Obstructive sleep apnea syndrome (OSA) is a common condition associated with major repercussions such as excessive daytime sleepiness and impaired quality of life as well as metabolic and cardiovascular complications. Continuous positive airway pressure (CPAP) remains the treatment of choice but its effectiveness remains limited, especially in reducing cardio-metabolic risk. Interventions to modify the lifestyle are therefore recommended in the management of OSA. The emergence of information and communication technologies is an opportunity for patients to have tools that promote self-management and behavioral changes. The recent development of telerehabilitation (TR) is a promising approach that has only been the subject of pilot studies. In a randomized, controlled and multicenter study, we propose to test the hypothesis according to which the use of a mobile TR solution, associated with continuous positive airway pressure (CPAP), will allow obese patients to adopt behavioral modifications to improve markers of severity of OSA. The analysis of big data (data-mining) will allow a better understanding of the motivational obstacles and levers.

Statistical analysis plan The number of patients solicited for the study and the number of subjects included and randomized in each group will be reported on the flow chart. Premature stops (lost to follow-up, death, study withdrawals) and their reasons will be reported as well as any deviations from the protocol.

An initial descriptive analysis of each of the groups will be carried out. For qualitative variables, this description will include the number as well as the frequency of the different modalities. Concerning the quantitative variables, the description will include the number, the mean, the standard deviation, the median as well as the extreme values according to the distribution.

The comparability of the 2 groups will be verified for all the initial characteristics likely to influence the results. In the event of non-comparability on one or more parameters, an adjustment will be made on this or these parameters for the comparisons between groups of judgment criteria.

Analysis of the primary judgment criterion The variation in the evolution of the various main criteria will be compared between the 2 groups using a nonparametric means comparison test (Mann-Whitney test) or if the distribution turns out to be Gaussian and the conditions application are verified, a parametric test (Student's T).

Analysis of secondary endpoints The qualitative variables will be compared by a Chi-square test. Otherwise, if the conditions for performing this test are not met, Fisher's exact test will be used.

The quantitative variables will be compared between the two groups:

• in the case of Gaussian variables, by a student's t test or an analysis of variance
• in the case of non-Gaussian variables, by a Wilcoxon-Mann-Withney test

Longitudinal data (quality of life scores, physical activities) will be modeled by a mixed regression with the patient effect as a random variable. A linear regression model modeling the variation of the longitudinal endpoints will be fitted to the covariates having at least one trend (p <0.15) to the difference between the two arms of the study. A forward type procedure will be implemented to obtain a final multivariate model.

Management of missing data:

The IPTW method will be used. The lost to follow-up will be considered using a weighting on the inverse of the probability of being censored

Inclusion Criteria:

• Severe obstructive sleep apnea (apnea-hypopnea index ≥30 / hour)
• Body mass index ≥ 30 kg / m²
• age between 30 and 75 years old
• written consent

Exclusion Criteria:

• Presence of contraindications to perform exercise training (neuromuscular disease, orthopedic cause, acute coronary syndrome or stroke in the previous three months)
• Active infection, chronic inflammatory disease, cancer under treatment. Systemic treatment with immunosuppressants or corticosteroids
• Obstructive Sleep Apnea treated by Continuous Positive Airway Pressure or mandibular advancement device in the 3 months preceding inclusion
• Participation in a rehabilitation program scheduled within six months of inclusion
• Predominantly central sleep apnea syndrome (AC> 50%)
• Heart failure with Left Ventricular Ejection Fraction <40% known
• Bariatric surgery in the previous 6 months or bariatric surgery project in the 6 months
• Alcohol> 14 drinks per week
• Inability to understand and / or answer questionnaires.
• Refusal to use a smartphone or digital device
• Inability to access an internet connection at home.
• Subject in a period of relative exclusion with respect to another protocol or for which the maximum annual amount of compensation of € 4,500 has been reached.
• Subject not affiliated with a social security scheme, or not beneficiary of such a scheme.
• Pregnant or breastfeeding woman, patient unable to give her protected adult consent, vulnerable people
• Subject deprived of liberty by judicial or administrative decision