Arthur Connolly, John Handren, and James Kane challenge (pursuant to G. L. c. 151B, § 6, and G. L. c. 30A, § 14) a judgment of the Superior Court that affirmed a decision of the Massachusetts Commission Against Discrimination (MCAD).3 The MCAD had found the Suffolk County sheriff’s department (department) liable for employment discrimination against the plaintiffs and awarded them partial relief, including damages for emotional distress.4 Dissatisfied with the extent of the relief granted, the plaintiffs contend that the MCAD failed to apply the correct legal standard in determining whether the department discriminated against Connolly and Handren in its refusal to promote them, and erred in not awarding compensatory damages to the plaintiffs for the department’s denial of overtime, on the stated ground that any damages would be speculative. We affirm.

1. Background. After being denied the opportunity to work overtime, the plaintiffs filed individual complaints of discrimination with the MCAD in August, 1991. The MCAD found probable cause that discrimination had occurred and held a public *189hearing on the plaintiffs’ allegations. By written decision on August 19, 1998, an MCAD hearing officer made the following undisputed factual determinations.5

The plaintiffs were correction officers employed by the department and assigned to the Deer Island house of correction. Each had been injured in incidents of inmate violence in the late 1980’s, resulting in his inability to work in his normal capacities.

Prior to 1991, injured officers were not permitted to work unless they could perform the full duties of their positions. They were required to remain out of work, and they received workers’ compensation benefits. This policy was changed early in 1991, and the plaintiffs then returned to work after receiving assurances that appropriate measures would be taken to accommodate their physical limitations.

Initially, Kane returned to work in a “transitional duty position,” in which he was assigned to write policies and procedures for the new prison facility at South Bay. When that position was eliminated, he was reassigned as a training coordinator, a position from which he was later transferred. Handren was originally assigned to work in a tower area where there was no inmate contact. Connolly returned to work but was unable to perform the necessary tasks, and was reassigned to work in the towers. After missing several morning roll calls, for which he received both oral and written warnings and two suspensions, Connolly was transferred from the morning shift to the less desirable evening shift despite his protests that his tardiness was a result of his injuries.

In the beginning, each of the plaintiffs received work assignments that accommodated his disabilities. Eventually, however, they were each transferred to positions that they felt did not meet their medical restrictions. Kane was sent to work in the medical observation unit at the South Bay facility, a posting that he feared might require inmate contact. He felt uncomfortable *190wearing his medical braces in this position because it exposed his vulnerability to the inmates. After voicing these concerns, he was again transferred, this time to the infirmary, which presented similar problems. Although Kane was instructed not to intervene in any inmate strife, he was concerned that he would not be able to protect himself or fellow officers in the event of a physical altercation. Kane consequently declined to accept a position supervising visitors and inmates in the visitors’ area. Shortly thereafter, Kane received notice that he was being transferred to the 3:00 p.m. to 11:00 p.m. shift, which was less desirable than the 7:00 a.m. to 3:00 p.m. shift. Sometime after receiving notice of that transfer, Kane collapsed while attempting to open a door in the infirmary, was hospitalized, and did not return to work.

Connolly was reassigned from the towers to the medical observation unit and later to the infirmary at the South Bay facility. He claimed that both positions exacerbated his injuries but that he did not express his dissatisfaction because he felt Kane was being retaliated against for his complaints. Connolly continued to work in the infirmary until he fell and injured his knee in May, 1992. After returning to work in August, 1992, he was placed in a unit that housed inmates who worked details. That position was not considered “light duty,” but Connolly said he did not complain for fear of retaliation. In May, 1993, he was injured when he attempted to intervene in an inmate fight and was knocked unconscious. He did not return to work.

Handren was initially assigned to the same medical observation unit at the South Bay facility. He complained that this new position did not accommodate his medical restrictions because of the potential for inmate contact. He was then assigned to other posts, all of which involved inmate contact. In August, 1992, he stayed out of work for approximately six weeks because of neck pain he began to experience while on the job. Upon his return, he was assigned to an upper-level gate control post, but when that job was eliminated, he was relocated to the ground level, which again required interaction with inmates. At the time of the hearing officer’s decision, he was still on that job.

Prior to 1991, “light duty” officers, such as the plaintiffs, *191had been allowed to work overtime. Officers would shift their positions to accommodate light duty officers who had volunteered for overtime. Prior to being injured, Kane and Connolly had worked significant amounts of overtime, while Handren had worked a moderate amount. On August 15, 1991, Kane was shown a memorandum from the superintendent of operations at Deer Island, which stated that light duty officers would no longer be permitted to work overtime. The plaintiffs continued to volunteer for overtime but were not selected for any overtime shifts.

