Patrick Brzoska
email: patrick.brzoska@uni-bielefeld.de

Sven Voigtländer

Jacob Spallek

Oliver Razum

Utilization and effectiveness of medical rehabilitation in foreign nationals residing in Germany

Keywords: Effectiveness, foreign nationals, Germany, migrants, rehabilitation, utilization

In Germany, the proportion of foreign national residents receiving an invalidity pension is higher than that of Germans. Lower utilization and effectiveness of medical rehabilitation are presumed to be the main reasons. We aimed to examine whether differences in utilization and effectiveness of medical rehabilitation between Germans and foreign nationals are attributable to differences in socio-demography, socioeconomic background and health status. Utilization of rehabilitation was analyzed for household members aged 18 years or above enrolled in the German Socio-Economic Panel in 2002-2004 (n=19,521). Effectiveness of rehabilitation was defined by the occupational performance at the end of rehabilitation. It was examined by using an 80% random sample of all completed medical rehabilitations in the year 2006 funded by the German Statutory Pension Insurance Scheme (n=634,529). Our study shows that foreign nationals utilize medical rehabilitation less often than Germans (OR=0.68; 95%-CI=0.50;0.91). For those who do, medical rehabilitation is less effective (OR for low occupational performance=1.50; 95%-CI=1.46;1.55). Both findings are only partially attributable to socio-demographic, socio-economic and health characteristics: After adjusting for these factors, ORs for utilization and low occupational performance were 0.66 (95%-CI=0.49;0.90) and 1.20 (95%-CI=1.16;1.24), respectively. It can be concluded that differences in the utilization and effectiveness of medical rehabilitation between Germans and foreign nationals cannot be explained only by socio-economic differences or poorer health before rehabilitation. In addition, factors such as the ability of the rehabilitative care system to accommodate clients with differing expectations, and migrant-specific characteristics such as cultural differences, seem to play a role.

Introduction

About 7.3 million people residing in Germany (9% of the total population) have a foreign nationality [START_REF]Bevölkerung und Erwerbstätigkeit. Bevölkerung mit Migrationshintergrund. Ergebnisse des Mikrozensus 2007[END_REF]. They or their parents have migrated to Germany, many as "guest workers", in the 1950s and 1960s. Foreign nationals differ from Germans in different health aspects, for example because of the more straining jobs many of them held, but also because they face barriers to access in the health care system [START_REF] Razum | Migration und Gesundheit[END_REF]. The consequences are evident from the higher proportion of foreign nationals receiving an invalidity pension, compared to the proportion of Germans of a similar age [START_REF] Höhne | Erwerbsminderungsrenten und medizinische Rehabilitation in Deutschland unter Berücksichtigung des Migrationshintergrunds[END_REF]. Aside from different working conditions, two explanations have been proposed for this finding. First, utilization of medical rehabilitationa measure of tertiary prevention aiming to mitigate consequences of disease and disabilityis considerably lower in foreign as compared to German nationals [START_REF] Meinlschmidt | Basisbericht 2006/2007. Daten des Gesundheits-und Sozialwesens[END_REF][START_REF] Rommel | Migration und Rehabilitation psychischer Erkrankungen -Perspektiven und Grenzen einer Gesundheitsberichterstattung mit Routinedaten[END_REF][START_REF] Korporal | Die Gesundheit von Migrantinnen und Migranten als Voraussetzung für Beschäftigungsfähigkeit im Alter. Expertise für den 5[END_REF][START_REF] Hackhausen | Arbeitsmigration und soziale Absicherung in Deutschland[END_REF]. Second, those foreign nationals who attend medical rehabilitation do not benefit by this health care intervention in the same way as Germans do: On average, effectiveness of medical rehabilitation is lower for foreign nationals than it is for Germans [START_REF] Mösko | Beeinflusst der türkische Migrationshintergrund das Behandlungsergebnis? Ergebnisse einer prospektiven Versorgungsstudie in einer stationären Rehabilitation von Patienten mit psychischen/psychosomatischen Störungen[END_REF][START_REF] Maier | Migration und rehabilitative Versorgung in Deutschland -Inanspruchnahme von Leistungen der medizinischen Rehabilitation und Rehabilitationserfolg bei Personen mit türkischem Migrationshintergrund[END_REF][START_REF] Nickel | Behandlungsergebnisse stationärer psychosomatischer Rehabilitation bei türkischen Migranten: Eine prospektive Studie[END_REF][START_REF] Schmeling-Kludas | Stationäre psychosomatische Rehabilitation für türkische Migranten: Was ist realisierbar, was ist erreichbar?[END_REF].

However, previous results have shortcomings which possibly limit their validity. As regards utilization, the studies either do not adjust for confounding factors such as age differences [6;7], do only use data from particular regions of Germany [START_REF] Meinlschmidt | Basisbericht 2006/2007. Daten des Gesundheits-und Sozialwesens[END_REF], or restrict their analyses to selected diagnostic groups [START_REF] Rommel | Migration und Rehabilitation psychischer Erkrankungen -Perspektiven und Grenzen einer Gesundheitsberichterstattung mit Routinedaten[END_REF]. Available studies on the effectiveness of rehabilitation face similar limitations. All have small sample sizes, and in some, patients were sampled from one rehabilitation clinic only [6;8;10]. Another study used representative routine data but operationalized the effectiveness of rehabilitation by means of treatment progression [START_REF] Maier | Migration und rehabilitative Versorgung in Deutschland -Inanspruchnahme von Leistungen der medizinischen Rehabilitation und Rehabilitationserfolg bei Personen mit türkischem Migrationshintergrund[END_REF] which is an indicator prone to social desirability bias and ceiling effects [START_REF]Rehabilitationen[END_REF]. In addition, these studies only controlled for a limited set of confounding variables.

Considering the shortcomings of previous research, our study had two objectives.

First, we wanted to assess possible differences in utilization and effectiveness of medical rehabilitation between Germans and foreign nationals by using a national sample and representative data as well as a more robust indicator of rehabilitation effectiveness. Second, we aimed to examine whether possible differences in these outcomes are solely attributable to differences in the socio-demography, socioeconomic background and health status of the two population groups or whether there are additional determinants that need to be considered.

Methods

We used two different data sources to examine determinants of the utilization and effectiveness of medical rehabilitation.

Utilization of medical rehabilitation

The utilization of medical rehabilitation was analyzed using data from the German Socio-Economic Panel (SOEP). The SOEP is a national longitudinal household survey which was set up in 1984. The initial sample contained approximately 12,000 adults in 6,000 households, among which households headed by a foreign national were oversampled. Later on, several additional samples were added. Wagner et al. provide further information on the SOEP, its data quality and the methods used [START_REF] Wagner | The German Socio-Economic Panel Study (SOEP) -Scope, Evolution and Enhancements[END_REF].

In the current analysis, we included household members aged 18 years or above who took part in the survey during the years 2002 and 2004 (n=19,521). Choosing a three-year study period was necessary in order to use information on the utilization of medical rehabilitation in the year before the survey (i.e. 2003) as well as to consider potentially influencing variables prior to rehabilitation (i.e. 2002). Our outcome measure was the utilization of medical rehabilitation in the year 2003.

Socio-demographic and socio-economic status (SES) was assessed by age (in years), sex as well as educational, professional and income status. Educational status was measured by the Comparative Analyses of Social Mobility in Industrial Nations (CASMIN) classification [START_REF] Brauns | The CASMIN educational classification in international comparative research[END_REF] and categorized as "high", "intermediate", "low", "still in school" and "not specified". The International Standard Classification of Occupations from 1988 (ISCO-88) [START_REF] Elias | Occupational Classification (ISCO-88): Concepts, Methods, Reliability, Validity and Cross-National Comparabilty[END_REF] was used to measure professional status, categorized as "white collar", "blue collar" and "not applicable". Members of armed forces were coded as not applicable and not included in the analysis. Income status was defined as the annual net equivalence income (adjusted for the rental value of owner occupied housing) according to the modified OECD-scale [START_REF] Grabka | Codebook for the $PEQUIV File 1984-2006. CNEF Variables with Extended Income Information for the SOEP[END_REF]. Missing values for income status had been imputed by the SOEP Study Group prior to the data release [START_REF] Grabka | Imputation of Item-Non-Response on Income Questions in the SOEP 1984-2002[END_REF]. For analysis, we log-transformed the income variable in order to obtain a normal distribution. Self-rated health (based on a five-point Likert scale ranging from "very good" to "bad" following the example of the SF-36 Health Survey questionnaire [START_REF] Ware | The MOS 36-item short-form health survey (SF-36). I. Conceptual framework and item selection[END_REF]) was used as a measure of health status and was dichotomized for analysis into "good" (comprising the categories "very good", "good" and "satisfactory" health) and "poor" health (comprising the categories "poor" and "bad" health).

