Catuscia Palamidessi

Marco Romanelli

Feature selection with Rényi min-entropy

We consider the problem of feature selection, and we propose a new information-theoretic algorithm for ordering the features according to their relevance for classification. The novelty of our proposal consists in adopting Rényi min-entropy instead of the commonly used Shannon entropy. In particular, we adopt a notion of conditional min-entropy that has been recently proposed in the field of security and privacy, and that avoids the anomalies of previouslyattempted definitions. This notion is strictly related to the Bayes error, which is a promising property for achieving accuracy in the classification. We evaluate our method on 2 classifiers and 3 datasets, and we show that it compares favorably with the corresponding one based on Shannon entropy.

Introduction

The identification of the "best" features for classification is a problem of increasing importance in machine learning. The size of available datasets is becoming larger and larger, both in terms of samples and in terms of features of the samples, and keeping the dimensionality of the data under control is necessary for avoiding an explosion of the training complexity and for the accuracy of the classification. Several authors have considered this problem, including [START_REF] Jain | Statistical pattern recognition: A review[END_REF][START_REF] Guyon | An introduction to variable and feature selection[END_REF][START_REF] Liu | Toward integrating feature selection algorithms for classification and clustering[END_REF][START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF][START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF][START_REF] Bennasar | Feature selection using joint mutual information maximisation[END_REF][START_REF] Sheikhpour | A survey on semisupervised feature selection methods[END_REF][START_REF] Nakariyakul | High-dimensional hybrid feature selection using interaction informationguided search[END_REF][START_REF] Cai | Feature selection in machine learning: A new perspective[END_REF][START_REF] Liu | Online multi-label group feature selection[END_REF].

The known methods for reducing the dimensionality can be divided in two categories: those which transform the feature space by reshaping the original features into new ones (feature extraction), and those which select a subset of the features (feature selection). The second category can in turn be divided in three groups: the wrapper, the embedded, and the filter methods. The last group has the advantage of being classifierindependent, more robust with respect to the risk of overfitting, and more amenable to a principled approach. In particular, several proposals for feature selection have successfully applied concepts and techniques from information theory [START_REF] Battiti | Using mutual information for selecting features in supervised neural net learning[END_REF][START_REF] Yang | Feature selection based on joint mutual information[END_REF][START_REF] Fleuret | Fast binary feature selection with conditional mutual information[END_REF][START_REF] Peng | Feature selection based on mutual information: Criteria of max-dependency, max-relevance, and min-redundancy[END_REF][START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF][START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF][START_REF] Bennasar | Feature selection using joint mutual information maximisation[END_REF]. The idea is that the smaller is the conditional (aka residual) entropy of the classes given a certain set of features, the more likely the classification of a sample is to be correct. Finding a good set of features corresponds therefore to identifying a set of features, as small as possible, for which such conditional entropy is below a certain threshold.

In this paper, we focus on the filter approach and we propose a new informationtheoretical method for feature selection. The novelty of our proposal consists in the use of Rényi min-entropy H ∞ rather than Shannon entropy. As far as we know, all the previous proposals are based on Shannon entropy, with the notable exception of [START_REF] Endo | Weighted Naïve Bayes Classifiers by Renyi Entropy[END_REF] who considered the Rényi entropies. However [START_REF] Endo | Weighted Naïve Bayes Classifiers by Renyi Entropy[END_REF] reported experimental results only on other orders of Rényi entropies. We will discuss more that work in Section 5.

For feature selection we need the conditional version of Rényi min-entropy. Rényi did not define it, but there have been various proposals, in particular by Arimoto [START_REF] Arimoto | Information measures and capacity of order α for discrete memoryless channels[END_REF], Sibson [START_REF] Sibson | Information radius[END_REF], Csiszár [START_REF] Csiszár | Generalized cutoff rates and Rényi's information measures[END_REF], and Cachin [START_REF] Cachin | Entropy Measures and Unconditional Security in Cryptography[END_REF]. In particular, [START_REF] Cachin | Entropy Measures and Unconditional Security in Cryptography[END_REF] defined the conditional minentropy of X given Y along the lines of conditional Shannon entropy, namely as the expected value of the entropy of X for each given value of Y . Such definition, however, violates the monotonicity property. Namely, knowing the value of Y could increase the entropy of X rather than diminishing it.

Recently, some advances in the fields of security and privacy have revived the interest for the Rényi min-entropy. The reason is that it models a basic notion of attacker: the (one-try) eavesdropper. Such attacker tries to infer a secret (e.g., a key, a password, etc.) from the observable behavior of the system, with the limitation that he can try only once. Naturally, a rational attacker will try to minimize the probability of error, so he will pick the secret with the highest probability given what he has observed. Note the similarity with the classification problem, where we choose a class on the basis of the observed features, trying to minimize the probability of mis-classification.

Driven by the motivation of providing an information-theoretic interpretation of the eavesdropper operational behavior, [START_REF] Smith | On the foundations of quantitative information flow[END_REF] proposed a definition of conditional minentropy H ∞ (X|Y ) which is consistent with the rest of the theory, models all the expected properties of an eavesdropper, and corresponds closely to the Bayes risk of guessing the wrong secret. (The formal definition of H ∞ (X|Y ) will be given in Section 2.) The definition proposed by [START_REF] Smith | On the foundations of quantitative information flow[END_REF] can be shown equivalent to the one of Arimoto [START_REF] Arimoto | Information measures and capacity of order α for discrete memoryless channels[END_REF]. It is then natural to investigate whether this new notion can be successfully applied also to the problem of feature selection.

We could state the problem of feature selection as finding a minimum-size subset S of the whole set of features F such that the min-entropy H ∞ (C|S) of the classification C given S is below a given threshold. Because of the correspondence with the Bayes risk, this would mean that the set S is optimal (i.e, minimal) among the subsets for which the Bayes classifier achieves the desired level of accuracy. However, is that the construction of such an optimal S would be NP-hard. This is not due to the kind of entropy that we choose, but simply to the fact that it is a combinatorial problem. In [START_REF] Kohavi | Wrappers for feature subset selection[END_REF] it was shown that the problem of feature selection can be modeled as search problem on a decision tree, and it was argued that finding the optimal subtree which is able to cover F is an NP-hard problem. The same intractability was claimed in [START_REF] Guyon | An introduction to variable and feature selection[END_REF] with respect to wrappers and embedded methods, on the basis of the proof of [START_REF] Amaldi | On the Approximability of Minimizing Nonzero Variables or Unsatisfied Relations in Linear Systems[END_REF].

We then adopt a greedy strategy to approximate the minimal subset of features: following [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF], we construct a sequence of subsets S 0 , S 1 , . . . , S t , . . ., where S 0 = ∅ and at each subsequent step S t+1 is obtained from S t by adding the next feature in order of relevance for the classification, taking into account the ones already selected. In other words, we select the feature f such that H ∞ (C|S t ∪ {f }) is minimal, and we define S t+1 as S t ∪ {f }. The construction of this series should be interleaved with a test on the accuracy of the intended classifier(s): when we obtain an S T that achieves the desired level of accuracy, we can stop. The difference with respect to [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF] is that the relevance is measured by Rényi min-entropy rather than Shannon entropy.

Note that, because of the relation between the conditional min-entropy and the Bayes risk, our method is locally optimal. Namely, for any other possible feature f (including the one that would be selected using Shannon entropy), the set S t+1 is at least as good as S t ∪ {f } in terms of accuracy of the Bayes classifier (the ideal classifier giving the best accuracy). This does not necessarily mean that the set S T is the smallest one: since we are not making an exhaustive search on all possible subsets of F , and we add the features one by one, we may not find the "shortest path" to achieve sufficient accuracy. The same applies to the analogous algorithms based on Shannon entropy. Hence we have no guarantee that our method is better than that of [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF], nor vice versa. In the experiments we have performed, however, our method outperforms almost always the one based on Shannon entropy (cfr. Section 4).

