Tjibbe Donker
email: tjibbe.donker@rivm.nl

Michiel Van Boven

W Marijn Van Ballegooijen

Tessa M Van 't Klooster

C C ) Wielders

Jacco Wallinga

Nowcasting pandemic influenza A/H1N1 2009 hospitalizations in the Netherlands

During emerging epidemics of infectious diseases, it is vital to have up-to-date information on epidemic trends, such as incidence or health care demand, because hospitals and intensive care units have limited excess capacity. However, real-time tracking of epidemics is difficult, because of the inherent delay between onset of symptoms or hospitalizations, and reporting. We propose a robust algorithm to correct for reporting delays, using the observed distribution of reporting delays.

We apply the algorithm to pandemic influenza A/H1N1 2009 hospitalizations as reported in the Netherlands. We show that the proposed algorithm is able to provide unbiased predictions of the actual number of hospitalizations in real-time during the ascent and descent of the epidemic. The real-time predictions of admissions are useful to adjust planning in hospitals to avoid exceeding their capacity.

Introduction

A hallmark of influenza A pandemics is their unpredictability, not only with respect to the timing of their occurrence but also with respect to their size, duration, and severity. With the benefit of hindsight it is now clear that the 2009 pandemic influenza A/H1N1 has been relatively mild, both in terms of the fraction of the population that has developed influenza-like illness, and the overall severity of the disease [START_REF] Fraser | Pandemic potential of a strain of influenza A (H1N1): early findings[END_REF][START_REF] Presanis | The severity of pandemic H1N1 influenza in the United States, from April to July 2009: a Bayesian analysis[END_REF][START_REF] Wu | School closure and mitigation of pandemic (H1N1) 2009, Hong Kong[END_REF]. Nevertheless, demand for high care hospital beds has been high compared to the available number of beds [START_REF] Domínguez-Cherit | Critically Ill patients with 2009 influenza A(H1N1) in Mexico[END_REF][START_REF] Kumar | Critically ill patients with 2009 influenza A(H1N1) infection in Canada[END_REF].

Excess high care hospital capacity is limited [START_REF] Carr | Variation in critical care beds per capita in the United States: implications for pandemic and disaster planning[END_REF]. For instance, in the Netherlands intensive care capacity is approximately 10 beds per 100,000 persons, of which less than 2 per 100,000 may be available to meet sudden increases in demand [START_REF] Hansen | Intensive Care Needs Assessment for Adults 2006-2016 [Behoefteraming Intensive Care voor Volwassenen[END_REF]. It is therefore vital that trends of rapidly increasing incidence and health care demand (especially hospitalizations requiring intensive care) are noticed early so that there may be time to increase operational capacity by strict triaging and by postponement of non-critical operations [START_REF] White | Preparing for the Sickest Patients With 2009 Influenza A(H1N1)[END_REF].

Real-time tracking of hospital admissions during epidemics is difficult because of the inherent delay in reporting of cases or hospital admissions.

Reasons for such delays include the time to complete diagnostic tests, logistics, and overwhelmed surveillance systems. A number of studies have addressed the problem of reporting delay, and recently the term 'nowcasting' has been coined for attempts to assess the current situation based on imperfect information [START_REF] Farrington | A Statistical Algorithm for the Early Detection of Outbreaks of Infectious Disease[END_REF][START_REF] Garske | Assessing the severity of the novel influenza A/H1N1 pandemic[END_REF][START_REF] Nicoll | Experience and lessons from surveillance and studies of the 2009 pandemic in Europe[END_REF].

Here we propose an algorithm to correct for delays in reporting, and infer the number of admissions from incoming reports. We have applied this nowcasting method to track the 2009 influenza A/H1N1 hospitalizations in the Netherlands, using a complete set of dates of hospital admissions and associated reporting delays. Now the pandemic has passed, we can assess retrospectively the precision of our nowcasting estimates of the number of hospital admissions during the pandemic.

Methods surveillance system

From April 25 2009, both general practitioners and hospitals were required to notify to the municipal health services patients with influenza like symptoms.

Laboratory tests were performed at the National Influenza Centre (represented by the National Institute for Public Health and the Environment, RIVM, and the Erasmus Medical Centre), using RT-PCR. Anonymized data about confirmed cases, including date of admission and travel history, was entered into a web-based database by the municipal health services and collected at the RIVM. On August 3 it was announced that the novel influenza A/H1N1 would no longer be a notifiable disease, which stopped the registration of cases. From that day on, only hospitalized cases that fulfilled the case definition [START_REF] Hahné | Epidemiology and control of influenza A(H1N1)v in the Netherlands: the first 115 cases[END_REF] were reported by the hospitals to the municipal health

Reporting probability

We reconstructed the number of hospitalized patients on each day of the epidemic. The reporting delay was measured as the difference between the date of admission of a patient and the date of reporting. We observed the reporting delays for all cases from June 5 to measure the distribution of reporting delays. The cumulative frequency distribution of reporting delays gives the probability of a case having been reported i days after the day of onset of symptoms, ρ i . The 95% reporting horizon is the delay where this cumulative distribution surpasses 0.95.

Estimation of the actual number of cases

We set the current day as day 0. Our goal is to estimate the number of admissions i days ago, N i . We denote the number of admissions on that day that have been reported up to the current day by C i . We denote the probability that an admission on that day has been reported before or during that day by ρ i . We are looking for the actual number of admissions i days ago, N i , given the number of reported cases for that day so far, C i .

We note that the number of observed cases is the product of the reporting probability ρ i and the actual number of cases N i :

C i = ρ i N i . Rearranging
gives an estimator for the actual number of cases

Ni = C i ρ i . ( 1
)

Maximum likelihood estimator and 95% confidence interval

In order to construct a likelihood function for the actual number of cases we assume that the number of observed cases follows a binomial distribution that is defined by a number of N i independent trials where the probability of success is ρ i . The probability of observing C i cases is

P (C i |N i , ρ i ) = N i C i ρ C i i (1 -ρ i ) N i -C i (2)
The corresponding likelihood function for N i given the number of observed cases C i and probability of reporting ρ i is, up to a constant, given by

L(N i ; C i , ρ i ) = N i C i (1 -ρ i ) N i -C i (3)
The value of the actual number of cases that maximizes the likelihood is

Ni = C i ρ i . ( 4
)
This confirms that the straightforward estimator derived earlier is a maximum likelihood estimator.

accept values of N i that have a likelihood ratio of λ = L(N i ) L( Ni ) in the acceptance region specified by the likelihood ratio test with α = 0.05, that is -2 log λ ≤ χ 2 1:0.05 . This means that the confidence interval includes all values N i that have a likelihood higher than 1/6.8 of the maximum likelihood.

Results

Figure 1 summarizes the information on the daily number of hospitalizations due to pandemic influenza A/H1N1 in the Netherlands. After a period from July up to early October during which approximately 15 hospitalizations were recorded per week, the number of hospitalizations started to increase steeply in the second week of October (Figure 1A). The frequency distribution of the reporting delay and the associated cumulative delay distribution are shown in Figure 1C and 1D, respectively. The delay distribution is sharply peaked around 3 days, but also has a long tail that extends to more than 25 days. 95% of all hospitalizations is reported within 14 days, hence we call this the 95% reporting horizon. The mean reporting delay was variable in the early stages of the epidemic (2-7 days) due to the small number of hospitalizations (Figure 1B). Later on the delay first stabilized to 6-7 days in the period from early September until mid October, and then slowly decreased to 5 days over the period from mid October to December (Fig- ure 1B). Hence, the daily number of reported admissions provides a poor estimate of the actual number of admissions.

With the number of reported cases and the cumulative delay distribution at hand, it is possible to estimate the number of hospitalizations that are still to be reported. Figure 2 shows for two specific dates the number of hospitalizations that were recorded up to that day (top panels), the cumulative delay distribution up to that day (middle panels), the expected number of hospitalizations (bottom panel, black lines), and the number of hospitalizations that were ultimately recorded (bottom panels, dark gray lines). The bottom panel of Figure 2 also shows the likelihood support for the estimates as confidence bounds (light grey shading). Both the increasing and decreasing trends in the early and late stages of the epidemic are well captured. Moreover, our method is even able to estimate the actual number of cases with fair precision (Figure 3).