Effectiveness of medical rehabilitation

In order to examine effectiveness of medical rehabilitation, we used a dataset provided by the German Statutory Pension Insurance Scheme (Deutsche Rentenversicherung) which accounts for about two-thirds of all medical rehabilitations provided in Germany [START_REF] Bengel | Grundlagen der Rehabilitationswissenschaften: Themen, Strategien und Methoden der Rehabilitationsforschung[END_REF] and is considered highly valid [START_REF] Bestmann | Datenquellen und Datenqualität der Reha-Statistik-Datenbasis[END_REF]. The dataset contained information on all individuals (n=794,163) who completed medical rehabilitation in the year 2006 granted by this insurer (pensioners and insurees not participating in the labor market are usually not provided with rehabilitation services from this insurance scheme). For reasons of data protection, an 80% random sample of this dataset has been made available for analysis (n=634,529). Aside from a comparison of German and foreign nationals, the data allowed to stratify foreign nationality by different countries of origin, comprising Turkey, Former Yugoslavia and the Mediterranean countries Portugal, Spain, Italy and Greece. The assessment of the outcome variable rehabilitation effectiveness was based on a medical judgment regarding the occupational performance at the time the rehabilitation was completed.

This judgment tells about the average number of hours per day the patient is able to work in his or her former occupation. It is documented using a categorical three-point scale: full performance (≥6 hours/day), medium performance (3 to <6 hours/day), low performance (<3 hours/day). The judgment is part of a medical discharge summary that is issued by a physician for each patient who completes medical rehabilitation [START_REF] Rentenversicherung | Der ärztliche Reha-Entlassungsbericht. Leitfaden zum einheitlichen Entlassungsbericht in der medizinischen Rehabilitation der gesetzlichen Rentenversicherung[END_REF].

The variables used to assess socio-demography, socio-economic (SES) and health status differed from those available in the SOEP dataset. Information on age (in years), sex, marital status (single, married, divorced/widowed), employment status (full-time, part-time, unemployed, not applicable), occupational position (skilled labor, semi-skilled/unskilled labor, trainee/unemployed) and type of occupation (15 types using the Blossfeld classification [START_REF] Blossfeld | Bildungsexpansion und Berufschancen[END_REF]: e.g. agricultural occupation, engineering) were used to operationalize socio-demographic characteristics and SES. ICD-10 major diagnostic categories and time absent from work due to illness in the last 12 months (0 months, <3 months, 3 to <6 months, ≥6 months, not employed) were used to account for differences in health status. There were no missing values within the data.

Statistical analysis

The datasets on utilization and effectiveness of medical rehabilitation were analyzed separately using similar approaches. First, for both datasets, descriptive statistics stratified by nationality were calculated. Second, block-wise logistic regression was used to analyze the relationship between nationality (exposition) and the utilization as well as the effectiveness of medical rehabilitation (outcomes). Model 1 presents the crude relationship between the exposition and the outcome. In a second block, sociodemographic and SES variables were entered into the model as potential confounding variables, resulting in model 2. In a third block, in addition to socio-demographic/SES variables, health variables were included to control for differences in health before rehabilitation, resulting in model 3. For all models, odds ratios (OR) and 95% confidence intervals (95%-CI) are provided. We applied the Hosmer-Lemeshow goodness-of-fit test as a measure of model calibration, comparing predicated and observed probabilities for each of the two outcomes. As a measure of discrimination, i.e. in order to test how well the two models distinguish between the two outcome groups, we assessed the area under the receiver operator characteristic curve (cstatistic) [START_REF] Hosmer | Applied logistic regression[END_REF]. All analyses were performed using SPSS Statistics 17 [24].

Results

Utilization of medical rehabilitation

Regarding utilization of medical rehabilitation, data for 19,521 SOEP household members aged 18 years or above was available. Of these, 7.4% had a foreign nationality. Tab. 1 shows the characteristics of the study sample stratified by nationality. Of the 19,521 respondents, 896 (4.6%) had used medical rehabilitation in the year 2003. The rate of utilization was 3.2% for foreign nationals as compared to 4.7% for German nationals. Furthermore, foreign nationals were on average about 5 years younger than Germans and overall had a lower social status. For instance, their average net equivalence income was €17,665 whereas it was €22,685 for German nationals. In terms of health status, 18.7% of foreign nationals rated their health as poor, while only 15.2% of the German nationals did so.

Tab. 2 shows the results of the block-wise logistic regression model with utilization of medical rehabilitation in 2003 as the dependent variable. In the crude model (model 1), the chance of using medical rehabilitation is decreased by 32% in foreign as compared to German nationals (OR=0.68). After adjusting for age, sex, education, profession and income, the chance slightly changes to 30% (OR=0.70) (model 2).

Aside from foreign nationality, significant effects can be identified for age (OR=1.02) as well as for being a blue collar worker (OR=1.46). Including self-rated health in addition to the variables in model 2, reduces the odds ratio for foreign nationality to 0.66 (model 3). In model 3, age (OR=1.02), being a blue collar worker (OR=1.47), and poor self-rated health (OR=2.76) are significantly associated with the utilization of medical rehabilitation.

Effectiveness of rehabilitation

Information on 634,529 subjects was available who completed medical rehabilitation funded by the German Statutory Pension Insurance Scheme in 2006. Of these, 5.5% held a foreign nationality. Tab. 3 summarizes their socio-demographic, SES and health characteristics stratified by nationality. The proportion of rehabilitation clients with a lower occupational position (semi-skilled and unskilled labor) as well as of those without employment is higher among foreign nationals than among Germans. Also, the distribution of underlying diseases that led to rehabilitation differs between the two populations. The proportion of mental diseases as well as diseases of the skeletal system is higher, while the proportion of neoplasms is considerably lower in foreign as compared to German nationals. Tab. 3 also shows that foreign nationals undergoing rehabilitation are quite heterogeneous since values of many independent variables differ between the nationalities included.

Effectiveness of medical rehabilitation as measured by the degree of occupational performance at the time the rehabilitation was completed is lower in foreign than in German nationals: 78.9% of all German patients who completed medical rehabilitation were judged by physicians as being fully able to work in the position they had occupied prior to rehabilitation. This proportion was considerably lower in foreign nationals (70.1%) and lowest for the subgroup of rehabilitation clients from Former Yugoslavia (66.3%). Correspondingly, the proportion of low and medium occupational performance was higher in all foreign national groups than in Germans.

Tab. 4 shows the results of the logistic regression model with low occupational performance as the outcome variable. Patients of all foreign nationalities, as compared to Germans, have a 1.3 to 1.8 higher chance of completing medical rehabilitation with less favorable results. The strength of this association is reduced in all groups when socio-demographic/SES and health variables are adjusted for. In all foreign nationality strata, the chance of completing medical rehabilitation with less favorable results is higher for men than women, for persons working in semi-skilled/unskilled positions and for persons working in part-time employment. Also, it increases with longer absence from work in the last 12 months. However, except for patients originating from Portugal/Spain/Italy/Greece, the effects of foreign nationality are still significant, ranging from OR=1.2 for Turks and 1.5 for rehabilitation clients from Former Yugoslavia.

The logistic model for utilization had a borderline acceptable discrimination between the outcomes groups. For the second and third model, the c-statistic was 0.61 and 0.66 respectively (Tab. 2). It was higher for the model on rehabilitation effectiveness (0.71 and 0.80 for the second and third model in each nationality stratum), suggesting an acceptable to excellent discrimination (Tab. 4). As regards model calibration, the observed and expected probabilities showed only slight differences in the full models on utilization and effectiveness of rehabilitation, indicating a sufficient model fit (the significant Hosmer-Lemeshow test statistic in the models on rehabilitation effectiveness is most likely the result of overpowering given the very large sample sizea well known limitation of this goodness-of-fit test [START_REF] Kramer | Assessing the calibration of mortality benchmarks in critical care: The Hosmer-Lemeshow test revisited[END_REF]).

Discussion

Migrants frequently have poorer health outcomes than the majority population of the host country. A pertinent question in migrant research is to what degree barriers in the health care system are contributing to this health inequality. In particular, health services research needs to establish whether migrants utilize existing services according to their health needs, and whether outcomes are equal to those of nonmigrant patients.

In this study, we analyzed the utilization and effectiveness of medical rehabilitation in foreign nationals living in Germany. Using national level and representative data, we confirmed the results of previous, smaller studies showing that both the utilization [4;5;9] as well as the effectiveness of medical rehabilitation [8;9;11] is lower in foreign as compared to German nationals. A new finding of our study is that these differences are only in part attributable to a different distribution of sociodemographic and socio-economic variables, and of health status before rehabilitation, between the two population groups. In addition, we showed that rehabilitation effectiveness does not only differ between Germans and foreign nationals but also among foreign nationals themselves. Our study indicates that the way rehabilitation services are organized plays a role in this difference; most likely, there is a mismatch between existing services and the needs of migrants.

Interpretation of result

The inclusion of variables on socio-demographic status/SES and health status in the regression model reduced the difference in the effectiveness of medical rehabilitation as defined by occupational performance between foreign and German nationals.

However, the difference remained significant for all foreign national groups except for rehabilitation clients from Portugal/Spain/Italy/Greece. This finding is in line with results from other studies. Maier et al. [START_REF] Maier | Migration und rehabilitative Versorgung in Deutschland -Inanspruchnahme von Leistungen der medizinischen Rehabilitation und Rehabilitationserfolg bei Personen mit türkischem Migrationshintergrund[END_REF] compared rehabilitative treatment outcomes between Turkish and non-Turkish migrants undergoing medical rehabilitation in North Rhine-Westphalia, Germany, and found that Turkish migrants diagnosed with skeletal or mental disorders have a higher chance of completing the program without any substantial improvement in their medical condition. Odds ratios, adjusted for age, sex and occupation, were 2.1 and 1.8, respectively. No differences were found among patients with diseases of the circulatory system. We can confirm these results using a more robust indicator of rehabilitation effectiveness and data that is representative for Germany. We also show that the difference between foreign nationals and Germans is substantially reduced when additional confounders are taken into consideration.