On August 16, 1991, the plaintiffs’ union filed a class action grievance challenging the new overtime policy on the ground that it constituted discrimination. On August 21, 1991, the plaintiffs also filed their individual discrimination complaints with the MCAD. The “no overtime” policy was eventually rescinded by the department. Light duty officers were again eligible to work overtime, but only where the available overtime positions accommodated the officers’ needs. After the new policy was implemented, the plaintiffs continued to request overtime positions but were rarely called upon for such work. The department claimed that the reduction in overtime availability was the result of the opening of the new South Bay facility and the addition of new officers.

In December, 1991, the department announced opportunities for promotions to the positions of sergeant and lieutenant. No specific criteria were listed in conjunction with either position. Kane applied for promotion to lieutenant, Connolly applied for promotion to sergeant and lieutenant, and Handren applied for promotion to sergeant. None of the plaintiffs was promoted. Kane and Connolly were senior to officers who were promoted to lieutenant.

Pursuant to a provision of their collective bargaining agreement, the plaintiffs requested the reasons why they were bypassed for promotion. The department issued a response stating that the individuals who received promotions were more qualified than the plaintiffs. Later, the department asserted that neither Kane nor Connolly had ever taken the civil service promotional exam, that Kane had not been promoted in the twelve years he worked prior to his injury, and that, prior to his *192injury, Connolly had resigned his sergeant’s position in order to remain on day shift. The department also noted that Handren had declined an earlier promotional opportunity and that Deputy Superintendent Brown was not impressed with Handren’s record or performance at the interview. Union representatives, however, testified to their understanding, based on comments by department superiors, that the department did not look favorably on promoting officers with disabilities until they returned to full duty.

Upon these facts, the hearing officer found that the department had discriminated against the plaintiffs by denying them the opportunity to work overtime. She also found that the department unreasonably failed to accommodate Connolly’s and Kane’s disabilities and that the decision not to promote Kane was discriminatory in nature. All three plaintiffs were awarded damages for the emotional distress caused them by the discrimination, and Kane and Connolly were also awarded damages for lost wages. The hearing officer did not, however, award any damages for the claimed lost overtime, because she determined that such an award “would be nothing short of speculative.” The department appealed the decision to the MCAD, and the plaintiffs cross-appealed. The MCAD affirmed the decision of the hearing officer and was in turn affirmed by a Superior Court judge on the plaintiffs’ petition for judicial review.

2. Standard of review. An administrative agency’s final decision is most commonly modified or set aside on judicial review where the court determines that the aggrieved party’s substantial rights were prejudiced because the agency’s decision was based upon an error of law, was unsupported by substantial evidence, or was arbitrary and capricious. G. L. c. 30A, § 14(7). See Wheelock College v. Massachusetts Commn. Against Discrimination, 371 Mass. 130, 133 (1976). The reviewing court “shall [however] give due weight to the experience, technical competence, and specialized knowledge of the agency, as well as to the discretionary authority conferred upon it.” G. L. c. 30A, § 14(7). “A court may not displace an administrative [agency’s] choice between two fairly conflicting views, even though the court would justifiably have made a different choice *193had the matter been before it de nova.” Embers of Salisbury, Inc. v. Alcoholic Bevs. Control Commn., 401 Mass. 526, 529 (1988), quoting from School Comm. of Wellesley v. Labor Relations Commn., 376 Mass. 112, 120 (1978). See Hotchkiss v. State Racing Commn., 45 Mass. App. Ct. 684, 695-696 (1998). When determining whether an agency decision is supported by substantial evidence, the standard of review is “highly deferential” to the agency. Id. at 695. See Boston Police Superior Officers Fedn. v. Labor Relations Commn., 410 Mass. 890, 892 (1991) (“We generally accord considerable deference to the commission’s disposition of a charge”). Where, as is the case here, the parties waived their right to submit a transcript, the scope of review is further narrowed. The plaintiffs are precluded from arguing that the hearing officer’s findings are not supported by substantial evidence. Therefore, unless it is clear that the hearing officer’s ultimate findings are not supported by her subsidiary findings, our review is limited to determining whether error of law occurred. See Children’s Hosp. Med. Center v. Assessors of Boston, 353 Mass. 35, 39 (1967); New Bedford Gas & Edison Light Co. v. Assessors of Dartmouth, 368 Mass. 745, 749-751 (1975); Sturdy Memorial Foundation, Inc. v. Assessors of N. Attleborough, 47 Mass. App. Ct. 519, 520 (1999), S.C., 60 Mass. App. Ct. 573 (2004). We discern no such failure of support or legal error.

