 3 and 4 were respectively of 78.1 ± 10.0, 78.2 ± 9.8 and 72.9 ± 9.6. All subjects declared a preference for the right handiness and performed MPT accordingly. The MPT measurement system was home-made and will be described in the following.

Manual-pointing task

Each subject sat on a chair behind a desk. In front of the subject two wooden parallelepipeds (5 cm in height each) were fixed one following the other upon the desk top board. The first parallelepiped was at a distance of 25 cm from the chest of the subject. At the center of the superior face of each parallelepiped a circular button was fixed. The distance between the centers of the two buttons was of 25 cm. During each cognitive task (Fig. 1,§ 2.1.) the subjects were asked to touch the button fixed on the first parallelepiped with the index finger, the middle finger and the ring finger of the right hand. At the end of each cognitive task the subjects were required to press the button of the second parallelepiped as fast as possible.

Performance measurement

The beginning of the movement was determined by a three-axis accelerometer (Vernier ® ; 13979 SW Millikan Way, Beaverton, OR 97005, USA) fixed on a mitten that each subject wore on the right hand. The Response Time (RT) was determined when the button on the distal parallelepiped was pressed. The acquisition and the analysis of the measures were made with a data acquisition card (DAQ ® ; data acquisition, National Instruments, 11500 Mopac Expwy, Austin, Texas) connected to Labview ® software [Laboratory Virtual Instrument Engineering Workbench 2009 (32-bit), National Instruments, Austin, Texas] and Matlab ® script (Matlab R2016a, MathWorks incorporation, Natick, Massachusetts).

Statistical analysis

The data collected from each experiment were analyzed by using a multilevel linear mixedeffect model [START_REF] Pinheiro | Mixed-effects models in S and S-PLUS[END_REF][START_REF] Finch | Multilevel Modeling Using R[END_REF]. The response variables in Series SVJ and MPT were respectively the height of jump (cm) and the RT (ms).

In each case three levels of variability were considered and represented by three nested random effects in the statistical model. The effect at the first level was the subject effect. The effect at the second level considered the performance of each series of three consecutive movements in the same experimental condition (see Fig. 1,§ 2.1.). At the third level the effect was a residual effect, considering the individual performance in the same experimental condition. All nested random effects followed non-correlated normal distributions with zero expectation.

The data obtained from the control experiment of each Series were analyzed by considering two fixed-effect and the corresponding interaction. A block fixed-effect was examined to control a possible effect of block repetition (i.e. effect of fatigue or of training) during the experiments. An intra-block fixed-effect was also examined to allow a comparison of the two successive series of three movements (black wall) in the same block. Finally the interaction effect between block and intra-block was considered.

The absence of statistically significant intra-block and interaction effects would show the stability of the baseline. A statistically significant block effect would lead to check such possible effect during experiments 1 to 4. In any case the statistical analysis of the data of each of the experiments 1 to 4 aimed at examining a fixed-effect regarding the experimental conditions (see Fig. 1,§ 2.1.), i.e. black screen (baseline), action verb reading, number reading and mental subtraction.

All multilevel linear mixed-effect models were fitted by using restricted maximum likelihood estimation (Pinheiro & Bates, 2000, p. 75). Fixed-effects were tested by using conditional Ftests (Pinheiro & Bates, 2000, p. 90) at the 5% level. The method of [START_REF] Bretz | Multiple Comparisons Using R[END_REF] was used to carry out post-hoc tests (single-step method; family-wise error rate set at 5%). All computations were performed by using the R statistical software 3.4.3 (R Core Team, 2017) and the packages nlme, lme4, multcomp and effects.

Results

Series SVJ

In the SVJ control experiment the test of the block fixed-effect reached statistical significance (F(2,100) = 3.61, p < .05) suggesting that the repetition of three blocks of SVJs (see Fig. 1,§ 2.1.) may influence jump height. From the first block to the third one SVJ heights in cm (mean ± SD) were respectively: 28.97 ± 2.04, 29.42 ± 1.85 and 29.31 ± 1.97. All-pairwise comparisons showed that only the difference between the first and the second blocks was significant. Although this difference was only 0.45 cm, its statistical significance imposed to control a possible effect of blocks repetition in experiments 1 and 2. No intra-block fixedeffect (controlling a possible order effect of the two series of three jumps in the same block) and no interaction between block and intra-block were found with respectively: F(1,100) = 2.06, p > .05 and F(2,100) = 1.00, p > .05. As a consequence the mean baseline of each block of both experiments 1 and 2 was used as reference.