Our study indicates that socio-demographic/SES and health variables do not fully explain the association between foreign nationality and utilization and effectiveness of rehabilitation, respectively. This can have two reasons. First, although we included variables in the model allowing to control for confounding by socio-demographic/SES and health factors, residual confounding by these factors cannot be ruled out. This might in particular be the case for the analysis of rehabilitation effectiveness because no information on the level of education and health status or disease severity prior to rehabilitation was available. Since many migrants experience barriers to rehabilitative services, those migrants utilizing rehabilitation despite these barriers might have a higher objective need and suffer from more severe conditions than non-migrants.

Consequently, their likelihood for less favorable outcomes and reduced effectiveness of rehabilitation might be higher. We tried to account for this distortion by controlling for different disease patterns and time absent from work.

Second, the results could indicate that factors related to the rehabilitative care system, as well as migrant-specific characteristics that go beyond differences in education, age, occupation and health patterns, may play a role. Particularly, this may be true for the utilization of rehabilitation, since the effect of foreign nationality did not decrease when socio-demographic/SES and health variables were accounted for in the respective multivariate model. For both utilization and effectiveness of rehabilitation, this assumption is supported by previous research. In a qualitative study using focus group discussions and expert interviews with patients, caregivers and decision makers within the rehabilitative care system, our group identified different obstacles to access and effectiveness of medical rehabilitation for migrants residing in Germany [START_REF] Razum | Medizinische Rehabilitation für Personen mit Migrationshintergrund -Zwischenergebnisse eines Forschungsprojektes im Auftrag des Bundesministeriums für Arbeit und Soziales[END_REF].

Among others, these obstacles comprise cultural differences, communication problems and a disadvantageous attitude of rehabilitative care institutions towards the diversity of migrants' subjective and objective needs [START_REF] Razum | Medizinische Rehabilitation für Personen mit Migrationshintergrund -Zwischenergebnisse eines Forschungsprojektes im Auftrag des Bundesministeriums für Arbeit und Soziales[END_REF]. Other studies showed that the expectations of patients attending rehabilitative care, their motivation, as well as their desire for early retirement, may influence outcomes of rehabilitation [10;27]. A strong desire for early retirement may reduce the motivation of individuals to complete rehabilitation services successfully since early retirement in Germany cannot be granted for persons unless their potential for rehabilitation is fully exhausted (the so-called principle of "rehabilitation before retirement" [Reha vor Rente]),

Strengths and limitations of the study

The strengths of our study are the use of a national sample and of representative routine data to examine utilization and effectiveness of medical rehabilitation. Unlike other studies, we were able to control for different confounding variables that may disguise differences in these outcomes between Germans and foreign nationals. Also, for the analysis of rehabilitation effectiveness, we were able to stratify for different countries of origin.

Our study also has weaknesses. As regards the analysis of utilization, it was not possible to consider heterogeneity in countries of origin, levels of acculturation and German language proficiency. Also, differences in utilization of medical rehabilitation between Germans and foreign nationals as well as among foreign nationals themselves may depend on the kind of treatment and may differ between in-patient and out-patient care. Furthermore, the possible role of the funding institution (statutory health insurance, pension fund or accident insurance) on the utilization of rehabilitation among foreign nationals could not be taken into account. As for the analysis of rehabilitation effectiveness, the level of education as an important SES factor is missing in our analysis. Finally, foreign nationality is not equivalent to migrant statusbut it is the best proxy available in the large datasets of the funders of rehabilitation in Germany. In the SOEP dataset, where it was possible to compare populations with migration background by nationality or country of origin, no relevant differences were found (data not shown).

Conclusion

Our study showed that foreign nationals living in Germany utilize medical rehabilitation less often than Germans, in spite of a higher expected need. For those who do, medical rehabilitation is less effective. Both findings are only partially attributable to differences between the two groups in socio-demographic variables, socio-economic status, and health status before undergoing rehabilitation. Our study supports suggestions of qualitative research indicating that both factors related to the rehabilitative care system and migrant-specific characteristics could influence differences in the utilization and effectiveness of medical rehabilitation between Germans and foreign nationals. Also, it gives evidence of the heterogeneity of foreign nationals. Together, these findings highlight the increasing need to consider diversity management that allows to address heterogeneity in subjective and objective needs an integral part of medical rehabilitation in Germany.

Tables

Tab. 1 :

 1 Study sample used to analyze the utilization of medical rehabilitation (sample of all household members aged 18 years or above living in Germany, German Socio-Economic Panel Study, wave 2002-2004, n=19,521).

  Note. SD: standard deviation. *All missing values are due to respondents not specifying the item. **Including 42 members of armed forces. Tab. 2: Logistic regression models with use of medical rehabilitation in 2003 as dependent variable (sample of all household members aged 18 years or above living in Germany, German Socio-Economic Panel Study, wave 2002-2004, n=19,521). Study sample used to analyze the effectiveness of medical rehabilitation (80% random sample of all individuals who completed medical rehabilitation in the year 2006 granted by the German Statutory Pension Insurance Scheme, n=634,529). Logistic regression models with low occupational performance (<3h/day) after rehabilitation completion as dependent variable. Odds ratios (OR) and 95% confidence intervals [95%-CI] (80% random sample of all individuals who completed medical rehabilitation in the year 2006 granted by the German Statutory Pension Insurance Scheme; cases with available data on occupational performance, n=519,831). .46;1.55] 1.35 [1.31;1.40] 1.20 [1.16;1.24] 1.63 [1.55;1.72] 1.54 [1.46;1.63] 1.23 [1.16;1.30] 1.82 [1.73;1.93] 1.59 [1.50;1.68] 1.47 [1.38;1.56] 1.33 [1.25;1.42] 1.18 [1.10;1.26] 1.06 [0.98;1.14] Note. Model 1: Crude; Model 2: Adjusted for socio-demographic and SES indicators; Model 3: Adjusted for socio-demographic, SES and health indicators. Models 2 and 3 are also adjusted for type of occupation (not shown); OR: odds ratio; CI: confidence interval; Ref.: reference category.

