Table 5. Explanation of each judgment for each domain in the risk of bias assessment of the included randomized controlled trials.

CT: Computed tomography, LED: Light-emitting diode, LLLI: Low-level laser irradiation, GCF: Gingival crevicular fluid

Study	Sequence generation	Allocation concealment	Blinding of participants and personnel	Blinding of outcome assessment	Incomplete outcome data	Selective reporting	Other bias
Charavet et al. 2016 Belgium [33]	High risk: No mention of the method used for randomization.	Unclear risk: No mention of the method used to conceal the allocation sequence. “Twenty-four adult patients presenting with mild overcrowdings were randomly allocated to either a control group that was treated with conventional orthodontics or a test group that received piezo-assisted orthodontics.”	High risk: No mention of this point.	High risk: No mention of this point.	Low risk: “All patients were followed until the completion of treatment. Two patients (1 in each group) failed to show up for the post-treatment CT scan and were excluded from the follow-up.” incomplete outcome data adequately. Addressed (number, reasons)	Low risk: The protocol for the study was registered in clinical trial.gov study ID: NCT03406130 and the outcomes that mentioned in the protocol have been reported (unless using CT instead of CBCT).	Low risk: No other forms of bias seemed to be found.
Nahas et al. 2016 United Arab Emirates [35]	High risk: No mention of the method used for randomization.	Low risk: “The patients were randomly assigned to one of the two groups utilizing simple randomization by asking them to draw a sealed envelope (n = 40) that indicated their allocation to a test group (n = 20) in which LED.	High risk: No mention of this point.	Low risk: “The alignment of the lower six anterior teeth was evaluated by a single investigator who was blinded regarding the patients’ group allocations.”.	Low risk: “a total of four patients were dropped out from the control group, three required interproximal reduction of the lower anterior teeth, and one failed to attend more than three consecutive appointments. In the test group, two patients were excluded from the study; the first patient did not commit to using the LED device with a compliance rate of above 80 % and the second patient experienced a malfunction with his LED device “ incomplete outcome data adequately. Addressed (number, reasons)	Low risk: The protocol was not registered. But the reported outcomes in the result section seemed to be corresponding with the pre-defined outcomes aforesaid in the method section.	Low risk: No other forms of bias seemed to be found.
Qammruddin et al. 2017 Pakistan [36]	High risk: No mention of the method used for randomization.	Low risk: “The maxillary arch was divided into experimental and placebo groups randomly by flipping a coin. LLLI was”.	High risk: No details of blinding of participants or personnel.	High risk: No details of blinding of outcome assessors.	Low risk: “Twenty-two patients were recruited for the study, of whom 2 were dropped because of dislodgement of the coil spring prematurely in 1 patient and analgesics taken by the other." (incomplete outcome data adequately addressed (numbers, reasons )	Low risk: The protocol was not registered. However, the reported outcomes in the result section seemed to be corresponding with the pre-defined outcomes aforesaid in the method section.	Low risk: No other forms of bias seemed to be found.
Kalemaj et al. 2017 Italy [34]	Low risk: “A list of block randomization with variable block size scheme of 3 and 6 was generated on Stata using the command “-ralloc” (StataCorp, College Station, Tex).”	Unclear risk: “Allocation concealment was obtained implementing a centralized assignment which did not involve trial investigators and staff.”	Low risk: “Blinding of patients and practitioner was not feasible”	Low risk: “however outcome assessment was blind because the GCF samples, models, and questionnaires were enumerated in a sequential order from the first to the last collection with no reference to pertaining patients or groups.”	Low risk: “No patient was excluded from the study and there were no lost to follow-up.”	Low risk: The protocol was not registered. But the reported outcomes in the result section seemed to be corresponding with the pre-defined outcomes aforesaid in the method section.	Low risk: No other forms of bias seemed to be found.
Lalnunpuii et al. 2020 India [38]	Low risk: “Randomization was carried out, using a computer-generated random allocation sequence to ensure equivalent distribution amongst the 3 groups.”	Low risk: “the sequences were concealed and were chosen by the patient.”	Low risk: “Blinding of the participants and the primary investigator was not possible due to the nature of the trial.”	Low risk: “Only the data analyser was blinded to the groups and the digital models presented to the data analyser were also coded.”	Low risk: The consort flow chart of participants through each stage of the trial shows there was no loss of participants.	Low risk: “the trial is registered at the National Trial Registry (CTRI/2018/04/013156)”	Low risk: No other forms of bias seemed to be found.
Kumar et al. 2020 India [37]	Low risk: “Randomization was carried out using a computer-generated random allocation sequence.”	Unclear risk: “The sequences were concealed and were chosen by the patient.”	Low risk: “Blinding of the participants and the primary investigator was not possible due to the nature of the trial”	Low risk: “Only the data analyser was blinded to the groups and the digital models presented were coded”	Low risk: “There was neither any loss of participants for any group nor a reported malfunction by any patient for the vibratory device”	Low risk: “the trial is registered at the National Trial Registry (CTRI/2018/04/013009).”	Low risk: No other forms of bias seemed to be found.
Al-Ibrahim et al. 2021 Syria [25]	Low risk: " SPSS for Windows, version 20 (IBM Corporation, Armonk, NY, USA) was used to determine the allocation of patients among the 3 groups, using a set of random numbers with an allocation ratio of 1:1:1.”	Low risk: “The allocation sequence was hidden using numbered, covered, and closed envelopes, which were opened only after performing all patients’ assessments and premolar extractions.”	Low risk: “It was not possible to make the treatment procedures blind for either patients or practitioners”	Low risk: ”so blinding was limited to data analysis”	Low risk: “No patient was lost to follow-up; therefore, 57 patients were included in the data analysis”	Low risk: “The current trial was registered at ClinicalTrials.gov (ID: NCT04950829)”	Low risk: No other forms of bias seemed to be found.