The number of hospital admissions was evenly distributed over all weekdays (∼300 per day), while the number of reports was high for working days (∼400 per day) and low (∼0) in weekends (Figure 4A). The delay between admission and reporting was highest for admissions on Thursday and Friday (∼6 days), and lower during the other days of the week (4-5 days) (Figure 4B). However, this difference in delays between weekdays did not greatly affect our nowcasting estimates (Figure 5). Overall, our nowcasting estimates were most precise on Wednesdays, while underestimating during the beginning of the week and slightly overestimating at the end of the week.

Discussion

In emerging outbreaks, it is important to have up-to-date information on the spread of the disease, and growth of the epidemic, because the number of cases can increase dramatically in a matter of days. Hospitalizations, and intensive care admissions in particular, should be tracked promptly, because excess capacity is small most of the time. However, surveillance systems suffer from delayed reporting of cases. This delay causes an apparent decrease in number of cases in the most recent part of the epidemic, and should therefore be taken into consideration when interpreting epidemic curves.

We have shown how routinely collected surveillance data can be used to obtain precise estimates of the actual number of hospitalized patients during an outbreak. Despite considerable reporting delays, the estimates were close to the actual numbers of daily hospitalized patients, up to 1 day before the observation.

The estimator has a number of limitations that should be addressed.

First, patients that are reported after a very long time have not yet been included in the delay distribution. This truncation of data can be adjusted for [START_REF] Hubbard | Nonparametric survival estimation when death is reported with delay[END_REF], but we believe that little additional precision can be obtained by such more complicated analyses. Second, we assume that the distribution of delays is at least approximately stationary. If there is evidence of significant changes in the delay distribution over time, the different phases of the epidemic should be analyzed separately to reduce bias in the estimation, at the expense of a loss of precision. Third, the reporting delay will typically differ between weekdays, because hospitalizations are generally not reported during the weekends. If the difference between weekdays is large, the number that is still to be reported for each day of the week should be analyzed separately. Again, this reduces bias at the expense of a loss of precision.

The application of the nowcasting algorithm to pandemic influenza A/H1N1

2009 hospitalizations in the Netherlands provides an example where the limitations as described above, have been checked carefully. The scale of the delay distribution is much shorter than the scale of the epidemic. The reporting delay differed between weekdays but not enough to cause a substantial bias. Our analysis of the pandemic influenza A/H1N1 2009 hospitalizations may have profited from the relatively long period between the first reported cases and the start of the epidemic growth in the Netherlands. The Dutch hospitals and health services had ample time to prepare for the epidemic, and diagnostic tests were available throughout the epidemic. These preparations resulted in the relatively stable reporting delay distribution throughout the epidemic. Whereas a shorter period, such as in the UK, USA or Australia, could have overwhelmed the health services, causing larger fluctuations in the reporting delay. This, in turn, could result in less precise estimations.

The method presented here enables estimation of the current number of cases, and is not intended to predict the development of the epidemic. To that end, real-time prediction models are available that use the numbers of reported cases in combination with simple mathematical models to project the trajectory of the epidemic [START_REF] Hall | Real-time epidemic forecasting for pandemic influenza[END_REF][START_REF] Ong | Real-time epidemic monitoring and forecasting of H1N1-2009 using influenza-like illness from general practice and family doctor clinics in Singapore[END_REF][START_REF] Nishiura | Real-time forecasting of an epidemic using a discrete time stochastic model: a case study of pandemic influenza (H1N1-2009)[END_REF]. These models usually assume that cases are reported instantaneously, which is hardly ever the case in practice. We believe that a two-pronged approach in which our nowcasting estimator is used in conjunction with real-time prediction models could substantially improve prospects for the practical application of predicting the future course of an epidemic.

Concluding, we combined surveillance data to estimate the number of hospitalizations during the pandemic influenza A/H1N1 2009 outbreak, to track the actual health care demand. The method reliably predicts both increasing and decreasing trends in the number of hospitalizations. The nowcasting tool holds considerable promise for gauging actual number of hospitalizations in the presence of reporting delays. Figure 2: Correcting for the reporting delay on October 28 (left) and December 2 (right). Top panels) The reported number of patients admitted to hospital with confirmed influenza A/H1N1 at each of the dates. Middle panels) The probability of having been reported, as a function of admission date. The dotted line shows the 95% threshold, used for the reporting horizon (dashed line). Bottom panels) The estimated number of cases (black line), including 1/6.8 likelihood support (light grey area) and 95% reporting horizon (black dashed line). The dark grey line denotes the final number of cases, reported until December 30. The initial number of cases shows a decline on both dates, but the compensation shows that the number of hospitalized patients is still increasing on the first date.

Figure 3: Accuracy of the estimates. A) Distribution of the difference between the estimated and actual number of admitted patients as a function of the time between the admission and observation, measured as a moving window over the entire epidemic. The solid black line shows the median, the shaded areas show the total range, 95% of the data (between the 2.5% and 97.5% percentile), and 75% of the data (between the 12.5% and 87.5% percentile) B) The percentage of estimations where the actual number of cases was below the lower or above the upper confidence bound.

Figure 1 :

 1 Figure 1: Hospitalizations of confirmed pandemic influenza A/H1N1 cases during the 2009 pandemic. A) Daily numbers of patients admitted between July 13 and December 30. The peak of the hospitalizations is on November 12. B) The mean reporting delay over all cases up to the indicated date of admission. C) The frequency distribution of the admission-to-reporting delay. D) The normalized cumulative delay distribution. The dotted line indicates the threshold level of 0.95 for the reporting horizon, the dashed line indicates that after 14 days more than 95% of the hospitalizations has been reported.

Figure 4 :

 4 Figure 4: Differences between weekdays. A) Total number of hospitalizations (dark grey) and incoming reports (light gray) by weekday. Only 8 reports were filed during the weekends. B) The mean (and 95%CI) reporting delay for each weekday. Delays are longest on Thursday and Friday, and shortest on Sunday and Monday.

Figure 5 :

 5 Figure5: Total difference between estimated and actual number of admitted patients over the entire time period, excluding the day of observation (0 days delay), and split up be weekday of observation. The results of the estimator clearly shifts during the week, with the best estimations on Wednesday.


Luc Brun
email: luc.brun@greyc.ensicaen.fr

Donatello Conte
email: dconte@unisa.it

Pasquale Foggia
email: pfoggia@unisa.it

Mario Vento
email: mvento@unisa.it

People re-identification by Graph Kernels Methods

People re-identification using single or multiple camera acquisitions constitutes a major challenge in visual surveillance analysis. The main application of this research field consists to reacquire a person of interest in different non-overlapping locations over different camera views. This paper present an original solution to this problem based on a graph description of each person. In particular, a recently proposed graph kernel is used to apply Principal Component Analysis (PCA) to the graph domain. The method has been experimentally tested on two video sequences from the PETS2009 database.

Introduction

Over a couple of decades, visual surveillance gained more and more interest due to its important role in security. Fundamental research issues in this context are object detection, tracking, shadow removal, an so on. However, recently, researchers draws much attention to high-level event detection, such as behaviour analysis, abandoned object detection, etc. An important task within this research field is to establish a suitable correspondence between observations of people who might appear and reappear at different times and across different cameras. This kind of problematic is commonly known as "people re-identification".

Several applications using single camera setup may benefit from information induced by people re-identification. One of the main appllications is loitering detection. Loitering refers to prolonged presence of people in an area. This behaviour is interesting in order to detect, for example, beggars in street corners, or drug dealers at bus stations, and so on. Beside this, information on these re-occurrences is very important in multi-camera setups, such as the ones used for wide area surveillance. Such surveillance systems create a novel problem of discontinuous tracking of individuals across large sites, which aims to reacquire a person of interest in different non-overlapping locations over different camera views.

Re-identification problem has been studied for last five years approximately. A first group [START_REF] Gandhi | Panoramic appearance map (pam) for multi-camera based person re-identification[END_REF][START_REF] De Oliveira | People reidentification in a camera network[END_REF][START_REF] Bak | Person re-identification using haarbased and dcd-based signature[END_REF][START_REF] Bazzani | Multiple-shot person re-identification by hpe signature[END_REF] deals with this problem by defining a unique signature which condenses a set of frames of a same individual; re-identification is then performed using a similarity measure between signatures and a threshold to assign old or new labels to successive scene entrances. In [START_REF] Gandhi | Panoramic appearance map (pam) for multi-camera based person re-identification[END_REF] a panoramic map is used to encode the appearance of a person extracted from all cameras viewing it. Such a method is hence restricted to multicamera systems. The signature of a person in [START_REF] De Oliveira | People reidentification in a camera network[END_REF] is made by a combination of SIFT descriptors and color features. The main drawback of this approach is that people to be added into the database are manually provided by a human operator. In [START_REF] Bak | Person re-identification using haarbased and dcd-based signature[END_REF] two human signatures, which use haar-like features and dominant color descriptor (DCD) respectively, are proposed while in [START_REF] Bazzani | Multiple-shot person re-identification by hpe signature[END_REF] the signature is based on three features, one capturing global chromatic information and two analyzing the presence of recurrent local patterns.

