Here is the design of a clinical trial:

Official Title: High Dose Vitamin-D Substitution in Patients With COVID-19: a Randomized Controlled, Multi Center Study

Included conditions:
- Covid19
- Vitamin D Deficiency
- Corona Virus Infection
- ARDS
- Coronavirus
- SARS-CoV Infection

Study Armgroups:
- {'label': 'High Dose Vitamin D', 'type': 'EXPERIMENTAL', 'description': "Patient will receive a single high dose of vitamin D (140'000) in addition to daily 800 IU of vitamin D.\n\nThe medication be administered orally", 'interventionNames': ['Drug: Single high dose vitamin D', 'Drug: Treatment as usual vitamin D']}
- {'label': 'Placebo', 'type': 'PLACEBO_COMPARATOR', 'description': 'Patient will receive a single dose of placebo, orally administered and then treatment as usual (daily 800 IU of vitamin D, orally administered)', 'interventionNames': ['Drug: Placebo', 'Drug: Treatment as usual vitamin D']}

Interventions:
- {'type': 'DRUG', 'name': 'Single high dose vitamin D', 'description': "Patient receives either one dose orally of 140'000 IU (7 ml) of this drug once as an intervention treatment additionally to TAU or the patient receives 7 ml of the placebo Solution (7 ml) in addition to TAU", 'armGroupLabels': ['High Dose Vitamin D'], 'otherNames': ['VITAMIN D3 oil 500 IU/drop']}
- {'type': 'DRUG', 'name': 'Placebo', 'description': 'Patient receives a single dose of a placebo solution', 'armGroupLabels': ['Placebo'], 'otherNames': ['Oily placebo solution']}
- {'type': 'DRUG', 'name': 'Treatment as usual vitamin D', 'description': 'Both groups receive the treatment as usual after the single high dose or the placebo which will be 800 IU per day', 'armGroupLabels': ['High Dose Vitamin D', 'Placebo'], 'otherNames': ['Vitamin D3 solution 4000 IU/ml']}

Primary Outcomes:
- {'measure': 'Length of hospitalization', 'description': 'Overall duration of the hospitalization from day of admission until the day of discharge or fatality', 'timeFrame': 'Administration to Discharge from hospital care (mean duration is between 14 and 22 days for Patients with COVID 19)'}

Please estimate the sample size based on the assumption: 
Alpha = 0.05, power = 0.80, sigma = 2.96, delta = 2, alternative hypothesis = two-sided, and a 10% dropout rate.
Sample size {14}
        To calculate the sample size, we looked at an interventional study of vitamin D, which examined the length of hospital stay as an outcome and what is known about the median length of hospital stay in patients hospitalized with COVID-19. A randomized, double-blind, placebo-controlled trial examined the effects of adjunctive vitamin D in adults hospitalized with community-acquired pneumonia (CAP) on time to recovery [28]. The primary outcome was the complete resolution of chest radiograph infiltrates at 6 weeks post-study treatment. Secondary outcomes included length of hospital stay, intensive care admission, and return to regular activity. At the same time, adjunctive vitamin D did not affect the primary outcome (OR 0.78, 95% CI 0.31 to 1.86, p = 0.548). There was evidence it increased the complete resolution of pneumonia in participants with baseline vitamin D levels <â25ânmol/L (OR 17.0, 95% CI 1.40â549.45, p = 0.043). Although not significant, the results indicated that patients in the intervention group had shorter length of hospital stay (mean = 3.8, SD = Â± 3.1, n = 60) when compared with patients in the placebo group (mean = 4.8, SD = Â± 6.5, n = 57), which indicates a small effect size (Cohenâs d) of 0.1980. The study indicated a 20% reduction in length of hospital stay.
        Considering the above results in the light of COVID-19, more extended hospital stays in COVID-19 patients are expected. In smaller observational studies describing the clinical course of hospitalized COVID-19 patients in China, a median hospital stay of 12âdays (IQR 9â15) (n = 137) [1] or of 10âdays (IQR, 7.0â14.0) among those discharged alive (n = 47) [29] were observed. Guan et al. presented the clinical characteristics of 1099 patients with laboratory-confirmed COVID-19, and the median length of hospital stay was reported to be 12âdays (IQR = 10â14) [30], which we will take as a basis for our sample size calculation.
        Sample size calculation was performed using the function âpower.t.testâ from the R package âstatsâ with the following parameters [31]: alpha = 0.05, power = 0.80, sigma = 2.96, delta (difference in the primary outcome between intervention and control group) = 2, and alternative hypothesis = two-sided, which resulted in a minimum of 35 patients per group, therefore a total of 70 patients. With a conservatively estimated 10% dropout rate, including the loss to follow-up, and a block size randomization of four, we decided to include 80 patients.