Preliminaries

In this section we briefly review some basic notions from probability and information theory. We refer to [START_REF] Cover | Elements of Information Theory[END_REF] for more details.

Let X, Y be discrete random variables with respectively n and m possible values: X = {x 1 , x 2 , . . . , x n } and Y = {y 1 , y 2 , . . . , y m }. Let p X (•) and p Y (•) indicate the probability distribution associated to X and Y respectively, and let p Y,X (•, •) and p Y |X (•|•) indicate the joint and the conditional probability distributions, respectively. Namely, p Y,X (x, y) represents the probability that X = x and Y = y, while p Y |X (y|x) represents the probability that Y = y given that X = x. For simplicity, when clear from the context, we will omit the subscript, and write for instance p(x) instead of p X (x).

Conditional and joint probabilities are related by the chain rule p(x, y) = p(x) p(y|x), from which (by the commutativity of p(x, y)) we can derive the Bayes theorem:

p(x|y) = p(y|x) p(x) p(y) (1)

Rényi entropies, Shannon entropy, and mutual information

The Rényi entropies ( [START_REF] Rényi | On Measures of Entropy and Information[END_REF]) are a family of functions representing the uncertainty associated to a random variable. Each Rényi entropy is characterized by a non-negative real number α (order), with α = 1, and is defined as

H α (X) def = 1 1 -α log i p(x i ) α
If p(•) is uniform then all the Rényi entropies are equal to log |X|. Otherwise they are weakly decreasing in α. Shannon and min-entropy are particular cases:

α → 1 H 1 (X) = -x p(x) log p(x) Shannon entropy α → ∞ H ∞ (X) = -log max x p(x) min-entropy
Shannon conditional entropy of X given Y represents the average residual entropy of X once the value of Y is known, and it is defined as

H 1 (X|Y ) def = - xy p(x, y) log p(x|y) = H 1 (X, Y ) -H 1 (Y ) (2)
where H 1 (X, Y ) represents the entropy of the intersection of X and Y . Shannon mutual information of X and Y represents the correlation of information between X and Y , and it is defined as

I 1 (X; Y ) def = H 1 (X) -H 1 (X|Y ) = H 1 (X) + H 1 (Y ) -H 1 (X, Y ) (3)
It is possible to show that I 1 (X; Y ) ≥ 0, with I 1 (X; Y ) = 0 iff X and Y are independent, and that I 1 (X; Y ) = I 1 (Y ; X). Rényi did not define conditional entropy and mutual information for generic α. Only recently, thanks to its applications in security and privacy, there has been interest in "completing the theory", and [START_REF] Smith | On the foundations of quantitative information flow[END_REF] proposed the following definition for the conditional min-entropy:

H ∞ (X|Y ) def = -log y max x (p(y|x)p(x)) (4)
It is possible to show that this definition closely corresponds to the Bayes risk, i.e., the expected error when we try to guess the exact value of X, once we know that of Y , formally the Bayes risk of X given Y is defined as:

B(X |Y ) def = 1 - y p(y) max x p(x|y) (5)
The "mutual information" is defined as:

I ∞ (X; Y ) def = H ∞ (X) -H ∞ (X|Y ) (6)
It is possible to show that I ∞ (X; Y ) ≥ 0, and that I ∞ (X; Y ) = 0 if X and Y are independent (the reverse is not necessarily true). It is important to note that, contrary to Shannon mutual information, I ∞ is not symmetric. The conditional mutual information is defined as

I ∞ (X; Y |Z) def = H ∞ (X|Z) -H ∞ (X|Y, Z) (7)
and analogously for Shannon conditional mutual information.

Our proposed algorithm

Let F be the set of features at our disposal, and let C be the set of classes. Our algorithm is based on forward feature selection and dependency maximization: it constructs a monotonically increasing sequence {S t } t≥0 of subsets of F , and, at each step, the subset S t+1 is obtained from S t by adding the next feature in order of importance (i.e., the informative contribution to classification), taking into account the information already provided by S t . The measure of the "order of importance" is based on conditional min-entropy. The construction of the sequence is assumed to be done interactively with a test on the accuracy achieved by the current subset, using one or more classifiers. This test will provide the stopping condition: once we obtain the desired level of accuracy, the algorithm stops and gives as result the current subset S T . Of course, achieving a level of accuracy

1 -ε is only possible if B(C | F ) ≤ ε.
Definition 1. The series {S t } t≥0 and {f t } t≥1 are inductively defined as follows:

S 0 def = ∅ f t+1 def = argmin f ∈F \S t H ∞ (C | f, S t ) S t+1 def = S t ∪ {f t+1 }
The algorithms in [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF] are analogous, except that they use Shannon entropy. They also define f t+1 based on the maximization of mutual information instead of the minimization of conditional entropy, but this is irrelevant. In fact

I ∞ (C; f | S t ) = H ∞ (C | S t ) -H ∞ (C | f, S t ) hence maximizing I ∞ (C; f | S t ) with respect to f is the same as minimizing H ∞ (C | f, S t
) with respect to f . The same holds for Shannon entropy.

Our algorithm is locally optimal, in the sense stated by the following proposition:

Proposition 1. At every step, the set S t+1 minimizes the Bayes risk of the classification among those which are of the form S t ∪ {f }, namely:

∀f ∈ F B(C | S t+1 ) ≤ B(C | S t ∪ {f }) Proof. Let v, v, v represent generic value tuples and values of S t , f and f t+1 , respec- tively. Let c represent the generic value of C. By definition, H ∞ (C | S t+1 ) ≤ H ∞ (C | S t ∪ {f }), for every f ∈ F . From (4) we then obtain v,v max c (p(v, v|c)p(c)) ≤ v,v max c (p(v, v |c)p(c))
Using the Bayes theorem (1), we get

v,v p(v, v) max c p(c|v) ≤ v,v p(v, v ) max c p(v, v |c)
Then, from the definition (5) we deduce

B(C | S t ∪ {f t+1 }) ≤ B(C | S t ∪ {f })
In the following sections we analyze some extended examples to illustrate how the algorithm works, and also compare the resulting sequence with the one produced by the algorithm of [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF]. Class

f 0 f 1 f 2 f 3 f 4 f 5 0 A C F I L O 1 A D F I L O 2 A E G I L O 3 A E H I L O 4 B E F J L O 5 B E F K L O 6 B E F I M O 7 B E F I N O 8 B E F I L P 9 B E F I L Q Fig. 1. The dataset
Let us consider the dataset in Fig. 1, containing ten records labeled each by a different class, and characterized by six features (columns f 1 , . . . , f 5 ). We note that column f 0 separates the classes in two sets of four and six elements respectively, while all the other columns are characterized by having two values, each of which univocally identify one class, while the third value is associated to all the remaining classes. For instance, in column f 1 value A univocally identifies the record of class 0, value B univocally identifies the record of class 1, and all the other records have the same value along that column, i.e. C. The last five features combined completely identify all classes, without the need of the first one. On the other hand, the last five features are also necessary to completely identify all classes, they cannot be replaced by f 0 (i.e., f 0 with whatever combination of other four or less features is not sufficient to completely identify all classes.) In fact, each pair of records which are separated by one of the features f 1 , . . . , f 5 , have the same value in column f 0 .