3. Discriminatory promotion policies. Although all three plaintiffs asserted that they had been denied promotions because of their handicaps, the hearing officer was persuaded that only Kane had been “eminently qualified for the promotion and was denied [it] on the basis of his handicap.” As to Connolly’s and Handren’s claims, the hearing officer was not persuaded that “they had distinguished themselves for promotion” or that the department’s contention, that Connolly and Handren were not the most qualified applicants for the available positions, was a pretext for discrimination.

Connolly and Handren fault the hearing officer’s conclusion, contending that she erred in her application of the relevant law. In particular, they argue that she improperly shifted to them the burden to prove that the department’s “legitimate” reasons for denying the promotions were pretextual and that she failed to *194apply the proper test for a “mixed-motive” discrimination case as announced by the Supreme Judicial Court (only days before the hearing officer issued her decision) in Wynn & Wynn, P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. 655, 666-667 (2000). We disagree.

A case will fall within the “rare class of cases, referred to as ‘mixed-motive’ cases, [when] the plaintiff, armed with some strong (direct) evidence of discriminatory bias, demonstrates that at least one factor motivating the employer’s decision is illegitimate.”6 Id. at 666. The analysis of a mixed-motive case is limited “to those infrequent cases in which. . . plaintiffs] can demonstrate with a high degree of assurance that the employment decision of which [they] complain[] ‘was the product of a mixture of legitimate and illegitimate motives.’ ” Fernandes v. Costa Bros. Masonry, Inc., 199 F.3d 572, 580 (1st Cir. 1999), quoting from Price Waterhouse v. Hopkins, 490 U.S. 228, 247 (1989).

A plaintiffs initial burden in a mixed-motive case may only be met by direct, or strong, evidence that discrimination played a motivating role in the employment decision. See Wynn & Wynn, P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. at 666-667; Lipchitz v. Raytheon Co., 434 Mass. 493, 500-501 (2001); Price Waterhouse v. Hopkins, 490 U.S. at 244. Direct evidence is evidence that “if believed, results in an inescapable, or at least highly probable, inference that a forbidden bias was present in the workplace.” Wynn & Wynn, P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. at 667, quoting from Johansen v. NCR Comten, Inc., 30 Mass. App. Ct. 294, 300 (1991).

*195The appellants contend that they introduced such strong, direct evidence before the hearing officer, but all they point to is evidence of hearsay statements allegedly made by Special Sheriff Brassil (who was only one of a panel of four decision makers in the promotional process, cf. Mole v. University of Mass., 442 Mass. 582, 598-600 [2004]) regarding his supposedly adverse feelings towards light duty officers, as subjectively interpreted by union officials manifestly biased in the plaintiffs’ favor.7 Specifically, the president of the plaintiffs’ union, one Gordon, testified that in discussions with Brassil regarding the plaintiffs’ grievances, he heard Brassil say that the officers on light duty did not deserve to be promoted and that he would consider them for promotions when they returned to full duty. Gordon’s successor, Russo, testified that he also discussed promotions with Brassil and was “led to believe” that officers with disabilities “were not looked upon favorably” for promotion or anything else and that the plaintiffs would never be promoted. Additionally, Russo inferred that “there was an attitude” that the injured officers would eventually “screw up” and the department would “get them.”

While it is questionable whether the hearsay statements and “attitudes” attributed to Brassil by presumptively hostile members of the union constituted either direct or strong evidence of discrimination,8 we nevertheless assume the applicability of the mixed-motive analysis announced in Wynn & *196Wynn, P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. at 666, as it does not influence the outcome of our ultimate conclusion.9 Since the plaintiffs can be presumed to have demonstrated that a forbidden motive played some part in the department’s decision not to promote them, the department may not prevail by merely demonstrating that a legitimate reason influenced their decision, but rather must clearly demonstrate that the legitimate reason articulated, standing alone, would have resulted in the same decision. See id. at 666, citing Price Waterhouse v. Hopkins, 490 U.S. at 252. “[0]noe the [plaintiffs] [meet the] initial burden of persuasion on the presence of an illegitimate motive, the decision whether the [department] has met its burden of proving that another legitimate, nondiscriminatory reason actually led it to make the decision is normally for the . . . finder of fact to decide.” Wynn & Wynn P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. at 670. See Price Waterhouse v. Hopkins, 490 U.S. at *197246 (the employer’s burden in a mixed-motive case is apdeemed to be an affirmative defense).

