A Study of the Pharmacokinetics and Safety of MK-8808 (MK-8808-002)

This is a study of the overall safety, tolerability, and pharmacokinetics (PK) of MK-8808 versus rituximab (MabThera® and Rituxan®) in participants with moderate to severe RA with an inadequate response or intolerance to methotrexate.

In Part A of the base study, participants are randomized to either MK-8808 or MabThera®. In Part B of the base study, participants are randomized to either MK-8808, MabThera®, or Rituxan®. Participants enrolled in Part A are not eligible to participate in Part B. In both Parts A and B, participants will receive one or two courses of therapy, with each course including two infusions of the study drugs.

The extension portion of the study (Part C) will sequentially follow the base study beginning at Week 52 and continue for an additional 54 weeks. All participants who meet eligibility criteria and continue into the study extension will be treated with open-label MK-8808. Participants randomized to MK-8808 in the base study will remain on the same therapy. Participants randomized to rituximab (MabThera® or Rituxan®) in the base study will be switched to MK-8808 for the extension study.

Eligibility Criteria:
Inclusion Criteria:

* Female participants of reproductive potential must demonstrate a serum β-human chorionic gonadotropin (hCG) level consistent with the nongravid state at the pre-study (screening) visit, and a negative urine pregnancy test within 24 hours prior to all doses and agree to use (and/or have their partner use) two acceptable methods of birth control beginning at least 2 weeks prior to administration of the first dose of study drug, throughout the study (including washout intervals between treatment periods/panels) and until at least 12 months after administration of the last dose of study drug in the last treatment period
* The participant has a Body Mass Index (BMI) ≤35 kg/m\^2 at the prestudy (screening) visit
* For Part A Only: The participant has a body surface are (BSA) ≤2.0 m\^2 at the prestudy (screening) visit.
* Has satisfied at least 4 of 7 American Rheumatology Association (ARA) 1987 revised criteria for the diagnosis of RA
* Is American College of Rheumatology (ACR) Functional Class I, II, or III
* Had a diagnosis of RA made at least 6 months prior to the prestudy (screening) visit, was ≥ 16 years of age when diagnosed, and has active disease
* Is on a stable oral, IM, or SC dose of methotrexate and is continuing to take methotrexate
* Has an inadequate response or intolerance to at least one disease-modifying antirheumatic drug (DMARD)
* For Part A: Participant is either naïve to biological therapy for RA or has had an inadequate response to previous or current treatment with an anti-tumor necrosis factor (TNF) treatment (patient could have failed up to three anti-TNF agents treatments) or participant has had intolerance up to three anti-TNF treatments.
* For Part B: Participant has had an inadequate response to previous or current treatment with an anti-TNF treatment (patient could have failed up to three anti-TNF agents treatments) or participant has had intolerance up to three anti-TNF treatments
* Participant has no clinically significant abnormality on electrocardiogram performed at the prestudy (screening) visit and/or prior to administration of the initial dose of study drug
* For Part B Only: Participant is positive for rheumatoid factor (RF) or, if negative for RF, is positive for anti-CCP at screening visit
* For Part C Only: Participant must have completed the first 52 weeks of treatment in the base study
* For Part C Only: Participant achieved a minimum 20% response from baseline on the American College of Rheumatology (ACR) Responder Index (ACR20) at Visit 19 (last visit for the base study)

Exclusion Criteria:

* Mentally or legally incapacitated, has significant emotional problems at the time of the prestudy (screening) visit or during the conduct of the study or has a history of a clinically significant psychiatric disorder over the last 5 years
* Creatinine clearance of ≤ 80 mL/min
* History of stroke, chronic seizures or major neurological disorder
* History of neoplastic disease, except treated basal cell carcinoma or carcinoma in situ of the cervix or other malignancies which have been successfully treated ≥ 5 years
* History of leukemia, lymphoma, malignant melanoma, or myeloproliferative disease regardless of the time since treatment
* History of coronary artery disease, congestive heart failure (New York Heart Association Class I-IV), or a history of clinically significant arrhythmia (including any history of atrial fibrillation, atrial flutter, or any sustained ventricular arrhythmia)
* Hypersensitivity or allergy to rituximab or any of the excipients of MK-8808 or rituximab (MabThera® or Rituxan® )
* History of a rheumatic autoimmune disease other than RA (e.g. systemic lupus erythematosus (SLE), polymyositis, etc.)
* Severe active infection of any type or history of a medically serious infection as defined by a history of treatment requiring hospitalization, long term IV outpatient treatment for systemic bacterial, viral or fungal infection, use of IV antibiotics within 30-days of screening, or use of antibiotic therapy three or more times in the last six months prior to screening
* History of opportunistic infection
* Active-virus vaccination within 4 weeks
* Active tuberculosis with or without adequate treatment, history of latent tuberculosis without written confirmation from health care provider of adequate prophylaxis or any evidence of tuberculosis on a chest X-ray performed within 3 months of dosing
* Chronic hepatitis B or hepatitis C infection or has human immunodeficiency virus (HIV) infection
* Previously treated with rituximab (MabThera® or Rituxan®) or any investigational anti-CD20 antibody
* Active use or planned use of a prohibited DMARD during the course of study participation, and/or insufficient washout from a prohibited DMARD at the time of the planned first dose of MK-8808/rituximab (MabThera® or Rituxan®)
* Had major surgery, donated or lost 1 unit of blood (approximately 500 mL) within 4 weeks
* Participated in another investigational study with length of time within at least 5 half-lives of the previous investigational study drug
* Pregnant or breastfeeding or expecting to conceive
* Allergy to murine proteins
* Allergy or sensitivity to components of the drug vial or any of the materials used for infusion