If we apply the discussed feature selection method and we look for the feature that minimizes H(Class|f i ) for i ∈ {0, . . . , 5} we obtain that:

-The first feature selected by the Shannon entropy is f 0 , in fact H 1 (Class|f 0 ) ≈ 2.35 and H 1 (Class|f =0 ) = 2.4. (The notation f =0 stands for any of the f i 's except f 0 .) In general, indeed, with Shannon entropy the method tends to choose a feature which splits the classes in a way as balanced as possible. The situation after the selection of the feature f 0 is shown in Fig. 2(a). -The first feature selected by the Rényi min entropy is either

f 1 or f 2 or f 3 or f 4 or f 5 , in fact H ∞ (Class|f 0 ) ≈ 2.32 and H ∞ (Class|f =0 ) ≈ 1.74.
In general, indeed, with Rényi min-entropy the method tends to choose a feature which divides the classes in as many sets as possible. The situation after the selection of the feature f 1 is shown in Fig. 2(b).

Going ahead with the algorithm, with Shannon entropy we will select one by one all the other features, and as already discussed we will need all of them to completely identify all classes. Hence at the end the method with Shannon entropy will return all the six features (to achieve perfect classification). On the other hand, with Rényi min entropy we will select all the remaining features except f 0 to obtain the perfect discrimination. In fact, at any stage the selection of f 0 would allow to split the remaining classes in at most two sets, while any other feature not yet considered will split the remaining classes in three sets. As already hinted, with Rényi we choose the feature that allows to split the remaining classes in the highest number of sets, hence we never select f 0 . For instance, if we have already selected f 1 , we have

H ∞ (Class|f 1 f 0 ) ≈ 1.32 while H ∞ (Class|f 1 f =0 ) = 1. If we have already selected f 4 , we have H ∞ (Class|f 4 f 0 ) ≈ 1.32 while H ∞ (Class|f 4 f =0 ) = 1. See Fig. 3.
At the end, the selection of features using Rényi entropy will determine the progressive splitting represented in Fig. 4. The order of selection is not important: this particular example is conceived so that the features f 1 , . . . , f 5 can be selected in any order, the residual entropy is always the same. Discussion It is easy to see that, in this example, the algorithm based on Rényi minentropy gives a better result not only at the end, but also at each step of the process. Namely, at step t (cfr. Definition 1) the set S t of features selected with Rényi minentropy gives a better classification (i.e., more accurate) than the set S t that would be selected using Shannon entropy. More precisely, we have

B(C | S t ) < B(C | S t ).
In fact, as discussed above the set S t contains necessarily the feature f 0 , while S t does not. Let S t-1 be the set of features selected at previous step with Rényi minentropy, and f t the feature selected at step t (namely, S t-1 = S t \ {f t }). As argued above, the order of selection of the features f 1 , . . . , f 5 is irrelevant, hence we have

B(C | S t-1 ) = B(C | S t \ {f 0 }
) and the algorithm could equivalently have selected S t \{f 0 }. As argued above, the next feature to be selected, with Rényi, must be different from f 0 . Hence by Proposition 1, and by the fact that the order of selection of f 1 , . . . , f 5 is irrelevant, we have:

B(C | S t ) = B(C | (S t \ {f 0 }) ∪ {f t }) < B(C | S t ).
As a general observation, we can see that the method with Shannon tends to select the feature that divides the classes in sets (one for each value of the feature) as balanced as possible, while our method tends to select the feature that divides the classes in as many sets as possible, regardless of the sets being balanced or not. In general, both Shannon-based and Rényi-based methods try to minimize the height of the tree representing the process of the splitting of the classes, but the first does it by trying to produce a tree as balanced as possible, while the second try to do it by producing a tree as wide as possible. Which of the method is best, it depends on the correlation of the features. Shannon works better when there are enough uncorrelated (or not much correlated) features, so that the tree can be kept balanced while being constructed. Next section shows an example of such situation. Rényi, on the contrary, is not so sensitive to correlation and can work well also when the features are highly correlated, as it was the case in the example of this section.

The experimental results in Section 4 show that, at least in the cases we have experimented with, our method based on Rényi min-entropy outperforms the one based on Shannon entropy. In general however the two methods are incomparable, and perhaps a good practice would be to construct both sequences at the same time, so to obtain the best result of the two.

An example in which Shannon entropy eventually gives a better feature selection than Rényi min-entropy

Consider a dataset containing samples equally distributed among 32 classes, indexed from 0 to 31. As usual, we denote by C the random variable ranging over the classes. Assume that the data have 8 features divided in 2 types F and F , each of which consisting of 4 features denoted as follows:

F = {f 1 , f 2 , f 3 , f 4 } and F = {f 1 , f 2 , f 3 , f 4 }.
Thus the total set of features is F ∪ F . The features f i , f i are also regarded as random variables.

Assume that the features of type Concerning the features of type F , assume that each of them can range in a set of 9 values, and that these sets are mutually disjoint. We use the following notation:

for 1 ≤ i ≤ 4 f i = {v i } ∪ {v ij | 0 ≤ j ≤ 7}
The association between the features of F and the classes is represented in the following figure :  All the v ij values allow to identify exactly one class. For instance, if in a certain sample the f 1 value is v 10 , it means that the sample can be univocally classified as belonging to class 0. If the f 2 value is v 22 , it belongs to class 9, etc. As for the v i 's, each of them is associated to a set of 24 classes, with uniform distribution. For instance, the value v 1 is associated to classes 1, 2, 3, 5, . . . , 29, 30, 31.

Let us select the first feature f in order of importance for the classification. We will consider at the same time our method, based on Rényi min-entropy, and the method of [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF]. In both cases, the aim is to choose f such that it maximizes the mutual information For the features of type F we have:

I α = H α (C) -H α (C | f ),
H 1 (C | f 1 ) = H 1 (C | f 2 ) = H 1 (C | f 3 ) = 3 while H 1 (C | f 4 ) = 4 > 3
On the other hand, with respect to the features of type F , we have

H 1 (C | f 1 ) = H 1 (C | f 2 ) = H 1 (C | f 3 ) = H 1 (C | f 4 ) ≈ 3.439 > 3
So, the first feature selected using Shannon entropy would be f 1 , f 2 or f 3 . Let us assume that we pick f 1 . Hence with Shannon the first set of the series is S 1 1 = {f 1 }. Let us now consider our method based on Rényi min-entropy. For the F feature the conditional entropy is the same as for Shannon:

H ∞ (C | f 1 ) = H ∞ (C | f 2 ) = H ∞ (C | f 3 ) = 3 H ∞ (C | f 4 ) = 4 > 3
But for the F features Rényi min-entropy gives a different value. In fact we get:

H ∞ (C | f 1 ) = H ∞ (C | f 2 ) = H ∞ (C | f 3 ) = H ∞ (C | f 4 ) ≈ 1.83 < 2
So, the first feature selected using Rényi min entropy would be f 1 , f 2 , f 3 , or f 4 . Let us assume that we pick f 1 . Hence with our method the first set of the series is S 1 ∞ = {f 1 }. For the selection of the second feature, we have

H 1 (C | f 1 , f 2 ) = H 1 (C | f 1 , f 4 ) = 2 while H 1 (C | f 1 , f 3 ) = 1 H 1 (C | f 1 , f 1 ) = H 1 (C | f 1 , f 2 ) = H 1 (C | f 1 , f 3 ) = H 1 (C | f 1 , f 4 ) > 2
Hence the second feature with Shannon can only be f 3 , and thus the second set in the sequence is S 2 1 = {f 1 , f 3 }. With Rényi min-entropy we have:

H ∞ (C | f 1 , f 4 ) > H ∞ (C | f 1 , f i ) for i = 1, 2, 3 H ∞ (C | f 1 , f i ) > H ∞ (C | f 1 , f j ) for i = 1, 2, 3 and j = 2, 3, 4
Hence with our method the second feature will be f 2 , f 3 , or f 4 . Let us assume that we choose f 2 . Thus the second set of the series is S 2 ∞ = {f 1 , f 2 }. Continuing our example, we can see that S 3 1 = {f 1 , f 3 , f 4 }, and S 3 ∞ = {f 1 , f 2 , f i } where i can be, equivalently, 3 or 4. At this point the method with Shannon can stop, since the residual Shannon entropy of the classification is H 1 (C | S 3 1 ) = 0, and also the Bayes risk is B(C | S 3 1 ) = 0, which is the optimal situation in the sense that the classification is completely accurate. S 3 ∞ on the contrary is not enough to give a completely accurate classification, for that we have to make a further step. We can see that S 4 ∞ = F , and finally we have H ∞ (C | S 4 ∞ ) = 0. Thus in this particular example we have that for small values of the threshold on the accuracy our method gives better results. On the other hand, if we want to achieve perfect accuracy (threshold 0) Shannon gives better results. This is not always the case however: for example, if we consider the initial set of features to be F ∪ F , with F = {f 1 }, with f 1 and F defined as before, we have that our method achieves perfect accuracy at step 4, giving as result S 4 ∞ = F as before, while with Shannon the method would still select as first feature f 1 , and would achieve perfect accuracy only at step 5, giving as result S 5 1 = F ∪ F .