A second group ( [START_REF] Truongcong | People reidentification by spectral classification of silhouettes[END_REF][START_REF] Bird | Detection of loitering individuals in public transportation areas[END_REF]) deals with re-identification of people by means of a representation of a person in a single frame. Each representation corresponds to a point in a feature space. Then a classification is performed by clustering these points using a SVM ( [START_REF] Truongcong | People reidentification by spectral classification of silhouettes[END_REF]) or a correlation module ( [START_REF] Bird | Detection of loitering individuals in public transportation areas[END_REF]). Both [START_REF] Truongcong | People reidentification by spectral classification of silhouettes[END_REF][START_REF] Bird | Detection of loitering individuals in public transportation areas[END_REF] use the so-called "color-position" histogram: the silhouette of a person is first vertically divided into 𝑛 equal parts and then some color features (RGB mean, or HSV mean, etc.) are computed to characterize each part.

This paper can be ascribed to the second group but with some significant novelty: first, we have a structural (graph-based) representation of a person; second, our classification scheme is based on graph kernels. A graph kernel is a function in graph space that shares the properties of the dot-product operator in vector space, and so can be used to apply many vector-based algorithms to graphs.

Many graph kernels proposed in the literature have been built on the notion of bag of patterns. Graphlets kernels [START_REF] Shervashidze | Efficient graphlet kernels for large graph comparison[END_REF] are based on the number of common sub-graphs of two graphs. Vert [START_REF] Mah | Graph kernels based on tree patterns for molecules[END_REF] and Borgwardt [START_REF] Shervashidze | Fast subtree kernels on graphs[END_REF] proposed to compare the set of sub-trees of two graphs. Furthermore, many graph kernels are based on simpler patterns such as walks [START_REF] Kashima | Marginalized kernel between labeled graphs[END_REF], trails [START_REF] Dupé | Tree covering within a graph kernel framework for shape classification[END_REF] or paths.

A different approach is to define a kernel on the basis of a graph edit distance, that is the set of operations with a minimal cost transforming one graph into another. Kernels based on this approach do not rely on the (often simplistic) assumption that a bag of patterns preserves most of the information of its associated graph. The main difficulty in the design of such graph kernels is that the edit distance does not usually corresponds to a metric. Trivial kernels based on edit distances are thus usually non definite positive. Neuhaus and Bunke [START_REF] Neuhaus | Bridging the Gap Between Graph Edit Distance and Kernel Machines[END_REF] proposed several kernels based on edit distances. These kernels are either based on a combination of graph edit distances (trivial kernel, zeros graph kernel), use the convolution framework introduced by Haussler [START_REF] Haussler | Convolution kernels on discrete structures[END_REF] (convolution kernel, local matching kernel), or incorporate within the kernel construction schemes several features deduced from the computation of the edit distance (maximum similarity edit path kernel, random walk edit kernel). Note that a noticeable exception to this classification is the diffusion kernel introduced by the same authors [START_REF] Neuhaus | Bridging the Gap Between Graph Edit Distance and Kernel Machines[END_REF] which defines the gram matrix associated to the kernel as the exponential of a similarity matrix deduced from the edit distance.

We propose in this paper to apply a recent graph kernel [START_REF] Brun | Symbolic learning vs. graph kernels: An experimental comparison in a chemical application[END_REF][START_REF] Gauzere | Graph edit distance and treelet kernels for chemoinformatic[END_REF] based on edit distance, together with statistical machine learning methods, to people reidentification. The remaining of this paper is structured as follows: we first describe in Section 2 our graph encoding of objects within a video. Moving objects are acquired from different view points and are consequently encoded by a set of graphs. Given such a representation we describe in Section 3 an algorithm which allows to determine if a given input graph corresponds to a new object. If this is not the case, the graph is associated to one of the objects already seen. The different hypotheses used to design our algorithm are finally validated through several experiments in Section 4.

Graph-based Object Representation

The first step of our method aims to separate pixels depicting people on the scene (foreground) from the background. We thus perform a detection of moving areas, by background subtraction, combined with a shadow elimination algorithm [START_REF] Conte | Performance evaluation of a people tracking system on pets2009 database[END_REF]. This first step provides a set of masks which is further processed using mathematical morphology operations (closing and opening) (Fig. 1a). Detected foreground regions are then segmented using Statistical Region Merging (SRM) algorithm [START_REF] Nock | Statistical region merging[END_REF] (Fig. 1c). Finally, the segmentation of the mask within each rectangle is encoded by a Region adjacency Graph (RAG). Two nodes of this graph are connected by an edge if the corresponding regions are adjacent. Labels of a node are: the RGB average color, the area, and the size 𝜂 normalized with respect to the overall image (Fig. 1d).

Comparisons between objects by means of Graph Kernels

Objects acquired by multiple cameras, or across a large time interval, may be subject to large variations. Common kernels [START_REF] Kashima | Marginalized kernel between labeled graphs[END_REF] based on walks, trails or paths are quite sensitive to such variations. On the other hand, graph edit distances correspond to the minimal overall cost of a sequence of operations transforming two graphs. Within our framework, such distances are parametrized by two sets of functions 𝑐(𝑢 → 𝑣), 𝑐(𝑢 → 𝜖) and 𝑐(𝑒 → 𝑒 ′ ), 𝑐(𝑒 → 𝜖) encoding respectively the substitution, and deletion costs for nodes and edges. Using such distances, small graph distortions may be encoded by small edit costs, hence allowing to capture graph similarities over sets having important within-class distance.

Unfortunately, the computational complexity of the exact edit distance is exponential in the number of involved nodes, which drastically limits its applicability to databases composed of small graphs. This paper is based on a sub optimal estimation of the edit distance proposed by Riesen and Bunke [START_REF] Riesen | Approximate graph edit distance computation by means of bipartite graph matching[END_REF][START_REF] Riesen | Bipartite graph matching for computing the edit distance of graphs[END_REF]. In this estimation, first a cost for matching two nodes is defined, taking into account also the edges incident to the nodes. This cost can be computed in a polynomial time (with respect to the number of incident edges) using the Hungarian Algorithm [START_REF] Riesen | Bipartite graph matching for computing the edit distance of graphs[END_REF]. The cost for matching a pair of nodes depends on 𝑐(𝑢 → 𝑣), 𝑐(𝑒 → 𝑒 ′ ) and 𝑐(𝑒 → 𝜖).

Then, the edit distance is estimated by finding a mapping between the nodes of the two graphs that minimizes the total cost of the mapped pairs, attributing the deletion cost 𝑐(𝑢 → 𝜖) to unmapped nodes. This problem can be solved in polynomial time with the Hungarian algorithm. It can be demonstrated that the estimate computed in this way lies below the true value of the edit distance, since it computes the edge-related costs in an optimistic way, assuming that the edge mapping can be performed using only local information, ignoring global constraints. Now we will discuss the four cost functions used for defining the edit distance. Within our framework, each node 𝑢 encodes a region and is associated to the mean color (𝑅 𝑢 , 𝐺 𝑢 , 𝐵 𝑢 ) and to the normalized size 𝜂 𝑢 of the region (Section 2). We experimentally observed that small regions have larger chances to be deleted between two segmentations. Hence, the normalized size of a region can be used as a measure of its relevance within the whole graph.

The cost of a node substitution is defined as the distance between the mean colors of the corresponding regions. We additionally weigh this cost by the maxi-mum normalized size of both nodes. Such a weight avoids to penalize the matching of small regions, which should have a small contribution to the global similarity of both graphs. Also, a term is added to account for the size difference between the regions:

𝑐(𝑢 → 𝑣) = max(𝜂 𝑢 , 𝜂 𝑣 ) ⋅ 𝑑 𝑐 (𝑢, 𝑣) + 𝛾 𝑁𝑜𝑑𝑒𝑆𝑖𝑧𝑒 ⋅ |𝜂 𝑢 -𝜂 𝑣 |
where 𝑑 𝑐 (𝑢, 𝑣) is the distance in the color space, and 𝛾 𝑁𝑜𝑑𝑒𝑆𝑖𝑧𝑒 is a weight parameter selected by cross validation. The distance 𝑑 𝑐 (𝑢, 𝑣) is not computed as the Euclidean distance between RGB vectors, but uses the following definition that is based on the human perception of colors:

𝑑 𝑐 (𝑢, 𝑣) = √ (2 + 𝑟 2 𝑘 )𝛿 2 𝑅 + 4𝛿 2 𝐺 + (2 - (2 𝑘 -1) -𝑟 2 𝑘 )𝛿 2 𝐵
where 𝑘 is the channel depth of the image, 𝑟 = 𝑅𝑢+𝑅𝑣 2 and 𝛿 𝑅 , 𝛿 𝐺 and 𝛿 𝐵 encode respectively the differences of coordinates along the red, green and blue axis.