	Total (n=19,521) 8,504 1.12 [0.97; 1.29] Germans (n=18,068) 1.4 1.04 [1.04;1.04] 1.03 [1.03;1.03] 8,676 1.4 1.04 [1.04;1.04] 1.03 [1.03;1.03] Foreign nationals 1.15 [1.00; 1.32] 172 0.6 43 (n=1,453) Model 1: Crude Model 2: Adjusted for socio-demographic and SES variables Model 3: Adjusted for socio-demographic, SES and health variables OR [95%-CI] OR [95%-CI] OR [95%-CI] 0.68 [0.51; 0.92] 0.70 [0.52; 0.95] 0.66 [0.49; 0.90] 250,064 39.4 235,658 39.1 14,406 46.2 3,812 56,221 8.9 52,665 8.7 3,556 11.4 961 26,889 4.2 25,757 4.3 1,132 3.6 359 Tab. 4: All foreign nationals Sex (Ref.: Male) Skin and subcutaneous tissue Age Nationality (Ref.: German) Diagnosis at rehabilitation entry (n, %) Skeletal system Circulatory system Endocrine, nutritional and metabolic Turkey Model 1 Model 2 Model 3 Model 1 Model 2 Model 3 Model 1 43.1 0.5 1.04 [1.04;1.04] 1.03 [1.03;1.03] 31 0.4 3,778 51.6 10.9 754 10.3 4.1 198 2.7 Former Yugoslavia Model 2 Model 3 Tab. 3: Total Germans Foreign nationals All Turkey Former Yugoslavia Respiratory system 24,896 3.9 24,091 4.0 805 2.6 224 2.5 150 2.0 OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] OR [95%-CI] n=634,529 N=603,352 n=31,177 n=8,854 n=7,327 Neoplasms 120,767 19.0 117,667 19.5 3,100 9.9 626 7.1 757 10.3 Genitourinary system 2,614 0.4 2,532 0.4 82 0.3 17 0.2 22 0.3 Nervous system 19,113 3.0 18,621 3.1 492 1.6 128 1.4 117 1.6 test (p-value) Nationality (Ref.: German) c-statistic 0.71 0.80 0.71 0.80 0.71 0.80 1.51 [1Sex (Ref.: male) 0.67 [0.66;0.68] 0.57 [0.56;0.58] 0.67 [0.65;0.68] 0.56 [0.55;0.58] 0.66 [0.65;0.67] 0.56 [0.55;0.57] Hosmer-Lemeshow goodness-of-fit <0.05 <0.05 <0.05 <0.05 <0.05 <0.05	33 3,018 720 271 Portugal/Spain/Italy/Greece 0.5 65 1.04 [1.04;1.04] 1.03 [1.03;1.03] 0.7 48.6 3,798 43.2 11.6 1,121 12.8 4.4 304 3.5 Model 1 Model 2 Model 3 P/E/I/GR 162 2.6 269 OR [95%-CI] OR [95%-CI] OR [95%-CI] 3.1 Other n=6,211 n=8,785 690 11.1 1,027 11.7 10 0.2 33 0.4 97 1.6 150 1.7 0.71 0.80 0.66 [0.65;0.68] 0.56 [0.55;0.57] <0.05 <0.05
	Age in years (mean; SD) Age Alcohol, drugs		34,573	5.4	48.3 (16.4) 1.02 [1.02; 1.03] 48.7 (16.5) 32,984 5.5 1,589	43.9 (14.8) 1.02 [1.01; 1.02] 5.1 470	5.3	177	2.4	243	3.9	699	8.0
	Mental disorders (without drugs) Gender (n, %) Male Female Education (Ref.: High) Intermediate Other Marriage status (Ref.: single) Education (n, %) Low Married Still in School Time absent from work in the last 12 Divorced/widowed		72,335 18,381 0.90[0.88;0.92] 0.86 [0.84;0.89] 11.4 9,345 10,176 2.9 0.88[0.86;0.91] 0.93 [0.90;0.96]	67,830 47.9 52.1 1.06 [0.87; 1.31] 11.2 8,632 9,436 17,043 2.8 0.99 [0.80; 1.22] 0.90 [0.87;0.92] 0.86 [0.84;0.89] 4,505 14.4 47.8 713 52.2 740 1,338 4.3 1.03 [0.83; 1.26] 1,876 49.1 338 50.9 0.90 [0.73; 1.11] 0.67 [0.09; 4.92] 0.68 [0.09; 4.97] 0.88 [0.85;0.91] 0.94 [0.91;0.96]	21.2 3.8 0.90 [0.87;0.92] 0.86 [0.84;0.89] 1,061 14.5 282 3.8 0.88 [0.85;0.91] 0.94 [0.91;0.97]	711 256	11.4 4.1 0.89 [0.87;0.92] 0.86 [0.84;0.89] 857 9.8 462 5.3 0.88 [0.85;0.91] 0.94 [0.91;0.97]
	High Intermediate Not specified months (n, %)* Occupational position Low None Still in school Missing* Profession (Ref.: White collar) Blue collar <3 months (Ref.: Skilled) Not applicable 3-6 months Semi-skilled/unskilled	3,944 7,208 7,854 67 448 1.21 [1.19;1.24] 1.29 [1.26;1.32] 116,035 22.5 250,711 48.5 70,884 13.7	20.2 36.9 1.04 [0.63; 1.74] 3,792 6,922 40.2 6,926 110,745 22.7 0.3 63 2.3 365 1.46 [1.17; 1.81] 238,373 48.8 1.09 [0.90; 1.32] 65,916 13.5 1.22 [1.20;1.25] 1.30 [1.27;1.34] 21.0 152 38.3 286 1.01 [0.60; 1.69] 10.5 19.7 38.3 928 5,290 18.5 1,583 63.9 0.3 4 0.3 2.0 83 12,338 43.2 2,865 1.47 [1.18; 1.83] 5.7 1.00 [0.82; 1.21] 4,968 17.4 1,535	19.7 35.7 19.1 1.22 [1.20;1.25] 1.31 [1.28;1.34] 1,041 15.4 3,067 45.3 1,266 18.7	959 2,630 1,027	16.8 46.1 18.0 1.23 [1.20;1.25]	1,707 21.3 3,776 47.0 1,140 14.2 1.31[1.28;1.34]
	Profession (n, %) 6+ months Trainee/not employed		79,133 1.94 [1.87;2.01] 1.07 [1.03;1.12] 15.3	73,201	15.0 1.95 [1.88;2.02] 1.08 [1.04;1.13] 5,932 20.8	2,041	25.4 1.95 [1.88;2.02] 1.09 [1.04;1.13] 1,400 20.7	1,087	19.1 1.95 [1.88;2.01] 1.08 [1.04;1.13] 1,404 17.5
	White collar Net equivalence income (log.) Blue collar Not applicable** Employment status (Ref.: Fulltime) Occupational performance (n, %) * Self-rated health (Ref.: Good) Full (6+ hours/day) Part-time	7,945 3,424 8,152 2.17 [2.10;2.24] 1.81 [1.74;1.88] 407,048 78.3	40.7 0.90 [0.78; 1.03] 7,606 17.5 2,925 41.8 7,537 386,250 78.8 2.18 [2.11;2.26] 1.82 [1.75;1.89] 42.1 339 0.95 [0.83; 1.10] 23.3 16.2 499 34.3 41.7 615 42.3 2.76 [2.37; 3.22] 20,798 70.6 5,791	69.2 2.20 [2.13;2.28] 1.82 [1.75;1.89] 4,667 66.3	4,274	72.8 2.20 [2.12;2.27] 1.82 [1.75;1.90] 6,066 74.0
	Net equivalence income in Euro c-statistic Low (<3 hours/day) (mean; SD) N Medium (3 to 6 hours/day) Unemployed Other	82,224 30,559 1.08 [1.05;1.12] 1.11 [1.08;1.15] 15.8 22,311 (19,740) 75,858 19,346 5.9 28,249 2.67 [2.61;2.73] 3.09 [3.02;3.17]	0.61 15.5 22,685 (19,879) 6,366 19,346 5.8 2,310 1.09 [1.06;1.13] 1.12 [1.09;1.16] 21.6 17,665 (17,267 ) 1,923 0.66 19,346 7.8 660 2.71 [2.65;2.77] 3.14 [3.06;3.22]	23.0 7.9 1.10 [1.06;1.13] 1.13 [1.09;1.16] 1,764 25.1 604 8.6 2.74 [2.68;2.80] 3.16 [3.08;3.24]	1,150 447	19.6 7.6 1.10 [1.06;1.13] 1.13 [1.09;1.16] 1,529 18.7 599 7.3 2.73 [2.67;2.80] 3.16 [3.08;3.24]
	Self-rated health (n, %) Good Hosmer-Lemeshow goodness-of-fit test (p-value) Note. *Restricted to applicable cases. P/E/I/GR: Country group "Portugal, Spain, Italy, Greece"; SD: standard deviation. No missing values existed. 16,470 84.4 15,290 84.6 1180 81.2 <0.05 0.59 Diagnosis at rehabilitation entry
	Poor Note. SES: socio-economic status; OR: odds ratio; CI: confidence interval; Ref.: reference category. 3,026 15.5 2755 15.2 (Ref: Skeletal system)		271	18.7
		Missing*				25	0.1	23	0.1		2	0.1
	Circulatory system				1.78 [1.73;1.83]				1.77 [1.72;1.82]		1.79 [1.74;1.84]	1.78 [1.73;1.84]
	Use of medical rehabilitation (n, %) Endocrine,nutritional, metabol.	0.92 [0.87;0.96]				0.91 [0.87;0.96]		0.91 [0.87;0.96]	0.91[0.87;0.96]
	Respiratory system	Yes			896 1.60 [1.52;1.68]	4.6	849	4.7	47 1.59 [1.51;1.68]	3.2	1.57 [1.49;1.66]	1.59 [1.51;1.68]
	Neoplasms	No Missing*			18,518 107 2.06 [2.01;2.12]	94.9 0.6	17,118 101	94.7 0.6	1,400 6 2.06 [2.01;2.12]	96.4 0.4	2.07 [2.01;2.12]	2.07 [2.01;2.12]

Conflict of interests

This study was part of a project funded by the German Federal Ministry for Labor and Social Affairs (BMAS). The Ministry had no role in analyzing the data, interpreting the results and drawing the conclusions.


Samaneh Chagheri
email: samaneh.chagheri@insa-lyon.fr

Catherine Roussey
email: catherine.roussey@liris.cnrs.fr

Sylvie Calabretto
email: sylvie.calabretto@insa-lyon.fr

Cyril Dumoulin
email: cyril.dumoulin@continew.fr

Technical Documents Classification

Keywords: Structural document, document classification, support vector machine, vector space model

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

I. INTRODUCTION

This research takes place in an industrial context: the CONTINEW Company. This company ensures the storage and security of critical data and technical documentation. The term "technical documentation" refers to different documents with product-related data and information that are used and stored for different purposes, like user manuals, product specifications, manufacturing documents and product presentations. Each purpose indeed present a category. Documentation is necessary in the life cycle of an innovative product, from its functional design to marketing. Indeed without such documents, the products can neither be manufactured nor used according to their complexity. Readers of technical documentation may be the end user of the product or the product administrator or technician. It describes specification and requirements for the product to function as designed and it contains more elements like tables and schema. In a general collection the documents are more textual and are strongly structured. Structure here means logical structure like title, chapters, sections, figures, paragraphs, etc. These documents are created by different authors using different styles and forms to present them. A classification system is needed for manipulating such a voluminous collection in order to facilitate information retrieval and to construct a standard model for each category. Traditional classification systems do not consider the structure of a document. They mostly focus on representing the document content as a bag of words without considering the structural elements in which the words appear. But taking into account the structure facilitates document presentation and improves the classification precision. These systems exploit the available structural information in documents, as marked up in XML, in order to implement a more accurate classification strategy and return document structural elements instead of complete documents. However, much of the information is contained in the text fields not just in tag labels, therefore, applying a method which exploits the structure of document and its content is promising.