In both experiments 1 (numbers written as words) and 2 (numbers written in Arabic digits) no influence of block repetition (block fixed-effect) was found with respectively: F( 2 .001 (see Fig. 2).

Insert Fig. 2 here

Regarding experiment 1 (numbers written as words; Fig. 2. a.) all-pairwise comparisons (Table 1. a.) showed that SVJ height was significantly higher (1.57 cm) after reading "jump" than in the baseline condition, p < .001. SVJ height was also significantly higher after reading a number written as word than in the baseline condition, p < .05. This difference was of 0.53 cm only and SVJ height after reading "jump" was significantly higher than SVJ height after numbers read as words, p < .001. In addition SVJ height after mental subtraction of numbers written as words was not significantly different than in the baseline condition, p > .05.

Insert Table 1 here

In experiment 2 (numbers written in Arabic digits; Fig. 2. b.) all-pairwise comparisons (Table 1. b.) showed again that SVJ height was higher after reading "jump" than in the baseline condition, p < .001. In contrast with experiment 1 (numbers written as words), in this experiment 2 it was found that SVJ height after reading a number written in Arabic digits was not significantly different than in the baseline condition, p > .05. It was also found that mental subtraction with numbers written in Arabic digits was higher than in the baseline condition, p < .001 and after the reading of a number written in Arabic digits, p < .001.

Series MPT

In the control experiment no statistically significant effect of block repetition (block effect) was found, F(2,59) = 1.19, p > .05. However considering that a significant block effect was found in the control experiment of Series SVJ it was decided to check again such possible effect in the experiments 3 and 4 of Series MPT. Similarly to the control experiment of Series SVJ the control experiment of Series MPT showed no statistically significant intra-block effect and interaction effect between block and intra-block. As a consequence the mean baseline of each block of both experiments 3 and 4 was used as reference.

In both experiments 3 (numbers written as words) and 4 (numbers written in Arabic digits) no influence of block repetition (block fixed-effect) was found with respectively: F(2,59) = 2.56, p > .05 and F(2,59) = 0.41, p > .05, while the condition (cognitive tasks) effect was significant with respectively: F(3,59) = 43.34, p < .001 and F(3,59) = 51.24, p < .001 (see Fig. 3).

Insert Fig. 3 here

In experiment 3 (numbers written as words; Fig. 3. a.) all-pairwise comparisons (Table 2. a.)

showed only the influence of reading the word "reach" upon MPT performance (RT) when compared to the baseline, p < .001. In addition the RT observed after reading "reach" was significantly faster than the RT measured after reading a number written as word and after mental subtraction with numbers written as words (Table 2.a).

Insert Table 2. here

In experiment 4 (numbers written in Arabic digits; Fig. 3. a.) all-pairwise comparisons (Table 2. b.) showed again that MPT RT was faster after reading "reach" than in the baseline condition, p < .001. Interestingly RT after reading a number written in Arabic digits was not significantly different than in the baseline while RT after mental subtraction with numbers written in Arabic digits was significantly faster than the baseline RT, p < .001. In addition the RTs after reading "reach" and after mental subtraction with numbers written in Arabic digits were significantly faster than the baseline RT and the RT after reading a number written in Arabic digits (Table 2. b.).

Discussion

The aim of the present study was to examine the possible effect of mental subtraction on the performance of a complex movement of squat vertical jump with maximal intensity (SVJ) and of a manual-pointing task with maximal velocity (MPT). Two series of experiments, respectively including SVJ (Series SVJ) and MPT (Series MPT), examined this possible effect when numbers were written as words or in Arabic digits. Each experiment allowed comparisons among motor performance after viewing a black screen (baseline), reading a specific action verb (control), reading a number and after mental subtraction.

In each experiment of Series SVJ (experiments 1 and 2) the subjects performed higher SVJs after reading the word "jump" than in the baseline condition (see Fig. 2 and Table 1, § 3.1.) as it has been previously shown [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF]. In each experiment of Series MPT (experiments 3 and 4) the subjects also performed faster RTs after reading "reach" than in the baseline condition (see Fig. 3 and Table 2,§ 3.2.). The present results thus confirmed the influence of reading a specific action verb on motor performance.