The cost of a node deletion should be proportional to its relevance encoded by the normalized size, and is thus defined as:

𝑐(𝑢 → 𝜖) = 𝛾 𝑁𝑜𝑑𝑒𝑆𝑖𝑧𝑒 ⋅ 𝜂 𝑢
Using the same basic idea, the cost of an edge removal should be proportional to the minimal normalized size of its two incident nodes.

𝑐((𝑢, 𝑢

′ ) → 𝜖) = 𝛾 𝐸𝑑𝑔𝑒 ⋅ 𝛾 𝐸𝑑𝑔𝑒𝑆𝑖𝑧𝑒 ⋅ min(𝜂 𝑢 , 𝜂 𝑢 ′ )
where 𝛾 𝐸𝑑𝑔𝑒𝑆𝑖𝑧𝑒 encodes the specific weight of the edge removal operation while 𝛾 𝐸𝑑𝑔𝑒 corresponds to a global edge's weight.

Within a region adjacency graph, edges only encode the existence of some common boundary between two regions. Moreover, these boundaries may be drastically modified between two segmentations. Therefore, we choose to base the cost of an edge substitution solely on the substitution's cost of its two incident nodes.

𝑐((𝑢, 𝑢

′ ) → (𝑣, 𝑣 ′ )) = 𝛾 𝐸𝑑𝑔𝑒 ⋅ (𝑐(𝑢 → 𝑣) + 𝑐(𝑢 ′ → 𝑣 ′ ))
Note that all edge costs are proportional to the weight 𝛾 𝐸𝑑𝑔𝑒 . This last parameter allows thus to balance the importance of node and edge costs.

From graph edit distance to graph kernels

Let us consider a set of input graphs {𝐺 1 , . . . , 𝐺 𝑛 } defining our graph test database. Our person re-identification is based on a distance of an input graph 𝐺 from the space spanned by {𝐺 1 , . . . , 𝐺 𝑛 }. Such a measure of novelty detection requires to embed the graphs into a metric space. Given our edit distance (Section 3), one may build a 𝑛×𝑛 similarity matrix 𝑊 𝑖,𝑗 = 𝑒𝑥𝑝(-𝐸𝑑𝑖𝑡𝐶𝑜𝑠𝑡(𝐺 𝑖 , 𝐺 𝑗 )/𝜎) where 𝜎 is a tuning variable. Unfortunately, the edit distance does not fulfill all the requirements of a metric; consequently, the matrix 𝑊 may be not semidefinite and hence does not define a kernel. As mentioned in Section 1, several kernels based on the edit distance have been recently proposed. However, these kernels are rather designed to obtain a definite positive matrix of similarity than to explicitly solve the problem of kernel-based classification or regression methods. We thus use a recent kernel construction scheme [START_REF] Brun | Symbolic learning vs. graph kernels: An experimental comparison in a chemical application[END_REF][START_REF] Gauzere | Graph edit distance and treelet kernels for chemoinformatic[END_REF] based on an original remark by Steinke [START_REF] Steinke | Kernels, regularization and differential equations[END_REF]. This scheme [START_REF] Brun | Symbolic learning vs. graph kernels: An experimental comparison in a chemical application[END_REF][START_REF] Gauzere | Graph edit distance and treelet kernels for chemoinformatic[END_REF] exploits the fact that the inverse of any regularised Laplacian matrix deduced from 𝑊 defines a definite positive matrix and hence a kernel on {𝐺 1 , . . . , 𝐺 𝑛 }. Thus, our kernel construction scheme first builds a regularised Laplacian operator L = 𝐼 + 𝜆𝐿, where 𝜆 is a regularisation coefficient and 𝐿 denotes the normalized Laplacian defined by: 𝐿 = 𝐼 -𝐷 -1 2 𝑊 𝐷 -1 2 and 𝐷 is a diagonal matrix defined by 𝐷 𝑖,𝑖 = ∑ 𝑛 𝑗=1 𝑊 𝑖,𝑗 . Our kernel is then defined as: 𝐾 = L-1 . Using a classification or regression scheme, such a kernel leads to map graphs having a small edit distance [START_REF] Brun | Symbolic learning vs. graph kernels: An experimental comparison in a chemical application[END_REF][START_REF] Gauzere | Graph edit distance and treelet kernels for chemoinformatic[END_REF] (and thus a strong similarity) to close values.

Novelty detection and person re-identification

Within our framework, each reappeared person is represented by a set of graphs encoding the different acquisitions of this person. Before assigning a new input graph to an already created class, we must determine if this graph corresponds to a person already encountered. This is a problem of novelty detection, with the specific constraint that each class of graphs encoding an already encountered person has a large within-class variation. Several methods, such as one class SVM [START_REF] Scholkopf | Estimating the support of a high-dimensional distribution[END_REF] or support vector domain description [START_REF] Tax | Support vector domain description[END_REF] have been used for novelty detection. However, these methods are mainly designed to compare an incoming data with an homogeneous data set. The method of Desobry [START_REF] Desobry | An online kernel change detection algorithm[END_REF] has the same drawback and is additionally mainly designed to compare two sets rather than one set with an incoming datum.

The method introduced by Hoffman [START_REF] Hoffmann | Kernel pca for novelty detection[END_REF] is based on kernel Principal Component Analysis (PCA). An input datum is considered as non belonging to a class if its squared distance from the space spanned by the first principal components of the class is above a given threshold. Note that this method is particularly efficient using high dimensional spaces such as the one usually associated to kernels. This method has the additional advantage of not assuming a strong homogeneity of the class.

Given an input graph 𝐺 and a set of 𝑘 classes, our algorithm first computes the set {𝑑 1 (𝐺), . . . , 𝑑 𝑘 (𝐺)} where 𝑑 𝑖 (𝐺) is the squared distance of the input graph 𝐺 from the space spanned by the first 𝑞 principal component of class 𝑖. Our novelty decision criterion is then based on a comparison of 𝑑(𝐺) = min 𝑘=1,𝑛 𝑑 𝑘 (𝐺) against a threshold.

If 𝑑(𝐺) is greater than the specified threshold, 𝐺 is considered as a new person entering the scene. Otherwise, 𝐺 describes an already encountered person, which is assigned to the class 𝑖 that minimizes the value of 𝑑 𝑖 (𝐺).

Experimental Results

We implemented the proposed method in C++ and tested its performance on two video sequences taken from the PETS2009 [1] database (Fig. 2). Each video sequence is divided in two parts so as to build the training and test sets. As shown on Fig. 3 we obtained around 85% of novelty detection accuracy, and 70% of total accuracy for both View001 and View005 sequences. These results were obtained with the Graph Laplacian Kernel using 𝜎 = 4.7 and 𝜆 = 10.0.

These results appear very promising. For a certain interval of threshold values the classification accuracy rate keeps to the value of 100% Furthermore the True Positive Rate curve has an high slope in correspondence of an high value of the threshold: this means that the method is quite robust. Finally, ROC curves (Fig. 4) show that the algorithm have a good true positives rate, with a quite low false positives rate.

Conclusions

This paper presents a novel method for people re-identification based on a graphbased representation and a graph kernel. It combines our graph kernel with a novelty detection method based on Principal Component Analysis in order to detect if an incoming graph corresponds to a new person and, if not, to correctly assign the identity of a previously seen person. Our future works will also extend the present method to people re-identification within groups. In such cases, a whole group is encoded by a single graph. Thus, the used kernel should be able to match subgraphs within larger graphs. We plan to study the ability of graphlet kernels to perform this task.

Fig. 1 .

 1 Fig. 1. a) Application of a suited morphological operator; b) Extraction of person appearance; c) Image segmentation; d) RAG construction.

Fig. 2 .