In this article, we have proposed a representation method which is an extension of the vector model of Salton [START_REF] Salton | Search and retrieval experiments in real-time information retrieval[END_REF] adjusting the calculation of the tf*idf by considering the structural element instead of the entire document. Document is represented as a tree in which nodes are structure, and leaves are document text. The rest of the article is organized as follows. Section 2 presents the different approaches proposed in the literature to XML document classification. Section 3 describes the basic methods for classification. Section 4 presents our proposal on document representation and feature construction. Section 5 describes the experiments and results, and section 6 presents the conclusion and further works.

II.RELATED WORKS

The continuous growth in XML documents has caused different efforts in developing classification systems based on document structure. Document representation has to be done before the classification process. The representation models can be divided into three groups: The first groups are the models which do not consider the structure of document. These works focus on representing the document content as a bag of words to classify them. They are the most studied classification methods. The second group are the models which take into account only the structure of a document in order to classify them without considering the document content. For example, Wisniewski [START_REF] Wisniewski | Classification automatique de documents structurés. Application au corpus d'arbres étiquetés de type XML[END_REF] uses Bayesien model to generate the possible DTDs of a documents collection. A class represents a DTD. They are interested only on document structure for classification. Reference [START_REF] Aïtelhadj | Classification de Structures Arborescentes: Cas de Documents XML[END_REF] and [START_REF] Dalamagas | Clustering XML Documents Using Structural Summaries[END_REF] have also classified the documents by only document structure trees similarities. Finally, the third group is composed of the models which consider both structure and content of XML documents in representation.

Mostly the classification systems use vector space model for document representation, the difference is based on selecting vector features and their weight computation. In [START_REF] Doucet | Naive Clustering of a Large XML Document Collection[END_REF] each vector feature can be a word or a tag of XML document. The tf*ief (Term Frequency * Inverse Element Frequency) is used for calculating the weight of words or tags in documents.

In [START_REF] Vercoustre | Classification de documents XML à partir d'une représentation linéaire des arbres de ces documents[END_REF] a feature can be a path of the XML tree or a path following by a text leaf. The term weight is based on tf*idf. This allows taking into account either the structure itself or the structure and content of these documents. Reference [START_REF] Yang | XML Document Classification Using Extended VSM[END_REF] presents also vector model containing terms or XML tree paths as vector elements. Ghosh [START_REF] Ghosh | Combining Content and Structure Similarity for XML Document[END_REF] has proposed a composite kernel for fusion of content and structure information. The paths from root to leaves are used as indexing elements in structure kernel weighted by tf*idf. The content and structure similarities are measured independently. A linear combination of these kernels is used finally for a content-structure classification by SVM. Wu [START_REF] Wu | A bottom-up approach for XML documents classification[END_REF] proposes a bottom up approach in which the structural elements are document tree nodes and the leaves are textual section of each element. First the terms in leaf nodes are identified, and their occurrences are extracted and normalized. Then the key terms are substantiated with the structural information included in the tags by the notion of key path. A key path ends at a leaf node that contains at least one key term for a class. By using the key terms set and key path the similarity is computed between new documents and class models. Yan [START_REF] Yan | Feature Matrix Extraction and Classification of XML Pages[END_REF] proposes a document representation by vectors of weighted words, a vector for each structural element in a document. In this method a weight is associated to each structural element according to its level in the document tree. Then the weight of words is calculated based on its frequency inside the element and the importance of this element. Yang in [START_REF] Yang | Extended VSM for XML Document Classification Using Frequent Subtrees[END_REF] and [START_REF] Yang | XML Document Classification Using Extended VSM[END_REF] has proposed an extension of the vector model called the Structured Link Vector Model (SLVM). In his model a document is represented by a set of term vectors, a vector for each structural element. The term weight is calculated by term frequency in each document element and the inverse document frequency of term.

The model that we propose belongs to the third group, constructing the document feature vector by structure and content. In which each vector feature is a couple of word and structure element. The weight of the word is calculated not only on its frequency in document and collection, but on its position in document. In our calculation we assign a value to each structural element according to its level in the XML document tree.

III.DOCUMENT CLASSIFICATION

Classification can be divided in two principal phases. The first phase is document representation, and the second phase is classification. The standard document representation used in text classification is the vector space model. The difference of classification systems is in document representation models. The more relevant the representation is, the more relevant the classification will be. The second phase includes learning from training corpus, making a model for classes and classifying the new documents according to the model. In this section we are going to explain the vector space model followed by a presentation of the SVM classification algorithm which is used in our approach.

A. Document Representation by VSM

The Vector Space Model (VSM) proposed by Salton [START_REF] Salton | Search and retrieval experiments in real-time information retrieval[END_REF]) is a common technique for document representation in classification. In this model each document is represented as a vector of features. Each feature is associated with a weight. Usually these features are simple words. The feature weight can be simply a Boolean indicating the presence or absence of the word in document, its occurrence number in document or it can be calculated by a formula like the well known tf*idf method. So, the feature vector of document d of collection D with n distinct terms is represented as follows: But in classical vector space model, document structure is not considered. For taking into account the notion of structure in document representation an extension of VSM seems promising.

B. SVM Classifier

The Support Vector Machine (SVM) proposed by Vapnik [START_REF] Cortes | Support-vector networks[END_REF], is a supervised learning algorithm that can be applied to

IV.GENERAL OVERVIEW OF OUR APPROACH

In this article we propose inputsa combination of content and structure of XML document in the vector model for document representation. In our vector, each feature is a couple of (tag: term). Tag corresponds to a structural element in an XML document. The XML document is represented as a tree in which the nodes are the XML tags and the leaves are the textual part in each document element. To construct our features we apply two processes. First, a process is performed on the logical structure document in order to construct the aggregated tree of XML document. Secondly, a lexical process is performed on document content to select the most informative terms. These processes aim to reduce the computational complexity and improve system performance. Then the weight of features is computed. Finally, the document vectors are used as the inputs of the classification system.

A. Aggregated Document Tree Construction

We consider the XML document as a tree in which the nodes are structural elements like title, chapter, etc. The arcs of this tree represent the inclusion relation between nodes, and the leaf nodes contain the document text. The depths of nodes in document tree are important. Thus we consider that two nodes with the same label localized at different depths are two different structural elements. The structural element represents a node type. We modify document tree organization by aggregating the nodes with the same label and localized at the same depth. For example, all paragraphs in a section are aggregated to a single node "paragraph" which contains all terms of these paragraphs. We take into account only the leaf nodes which contain text. We assume that all documents of collection are homogeneous. So, a label becomes an adequate identifier of the structural element. Also, we filter some tags which are not representative for document semantic based on a tag list made manually.

B. Feature Extraction

After constructing the aggregated tree, we extract the terms in each node. We apply a series of linguistic analysis. First we extract the terms lemma and their part of speech using TreeTagger. Secondly we filter the words, just nouns and verbs are kept and the other words are removed. Then the words are replaced by their lemma. The results of such analysis are called term. Therefore, each feature is constructed by combining the node label and the term, for example (title: technical) or (p: cleaning). Such feature represents the document structure and content. Our hypothesis is that a term appearing in two different structural elements should be considered as two different features with different weights. Indeed, moving down in document tree, from root to leaf, decreases the elements importance. For example the word "classification" appearing in the document title and in a paragraph of a document composes two different features: (title: classification) and (paragraph: classification). And the weight of the first feature is more important than the second.

C. Weight Computation

We assume that terms occurring in different structural elements have different importance. For calculating the weight of features we use an extension of traditional tf*idf on structural element level instead of document level. We alsot take into account the importance of the structural element in which the term has appeared. We assume that the deeper a node is in the document tree, the less it will be important. Therefore, the weight of feature will be calculated as below:

(3) After extracting all features in documents and calculating their weight, document vector is constructed and SVM is used for learning and classifying.

V.EXPERIMENTS AND RESULTS

For experimentation we have used SVM light [START_REF] Joachims | Making large-Scale SVM Learning Practical[END_REF] for learning and classifying the documents. It is an implementation of the SVM algorithm. This algorithm has scalable memory requirements and can handle problems with many thousands of support vectors efficiently.

A. Test Collection

Experiments were performed on the Reuters Corpus Volume 1 (RCV1) which includes over 800,000 English language news stories. Reuters is a leading global provider of financial information, news and technology to financial institutions, the media, businesses and individuals. The stories in this collection are formatted using a consistent XML schema. Reuters collection has been used by many researchers in the field of information retrieval and machine learning. All the stories in RCV1 have been coded for topic, region (geography) and industry sector and they are multiclass.

B. Collection Pre-processing

In order to perform a mono classification we have used the first topic code in documents as the class of documents. Also the unclassified documents are removed from collection. We have used 800 documents of collection by considering 400 positive and 400 negative examples for a topic class called "GCAT". After analyzing the structural elements in documents, we have selected the header and paragraphs tags in stories. These tags seem more representative for document topic. Other tags are removed from document tree. TreeTagger and Tokenizer of platform GATE has been used to extract the terms.

C. Experimentation

We performed two experiments on chosen collection by using SVM. The first one called "content and structure" is an implementation of our proposed method in which the document vector is constructed by tag and term, and where the features weight is also calculated using our proposed formula. The second one called "content only" uses a vector of simple terms weighted by standard tf*idf. The application has been implemented in java except SVM light algorithm which is on C. K-fold Cross validation has been used as the method for evaluating the system performance.