Evaluation

In this section we evaluate the method for feature selection that we have proposed, and we compare it with the one based on Shannon entropy by [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF].

To evaluate the effect of feature selection, some classification methods have to be trained and tested on the selected data. We decided to use two different methods so to avoid the dependency of the result on a particular algorithm. We chose two widely used classifiers, which in many cases represent the state of the art in the machine learning field, namely Support Vector Machines (SVM) and Artificial Neural Networks (ANN).

Even though the two methods are very different, they have in common that their efficiency is highly dependent on the choice of certain parameters. Therefore, it is worth spending some effort to identify the best values. Furthermore, we should take into account that the particular paradigm of SVM we chose only needs 2 parameters to be set, while for ANN the number of parameters increases (at least 4).

It is very important to choose values as robust as possible for the parameters. It goes without saying that the strategy used to pick the best parameter setting should be the same for both Shannon entropy and Rényi min-entropy. On the other hand for SVM and ANN we used two different hyper-parameter tuning algorithms, given that the number and the nature of the parameters to be tuned for those classifiers is different.

In the case of SVM we tuned the cost parameter of the objective function for margin maximization (C-SVM) and the parameter which models the shape of the RBF kernel's bell curve (γ). Grid-search and Random-search are quite time demanding algorithms for the hyper-parameter tuning task but they are also widely used and referenced in literature when it comes to SVM. Following the guidelines in [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF] and [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF], we decided to use Grid-search, which is quite suitable when we have to deal with only two parameters. In particular we performed Grid-search including a 10 folds CV step.

Things are different with ANN because many more parameters are involved and some of them change the topology of the network itself. Among the various strategies to attack this problem we picked Bayesian Optimization [START_REF] Snoek | Practical bayesian optimization of machine learning algorithms[END_REF]. This algorithm combines steps of extensive search for a limited number of settings before inferring via Gaussian Processes (GP) which is the best setting to try next (with respect to the mean and variance and compared to the best result obtained in the last iteration of the algorithm). In particular we tried to fit the best model by optimizing the following parameters:

number of hidden layers number of hidden neurons in each layer learning rate for the gradient descent algorithm size of batches to update the weight on network connections number of learning epochs To this purpose, we included in the pipeline of our code the Spearmint Bayesian optimization codebase. Spearmint, whose theoretical bases are explained in [START_REF] Snoek | Practical bayesian optimization of machine learning algorithms[END_REF], calls repeatedly an objective function to be optimized. In our case the objective function contained some tensorflow machine learning code which run a 10 folds CV over a dataset and the objective was to maximize the accuracy of validation. The idea was to obtain a model able to generalize as much as possible using only the selected features before testing on a dataset which had never been seen before.

We had to decide the stopping criterion, which is not provided by Spearmint itself. For the sake of simplicity we decided to run it for a time lapse which has empirically been proven to be sufficient in order to obtain results meaningful for comparison. A possible improvement would be to keep running the same test (with the same number of features) for a certain amount of time without resetting the computation history of the package and only stop testing a particular configuration if the same results is output as the best for k iterations in a row (for a given k).

Another factor, not directly connected to the different performances obtained with different entropies, but which is of the highest importance for the optimization of ANN, is the choice of the activation functions for the layers of neurons. In our work we have been using ReLU activation function for all layers because it is well known as a function

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy Fig. 5. Accuracy of the ANN and SVM classifiers on the BASEHOCK dataset which works well for this aim, is quite easy to compute (the only operation involved is the max) and avoids the sigmoid saturation issue.

Experiments

As already stated, at the i-th step of the feature selection algorithm we consider all the features which have already been selected in the previous i -1 step(s). For the sake of limiting the execution time, we decided to consider only the first 50 selected features with both metrics. We tried our pipeline on the following datasets:

-BASEHOCK dataset: 1993 instances, 4862 features, 2 classes. This dataset has been obtained from the 20 newsgroup original dataset. -SEMEION dataset: 1593 instances, 256 features, 10 classes. This is a dataset with encoding of hand written characters. -GISETTE dataset: 6000 instances, 5000 features, 2 classes. This is the discretized version of the NIPS 2003 dataset which can be downloaded from the site of Professor Gavin Brown, Manchester University.

We implemented a bootstrap procedure (5 iterations on each dataset) to shuffle data and make sure that the results do not depend on the particular split between training, validation and test set. Each one of the 5 bootstrap iterations is a new and unrelated experimental run. For each one of them a different training-test sets split was taken into account. Features were selected analyzing the training set (the test set has never been taken into account for this part of the work). After the feature selection was executed according to both Shannon and Rényi min-entropy, we considered all the selected features adding one at each time. So, for each bootstrap iteration we had 50 steps, and in each step we added one of the selected features, we performed hyper-parameter tuning with 10 folds CV, we trained the model with the best parameters on the whole training set and we tested it on the test set (which the model had never seen so far). This procedure was performed both for SVM and ANN.

We computed the average performances over the 5 iterations and we got the results showed in Figures 5,6

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy 0 10

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy

Shannon and minEntropy accuracy on test set

Mean value over 5 bootstrap iterations q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Accuracy Shannon Accuracy minEntropy Fig. 7. Accuracy of the ANN and SVM classifiers on the SEMEION dataset using Rényi min-entropy usually gave better results than Shannon, especially with the BASEHOCK dataset.

Related works

In the last two decades, thanks to the growing interest in machine learning, many methods have been setup to tackle the feature reduction problem. In this section we discuss those closely related to our work, namely those which are based on information theory. For a more complete overview we refer to [START_REF] Bennasar | Feature selection using joint mutual information maximisation[END_REF], [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF] and [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF].

The approach most related to our proposal is that of [START_REF] Brown | Conditional likelihood maximisation: A unifying framework for information theoretic feature selection[END_REF] and [START_REF] Vergara | A review of feature selection methods based on mutual information[END_REF]. We have already discussed and compared their method with ours in the technical body of this paper.

As far as we know, Rényi min-entropy has only been used, in the context of feature selection, by [START_REF] Endo | Weighted Naïve Bayes Classifiers by Renyi Entropy[END_REF]. In the experiments they only show results for other Rényi entropies, not for the min one. But they mention also Rényi min-entropy, though. The definition of conditional min-entropy they consider, however, is that of [START_REF] Cachin | Entropy Measures and Unconditional Security in Cryptography[END_REF]. This notion, as we have already mentioned, has unnatural consequences. In particular, under this definition, a feature may increase the entropy of the classification instead of decreasing it. It is clear, therefore, that basing a method on this notion of entropy could lead to strange results.

