Article 10 of the Family Court Act defines an “abused child” as, inter alia, “a child under the age of 18 whose parent or other person legally responsible for the child’s care ‘commits, or allows to be committed, a sex offense against such child’ ” (Matter of Philip M., 82 NY2d 238, 243 [1993], quoting Family Ct Act § 1012 [e] [iii]). The Family Court Act “authorizes a method of proof which is closely analogous to the negligence rule of res ipsa loquitur,” and “once a petitioner in a child abuse case has established a prima facie case, the burden of going forward shifts to respondents to rebut the evidence of parental culpability” (Matter of Philip M., 82 NY2d at 244; see Matter of Fantaysia L., 36 AD3d 813, 814 [2007]). Here, the petitioner sustained its burden of proving by a preponderance of the evidence that the child Keiajae W. was an abused child, and the mother failed to rebut the petitioner’s prima facie case of abuse with respect to that child (see Family Ct Act § 1046 [b] [i]; Matter of Nicole V, 71 NY2d 112 [1987]; Matter of Aliyah G. [Arlenle G.], 95 AD3d 885 [2012]; Matter of Lisbeth H. [Noemy H.], 83 AD3d 836 [2011]). The proof of abuse of Keiajae W. was sufficient to establish that the mother derivatively abused the four other subject children, who were either whole or half siblings of *892Keiajae W. (see Family Ct Act § 1046 [a] [i]; Matter of Aliyah G. [Arlenie G.], 95 AD3d at 887; Matter of Tristan R., 63 AD3d 1075, 1078 [2009]).
Further, a preponderance of the evidence supports a finding that the mother neglected all the subject children (see Family Ct Act § 1012 [f] [i] [B]; Matter of Tristan R., 63 AD3d at 1078). The hearsay admitted into evidence at the fact-finding hearing, which consisted, inter alia, of caseworker progress notes and a child abuse evaluation redacted to contain only the statements of the subject children, was allowable pursuant to specific statutory provisions (see Family Ct Act § 1046 [a] [iv], [v], [vi]; Matter of Sanaia L. [Corey W.], 75 AD3d 554 [2010]). That evidence, together with a negative inference drawn from the mother’s failure to testify, was sufficient to support the Family Court’s findings (see Matter of Zaire D. [Benellie RJ, 90 AD3d 923 [2011]; Matter of Amanda Ann B., 38 AD3d 537 [2007]).
The Family Court providently exercised its discretion in denying the mother’s motion, made pursuant to Family Ct Act § 1061, to vacate the fact-finding order. Under the circumstances of this case, the court properly concluded that the mother failed to demonstrate “good cause” to warrant such relief (Family Ct Act § 1061; see Matter of Commissioner of Social Servs. v Amine B., 223 AD2d 703 [1996]).
The mother’s remaining contentions are without merit. Dillon, J.P., Chambers, Sgroi and Miller, JJ., concur.