Cross-validation is one of the several approaches to estimate how well the model we've just learned from some training data is going to perform on new arriving data. We have applied 4 folds in cross validation, each fold including 200 documents (positives and negatives examples). The results of the classifications are shown in table I. Precision and recall as the most widely used metrics for evaluating the correctness of classification have been used. Another used metric is F-Measure that combines precision and recall. The results demonstrate that including structure improves the classification accuracy.

TABLE I. CLASSIFICATION RESULTS

Vector features Results

Precision

VI.CONCLUSION AND FUTURE WORKS

In this article we have proposed a model for XML document classification. We proposed a combination of structure and content in document representation by the vector model. Our features are couples of (tag: term), all weighted by an extension of tf*idf taking into account the terms position in the documents tree. Document tree is achieved by filtering non relevant structural elements and aggregating the nodes with same label and same path in tree. Classification is performed in order to make a model for each category of technical documents. This model is used to evaluate the technical documents quality and adapting them with standard models.

We have done our experimentation on Reuters XML news collection which is a particular collection with homogenous documents. In this collection all documents share same logical structure. In future works we are going to improve our proposition in order to execute the test on CONTINEW corpus which contains heterogeneous documents created by different industries for different categories, and generating document model.

  dd = [w1,d, w2,d ,..,wn,d] . (1) wi,d = tfi,d٭ idfi (2) w i,d is the weight of term i in the document d, where tf is the frequency of term i in document d and idfi = log (|D|/|Di|) is inverse document frequency, |D| is the total number of the documents in collection D, and |D i | is the number of documents in collection containing the term i.

Figure 1 .

 1 Figure 1. A document feature vector: a & b represent two documents of the collection.

  classification. It is a binary linear classifier which separates the positives and negatives examples in a training set. The method looks for the hyperplane that separates positive examples from negative examples, ensuring that the margin between the nearest positives and negatives is maximal. The effectiveness of SVM is superior to other methods of text classification. SVM makes a model representing the training examples as the points in a dimensional space separated by the hyperplane, and it uses this model to predict a new example belongs to which side of this hyperplane. The examples used in searching the hyperplane are no longer used and only these support vectors are used to classify a new case. This makes a very fast method.

Figure 2 .

 2 Figure 2. Maximum-margin hyperplane

Figure 3 .

 3 Figure 3. XML document tree

Figure 4 .

 4 Figure 4. XML document aggregated tree

  Margin in : 13th International Symposium on the Management of Industrial and Corporate Knowledge, 8-9 juin 2011 e, and d represent respectively term i, node label e, like "paragraph" and document d in the collection. The Term Frequency TFi,e,d is the number of occurrences of the term i in structural element type e inside the document d. IDEF d,e is the Inverse Document Element Frequency with |D e | as the number of documents in the collection having a node label e, and |De,i | as the number of documents having node e containing the term i. IED e is the Inverse Element Depth that represents the importance of the node e in the collection. Where L d is the depth of document tree, and l d,e is the depth of the node e in this document.

in : 13th International Symposium on the Management of Industrial and Corporate Knowledge, 8-9 juin 2011


Jean-Yves Dauxois
email: jean-yves.dauxois@univ-fcomte.fr

Alexis Flesch
email: alexis.flesch@univ-fcomte.fr

Davit Varron
email: dvarron@univ-fcomte.fr

Empirical likelihood confidence bands for mean functions of recurrent events with competing risks and a terminal event

Keywords: Censoring, competing risks, confidence bands, empirical likelihood, empirical processes, recurrent events, terminal event

In this paper, we study recurrent events with competing risks in the presence of a terminal event and a censorship. We focus our attention on the mean functions which give the mean number of events of a specific type that have occured up to a time t. Using empirical likelihood ratio techniques, we are able to build confidence bands for these functions. We have a data set of nosocomial infections in an intensive care unit of a french hospital. For each patient, we know if and when he caught an infection, what infection it was (septicemia, urinary tract infection...), if and when he died and when he left the hospital. Our model fits this context and will be used to build confidence bands for one type of nosocomial infection and even a confidence tube for two types.

Introduction

In this paper we are considering a data set on nosocomial infections contracted by more than 7867 patients in an intensive care unit of a French hospital over a decade. This data set has already been introduced and studied by [START_REF] Dauxois | Non-parametric tests for recurrent events under competing risks[END_REF]. For each patient, we know if and when he contracted a nosocomial infection and what type of infection it was: pneumonia, septicemia, urinary tract infection, etc... Each type of infection can affect the same patient several times. We also know if and when he died, and if not, when he left the hospital. This leads us to working on recurrent events with competing risks under random censorship and with a terminal event. The aim of this paper is to build confidence bands for the mean number of infections of one or more types over time. This will be achieved using empirical likelihood and empirical processes methods.

Recurrent events models are useful in many fields, like in Social Science for recurrent periods of unemployment, in Reliability for recurrent occurences of failure on the same device or also in Biostatistics as in our case study of nosocomial infections, etc... Many statistical methods have been used over the last decades to study recurrent events based on Markov models, Martingale theory or Poisson processes. [START_REF] Andersen | Statistical models based on counting processes[END_REF] give a tour of these methods. [START_REF] Lawless | Some simple robust methods for the analysis of recurrent events[END_REF] and [START_REF] Lawless | The analysis of recurrent events for multiple subjects[END_REF] considered the mean function, which gives the mean number of events that have occurred up to a time t. [START_REF] Wang | Nonparametric estimation of successive duration times under dependent censoring[END_REF], [START_REF] Lin | Nonparametric estimation of the gap time distributions for serial events with censored data[END_REF], [START_REF] Lin | Nonparametric tests for the gap time distributions of serial events based on censored data[END_REF], Cai & Schaubel (2004a) and more recently [START_REF] Du | Nonparametric modeling of the gap time in recurrent event data[END_REF] focused on the gap time distribution. More recently, [START_REF] Cook | The statistical analysis of recurrent events[END_REF] issued a book on the analysis of recurrent events.

In some situations, a terminal event may stop the recurrent event process, as in our data set where the death of the patient is clearly dependent on the infections he suffered from, or when a patient leaves the hospital. The end of observation can also be due to an independent right censoring mechanism, specifically, as in our case, when the study comes to an end. Note that in other contexts, other causes of independent censoring can be observed like in the case of lost to follow up. Papers considering recurrent events with a terminal event and independent censoring are among others [START_REF] Cook | Robust estimation of mean functions and treatment effects for recurrent events under event-dependent censoring and termination: application to skeletal complications in cancer metastatic to bone[END_REF] and [START_REF] Cai | Marginal means/rates models for multiple type recurrent event data[END_REF].

Finally, there are some situations where the observed events are of multiple types. It is the case in our data set since several types of nosocomial infections can affect a patient: pneumonia, septicemia, urinary tract infection, etc... In the same manner, the failure of a device can be caused by multiple mechanical parts. As an example, the engine, the gearbox or the tires can be a cause of failure for cars. This is also true for recurrent unemployment: one can loose his job for numerous different reasons. It is thus interesting to consider competing risks in a setup of recurrent events, as seen in [START_REF] Dauxois | Non-parametric tests for recurrent events under competing risks[END_REF].

Empirical likelihood was first introduced by [START_REF] Thomas | Confidence interval estimation of survival probabilities for censored data[END_REF] in a setup of survival analysis. [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] generalized their concept to obtain confidence regions for the mean of an i.i.d. sample with finite variance. One of the advantages of this approach is that one doesn't need to estimate the variance to build the region. Moreover, the shape of those regions strongly depend on the geometry of the data whereas classical central limit theorem gives ellipse-shaped confidence regions. Since [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF], a lot of results on empirical likelihood have been published, and a few concern survival analysis. For example, [START_REF] Adimari | Empirical likelihood type confidence intervals under random censorship[END_REF] and [START_REF] Ren | Weighted empirical likelihood ratio confidence intervals for the mean with censored data[END_REF] used empirical likelihood to study the mean under random censorship. [START_REF] Owen | Empirical likelihood[END_REF] issued a book on empirical likelihood, and, more recently, [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF] gave a generalization of Owen's result which will be used in this article.

Our aim in this paper is to build confidence bands for the mean functions of a multiple-type recurrent events process with terminal event and under right censoring. In order to do so, we use results from [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF] and empirical processes methods, available for example in [START_REF] Pollard | Empirical processes: theory and applications[END_REF], [START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF] and [START_REF] Bilias | Towards a general asymptotic theory for Cox model with staggered entry[END_REF].

In section 2, we introduce the model and the empirical likelihood ratio for a specific mean function of a recurrent event process. The asymptotic distribution of the empirical likelihood ratio is obtained in section 3 and then we generalize this result to the empirical likelihood ratio for two specific mean functions. Finally, in section 4, we apply our results to the data set of nosocomial infections and obtain confidence bands for the mean functions.

Notations

For seek of simplicity, we shall suppose that only two types of events are observed, as the generalization is straightforward. For j = 1, 2, we write N ⋆ j (t) the total number of events of type j that have occured up to time t. We suppose that the counting processes N ⋆ j are almost surely bounded by a constant B. We also suppose the presence of a terminal event D a priori dependent on the N ⋆ j 's after which the counting processes can't jump. Finally, we suppose that the observation of the processes is subject to a censorship C which is independant on the N ⋆ j 's and D. Write X = D ∧ C the time after which the process doesn't jump anymore and δ = I(D ≤ C), where I(•) denotes the indicator function. This last random variable informs us on whether it was the terminating event that stopped the process or the censorship. If we write Y (t) = I(X ≥ t) and N j (t) = N ⋆ j (t∧C) for t ≥ 0 and j = 1, 2, then the observed data are i.i.d. replicates (N i,j (t), X i , δ i ), for i = 1, . . . , n and j = 1, 2, of (N j (t), X, δ), for t ∈ [0, τ ] where τ is a fixed constant chosen so that P(C > τ )P(X > τ ) > 0.