Two key concepts that have been widely used are relevance and redundancy. Relevance refers to the importance for the classification of the feature under consideration f t , and it is in general modeled as I(C; f t ), where I represents Shannon mutual information. Redundancy represents how much the information of f t is already covered by S. It is often modeled as I(f t , S). In general, we want to maximize relevance and minimize redundancy.

One of the first algorithms ever implemented was proposed by [START_REF] Battiti | Using mutual information for selecting features in supervised neural net learning[END_REF] and it is called MIFS algorithm. This algorithm is based on a greedy strategy. At the first iteration step it selects the feature f 1 = argmax fi∈F I(C; f i ), and at step t it selects f t = argmax fi∈F \S t-1 [I(C, f i ) -β fs∈S t-1 I(f i , f s )] where β is a parameter that controls the weight of the redundancy part.

The mRMR approach (redundancy minimization and relevance maximization) proposed by [START_REF] Peng | Feature selection based on mutual information: Criteria of max-dependency, max-relevance, and min-redundancy[END_REF] is based on the same strategy as MIFS. However the redundancy term is now substituted by its mean over the | S | elements of subset S so to avoid its value to grow when new attributes are selected.

In both cases, if relevance outgrows redundancy, it might happen that many features highly correlated and so highly redundant can still be selected. Moreover, a common issue with these two methods is that they do not take into account the conditional mutual information I(C, f t | S) for the choice of f t , the next feature to be selected.

More recent algorithms involve the ideas of joint mutual entropy I(C; f i , S) (JMI, [START_REF] Bennasar | Feature selection using joint mutual information maximisation[END_REF]) and conditional mutual entropy I(C; f i | S) (CMI, [START_REF] Fleuret | Fast binary feature selection with conditional mutual information[END_REF]) The deductive step for choosing the next feature for [START_REF] Bennasar | Feature selection using joint mutual information maximisation[END_REF] is f t = argmax fi∈F \S t-1 min fs∈S t-1 I(C; f i , f s ) , while for [START_REF] Fleuret | Fast binary feature selection with conditional mutual information[END_REF] is f t = argmax fi∈F \S t-1 min fs∈S t-1 I(C; f i | f s ) . In both cases the already selected features are taken into account one by one when compared to the new feature to be selected f t . The correlation between JMI and CMI is easy to prove [START_REF] Yang | Feature selection based on joint mutual information[END_REF]:

I

3. 1

 1 An example in which Rényi min-entropy gives a better feature selection than Shannon entropy Selection with Rényi.

Fig. 2 .

 2 Fig. 2. Classes separation after the selection of the first feature.

  4 f 0 ) ≈ 1.32.

Fig. 3 .

 3 Fig. 3. Selection of the second feature with Rényi.

Fig. 4 .

 4 Fig. 4. Sequence of class splitting with Rényi.

F

  , and their relation with C are as follows: Given the binary representation of the classes b 4 b 3 b 2 b 1 b 0 , f 1 consists of the bits b 1 b 0 , f 2 consists of the bits b 2 b 1 , f 3 consists of the bits b 3 b 2 , and f 4 consists of the bit b 4 only. The situation is represented by the figure below.

  or equivalently, that minimizes the conditional entropy H α (C | f ), where α is the entropy parameter (α = 1 for Shannon, α = ∞ for Rényi min-entropy).

Fig. 6 .

 6 Fig. 6. Accuracy of the ANN and SVM classifiers on the GISETTE dataset

  (C; f i , S) = H(C) -H(C | S) + H(C | S) -H(C | S) = I(C; S) + I(C; f i | S)

  , and 7. As we can see, in all cases the feature selection method

		0.94
		0.92
		0.90
	Accuracy	0.88
		0.86
		0.84
		0	10	20	30	40	50
					# of features


Edith Salès-Wuillemin
email: edith.sales-wuillemin@univ-paris8.fr

Rachel Morlot

Aurélie Fontaine

Daniel Talon

Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices

Keywords: Hospital Hygiene, Hospital Acquired Infection, Hand Hygiene

Hospital-Acquired Infections (HAI) significantly affect patients, prolong hospital stays and present a major challenge for the quality of patient care. In France, around 4,500 individuals die every year from an HAI (Government Statistics, 2010). Hand hygiene studies focusing on beliefs and practices may

Introduction

In France, 4,500 people die every year as a result of a hospital-acquired infection. The fight against hospital-acquired infections is one of the key priorities of the Haute Autorité de Santé (HAS, High Health Authority). Recently, the emphasis has been on the provision of training for healthcare professionals. Yet despite these efforts, there remains a significant gap between official Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices recommendations and actual hygiene practices1 . The purpose of this research is to examine the explanatory factors that may account for this gap. This chapter presents the findings of three studies. Study 1 is an analysis of social representations among healthcare workers based on a verbal association questionnaire. Study 2 aims to refine the results of the analysis performed in study 1 by using the results of explanatory interviews. Finally, study 3 is based on an observation of the behaviors exhibited by healthcare workers in a work context.

Theoretical Framework

Social Representations and Professional Representations

Professional representations are a subset of social representations. Social representations are common sense forms of knowledge that have a practical aim and differ from scientific knowledge. Social representations are developed by a social group as a result of its history, culture, and practices, but also its attitudes. They serve to decipher information from the environment, to shape behaviors, and to facilitate communication between individuals [START_REF] Flick | Social Representations and the social construction of everyday knowledge. Theoretical and methodological Queries[END_REF][START_REF] Jodelet | Madness and Social Representations[END_REF][START_REF] Moscovici | Social Representations: Explorations in Social Psychology[END_REF][START_REF] Wagner | Everyday Discourse and Common Sense. The Theory of Social Representations[END_REF]. The structural approach to representations posits that these kinds of knowledge are structured around a limited number of knowledge items defined as central or core elements. The core elements serve to structure and direct the representation, and need to be distinguished from the peripheral elements of the representation, which enable concrete expression and adaptation [START_REF] Flament | Consensus, salience and necessity in social representations Papers on Social Representations[END_REF][START_REF] Abric | Central System, Peripheral System: their functions and roles in the dynamics of social representations[END_REF][START_REF] Moliner | La Dynamique des Représentations Sociales[END_REF]. This explains why it is necessary to compare items appearing in the target area of the core in order to compare two distinct representations [START_REF] Abric | L'organisation interne des représentations sociales : système central et système périphérique[END_REF]. Professional representations operate like social representations, but pertain specifically to objects belonging to a particular professional context or milieu. The social visibility and implications of the object are connected to the position of the group within a work organization, as well as the place of the object within the professional activity. The analysis of professional representations helps to understand how knowledge is incorporated by individuals and teams in processing information and organizing work activity [START_REF] Bataille | Représentations sociales, représentations professionnelles, système des activités professionnelles[END_REF][START_REF] Labbé | Dynamique des implications et des représentations professionnelles : le cas d'une recherche action dans le milieu Edith Salès-Wuillemin[END_REF][START_REF] Salès-Wuillemin | Psychologie Sociale expérimentale de l'usage du langage. Représentations sociales, Catégorisation et attitudes : perspectives nouvelles, Paris : L'Harmattan, collection Figures de l'interaction[END_REF].