 2 Fig. 2. Sample frames from the PETS2009 dataset.

  In this experiment we have used one frame every 2 seconds from each video, in order to have different segmentations of each person. The training set of the first sequence (View001) is composed of 180 graphs divided into 8 classes, while the test set contains 172 graphs (30 new and 142 existing). The second sequence (View005) is composed of 270 graphs divided into 9 classes for the training set, and 281 graphs (54 new and 227 existing) for the test set. In order to evaluate the performances of the algorithm, we have used the following measures: -The true positives rate (TP), i.e the rate of test patterns correctly classified as novel (positive): 𝑇 𝑃 = true positive/total positive -The false positives rate (FP), i.e the rate of test patterns incorrectly classified as novel (positive): 𝑇 𝑃 = false positive/total negative -The detection accuracy (DA): 𝐷𝐴 = (true positive + true negative)/(total positive + total negative) -The classification accuracy (CA), i.e the rate of samples classified as negatives which are then correctly classified with multi-class SVM -The Total Accuracy: 𝑇 𝐴 = 𝐷𝐴 × 𝐶𝐴.

Fig. 3 .

 3 Fig. 3. Performances result on the view001 (a) and view005 (b) of the PETS2009 dataset.

Fig. 4 .

 4 Fig. 4. ROC curves for the two sequences from the PETS2009 dataset.


PhD, assistant professor of epidemiology Mirjam J Knol
email: m.j.knol@umcutrecht.nl

PhD, assistant professor of biostatistics Wiebe R Pestman

PhD Diederick E Grobbee

The (mis)use of overlap of confidence intervals to assess effect modification

Keywords: Confidence interval, overlap, type 1 error probability, effect modification

In randomized controlled trials as well as in observational studies, researchers are often interested in effects of treatment or exposure in different subgroups, i.e. effect modification [1;2]. There are several methods to assess effect modification and the debate on which method is best is still ongoing [START_REF] Wang | Statistics in medicine--reporting of subgroup analyses in clinical trials[END_REF][START_REF] Greenland | Interactions in epidemiology: relevance, identification and estimation[END_REF][START_REF] Pocock | Issues in the reporting of epidemiological studies: a survey of recent practice[END_REF][START_REF] Assmann | Subgroup analysis and other (mis)uses of baseline data in clinical trials[END_REF]. In this article we focus on an invalid method to assess effect modification, which is often used in articles in health sciences journals [START_REF] Schenker | On judging the significance of differences by examining the overlap between confidence intervals[END_REF]: Concluding that there is no effect modification if the confidence intervals of the subgroups are overlapping [START_REF] Ryan | On the misuse of confidence intervals for two means in testing for the significance of the difference between the means[END_REF][START_REF] Altman | Interaction revisited: the difference between two estimates[END_REF][START_REF] Austin | A brief note on overlapping confidence intervals[END_REF].

When assessing effect modification by looking at overlap of the 95% confidence intervals in subgroups, a type 1 error probability of 0.05 is often mistakenly assumed. In other words, if the confidence intervals are overlapping, the difference in effect estimates between the two subgroups is judged to be statistically insignificant. By using mathematical derivation, we calculated that the chance of finding non-overlapping 95% confidence intervals under the null hypothesis is 0.0056 if the variance of both effect estimates is equal and the effect estimates are independent (see Supplemental material for derivation of this probability). If the variance of the effect estimates is not equal, the chance of finding non-overlapping 95% confidence intervals can be calculated by taking into account ρ, i.e. the ratio between the standard deviations in the subgroups, σ 2 / σ 1 (Supplementary material, formula (3)).

Figure 1 shows the relation between ρ and the type 1 error probability if the effect estimates are independent. If the effect estimates are not independent, the correlation coefficient between the effect estimates can also be accounted for (Supplementary material, formula (3)).

To arrive at a type 1 error probability of 0.05, 83.4% confidence intervals should be calculated around the effect estimates in subgroups if the variance is equal and the effect estimates are independent (see Supplementary material for derivation of this percentage). If the variance is not equal, ρ should be taken into account (Supplementary material, formula (11)). Figure 2 shows the relation between ρ and the level of the confidence interval. If the effect estimates are not independent, the correlation coefficient should be taken into account (Supplementary material, formula (11)).

Adapting the level of the confidence interval can be especially useful for graphical presentations, for example in meta-analyses [START_REF] Goldstein | The graphical presentation of a collection of means[END_REF]. However, it is necessary to explicitly and clearly state which percentage confidence interval is calculated and its meaning should be thoroughly explained to the reader. Many readers will still interpret this 'new' confidence interval as if it were a 95% confidence interval, because this percentage is so commonly used. To prevent such confusion, other methods to assess effect modification could be used, such as calculating a 95% confidence interval around the difference in effect estimates [START_REF] Altman | Interaction revisited: the difference between two estimates[END_REF].

The assumption used in the formulas presented in the appendices is that the effect estimators in the subgroups are normally distributed. Assuming that epidemiologic effect measures, such as the odds ratio, risk ratio, hazard ratio and risk difference, follow a normal distribution, the methods presented can also be used for these epidemiologic measures. Note that the assumption for normality is generally unreasonable in small samples, but a satisfactory approximation in large samples.

Example

As an example, imagine a large randomized controlled trial that investigates the effect of some intervention on mortality and that includes 10,000 men and 5,000 women. Besides the main effect of treatment, the researchers are interested in assessing whether the treatment effect is different for men and women. Suppose that the risk ratio in men is 0.67 (95% CI: 0.59-0.75) and in women is 0.83 (95% CI: 0.71-0.98). The confidence intervals are partly overlapping, which the researchers may wrongly interpret as no effect modification by sex. Filling in formula (3) (Supplementary material) results in a probability of non-overlapping 95% confidence intervals under the null hypothesis of 0.006. A confidence level of 83.8% could have been calculated to arrive at a type 1 error probability of 0.05, resulting in a confidence interval of 0.61-0.73 for men and 0.74-0.93 for women. Now, the confidence intervals do not overlap, so the p-value is at least smaller than 0.05, indicating statistically significant effect modification. Calculating the difference in risk ratios with a 95% confidence interval results in a ratio of risk ratios of 0.80 with a 95% confidence interval of 0.66-0.98, corresponding to a p-value of 0.028. This confirms our earlier observation of statistically significant effect modification.

Figure legends

Figure 1 Relation between ρ, which is the ratio of σ 2 and σ 1 , and the probability of non-overlapping confidence intervals under the null hypothesis (type 1 error)

Figure 2

Relation between ρ, which is the ratio of σ 2 and σ 1 , and the percentage confidence intervals to be calculated to arrive at a type 1 error probability of 0.05

Acknowledgements

This study was performed in the context of the Escher project (T6-202), a project of the Dutch Top Institute Pharma.


Janne Jane Lindschou Hansen

Majken Schurmann Tolstrup

Karoline Jensen

Morten Grønbaek

Anne Tjønneland

Erik Berg Schmidt

Kim Overvad

Janne S Tolstrup

Majken K Jensen

Alcohol intake and risk of acute coronary syndrome and mortality in men and women with and without hypertension

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

Introduction

Hypertension is a serious public health problem, causing almost 13% of all deaths worldwide and 17% of all deaths in high income countries [START_REF]Mortality and burden of disease attributable to selected major risks[END_REF]. Heavy alcohol consumption is associated with an increased risk for hypertension [START_REF] Keil | Alcohol, blood pressure and hypertension[END_REF], and hypertension in itself is a major risk factor for coronary heart disease [START_REF] Yusuf | Effect of potentially modifiable risk factors associated with myocardial infarction in 52 countries (the INTERHEART study): case-control study[END_REF]. On the other hand, substantial epidemiological evidence suggests that alcohol intake has beneficial effects on the cardiovascular system, causing a lower risk of coronary heart disease among light to moderate alcohol drinkers compared with abstainers [START_REF] Collins | Alcohol in moderation, cardioprotection, and neuroprotection: epidemiological considerations and mechanistic studies[END_REF][START_REF] Corrao | Alcohol and coronary heart disease: a meta-analysis[END_REF]. The beneficial effect is believed mainly to be due to increasing levels of high-density lipoprotein cholesterol, decreased fibrinogen and increased insulin sensitivity [START_REF] Agarwal | Cardioprotective effects of light-moderate consumption of alcohol: a review of putative mechanisms[END_REF][START_REF] Mukamal | Drinking frequency, mediating biomarkers, and risk of myocardial infarction in women and men[END_REF][START_REF] Rimm | Alcohol and Coronary Heart Disease: Drinking Patterns and Mediators of Effect[END_REF].