In order to build confidence regions for the mean functions t → µ j (t) = E(N ⋆ j (t)), we use the estimators µ j (t) from [START_REF] Dauxois | Non-parametric tests for recurrent events under competing risks[END_REF]:

µ j (t) = 1 n n i=1 t 0 S(u -) Ȳ (u) dN i,j (u),
where S is the Kaplan-Meier estimator of S, the survival function of D (see, for example, [START_REF] Andersen | Statistical models based on counting processes[END_REF] for a definition of S) and

Ȳ (t) = 1 n n i=1 Y i (t).
We then apply empirical likelihood methods. Formally, write for all t in [0, τ ]:

EL n (µ j , t) := max n i=1 np i , µ j,p (t) = µ j (t), p i ≥ 0 and n i=1 p i = 1 ,
where:

µ j,p (t) := t 0 S(u -)d R j,p (u), R j,p (t) := n i=1 t 0 p i dN i,j (u) Ȳ (u) .
Straight calculus shows that:

µ j,p (t) = µ j (t) ⇐⇒ n i=1 p i t 0 S(u -) Ȳ (u) dN i,j (u) -µ j (t) = 0.
Following notations in [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF], we write:

h(u) = S(u -) Ȳ (u) and m i,j (N i,j , µ j , h, t) = t 0 h(u)dN i,j (u) -µ j (t).
EL n can now be rewritten as in [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF]:

EL n (µ j , h, t) = max n i=1 np i , n i=1 p i m i,j (N i,j , µ j , h, t) = 0, p i ≥ 0 and n i=1 p i = 1 .
Finally, define:

M n,j (µ j , h, t) = 1 n n i=1 m i,j (N i,j , µ j , h, t) and S n,j (µ j , h, t) = 1 n n i=1 m i,j (N i,j , µ j , h, t) 2 .
When no confusion is possible, we will denote by m i,j (t) and m i,j (t) the functions:

m i,j (t) = m i,j (N i,j , µ j , h, t) = t 0 S(u -) EY (u) dN i,j (u) -µ j (t), m i,j (t) = m i,j (N i,j , µ j , h, t) = t 0 h(u)dN i,j (u) -µ j (t).
One can easily show that E m i,j (t) = 0 for all t ∈ [0, τ ].

3 Main result

Convergence of the margins

We will need the following result which is easily obtained from Theorem 2 of [START_REF] Dauxois | Non-parametric tests for recurrent events under competing risks[END_REF]:

Theorem 1. For j = 1, 2, as n → ∞, √ n ( µ j (•) -µ j (•)) D -→ G j (•),
in the Skorohod space D[0, τ ], where G j (•) is a mean-zero Gaussian process.

We can now state our main result: under a classical variance condition we establish the weak convergence of the empirical likelihood ratio.

Theorem 2. Suppose that both V j (t) := Var m i,j (t) and E|m i,j (t)| are bounded away from zero and infinity on [τ 1 , τ ], for some real constant τ 1 > 0. Then:

-2 log EL n (µ j , h, •) D -→ G 2 j (•)V -1 j (•), when n → +∞, in the Skorohod space D[τ 1 , τ ].
Proof. This result is mainly based on the section 2.3 in [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF]. It is then sufficient to show that their conditions (A0 ⋆ ) to (A3 ⋆ ) are true, that is:

(A0 ⋆ ) : P(∃t ∈ [τ 1 , τ ] : EL n (µ j (t), h, t) = 0) -→ 0, as n → ∞. (A1 ⋆ ) : √ n M n,j (µ j , h, •) D -→ G(•), as n → ∞. (A2 ⋆ ) : sup t∈[τ1,τ ] |S n,j (µ j , h, t) -V j (t)| P -→ 0, as n → ∞. (A3 ⋆ ) : max 1≤i≤n m i,j (N i , µ j , h, t) ∞ = o p (n 1 /2 ),
where

• ∞ denotes the supremum norm on D[τ 1 , τ ].
Proof of (A0 ⋆ ). It is sufficient to show that almost surely:

lim n→∞ inf t∈[τ1,τ ] 1 n n i=1 I( m i,j (t) > 0) > 0 and lim n→∞ sup t∈[τ1,τ ] 1 n n i=1 I( m i,j (t) > 0) < 1. (1)
Let us first consider the left inequality. Note that we have almost surely, for all η > 0 and t ∈ [τ 1 , τ ],

I(| m i,j (t) -m i,j (t)| > η) ≤ I B S(u -) Ȳ (u) - S(u -) EY 1 (u) ∞ > η .
Thus, almost surely for all t in [τ 1 , τ ]:

1 n n i=1 I( m i,j (t) > 0) ≥ 1 n n i=1 (I(m i,j (t) > η) -I(| m i,j (t) -m i,j (t)| > η)) ≥ 1 n n i=1 I(m i,j (t) > η) -I B S(u -) Ȳ (u) - S(u -) EY 1 (u) ∞ > η .
We know that EY (u) -Ȳ (u) ∞ and S(u -) -S(u -) ∞ converge to 0 when n → +∞ from Glivenko-Cantelli Theorem and the uniform consistency of the Kaplan-Meier estimator (see e.g. Theorem IV.3.1 of [START_REF] Andersen | Statistical models based on counting processes[END_REF]) respectively. Thus, the second indicator function tends to zero and it is sufficient to show that there exists η > 0 such that inf t∈[τ1,τ ] P(m i,j (t) > η) > 0.

To this effect, let:

η = 1 2 inf t∈[τ1,τ ] E(m i,j (t)I(m i,j (t) ≥ 0)) = 1 4 inf t∈[τ1,τ ] E|m i,j (t)| > 0.
This last equality is a consequence of E m i,j (t) = 0 for all t ∈ [τ 1 , τ ]. Cauchy-Schwarz inequality yields:

P(m i,j (t) ≥ η) ≥ 1 E(m i,j (t) 2 ) E 2 (m i,j (t)I(m i,j (t) ≥ η)) = 1 E(m i,j (t) 2 ) E(m i,j (t)I(m i,j (t) ≥ 0)) -|E(m i,j (t)I(m i,j (t) ≥ 0)) -E(m i,j (t)I(m i,j (t) ≥ η))| 2 ≥ 1 E(m i,j (t) 2 ) (2η -η) 2 = η 2 E(m i,j (t) 2 ) ,
which is, from our assumptions on E |m i,j (t)| and Var (m i,j (t)), bounded away from 0 for all t ∈ [τ 1 , τ ].

To show the second inequation in (1), remark that:

lim n→∞ sup t∈[τ1,τ ] 1 n n i=1 I( m i,j (t) > 0) < 1 ⇐⇒ lim n→∞ inf t∈[τ1,τ ] 1 n n i=1 I( m i,j (t) ≤ 0) > 0,
and conclude by using the same arguments as previously.

Proof of (A1 ⋆ ). Recall that M n,j (µ j (t), h, t) = 1 n n i=1 m i,j (N i,j , µ j , h, t). Then, by Theorem 1:

√ n M n,j (µ j , h, •) = √ n ( µ j (•) -µ(•)) D -→ G j (•), as n → +∞.
Proof of (A2 ⋆ ). Recall that S n,j (µ j , h, t) = 1 n n i=1 m i,j (N i , µ j , h, t) 2 . Hence:

S n,j (µ j , h, t) = 1 n n i=1 A i,j (t) 2 + 2A i,j (t)m i,j (t) + m i,j (t) 2 ,
where:

A i,j (t) = t 0 S(u -) Ȳ (u) - S(u -) EY (u) dN i,j (u)
.

• Step 1. We shall show that:

1 n n i=1 m i,j (t) 2 -V j (t) ∞ P → 0 as n → ∞. Write f i,j (ω, t) := m i,j (t) 2 and F j nω = {(f 1,j (ω, t), . . . , f n,j (ω, t)), t ∈ [τ 1 , τ ]}.
Write also:

g i,j (ω, t) = t 0 S(u -) EY 1 (u) dN i,j (ω, u),
and

G j nω = {(g 1,j (ω, t), . . . , g n,j (ω, t)), t ∈ [τ 1 , τ ]}.
Then, almost surely:

|g i,j (ω, t)| = t 0 1 P(C ≥ u) dN i,j (ω, u) ≤ 1 P(C ≥ τ ) τ 0 dN i,j (ω, u) ≤ B P(C ≥ τ ) .
Hence, the class G j nω is euclidean (see [START_REF] Pollard | Empirical processes: theory and applications[END_REF] pp. 38 for a definition of euclidean). Indeed, as the g i,j 's are non decreasing, the class is of pseudo-dimension 1 (see [START_REF] Bilias | Towards a general asymptotic theory for Cox model with staggered entry[END_REF]). Conclude by using Theorem 4.8 from [START_REF] Pollard | Empirical processes: theory and applications[END_REF]. The same arguments apply to the class

H j n = {(µ j (t), . . . , µ j (t)), t ∈ [τ 1 , τ ]}, as |µ j (t)| ≤ B. Now, notice that F j nω ⊂ (G j nω -H j n ) 2 .
As euclidean classes are stable under addition and multiplication (see [START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF]), F j nω is euclidean. Thus, since m i,j (t) is a mean zero process:

sup t∈[τ1,τ ] 1 n n i=1 f i,j (t, ω) -Ef i,j (t, •) a.s.
-→ 0, as n → +∞.

