From Powershare, Inc. v. Syntel, Inc., 597 F.3d 10 (1st Cir. 2010):

The Federal Magistrates Act confers authority upon district judges to designate magistrate judges to hear pretrial motions. 28 U.S.C. § 636(b)(1). Magistrate judges serve as aides to, and under the supervision of, district judges; but magistrate judges are not themselves Article III judicial officers. Given their status as Article I judicial officers, magistrate judges ordinarily [edit. note: i.e., without consent of the parties] may not decide motions that are dispositive either of a case or of a claim or defense within a case. This is so because "[t]he Constitution requires that Article III judges exercise final decisionmaking authority." Ocelot Oil Corp. v. Sparrow Indus., 847 F.2d 1458, 1463 (10th Cir. 1988); see Stauble v. Warrob, Inc., 977 F.2d 690, 693-94 (1st Cir. 1992). Dispositive motions include those enumerated in 28 U.S.C. § 636(b)(1)(A), but this list is not exhaustive; rather, it simply "informs the classification of other motions as dispositive or nondispositive." Phinney v. Wentworth Douglas Hosp., 199 F.3d 1, 5-6 (1st Cir. 1999).

Consistent with this dichotomy between dispositive and non-dispositive motions, Federal Rule of Civil Procedure 72 sets out two separate standards of review to be employed by a district judge in reviewing a magistrate judge's determinations. When a magistrate judge decides a non-dispositive motion, the district judge may, given a timely appeal, set aside the order if it "is clearly erroneous or is contrary to law." Fed. R. Civ. P. 72(a). Absent a timely appeal, the order stands. Id. When, however, a magistrate judge passes upon a dispositive motion, he or she may only issue a recommended decision, and if there is a timely objection, the district judge must engage in de novo review. Fed. R. Civ. P. 72(b). Absent a timely objection, the recommended decision ripens into an order. Id.

Here, the district judge employed the "clearly erroneous or contrary to law" standard applicable to non-dispositive motions under Rule 72(a). Syntel protests that its motion to stay the litigation to allow resolution of the parallel arbitration proceeding was, in effect, dispositive of the court case and, thus, should have engendered de novo review by the district judge pursuant to Rule 72(b).

No court of appeals has decided this precise question. Nevertheless, a number of district courts have held that motions to stay litigation and compel related arbitration are non-dispositive motions under Rule 72(a). [Citations omitted.].

Motions to stay litigation pending the resolution of parallel arbitration proceedings are not among the motions enumerated in 28 U.S.C. § 636(b)(1)(A). Nor are they of the same character as the listed motions. A federal court's ruling on a motion to stay litigation pending arbitration is not dispositive of either the case or any claim or defense within it. Although granting or denying a stay may be an important step in the life of a case (lawyers are keenly aware that there are substantive consequences to whether or not a stay is granted), in the last analysis a stay order is merely suspensory. Even if such a motion is granted, the court still retains authority to dissolve the stay or, after the arbitration has run its course, to make orders with respect to the arbitral award. See Federal Arbitration Act, 9 U.S.C. § 9 (permitting parties to apply to the court for an order confirming the award); id. § 10 (providing district courts with authority to vacate an arbitral award); id. § 11 (providing district courts with authority to modify an arbitral award). We acknowledge that the scope of judicial review of arbitral awards is very narrow, but that does not extinguish such review.... Thus, there is no final exercise of Article III power at the time the court acts on the motion to stay. ***

In light of these realities, we conclude that, from a procedural standpoint, the district judge acted appropriately in reviewing the magistrate judge's denial of Syntel's motion to stay under the "clearly erroneous or contrary to law" standard elucidated in Rule 72(a).