4

Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… Although a number of studies have examined professional representations in nursing, no studies have so far been conducted on hospital hygiene. [START_REF] Guimelli | Contribution du modèle associatif des schèmes cognitifs de base à l'analyse structurale des représentations sociales[END_REF] examined nurses' representation of their work and showed that representations were structured around the role proper, i.e. the nursing diagnosis. It was found that hygiene did not appear in the representation. Zérillo (1998) conducted interviews with nursing students on hygiene-related matters and showed that representations were structured around the prevention of illnesses, well-being, the respect of others and avoiding bad smells, i.e. the objectives of the implementation of hygiene practices. However, students' representations tended to emphasize daily body hygiene, which needs to be distinguished from hospital hygiene. It is also important to note that the study conducted by Zérillo was not realized with healthcare professionals. Of more direct relevance are a number of recent studies examining the representation of hospital-acquired infections among healthcare workers conducted by [START_REF] Quintard | Les déterminants du changement des comportements en milieu professionnel[END_REF][START_REF] Quintard | Représentation sociales des soignants et prévention des infections nosocomiales. 1ers Entretiens de la Psychologie[END_REF]. The studies by Quintard et al. showed that a distinction must be drawn between two sub-groups within the targeted population: doctors and nurses on the one hand, healthcare assistants and hospital agents on the other. In the first sub-group, representations were found to be structured around the core elements staff training and prevention, while the second sub-group structured their representations around three core elements: training, questioning of healthcare workers and management issues. The study showed that the former tended to focus on issues related to prevention objectives, while the latter tended to emphasize issues involved in implementing established procedures.

A greater number of studies have examined the question of attitudes toward hygiene. Their aim is to determine whether attitudes account for the gap between a proper knowledge of hygiene protocols and the implementation of established hygiene practices by healthcare staff.

Hygiene Attitudes and Behaviors Observed among Healthcare Staff

The emphasis of this study is on the chief vector of hospital-acquired infections -i.e. hands. The measurements focused on the perception of hand hygiene and hand hygiene practices.

There is significant evidence of a gap between attitudes (largely favorable) and behaviors (generally not in compliance with established practices). Pittet, Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices Simon, Hugonnet, Pessoa-Silva, [START_REF] Pittet | Hand Hygiene among Physicians: Performance, Beliefs, and Perceptions[END_REF]), Akyol (2007[START_REF] Pessoa-Silva | Attitudes and perceptions toward hand hygiene among healthcare workers caring for critically ill neonates[END_REF] conducted interviews with healthcare professionals to determine the explanatory factors of non-compliance. The results highlighted five main factors reported by participants: working conditions (lack of time), infrastructures (lack of equipment), training (insufficient or inadequate), the human environment (unscrupulous superiors, colleagues, patients…) and staff health (skin irritation caused by frequent hand-washing). The presence of a conscientious supervisor was found to be the most significant determinant of compliance with protocols. Other factors were also highlighted, including an awareness of being observed, the feeling of serving as a model for other colleagues, the need to implement hand hygiene protocols after contact with a patient, and finally ease of access to hydroalcoholic solutions. Other studies based on observations of actual behaviors have gone even further. McGuckin, Waterman & Govednik (2009) found a generally low rate of compliance with hand-hygiene practices (< 30%). [START_REF] Pittet | Hand Hygiene among Physicians: Performance, Beliefs, and Perceptions[END_REF] argued that a lack of equipment at hand-washing facilities or of hydro-alcoholic solution is not the only explanatory factor since the rate of compliance remains low even when hospitals are correctly equipped (57%).

The studies presented above suggest that the work environment is not the only factor determining non-compliance since even under favorable conditions, there is no significant increase of the rate of compliance. It is also important to note that compliance is not governed by a belief in the legitimacy or validity of the norms embodied by protocols, but is driven instead by social desirability -i.e. when an individual produces the expected behavior, it is because s/he is adjusting to a social norm (in this instance, the expectations of superiors).

It is important therefore to consider the reasons that may account for noncompliance by examining naïve knowledge, a potential obstacle hindering the implementation of hygiene protocols. These issues pertain more specifically to the articulation of scientific knowledge and naïve knowledge and the dynamics subtending the relation between these two kinds of knowledge [START_REF] Moscovici | Social representations and social explanations: from the « naive » to « amateur[END_REF].

Two specific groups of healthcare staff were targeted in this research: nurses and healthcare assistants. Nurses and healthcare assistants are the two groups most frequently in contact with patients and may therefore be said to play a determining role in the provision of healthcare and the transmission of hospital-acquired infections. However, it is important to note that the work performed by nurses and healthcare assistants (i.e. staff practices) and their level of training (staff Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… knowledge) are not comparable. It is hypothesized that social representations of hygiene are likely to differ between the two groups.

Study 1: Questionnaires assessing Representations of Hygiene among Nurses and Healthcare Assistants

Method and Design

The study was based on verbal association task included in questionnaires. The aim was to highlight the key concepts structuring representations of hygiene.

Population

114 nurses and 35 healthcare assistants were interviewed as part of this study. At the time of the study, all participants had been working in a healthcare institution for at least 3 years and were aged between 26 and 57 (average age: 37.2). 95% of participants were women. Participants practiced hygiene protocols on a regular basis and had taken part in a hygiene training session at least once since entering the profession.

Questionnaires

The questionnaire was a 3-page document divided into two sections. Section one focused on questions concerning the identity of participants (age, gender, number of years in the profession, etc.). Section two involved a verbal association question. Participants were asked to write down as quickly as possible the first words that came to mind when seeing the term 'hygiene' (the inductor) and to cite a maximum of ten associated words. Verbal association involves the activation of knowledge present in the memory and organized into semantic networks. The activation is triggered by the inductor, which acts as a stimulus. Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices 2.4 Method used to analyze the Associated Words: Prototypical Analysis Prototypical analysis was initially developed by [START_REF] Vergès | L'évocation de l'argent : Une méthode pour la définition du noyau central d'une représentation[END_REF] and is used to highlight the most frequently and most rapidly cited items. Prototypical analysis involves two key criteria: citation frequency (number of individuals citing the term) and average citation rank (average order of citation in the chain of association). The most frequently cited words (high frequency) and the earliest words in the chain of association (low rank) are retained on the grounds that they refer to the most typical items of the representation. These words constitute target elements with a strong probability of being central.

A probability calculation based on the binomial law was performed to establish the thresholds used to distinguish between low and high citation frequencies [START_REF] Salès-Wuillemin | Psychologie Sociale expérimentale de l'usage du langage. Représentations sociales, Catégorisation et attitudes : perspectives nouvelles, Paris : L'Harmattan, collection Figures de l'interaction[END_REF][START_REF] Morlot | Effet des pratiques et des connaissances sur la représentation sociale d'un objet : application à l'hygiène hospitalière[END_REF]. For every population, the test considers the following factors: the number of subjects, the average number of words that can potentially be associated, and the number of words actually associated by subjects. The probability p that k individuals will cite the same associated word can be measured based on these parameters. Only the words with a probability of p<.000001 were retained. In table 1, the k threshold is 28.27% (32 participants). In table 2, the k threshold is 48.4% (17 participants). This means that the words associated by at least N participants are cited at a level significantly different from chance. We may therefore conclude that these words are shared socially by the members of the group.

An arbitrary threshold was determined to establish the citation thresholds used to distinguish between high and low average ranks. Since participants were allowed to cite up to 10 associated words, it was deemed that the words with an average rank below 4 could be viewed as precocious in the chain of association and therefore quickly activated.

Results of the Prototypical Analysis

144 different words were cited by the two groups of participants (129 words by the group of nurses and 66 words by the group of healthcare assistants). The words were divided into four areas. The target area is the area containing the most typical words.

Tables 1 and2 clearly highlight a common core of knowledge in the target area shared by both groups of healthcare professionals -i.e. cleanliness. However, Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… a number of differences are also apparent -for example 'disinfection' (healthcare assistants) and 'sanitization' and 'hand-washing' (nurses). The results of study 1 suggest that there is a shared concept structuring representations of hygiene: 'cleanliness'. However, in the group of nurses the term is associated with 'sanitization' in the target area, while in the group of healthcare assistants the term is associated with 'disinfection'. The results suggest that the term 'cleanliness' has a different meaning in the two populations.