A statistically significant influence of number reading upon motor performance was in addition found in experiment 1 but not in experiments 2, 3 and 4. This influence, found in the case of SVJ and when numbers were written as words, was rather small as the performance of SVJ after number reading was only 0.53 cm higher than the baseline, p < .05 (see Table 1. a., § 3.1.). Especially so when considering that SVJ performance after reading the word "jump" was: 1.57 cm higher than the baseline, p < .001, and 1.04 cm higher than after reading a number written as word, p < .001 (see Table 1. a., § 3.1.). The present study thus shows that simple number reading may have either no effect, or a weak effect, on motor performance. It should be mentioned that previous results have shown indirectly that a link might exist between number processing and lateral and/or vertical movement, but a real motor performance was not actually measured (e.g. [START_REF] Loetscher | Head turns bias the brain's internal random generator[END_REF][START_REF] Winter | More is up... and right: Random number generation along two axes[END_REF].

The main finding of the present study was that mental subtraction with numbers written in Arabic digits influenced both SVJ and MPT performance while mental subtraction with numbers written as words did not (see Table 1, § 3.1., and Table 2, § 3.2.). After mental subtraction with Arabic digits the subjects performed SVJ 1.06 cm higher than the baseline, p < .001, and MPT RTs 30.26 ms faster than the baseline, p < .001.

The positive effect of mental subtraction with numbers written in Arabic digits upon SVJ performance confirmed the findings of [START_REF] Rabahi | Effect of action verbs on the performance of a complex movement[END_REF] and showed that such an effect on motor performance was not limited to a complex movement of SVJ as it also influenced the simpler MPT performance. This effect might be linked to the difficulty level of the calculations to be done that were presented in a classical form (e.g. 12 -4) and were of medium difficulty level according to [START_REF] Thevenot | Mental subtraction in high-and lower skilled arithmetic problem solvers: Verbal report versus operand-recognition paradigms[END_REF]. The calculations may thus have fostered a feeling of accomplishment in the experimental undergraduate subjects possibly favoring their attentional control [START_REF] Derakshan | Anxiety, processing efficiency, and cognitive performance: New developments from attentional control theory[END_REF]. On the other hand mental subtractions with numbers written as words were certainly unusual to the subjects increasing thus a state of anxiety that might hamper the stimulation effect of the subtractions on performance.

In the case of the more usual operation with Arabic digits the mental subtraction might instead have directed the subjects' attention on the optimal path of movement in SVJ and MPT as the operation may involve a spatial representation of numbers oriented vertically (as in SVJ; e.g. [START_REF] Wiemers | Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect[END_REF] or horizontally (as in MPT; e.g. [START_REF] Anelli | Walking boosts your performance in making additions and subtractions[END_REF].

However previous studies have suggested that the performance of a mental subtraction may be either impaired by upward movements [START_REF] Lugli | Counting is easier while experiencing a congruent motion[END_REF][START_REF] Wiemers | Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect[END_REF] or forward ones [START_REF] Anelli | Walking boosts your performance in making additions and subtractions[END_REF], but only when they were performed concomitantly to the calculation. In the present study the subjects were initially aware of the movement that they would have to perform. They were also required to execute it with maximal intensity after mental calculation [contrary to the studies of [START_REF] Anelli | Walking boosts your performance in making additions and subtractions[END_REF], [START_REF] Lugli | Counting is easier while experiencing a congruent motion[END_REF], and [START_REF] Wiemers | Spatial interferences in mental arithmetic: Evidence from the motion-arithmetic compatibility effect[END_REF]]. This might have favored, during calculation, a mental representation of numbers (see § 1.) in a spatial a plane congruent with the plane in which the subsequent movement would be optimally executed (i.e. vertical plane in SVJ and horizontal plane in MPT).

Concerning the presentation of the operation with numbers written as Arabic digits or as words McCloskey and colleagues (e.g. McCloskey, 1992) have proposed a model in which numbers, regardless the numerical modality used to present them, are encoded into a unique abstract format to allow the mental calculation. Conversely Campbell and colleagues (e.g. [START_REF] Clark | Integrated versus modular theories of number skills and acalculia[END_REF] have proposed a model composed of a network of mechanisms specific to different modalities of number presentation each of them supporting number comprehension and calculation.

The results observed in the present study in this respect with Arabic and written numbers might only be explained, if interpreted with the McCloskey's model, by the influence of distinct encoding mechanisms of Arabic and of written numbers before calculation (see also [START_REF] Dehaene | Varieties of numerical abilities[END_REF]. Interpreted with the Campbell's model instead the present results might also be explained by the involvement of a memory specific for Arabic digits during calculation (see also [START_REF] Myers | Arithmetic memory is modality specific[END_REF], differently located from the language areas of the brain.