The association between alcohol intake and coronary heart disease and total mortality with respect to hypertension has been examined in earlier studies [START_REF] Beulens | Alcohol consumption and risk for coronary heart disease among men with hypertension[END_REF][START_REF] Bos | Alcohol consumption and risk of cardiovascular disease among hypertensive women[END_REF][START_REF] Britton | Relation of alcohol consumption and coronary heart disease in hypertensive male physicians (from the Physicians' Health Study)[END_REF][START_REF] Freiberg | Alcohol consumption, hypertension, and total mortality among women[END_REF][START_REF] Malinski | Alcohol consumption and cardiovascular disease mortality in hypertensive men[END_REF][START_REF] Palmer | Alcohol intake and cardiovascular mortality in hypertensive patients: report from the Department of Health Hypertension Care Computing Project[END_REF][START_REF] Reims | Alcohol consumption and cardiovascular risk in hypertensives with left ventricular hypertrophy: the LIFE study[END_REF], but important questions remain. First, apart from one these studies were restricted to individuals with hypertension, thus risks comparison with individuals without hypertension is not possible [START_REF] Reims | Alcohol consumption and cardiovascular risk in hypertensives with left ventricular hypertrophy: the LIFE study[END_REF]. Second, none of the studies investigating the risk of coronary heart disease included both men and women. Women differ from men in alcohol pharmacokinetics (i.e., processing and elimination of alcohol in the body) and the level of alcohol intake beyond which the risk of adverse effects increases is lower in women than in men [START_REF] White | Alcohol consumption and mortality: modelling risks for men and women at different ages[END_REF][START_REF] Mumenthaler | Gender differences in moderate drinking effects[END_REF]. Also, sex differences in the effect of a range of cardiovascular risk factors, including hypertension, have been observed, which further underlines the importance of studying the differences between men and women [START_REF] Anand | Risk factors for myocardial infarction in women and men: insights from the INTERHEART study[END_REF].

Our aim was to address if the current sensible drinking guidelines for moderate drinking apply to middle-aged women and men regardless of whether they have hypertension or not. Therefore, we investigated the association between alcohol intake and the incidence of acute coronary syndrome and total mortality in a large Danish cohort of men and women with and without hypertension.

Methods

The Diet, Cancer and Health study Between December 1993 and May 1997, 160,725 Danish adults aged 50-64 years were invited to participate in the Diet, Cancer and Health study. Eligible participants were born in Denmark and had no previous cancers. In total, 35% of the invited agreed to participate (27,178 men and 29,875 women).

Prior to a clinic visit, the participants filled in a comprehensive 192-item food frequency questionnaire (FFQ) including questions about amount of alcohol intake. While at the clinic, they filled in an additional questionnaire concerning health behavioural and social factors, as well as questions regarding the frequency of alcohol intake. Both questionnaires were checked for misunderstandings and inconsistencies by an interviewer. Anthropometrical measures and blood pressure were taken on site [START_REF] Tjonneland | Study design, exposure variables, and socioeconomic determinants of participation in Diet, Cancer and Health: a population-based prospective cohort study of 57,053 men and women in Denmark[END_REF].

Alcohol intake

In the food frequency questionnaire, participants reported average alcohol intake during the past year. Intake was reported for 6 different types of alcoholic beverages: Light, regular and strong beer (in bottles), wine (in glasses), fortified wine (in drinks) and spirits (in drinks) in predefined categories from "never" to "8 or more" drinks per day. The different types of alcoholic beverages had different alcohol content: light beer, 8.9 g ethanol; regular beer, 12.2 g ethanol; strong beer, 17.5 g ethanol; wine, 12.2 g ethanol; fortified wine, 9.3 g ethanol; and spirits 9.9 g ethanol [START_REF] Tjonneland | Alcohol intake, drinking patterns and risk of postmenopausal breast cancer in Denmark: a prospective cohort study[END_REF].

The alcohol content for each type was summed up, and a weekly amount of alcohol intake was calculated in number of drinks containing 12 g alcohol each. Drinking frequency was reported in the background questionnaire in predefined categories; "never", "less than once a month", "1-3 times monthly", "once a week", "2-4 times weekly", "5-6 times weekly" and "daily".

Hypertension

In the background questionnaire, participants were asked if they had been diagnosed with hypertension. Participants, who had answered yes, were categorized as having "self-reported hypertension". Among these, participants who additionally answered that they were or had been taking any medication for hypertension were categorized as "using medication". In addition, participants had their blood pressure measured during the clinic visit. Blood pressure was measured once in a lying position after 5 minutes of rest [START_REF] Tjonneland | Study design, exposure variables, and socioeconomic determinants of participation in Diet, Cancer and Health: a population-based prospective cohort study of 57,053 men and women in Denmark[END_REF]. Participants who had a systolic blood pressure above 140 mm Hg and diastolic blood pressure above 90 mm Hg were categorized as having "measured hypertension" [START_REF] Pickering | Recommendations for blood pressure measurement in humans and experimental animals: Part 1: blood pressure measurement in humans: a statement for professionals from the Subcommittee of Professional and Public Education of the American Heart Association Council on High Blood Pressure Research[END_REF]. We used both selfreports and measures of high blood pressure to define hypertension. In total, 8,722 (34%) men and 8,216 (28%) women were categorized as having hypertension (1,506 men and 2,860 women by selfreport only, 4,966 men and 3,171 women by measurement of blood pressure only, and 2,250 men and 2,185 women by both criteria).

Follow-up

Information about incident cases of acute coronary syndrome (ACS), including fatal and non-fatal myocardial infarction (MI), sudden cardiac death and unstable angina pectoris was obtained by linkage with central Danish registries of hospital discharges and causes of death (ICD-8 codes 410-410.99, 427.27 and ICD-10 codes I20.0, I21.0-21.9, I46.0-46.9). ACS cases were identified between baseline and January 1, 2004, which is the date of the last available update from the Hospital Discharge Register. As described elsewhere [START_REF] Joensen | Predictive values of acute coronary syndrome discharge diagnoses differed in the Danish National Patient Registry[END_REF], all discharge diagnoses were retrieved and reviewed to ensure that they met the criteria for ACS, proposed by the American Heart Association and the European Society of Cardiology [START_REF] Luepker | Case definitions for acute coronary heart disease in epidemiology and clinical research studies: a statement from the AHA Council on Epidemiology and Prevention; AHA Statistics Committee; World Heart Federation Council on Epidemiology and Prevention; the European Society of Cardiology Working Group on Epidemiology and Prevention; Centers for Disease Control and Prevention; and the National Heart, Lung, and Blood Institute[END_REF]. Our validation study confirmed that MI is recorded with a high degree of validity in this register (positive predictive value for diagnoses received upon discharge from a ward =92%), whereas unstable angina and cardiac arrest were less accurate (<=50%) [START_REF] Joensen | Predictive values of acute coronary syndrome discharge diagnoses differed in the Danish National Patient Registry[END_REF]. Through the Causes of Death Registry, we additionally identified 111 participants with fatal MI.

In a sub-analyses, we excluded participants who had received the ACS diagnosis based on reports of unstable angina or cardiac arrest (n=235). This analysis did not change the results using the global ACS diagnosis, which we report in the present paper.

Person-years are recorded from the date of participant enrolment to the date of coronary event (n=1,131), death from other causes (n=2,410), emigration (n=215), loss to follow-up (n=5) or January 1, 2004 which was the last day of follow-up, whichever came first.

Total mortality was the secondary outcome in this study. Cases were identified through the Danish Civil Registration System where vital status is registered for every citizen. The cohort is currently followed up until July 2006, and person-years are recorded from the date of enrolment to this date or date of censoring. Prior to this date, 4,210 people died, 311 emigrated and 6 were lost to follow-up.

Statistical analysis

We calculated hazard ratios (HR) by Cox proportional hazard regression analysis using the statistical package STATA version 10. Age was used as time axis with delayed entry implemented to ensure maximal adjustment for confounding by age. We addressed the joint effects of alcohol intake and BP status on risk of ACS by using light drinking participants (1-6 drinks per week) with normal blood pressure as reference category. Analyses were performed for the exposures weekly alcohol intake and drinking frequency, and for the outcomes ACS and total mortality.