•

Step 2. Let us show that:

1 n n i=1 A i,j (t) 2 ∞ P → 0, as n → +∞.
Rewrite A i,j (t) as:

A i,j (t) = t 0 1 Ȳ (u)EY (u) S(u -)(EY (u) -Ȳ (u)) -Ȳ (u)(S(u -) -S(u -)) dN i,j (u).
But, from our hypothesis on τ we have:

inf t∈[τ1,τ ] EY (t) = P(D ≥ τ )P(C ≥ τ ) > 0,
and:

Ȳ (u) = 1 n n i=1 I(X i ≥ u) ≥ Ȳ (τ ),
where the last term is well-known to converge almost surely to P(D ≥ τ )P(C ≥ τ ) > 0. Thus, almost surely:

1 n n i=1 A i,j (t) 2 ∞ ≤ EY (u) -Ȳ (u) ∞ + S(u -) -S(u -) ∞ Ȳ (τ )EY (τ ) 2 B 2 ,
since N i,j is bounded by B. The above inequality and the convergence of both EY (u) -Ȳ (u) ∞ and S(u -) -S(u -) ∞ yield the expected convergence of step 2.

• Step 3. Show that: 2 n n i=1 A i,j (t)m i,j (t) ∞ P → 0, as n → +∞.
The same arguments as in Step 2 apply to show that

1 n n i=1 A i,j (t) ∞ P → 0.
Conclude by noticing that:

max 1≤i≤n m i,j (t) ∞ ≤ max 1≤i≤n τ 0 S(u -) EY (u) dN i,j (u) + µ j (τ ) ≤ B EY (τ ) + B.
Proof of (A3 ⋆ ). First, notice that:

m i,j (N i , µ j , h, t) ∞ = t 0 S(u -) Ȳ (u) dN i,j (u) -µ j (t) ∞ ≤ τ 0 1 Ȳ (u) dN i,j (u) + µ j (τ ) ≤ B Ȳ (τ ) + B.
It is easily seen that this last sum of two terms is almost surely o p (n 1 /2 ) since Ȳ (τ ) converges almost surely to E Y (τ ) > 0 when n → +∞. Thus, condition (A3 ⋆ ) is fullfilled which completes the proof of Theorem 2.

Joint convergence

In this section we will state a corollary of our Theorem 2 where we consider the profile empirical likelihood ratio function for both mean functions µ 1 and µ 2 . This latter is defined as:

EL n (µ, h, t) = max n i=1 np i , n i=1 p i m i (N i , µ, h, t) = 0, p i ≥ 0 and n i=1 p i = 1 , where m i (N i , µ, h, t) = (m i,j (N i,j , µ j , h, t)) j=1,2 .
As before, when no confusion is possible, let us write:

m i (t) = (m i,j (N i,j , µ j , h, t)) j=1,2 = t 0 S(u -) EY (u) dN i,j (u) -µ j (t) j=1,2 , m i (t) = (m i,j (N i,j , µ j , h, t)) j=1,2 = t 0 h(u)dN i,j (u) -µ j (t) j=1,2
.

Before stating the corollary of our Theorem 2, recall the following result from [START_REF] Dauxois | Non-parametric tests for recurrent events under competing risks[END_REF]:

Theorem 3. As n → ∞, √ n µ 1 -µ 1 µ 2 -µ 2 D -→ G 1 G 2 =: G.
Corollary 4. Suppose that the eigenvalues of V (t), the covariance matrix of m i (t), are uniformly bounded away from zero and infinity on [τ 1 , τ ] for some real constant τ 1 > 0. Suppose also that E|m i (t)| is uniformly bounded away from zero and infinity on the same interval (where the absolute value is taken composant-wise). Then:

-2 log EL n (µ, h, •) D -→ G(•) t V -1 (•)G(•), as n → +∞, in the Skorohod space D[τ 1 , τ ].
Proof. Although the quantities M n and S n from last section have to be adapted to the vectorial case, the proof is very similar to the one of Theorem 2. Let:

M n (µ, h, t) = 1 n n i=1 m i (N i , µ, h, t) and S n (µ, h, t) = 1 n n i=1 m i (N i , µ, h, t) ⊗2 ,
where x ⊗2 := xx t . It is then easily seen that hypotheses (A1 ⋆ ) and (A3 ⋆ ), after straightforward adaptation to the multivariate case, are verified.

(A0 ⋆ ): P(∃t ∈ [τ 1 , τ ] : EL n (µ, h, t) = 0) -→ 0, as n → ∞.
It is sufficient to show that almost surely, as n → ∞, every half-space around 0 is occupied by the m i 's. Formally, it is sufficient to show that, almost surely:

lim n→∞ inf t∈[τ1,τ ],θ∈Θ 1 n n i=1 I( m i (t), θ > 0) > 0, (2)
where Θ is the unit sphere of R 2 . To show (2), remark that for η > 0, Cauchy-Shwarz's inequality yields:

| m i (t) -m i (t), θ | > η ⇒ m i (t) -m i (t) 2 > η,
where • 2 denotes the euclidean norm. Thus:

1 n n i=1 I( m i (t), θ > 0) ≥ 1 n n i=1 I( m i (t), θ > η) - 1 n n i=1 I( m i (t) -m i (t) 2 > η). (3)
But, we have shown that the last term in (3) converges to 0 as n → ∞ in probability. Let now T i (t, θ) := m i (t), θ . We know from our hypotheses that Var T i (t, θ) = V (t)θ, θ is uniformly bounded away from 0 and infinity on [τ 1 , τ ] × Θ. It is also true for E |T i (t, θ)|. We can then conclude by applying the same proof as in Theorem 2 to the random variable T i (t, θ) instead of m i,j (t).

(A2 ⋆ ): sup t∈[τ1,τ ] |S n (µ, h, t) -V (t)| P -→ 0. Notice that: S n (µ, h, t) = 1 n n i=1 m 2 i,1 m i,1 m i,2 m i,2 m i,1 m 2 i,2
, where m i,j is short for m i,j (N i , µ j , h, t). We have already shown the convergence of the diagonal terms in the proof of Theorem 2. Using similar arguments one can show that, in probability:

sup t∈[τ1,τ ] 1 n n i=1 m i,1 (t)m i,2 (t) -Cov(m 1 (t), m 2 (t)) -→ n→∞ 0.
4 Application to nosocomial infections

Algorithm in dimension 1

In this section, we will apply our results to the data set of nosocomial infections mentioned in section 1. We will restrict our attention to pneumonies, septicemias and urinary tract infections, as those are the most frequent infections in the data set: for example, 373 patients experienced a total amount of 463 pneumonies during their stay. Using Theorem 2, one can build simultaneous confidence band for the mean function of each type of infection. We shall then describe the algorithm used to obtain a 95% confidence band for the mean function of one type of infection.

Using the same notations as before, we have n = 7867 and max 1≤i≤n X i = 380. First of all, we approximate the 95%-quantile of the limiting law by bootstrap. To this effect, we draw (Z ⋆ 1 , . . . , Z ⋆ n ) uniformly in {Z 1 , . . . , Z n }, where Z i := (N i , X i , δ i ). Then, for each day k in {0, . . . , 380}, we compute -2 log EL n (μ(k), ĥ⋆ , k) and we store the supremum (over k) of those values. Finally, we repeat this 1000 times, sort the values and take the 950-th as our 95%-estimated quantile û95 . Then, for each day k in 0, 380 , we determine by dichotomy the convex set C k for which we have:

∀y ∈ C k , -2 log EL n (y, ĥ, k) ≤ û95 .
Notice that the confidence band stays the same after the last jump: for example, no patient contracted a pneumonia after the 125-th day as can be seen on the Figure 1. Notice also that the confidence band is not centered around μ: unlike central limit theorem, empirical likelihood builds confidence regions whose shape strongly depends on the geometry of the data.

Algorithm in dimension 2

As pointed out in section 3.2, our result still holds for more than one type of events. We thus built a confidence tube for (µ 1 , µ 2 ) where µ 1 is the mean number of pneumonies and µ 2 the mean number of septicemias. In this context, the number of patients who experienced either pneumonies or septicemias is 547, with 71 patients experiencing both types of infections at least one time. The total number of pneumonies is 463 (373 patients experienced it) and the total amount of septicemias is 289. For the 3D graph to be more readable, we projected it on a plane, eliminating the dimension corresponding to time. Each convex set corresponds to a day, the first day being at the bottom left corner and the last day at the upper right corner. As in dimension 1, one can notice that this confidence tube is not elliptic as would be expected had we used the central limit theorem. Once again, this is due to empirical likelihood.

Figure 1 :Figure 2 :

 12 Figure 1: Nosocomial data set. Estimated specific mean function and associated 95% empirical likelihood confidence bands for a) pneumonia, b) septicemia and c) urinary tract infection.