In conclusion the present study showed that mental subtraction, rather than number reading, may influence SVJ and MPT performance when numbers are written in Arabic digits (and not when numbers are written as words). Mental subtractions of moderate difficulty presented in a usual format (i.e. Arabic digits) possibly led subjects to an emotional state favoring attention to elements relevant to perform SVJ and MPT. Among such elements attention to the optimal path of movement might have been favored by the spatial representation of numbers used to calculate. The influence of mental subtraction with Arabic digits on motor performance might also be linked to mechanisms of encoding and/or memorization specific to this numerical format. The results of the present study are in any case in accordance with neuroimaging studies showing an increased functional activation of the Premotor Cortex during calculation with Arabic digits (e.g. [START_REF] Hanakawa | The role of rostral Brodmann area 6 in mental-operation tasks: An integrative neuroimaging approach[END_REF]. Whether attention, numbers encoding and/or memorization might contribute to such activation remains an opened question. Before the start of the experiment the subjects were requested to realize a warm-up during ∼10 mn.

Figure captions

They further performed three series of six movements (M = SVJ in Series SVJ and MPT in Series MPT), i.e. three blocks of movements separated by 3 mn rest. Each block began by the realization of three movements (M1 to M3) without any previous cognitive task (subjects in front of a black screen during 10 sec). The performance of these movements constituted a baseline. Each of the ensuing three movements was realized after a cognitive task, i.e. for each of M4 to M6: either after a mental subtraction, or after reading loudly a number during 10 sec, or after reading loudly an action verb during 10 sec ("jump" in Series SVJ, "reach" in Series MPT).

Fig. 2. Effect of cognitive tasks on SVJ height

The graphical display, called "effect plot" by [START_REF] Fox | Effect displays in R for multinomial and proportional-odds logit models: Extensions to the effects package[END_REF], shows the adjusted mean heights of jump in SVJ after: watching a black screen (Base: baseline performance), reading the verb "jump" (Jump), reading a number (N) and mental subtraction (S). In experiments 1 (Fig. The results of the multiple comparisons of MPT response times (RTs) among Base (baseline performance), Reach (RT after reading the word "reach"), N (RT after reading a number) and S (RT after mental subtraction) are shown. The multiple comparisons were carried out using the method of [START_REF] Bretz | Multiple Comparisons Using R[END_REF] and the family-wise error rate was set at 5%. Regarding Estim. (Estimate, differences in ms) differences between two SE (standard error) may occur after the second decimal. Regarding z-scores the adjusted p-values [single-step method; p (> |z|)] of the statistically significant differences are indicated by *** = p < .001.

  ,195) = 0.29, p > .05 and F(2,195) = 0.65, p > .05. The condition (cognitive tasks) effect reached statistical significance with respectively: F(3,195) = 24.51, p < .001 and F(3,195) = 23.22, p <

Fig. 1 .

 1 Fig. 1. Experimental protocol.

  2. a) and 2 (Fig. 2. b) the numbers were respectively written as words and in Arabic digits. The vertical bars indicate ± 0.95 confidence interval.

Fig. 3 .

 3 Fig. 3. Effect of cognitive tasks on MPT response timeThe graphical display, called "effect plot" by[START_REF] Fox | Effect displays in R for multinomial and proportional-odds logit models: Extensions to the effects package[END_REF], shows the mean response time (RT) in MPT after: watching a black screen (Base, baseline performance), reading the verb "reach" (Reach), reading a number (N) and mental subtraction (S). In experiments 3 (Fig.3. a.) and 4 (Fig. 3. b.) the numbers were respectively written as words and in Arabic digits. The vertical bars indicate ± 0.95 confidence interval.

Table 2 -All-pairwise comparisons in experiments 3 and 4 a. Experiment 3 (numbers as words) b. Experiment 4 (Arabic numerals)

 2

	Differences	Estim. (ms)	SE	z-score	Estim. (ms)	SE	z-score
	Reach -Base	-30.92	3.06	-10.09 ***	-30.26	3.06	-9.88 ***
	N -Base	1.43	3.06	0.47	5.21	3.06	1.70
	S -Base	6.97	3.08	2.26	-22.91	3.06	-7.49 ***
	N -Reach	32.35	3.75	8.63 ***	35.47	3.76	9.42 ***
	S -Reach	37.89	3.79	10.00 ***	7.35	3.75	1.96
	S -N	5.55	3.79	1.46	-28.12	3.75	-7.50 ***

Acknowledgment

The authors express their grateful thanks to Emeritus Professor Raphaël Massarelli for his key support in this study.