For ACS we calculated crude risk estimates and risk estimates adjusted for the following potential confounders: Smoking (never, ex (stop before/after age 30), 1-14, 15-24, or 25+ g tobacco per day), body mass index (BMI, <18.5, 18.5-25, 25-30, 30+ kg/m 2 ), waist circumference (cm), school education ( 7, 8-10, or 10+ years in school), physical activity in leisure time (hours per week), intake of fruit (quartiles, grams), vegetables (grams), fish (grams), fat (quartiles, % of total energy intake), saturated fat (% of total energy intake), and menopausal status and use of hormone replacement therapy (HRT) in women (pre-, unsure, postmenopausal not using HRT and postmenopausal using HRT), self-reported diabetes mellitus (yes/no) and self-reported hypercholesterolemia (yes/no).

We tested the proportional hazards assumption and no violations were found. When testing for linear trend, we analyzed the median value within categories continuously. Statistical interactions between alcohol intake and hypertension, and alcohol intake and sex were assessed on the multiplicative scale by comparing the log likelihoods in nested models with and without crossproduct terms.

To test whether participants taking medication for hypertension had similar risks of ACS and total mortality as those with normal blood pressure, we compared the relative risks according to alcohol intake in participants with hypertension that were medicated to those classified as normotensives. We also compared those with self-reported hypertension and those with measured hypertension to those with normal blood pressure.

To examine the possible bias due to undiagnosed cardiovascular illness at baseline, which might have influenced the participants" alcohol intake, we carried out analyses for the first 2 and 4 years of follow-up, which we compared with analyses of the subsequent years of follow-up.

We excluded participants with missing information on relevant questionnaire data (n=484), ACS diagnosis before baseline (n=994) and participants who were registered with an ACS diagnosis during follow-up, but whose medical record could not be retrieved and whose endpoint therefore could not be validated were excluded (n=113). In total, 26,035 men and 29,427 women were eligible for further analyses.

Results

Five percent of the men and 16% of the women were nondrinkers (<1 drink/week). Among the remaining participants the median alcohol intake was 12.1 drinks/week for men and 6.6 drinks/week for women. Participants in the lowest and in the highest drinking categories were more likely to be smokers and to suffer from hypercholesterolemia than participants in the inbetween drinking categories (Table 1). The prevalence of diabetes and a short school education were highest among the nondrinkers. The above trends were the same for women and men.

During a mean follow-up of 7.6 years, 860 and 271 incident cases of ACS occurred among men and women. In adjusted analyses, the amount of alcohol intake was inversely associated with risk of ACS, however only statistically significant among men (p for trend: men<0.001, women=0.11) (Table 2). The lowest risk was observed among the men and women in the highest alcohol drinking categories (28+ and 21+ drinks/week, corresponding to 48+ and 36+ grams/day).

Compared to light drinkers (1-6 drinks/week), the risks among men and women in the top alcohol consumption groups were 0.68 (95% CI: 0.53-0.91) and 0.64 (95% CI: 0.38-1.09), respectively.

We next studied the associations between alcohol intake and ACS among men and women with and without hypertension. In men and women, a higher alcohol intake was associated with a lower risk of ACS regardless of hypertensive status, however only statistically significant among men (Table 3). Approximately half of the ACS events occurred among participants with hypertension (47% (n=402) among men and 51% (n=137) among women). We found no evidence of interaction between sex and alcohol intake on risk of ACS in participants with or without hypertension (p=0.40 and p=0.56). To study the associations between alcohol intake and ACS risk according to different definitions of hypertension, we modelled men and women together with a joint reference group consisting of light drinkers without hypertension (Figure 1). Hypertension was investigated by selfreports (all), by blood pressure measurements (alone), and by selfreported use of antihypertensive medication. For all definitions of hypertensive status, we observed an inverse relationship between alcohol intake and risk of ACS (Figure 1). Compared with participants in the same alcohol category without hypertension, participants in the hypertension group consistently had a higher risk of ACS, at all levels of alcohol intake. There was no evidence of interaction between alcohol intake and hypertension in any of the models. Compared with participants with normal blood pressure, the relative risks for ACS among participants according to different definitions of hypertension were 1.63 (95% CI, 1.44-1.85; self-report and measurements combined), 1.46 (95% CI, 1.18-1.81; self-report (all)), 1.47 (95% CI, 1.26-1.72; measured (only)), and 1.79 (95% CI, 1.52-2.11; use of antihypertensive medication).

All analyses were repeated incorporating alcohol drinking frequency instead of the average amount of alcohol intake. Results were comparable to results reported above using quantity and we saw no signs of interactions between alcohol intake frequency and hypertension status.

All-cause mortality risk according to alcohol intake was J-shaped in men and women (data not shown). In analyses including hypertension, the association between alcohol and risk of mortality was also J-shaped and the risk among participants with hypertension was higher than for participants without hypertension at all levels of alcohol intake (Figure 2). Finally, to examine the possibility that baseline symptoms of preclinical disease might have caused a reduction in alcohol intake, and thereby biasing the results, we conducted analyses comparing the association between alcohol intake and ACS only including early cases (i.e. cases that occurred within the first two years of follow up, n=212) with the association including only later cases (cases n=919). There was an inverse association between drinking frequency and coronary heart disease in both groups. For early cases, multivariate hazard ratios for men and women who drank 21-27 drinks/week and 28+ drinks/week were 0.59 (0.31-0.89) and 0.53 (0.31-0.89) compared with drinking 1-6 drinks/week, and for later cases, multivariate hazard ratios for drinking 21-27 drinks/week and 28+ drinks/week were 0.74 (0.57-0.95) and 0.74 (0.58-0.93).

Discussion

In this prospective study of middle-aged men and women, moderate alcohol intake was associated with a lower risk of acute coronary syndrome (ACS) in men and women with and without hypertension. For all studied associations (amount and frequency of alcohol intake), the risk of ACS and of all-cause mortality among participants with hypertension was higher than among participants without hypertension. There was no indication of interaction between alcohol intake and hypertension on the risk of ACS.

To our knowledge, this is the first study that includes both men and women, includes both normotensives and hypertensives, and assesses both incident ACS and total mortality as outcome measures.

In agreement with our results, previous studies including individuals with hypertension reported inverse associations between alcohol intake and coronary heart disease [START_REF] Beulens | Alcohol consumption and risk for coronary heart disease among men with hypertension[END_REF][START_REF] Bos | Alcohol consumption and risk of cardiovascular disease among hypertensive women[END_REF][START_REF] Britton | Relation of alcohol consumption and coronary heart disease in hypertensive male physicians (from the Physicians' Health Study)[END_REF][START_REF] Reims | Alcohol consumption and cardiovascular risk in hypertensives with left ventricular hypertrophy: the LIFE study[END_REF]. No comparisons were made with normotensive individuals. Total mortality has been investigated in prior studies [START_REF] Beulens | Alcohol consumption and risk for coronary heart disease among men with hypertension[END_REF][START_REF] Freiberg | Alcohol consumption, hypertension, and total mortality among women[END_REF][START_REF] Malinski | Alcohol consumption and cardiovascular disease mortality in hypertensive men[END_REF][START_REF] Palmer | Alcohol intake and cardiovascular mortality in hypertensive patients: report from the Department of Health Hypertension Care Computing Project[END_REF][START_REF] Renaud | Moderate wine drinkers have lower hypertension-related mortality: a prospective cohort study in French men[END_REF], but with inconsistent conclusions about the potential beneficial effects of a light-to-moderate alcohol intake in men with hypertension [START_REF] Beulens | Alcohol consumption and risk for coronary heart disease among men with hypertension[END_REF][START_REF] Palmer | Alcohol intake and cardiovascular mortality in hypertensive patients: report from the Department of Health Hypertension Care Computing Project[END_REF]. We found that moderate drinkers had a lower all-cause mortality than non-drinkers and heavy drinkers, which seems to apply to both men and women with normal blood pressure and hypertension. Furthermore, we did not find any evidence of interaction between alcohol intake and hypertension which is in agreement with results of another study [START_REF] Bos | Alcohol consumption and risk of cardiovascular disease among hypertensive women[END_REF]. This was consistently observed among men and women, for ACS and total mortality, for different definitions of hypertension and for alcohol intake, evaluated both as average quantity and frequency. This consistency strengthens the findings, and is therefore less likely to be caused by bias.

A strength of our study is the Danish complete registration system, which allows us to have complete follow-up and that we used validated information on ACS. The coverage of the Danish registration system is very high, with virtually every event recorded [START_REF] Andersen | The Danish National Hospital Register. A valuable source of data for modern health sciences[END_REF][START_REF] Juel | The Danish registers of causes of death[END_REF]. All discharge diagnoses of ACS were validated and only confirmed cases were included in the present analyses.