Study 2: Interviews Focusing on the Representation of Hygiene among Nurses

and Healthcare Assistants

3.1Method and Design

The study was based on explanatory interviews aimed at highlighting the connections between the concepts structuring representations of hygiene among nurses and healthcare assistants. The interviews focused on the four concepts located in the target area -i.e. 'cleanliness', 'hand-washing', 'sanitization', and 'disinfection'.

Population

32 participants distributed into two groups (16 nurses and 16 healthcare assistants) were interviewed as part of this research. At the time of the study, all participants had been working in a healthcare institution for at least 3 years and were aged between 32 and 57 (average age: 33.5). 93% of participants were women.

Interviews

All of the interviews conducted in study 2 were performed by the same interviewer and involved a specific interview technique: the explanatory interview (see Salès-Wuillemin, [START_REF] Salès-Wuillemin | La représentation sociale de l'hygiène chez les professionnels de santé : Intérêt du recueil par entretien et de l'analyse discursive des opérateurs de liaison issus du modèle des Schèmes Cognitifs de Base (SCB)[END_REF][START_REF] Salès-Wuillemin | Psychologie Sociale expérimentale de l'usage du langage. Représentations sociales, Catégorisation et attitudes : perspectives nouvelles, Paris : L'Harmattan, collection Figures de l'interaction[END_REF]. The interviews were conducted in two stages. Every interview began with a free stage during which interviewees were encouraged to express themselves Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… spontaneously after reading an instruction asking them to describe their image of hygiene in the context of their work. Every interviewee was invited to position themselves from their particular professional perspective, i.e. as healthcare assistants or nurses. In the second stage, the interviewer returned systematically to each of the four target items identified in study 1. To introduce the target items, the interview followed a pre-defined interview guide that included as many questions as there were target items. The questions introduced every item in a general manner, for example: 'Now, if I say to you 'cleanliness' in referring to hygiene, what does the word evoke for you as a healthcare assistant (as opposed to a nurse)?'. Participants' responses were submitted to systematic questioning aimed at generating explanations about: 1/ the relation between two items (e.g. 'You said that hygiene automatically involves cleanliness; what exactly do you mean by automatically?'); 2/ the referents cited by participants (e.g. 'You said that cleanliness involves sanitization practices; what exactly do you mean by sanitization?'); or 3/ the rules of practice, reasoning or decision-making stated by participants, with a view to understanding the subtending logic (e.g. 'You said that if you provide care to the same patient, you might easily apply a dose of hydro-alcoholic solution to continue with another care procedure rather than using hand-washing; could you tell me why it is that in this particular case, i.e. when you provide care to the same patient, you may use the hydro-alcoholic solution?').

Method used to analyze the Interviews: Basic Cognitive Schemas

An analysis based on basic cognitive schemas was performed on the material generated by the interviews. The 28 connectors2 described by [START_REF] Guimelli | Contribution du modèle associatif des schèmes cognitifs de base à l'analyse structurale des représentations sociales[END_REF] in a study conducted among nurses and by [START_REF] Guimelli | Locating the central core of a representation : toward a method[END_REF] in a study conducted on a group of friends were used as an analytical framework. The key difference with these studies is that participants were required to respond to a questionnaire. In the present study, a coding was carried out based on the transcribed interviews. A valence index was calculated for each target item in order to highlight the items of the representation with the highest probability of being central. The index takes account of the number of times participants Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices connected the target item with another item in the representation, whatever its position (in the central or peripheral area), using one of the 28 possible connectors.

Results of Interviews on Valence

In the group of nurses, the words 'sanitization' and 'cleanliness' were found to be the items with the strongest valence, i.e. the strongest probability of being central, while the item with the strongest valence among healthcare assistants was 'disinfection' (see tables 3 and 4).

Results of the Semantic Analysis of 'Cleanliness'

A semantic analysis of the common items found in the representations generated by both groups ('cleanliness') was performed based on the interviews. All synonyms and associated definitions were identified to conduct the analysis.

A qualitative analysis indicates that among healthcare assistants, cleanliness is primarily synonymous with 'clean and without trace', i.e. 'impeccable'. By Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… contrast, among nurses, cleanliness refers above all to a whole range of technical gestures complying with protocols ('working within the norms', 'complying with rules') and involving the use of adequate equipment ('wearing gloves', 'using sterile gauze-dressing'). The results suggest that while 'cleanliness' occupies a central position in the representation of both groups, a significant difference is nevertheless apparent: for healthcare assistants, cleanliness is a state and a characteristic ('being clean') that can be assessed based on visual indicators. By contrast, for nurses, cleanliness refers above all to a specific procedure and the equipment used to 'work cleanly'. A third study was conducted to highlight the relations between the representation of hygiene and the actual practices implemented by healthcare professionals. Study 3 was based on observations of professionals at work. Based on the assumption that hand hygiene is one of the key techniques for working cleanly, the observations made in study 3 focused more specifically on the techniques used to ensure hand hygiene.

Study 3: Hygiene Practices among Healthcare Professionals, Nurses and

Healthcare Assistants

Objectives

The aim of the study was to understand hand-hygiene practices among healthcare professionals. A comparative analysis of a simple washing technique (soap and water) and friction (using hydro-alcoholic solution) was performed.

Studies examining hygiene have found that lack of time is a significant obstacle to compliance with hand-washing protocols. The effect of noncompliance is a decrease of washing time. Furthermore, traditional hand-washing presupposes access to washing facilities. However, patient rooms are often not appropriately equipped. For instance, washing facilities may be located in corridors or treatment rooms, causing significant time loss. This explains why the emphasis in recent years has been on the use of hydro-alcoholic solutions. Through rapid friction (lasting less than a minute), better results can be achieved using hydro-alcoholic solutions than through simple washing (lasting 1 minute). Hydro-alcoholic solutions also produce the same results as antiseptic cleaning (lasting 1 minute) and surgical cleaning (lasting 1 minute), without requiring the use of washing facilities. However, given its specific conditions of use, friction Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices can serve as a substitute for traditional hand-washing. Friction can only be applied on clean hands (i.e. without any trace of biological or other products) without talc (after wearing gloves). This explains why healthcare professionals are required to use both techniques and to select a particular technique based on the specific parameters of the situation.

The aim of study 3 was to conduct an error assessment of the hand-hygiene technique used and to identify the decision rules governing the choice of handhygiene techniques.

Method and Design

The study was based on systematic observations conducted using an observation chart. The observations were accompanied by concomitant verbalizations. Healthcare professionals were observed at work while delivering treatment and care to patients.

Population

Two groups of healthcare professionals participated in the study: 15 nurses and 13 healthcare assistants. At the time of the study, all participants had been working in a healthcare institution for at least three years and were aged between 30 and 46 (average age: 32.8). 91% of participants were women.

Observation Chart

An observation chart was used to record the number of patients and the type of treatment dispensed. The chart included the following information: whether treatment was interrupted; the hygiene technique used before and after the treatment was delivered; whether there was a mistake committed in applying the technique, and if so, the type of mistake; finally, whether gloves were used.

Results of Observations

No significant differences in the number of mistakes committed by nurses (m = 3.4) and healthcare assistants (m = 2.9) were found in the first analysis.

Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… F(1.24) = 2.49, p<.12. The two groups of healthcare professionals were therefore grouped together for the remainder of the treatments. For both groups, the aim was to highlight the impact of the specific hand-hygiene technique used on the number of mistakes committed and the impact of the contamination risk factor on the choice of technique. Finally, the decision rules governing the choice of hand-hygiene technique were analyzed. The results presented in figure 1 show that more mistakes were committed using friction based on a hydro-alcoholic solution than in the use of simple handwashing: (χ²=3,56, ddl=1, p<.06). Analysis indicates that dosage errors were the most common type of mistake committed by healthcare professionals. Healthcare professionals used a lower amount of hydro-alcoholic than necessary. By contrast, participants using a simple hand-washing technique tended to use more product than was strictly necessary. Based on the dosage results, it is hypothesized that dosage errors may result from Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices a feeling that simple hand-washing offers less protection than friction. Two different types of situation were therefore compared: situations with a significant risk of contamination (high-risk treatment) and situations with a low risk of contamination (low-risk treatment). In both cases, the technique chosen by healthcare professionals was analyzed.