We agree with the hearing officer and the judge below that the department met its burden of proving that there were legitimate, nondiscriminatory reasons for refusing to promote Connolly and Handren which, standing alone, validated the department’s decision. The hearing officer emphatically stated that she was “not persuaded that [Connolly and Handren] had distinguished themselves for promotion.” As to Connolly, she found that he had not even requested an interview and, therefore, was not interviewed for the position of sergeant10; that he had also previously resigned a sergeant’s position in order to remain on the day shift; and that he had been disciplined several times for tardiness and transferred to the night shift as a result (all facts known to Brassil). As to Handren, the hearing officer found that he had been interviewed for the promotion, but she credited Deputy Superintendent Brown’s testimony that he had not been impressed by Handren’s performance during that interview and that Handren possessed no particular qualifications that would place him ahead of the successful candidates. Handren had in fact declined an earlier promotion because he was active in the union and had never previously been offered or, so far as the record indicates, sought another promotion.

The hearing officer’s findings are consistent with a mixed-motive analysis and support a conclusion that the department’s legitimate reasons for not promoting Connolly and Handren would have produced the same outcome regardless of their handicaps. See Wynn & Wynn, P.C. v. Massachusetts Commn. Against Discrimination, 431 Mass. at 670; Lipchitz v. Raytheon Co., 434 Mass. at 502.

4. Overtime compensatory damages. All three plaintiffs argue that the hearing officer erred in denying them an award of compensatory damages caused by the department’s implementation of a discriminatory overtime policy. The denial was *198grounded on the hearing officer’s determination that “it would be nothing short of speculative to award damages for lost overtime shifts since it is impossible to approximate with any certainty the number of shifts the [plaintiffs] would have volunteered for or been rejected from. The amount of overtime available and the number of officers who volunteered varied considerably over the relevant period.” There was no error.

The hearing officer correctly concluded that although the facts supported a finding of discrimination, such a finding does not per se support an award of compensatory damages where the plaintiffs’ proffered method of damage calculation failed to reflect the reality that a host of unpredictable factors would have affected their opportunity to receive any overtime shifts. See Conway v. Electro Switch Corp., 402 Mass. 385, 388 (1988) (“damages may not be determined by speculation or guess,. . . must be causally related to the defendant’s wrongdoing, and the plaintiff should not be made more than whole”).

“While it is true that a plaintiff need not prove damages with mathematical certainty, ‘damages cannot be recovered when they are remote, speculative, hypothetical, and not within the realm of reasonable certainty.’ ” Kitner v. CTW Transport, Inc., 53 Mass. App. Ct. 741, 748 (2002), quoting from Lowrie v. Castle, 225 Mass. 37, 51 (1916). See G.S. Enterprises, Inc. v. Falmouth Marine, Inc., 410 Mass. 262, 276 (1991); Handrahan v. Red Roof Inns, Inc., 43 Mass. App. Ct. 13, 24 (1997). Even though the plaintiffs were not obliged to establish exactly what the overtime damages would have been, they were required to satisfy an initial burden of proving that the department did in fact owe them some amount of back wages for lost overtime opportunities during the time the discriminatory policy was in place. To do that, they needed to demonstrate with reasonable certainty that they would have received some overtime shifts and attendant wages had it not been for the department’s discriminatory policy. See Jinks v. Mays, 464 F.2d 1223, 1226 (5th Cir. 1972).

We agree with the hearing officer that the plaintiffs failed to make such a demonstration and that any amount of damages claimed to be owed would have been entirely speculative. The plaintiffs simply could not prove that they would likely have *199worked any overtime shifts during the short period in question. First and foremost, the hearing officer found that it was impossible even to determine the number of light duty overtime shifts that would have been available to the plaintiffs, nor was it possible to predict how many such shifts they would have been in a position to volunteer for or might have been legitimately precluded or preempted from taking. The gross number of overtime shifts of all kinds relied upon by the plaintiffs reflected their pattern of overtime worked prior to their injuries and failed to take into consideration their changed working conditions, primarily the impact of their injuries on their ability to work longer hours. Additionally, while the discriminatory policy was in place the plaintiffs were moved to the new facility at South Bay at a time when many new officers were being hired, with the result that there were many more employees competing for a more limited number of overtime opportunities. Such variables, taken together and in conjunction with the relatively short period of time in which the overtime policy was in place (little more than a week), made any award of damages for overtime compensation conjectural.11

Conclusion. We affirm the Superior Court’s judgment upholding the decision of the hearing officer and the MCAD. Consequently, we deny the plaintiffs’ request for appellate attorney’s fees and costs.

So ordered.