Due to limited information available, the 111 ACS diagnoses obtained from the Cause of Death Register are less accurate.

Another strength in our study is the large variation in alcohol intake. This means that we were able to describe the risk of ACS and total mortality associated with a wide range of alcohol intake and making results applicable to the whole Danish population.

A limitation in our study is the study participation rate of only 35%. People who chose to participate may have a different risk profile and be at better health than those who declined participation. A comparison between study participants and those who declined, showed that study participants on average had a higher income, higher level of education, larger size housing and were more likely to be married [START_REF] Tjonneland | Study design, exposure variables, and socioeconomic determinants of participation in Diet, Cancer and Health: a population-based prospective cohort study of 57,053 men and women in Denmark[END_REF]. This confirms the assumption that lower socioeconomic groups are underrepresented in epidemiological studies. However, the observed incidence of coronary heart disease within the study cohort did not differ from that of the general Danish population [START_REF] Tolstrup | Prospective study of alcohol drinking patterns and coronary heart disease in women and men[END_REF]. We observed similar tendencies among men and women, although, the number of cases was substantially lower among women than among men, and hence results for women are less certain and warrant further study.

Alcohol intake was reported in a food frequency questionnaire. The food frequency questionnaire has been found valid when comparing it to formal dietary interviews in Danish populations [START_REF] Grønbaek | Validity of self-reported intakes of wine, beer and spirits in population studies[END_REF]. However, alcohol intake in other studies has shown to change over time. One study reported that the U-shaped association between alcohol and total mortality shifted after adjusting for intake variation [START_REF] Emberson | Alcohol intake in middle age and risk of cardiovascular disease and mortality: accounting for intake variation over time[END_REF], while another study reported the U-shaped association to be consistent for those with a stable alcohol intake [START_REF] Grønbaek | Changes in alcohol intake and mortality: a longitudinal population-based study[END_REF]. In our study, we applied a single measure of alcohol intake, which could potentially bias our results. Also, alcohol intake could be misclassified among the participants who were diagnosed with high blood pressure during the clinic visit: it is possible, that such participants change their lifestyle and habits, including alcohol intake, due to this information. However, Beulens et al. measured alcohol intake several times, and they found that alcohol intake before and after diagnosis of hypertension was highly correlated [START_REF] Beulens | Alcohol consumption and risk for coronary heart disease among men with hypertension[END_REF]. Due to the low numbers of cases in the non-drinking category, it was not possible to analyse never drinkers and ex-drinkers separately. If the group of non-drinkers primarily consists of exdrinkers, the excess risk associated with non-drinking might be due to the fact that ex-drinkers could have been "sick quitters" [START_REF] Wannamethee | Lifelong teetotallers, ex-drinkers and drinkers: mortality and the incidence of major coronary heart disease events in middle-aged British men[END_REF]. The analyses were not adjusted for participants" medical history, which means that the increased risk of ACS and total mortality among nondrinkers could be caused by latent illness. However, participants with a history of ACS were excluded so this is unlikely to be the case. Also, we did measure alcohol intake in many different categories, and the inverse associations were not only driven by a relatively high risk of ACS and death among nondrinkers. Furthermore, when analysing early and later cases separately (defined as cases occurring before and after two years of follow up), inverse associations between alcohol and ACS were consistently observed, and it is thus unlikely that the sick-quitting are causing the observed results.

We avoided the term "binge drinking", which in most studies is defined as drinking a minimum number of drinks per occasion such as 6 or 13 [START_REF] Laatikainen | Increased mortality related to heavy alcohol intake pattern[END_REF][START_REF] Malyutina | Relation between heavy and binge drinking and all-cause and cardiovascular mortality in Novosibirsk, Russia: a prospective cohort study[END_REF], because the participants were not asked directly about occasional heavy drinking, and we can therefore not comment on this.

We used very similar definitions of hypertension as Bos et al. and Britton [10,[START_REF] Britton | Relation of alcohol consumption and coronary heart disease in hypertensive male physicians (from the Physicians' Health Study)[END_REF]. However, obtaining information about participants" hypertension status can be problematic. The measurements can be misclassified because it was only measured once, and some people tend to have elevated blood pressure in the presence of a physician, so called white coat hypertension [START_REF] Pickering | Recommendations for blood pressure measurement in humans and experimental animals: Part 1: blood pressure measurement in humans: a statement for professionals from the Subcommittee of Professional and Public Education of the American Heart Association Council on High Blood Pressure Research[END_REF].

Thus some individuals might be wrongly categorized as hypertensive at baseline. Despite these possibilities of measurement error, we found consistent results, when analyzing people with measured hypertension, self-reported hypertension and self-reported treatment for hypertension separately. When studying alcohol intake and risk of CHD, it is a concern that any decrease in risk of coronary heart events due to alcohol intake will be outweighed by an increase in other diseases or problems related to alcohol intake, e.g., liver diseases, cancers and road crashes. Our study showed that the overall risk of mortality was lowest for participants with a moderate alcohol intake, and it therefore appears that the cardioprotective effect outweighs the damaging effects associated with moderate alcohol intake. This result is, however, restricted to middle-aged men and women.

Conclusions

We found inverse associations between alcohol intake and ACS and a J-shaped association for total mortality among both hypertensives and individuals with normal blood pressure. This implies that the current sensible drinking guidelines for moderate drinking also apply to middle-aged women and men regardless of whether they have hypertension or not. This result further implies that people with hypertension who drink alcohol lightly to moderately do not need to reduce their alcohol intake in order to reduce their risk of ACS. It also implies that those who drink heavily should reduce their alcohol intake in order to reduce their risk of mortality from all causes. However, this does not mean that nondrinkers should be encouraged to drink alcohol in order to prevent ACS and prolong life: many detrimental consequences are associated with alcohol intake [START_REF] Pickering | Recommendations for blood pressure measurement in humans and experimental animals: Part 1: blood pressure measurement in humans: a statement for professionals from the Subcommittee of Professional and Public Education of the American Heart Association Council on High Blood Pressure Research[END_REF], and it would be unethical to advise people to increase their alcohol intake or even to start drinking alcohol if they do not do so already.

Figure legends

Fig. 1 Hazard ratios of acute coronary syndrome (ACS) relative to joint reference category of no hypertension and alcohol intake of 1-6 drinks/week in men and women combined and according to different definitions of hypertension. Multivariable adjusted for age, sex, smoking, school education, BMI, waist circumference, physical activity, intake of fruit, vegetables, fish, fat and saturated fat, hypercholesterolemia, diabetes, and for women: menopausal status and use of HRT.

 Squares represent participants with hypertension  Circles represent participants with normal blood pressure Fig. 2 Hazard ratios of all-cause mortality relative to joint reference category of no hypertension and alcohol intake of 1-6 drinks/week in men and women and according to hypertension.

Multivariable adjusted for age, sex, smoking, school education, BMI, waist circumference, physical activity, intake of fruit, vegetables, fish, fat and saturated fat, hypercholesterolemia, diabetes, and for women: menopausal status and use of HRT.

 Squares represent participants with hypertension

 Circles represent participants with normal blood pressure Adjusted for age, smoking, school education, BMI, waist circumference, physical activity, intake of fruit, vegetables, fish, fat and saturated fat, hypercholesterolemia, diabetes, and for women: menopausal status and use of HRT

Fig. 1

 1 Fig.1

Fig. 2 0

 2 Fig.2

Table 2

 2 Hazard ratios (95% confidence intervals) for acute coronary syndrome according to amount of alcohol

	Alcohol Intake (drinks per week a )

a 1 drink corresponds to 12 grams of pure alcohol b Upper category for women was 21+ drinks per week c P for trend: Median values within categories analyzed continuously d

Table 3

 3 Hazard ratios (95% Confidence Intervals) for acute coronary syndrome according to hypertension status and amount of alcohol intake in men and women 1 drink corresponds to 12 grams of pure alcohol; b Upper category for women is 21+ drinks per week c P for trend: Median values within categories analyzed continuously d Adjusted for age, smoking, school education, BMI, waist circumference, physical activity, intake of fruit, vegetables, fish, fat and saturated fat, hypercholesterolemia, diabetes, and for women: menopausal status and use of HRT

	Alcohol Intake (drinks per week a )

a

Acknowledgements

This study was supported by grants from the Health Insurance Foundation and the Danish Cancer Society.