Analysis of figure 2 indicates that the use of friction involving a hydroalcoholic solution is significantly more frequent when there is a low risk of contamination than when there is a high risk of contamination (χ²=7,36, ddl=1, p<.007). By contrast, there is no difference in the case of simple washing (an unexpected result). Friction is included in the category of antiseptic and surgical cleaning, i.e. techniques combining products that involve antimicrobial activity. Therefore, healthcare professionals might have been expected to use these techniques more frequently in situations involving a high risk of contamination. The verbalizations of healthcare professionals were analyzed to highlight the decision rules governing the choice of hand-washing technique.

Qualitative Analysis of Verbalizations

A qualitative analysis of verbalizations serves to explain this result, indicating that friction based on a hydro-alcoholic solution is viewed negatively by the majority of healthcare professionals. Participants expressed a marked preference for simple hand-washing (which was deemed to be safer, more efficient, and more in compliance with their idea of (true) hand-washing). Soap lathers and removes impurities, while water serves to rinse hands and eliminate impurities. Despite being described as quick and practical, the use of a hydro-alcoholic solution tends to be perceived negatively since healthcare professionals feel that impurities remain on the skin and that their skin is not clean. Some solutions may also leave a sticky deposit, increasing the feeling that hands are dirty.

This explains why, since friction using a hydro-alcoholic solution can be used to perform the necessary hygiene procedures required by healthcare practice without significant time loss, it is used in all situations in which the level of perceived risk is low. However, whenever there is a perceived risk of contamination, healthcare professionals tend to favor traditional hand-washing using water and soap -a washing technique that is deemed to be more efficient.

Interpretation

The three studies conducted as part of this research indicate that the concept of cleanliness is central in representations of hygiene observed among healthcare professionals. However, significant differences were also apparent in the network of meanings associated with the term 'cleanliness'. Analysis of the hygiene Edith Salès-Wuillemin, Rachel Morlot, Aurélie Fontaine et al… practices used by healthcare professionals improves our understanding of the factors determining non-compliance with established practices. The use of hydroalcoholic solutions is clearly an issue among healthcare professionals. Hydroalcoholic solutions are used more frequently than simple washing -in itself a positive finding. However, their use also involves a high rate of error. The most common mistakes are under-dosage errors, which makes them inefficient. Finally, a qualitative analysis indicated that healthcare professionals place more trust in simple hand-washing techniques than friction, a technique that appears not to comply with their idea of a 'true' wash. Healthcare professionals use friction more frequently than simple hand-washing when a care procedure is deemed to involve a low level of risk. However, when the level of perceived risk is high, simple hand-washing tends to be the favored technique.

This final result merits particular attention, as do its effects on hospitalacquired infections. The consumption of hydro-alcoholic solutions is not enough to guarantee its acceptance among healthcare professionals. The study showed that while hydro-alcoholic solutions are used more frequently than in the past, they are not always used for the right reasons.

This finding also raises a number of questions concerning staff training. The initial measures implemented between 2000 and 2008 were designed to encourage healthcare professionals to apply a simple hand-washing protocol before using hydro-alcoholic friction, in the case of antiseptic cleaning. However, official recommendations have changed since 2008. Healthcare professionals are now required to use friction only as long as their hands are not dirty. This study has shown that it is precisely in such instances that the obstacle is greatest. Healthcare professionals tend to feel that hand hygiene protocols are not properly implemented because they are applied on hands that are not strictly speaking clean.

Conclusion

Beyond these results and their usefulness in developing measures aimed at promoting prevention and hygiene education in hospitals, this research also has a specific theoretical and methodological value. In theoretical terms, this research advances our understanding of professional representations through increased understanding of the decision rules resulting in a specific behavior. In methodological terms, the research was based on the application of a method involving a comprehensive process of data collection, starting from an assessment Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices of the representation as evoked by interviewees, and ending with an analysis of the impact of the representation on the actual practices implemented by individuals.

Further research will need to be conducted among nursing students using a method similar to the method applied among healthcare professionals. This research in progress has shown that since students have been trained by IFSI in the use of hydro-alcoholic friction, it is predicted that fewer obstacles will impede the use of this particular hand-hygiene technique.

Figure 1 .

 1 Figure 1. Impact of the hand-washing technique used Number of mistakes according to the hand-hygiene technique used -comparison of the use of a hydro-alcoholic solution and simple hand-washing.

Figure 2 .

 2 Figure 2. Impact of the risk of contamination on the technique used Number of uses of friction and simple hand-washing according to the risk of contamination.

Table 1 . Words associated by nurses based on average citation frequency and average citation rank.

 1

	Citation	Low average citation rank	High average citation rank
	frequency/Citation	Words with a rank < general	Words with a rank ≥
	rank	average rank (<4)	general average rank (≥4)
	High average citation	Cleanliness (71.9% ; 2.17) ;
	frequency	Hand-washing (58.7% ;
	> Binomial Threshold BT=28.27%	3.20) ; Sanitization (44.73% ; 3.43) TARGET AREA
	Low average citation		Disinfection (28.07% ;
	frequency	Washing (20.17% ; 3.5);	4,3) ; Isolation (24.5% ;
		Precautions (16.6% ; 3.3);	4.86) ; Prevention (22.8% ;
	≤ 28.27%	Decontamination (17.1% ;	5); Cleaning (18.4% ;
	and > 13%	3.7); Security (13.15% ; 3.86)	4.43); Protocol (14.03% ;
			4.87%); Soap (13.15% ; 4)

Table 2 . Words associated by healthcare assistants based on average citation rank and average citation frequency.

 2 ≤ 48.4% et 28.5% Effects of Healthcare Worker's Perception of Hygiene on Hand Hygiene Practices

	Citation frequency/Citation rank	Low average citation rank Words with a rank < general average rank (<4)	High average citation rank Words with a rank ≥ general average rank (≥4)
	High average citation frequency	Cleanliness (80% ; 1.75) ; Disinfection (48.6% ;
	> Binomial Threshold	2.82)
	BT=48.4%	TARGET AREA
	Low average citation
	frequency	Hand-washing (37.2%; 3.8).	Sanitization (28.6% ; 4.3)

Table 3 . Valence of target items, average number of connectors associated by nurses with each of the target elements.

 3

	Target items	Valence	ANOVA comparisons 1/2/3 F = ; p<
	1-Hand-washing	0.06 (σ=0.007)
	2-Sanitization	0.12** (σ=0.05)	1-3 F(1.15)=19.34, p<.000001 1-2 NS
	3-Cleanliness	0.18** (σ=0.09)	2-3 F(1.15)=8.25, p<.012

Table 4 . Valence of target items, average number of connectors associated by healthcare assistants, with each of the target items.

 4

	Target items	Valence	ANOVA comparisons 1/2 F = ; p<
	1-Disinfection	0.13 (σ=0.08)
	2-Cleanliness	0.18** (σ=0,07)	1-2 F(1.15)=12.80, p<.003

Government statistics -see http://www.sante.gouv.fr/htm/pointsur/nosoco/34_980901.htm

The 28 connectors are grouped into three metaschemas: Descriptive: connectors that indicate a descriptive relation between two items (definition, synonymy, part-whole relation, etc.); Praxis: connectors that indicate a relation of action between two items (use, action, etc.); Evaluation: connectors that indicate a relation of judgement between two items (characterization, evaluation, etc.).