SPINA, J.

The defendant was convicted of the aggravated rape and murder of four women in Springfield between November 1, 1997, and March 11, 1998. The jury returned verdicts under all three theories of murder in the first degree in each case. On appeal, the defendant asserts error in certain pretrial rulings, including (1) the denial of his motions to suppress evidence where (a) his consent to the taking of a blood sample allegedly was obtained by trickery, and therefore not voluntary, (b) the scope of the search of his blood exceeded the scope of his consent, and (c) the second of two statements he gave police allegedly was obtained in violation of his assertion of the right to counsel; (2) change of trial venue to a community that is not demographically representative of the community where the crimes occurred; (3) joinder of four cases that he contends are not related within the meaning of Mass. R. Crim. P. 9, 378 Mass. 859 (1979); and (4) rulings that deoxyribonucleic acid (DNA) evidence taken from mixed DNA samples was sufficiently reliable to be admitted in evidence, and that the database on which Cellmark Diagnostics Laboratories (Cellmark) based its frequency calculations was adequate. The defendant also alleges error at trial from (1) the admission of hearsay testimony to impeach his first statement to police; (2) the admission of evidence that constituted improper comment on his right to remain silent; (3) the denial of his motion to limit the prosecutor's cross-examination of the defendant to the case on which the defendant intended to offer alibi testimony; and (4) the prosecutor's closing argument. Finally, the defendant asks us to grant him a new trial under G.L. c. 278, § 33E. We affirm the convictions and decline to grant relief under § 33E.

1. Facts. The jury could have found the following facts.

a. Victim no. 1. The victim put her children to bed at 10 P.M. on October 31, 1997. Sometime between midnight and 12:30 A.M., the first-floor tenant at 866 Worthington Street in Springfield heard a "scream of pain" followed by a "thud" from the victim's second-floor apartment. The victim's son discovered her body on the living room couch after he awoke at 7 A.M. on November 1. Her naked body was covered by a blanket and a towel had been placed over her head. Her hands were bound behind her back.

An autopsy revealed that death was caused by asphyxia due to manual strangulation. The victim's blood tested positive for metabolites of cocaine and alcohol. Her anus was widely dilated, consistent with penetration.

The defendant's thumb print was identified on a broken ashtray found in the living room, and his palm print was identified on a hair gel container, also in the living room. Fecal matter was found on a sock and inside the rim of a small vase recovered from the victim's living room. The defendant told his brother that he had had anal intercourse with women he met in the Worthington-Federal Streets section of Springfield, an area known for prostitutes and "crack" cocaine. The defendant's brother had driven him several times in October and November, 1997, to the neighborhood where the victim lived.

Several items collected from the scene and all biological evidence recovered from the first victim's body, as well as blood samples given by the defendant were sent to Cellmark for DNA analysis. Cellmark performed two series of polymerase chain reaction (PCR) tests on DNA samples taken from items submitted in all four cases. The first series of tests were performed at nine genetic loci: DQ Alpha, five polymarker (PM) loci, and three (TPOX, THO1, and CSF) short tandem repeat (STR) loci. In some instances testing was done at a tenth locus, identified as D1S80. Some of these tests were witnessed by a defense expert. A second series of PCR tests was done on some samples at thirteen core STR loci (including the three STR loci where testing previously had been done) designated for inclusion in the national database that contains DNA profiles of convicted felons, known as the combined DNA index system, or CODIS. See G.L. c. 22E, § 1 (definitions). A defense expert was present during the second series of tests. DNA samples from seven other suspects were examined and all seven were excluded as contributors in each case.

In the case of the first victim, sperm fractions, the part of sperm that contains DNA, were extracted from sperm cells found on several items taken from the scene, including a sock, the victim's shirt, the rim and side of the small vase, a carpet section and a towel. They were also extracted from sperm cells on an anorectal swab taken at autopsy. After the first series of tests on DNA samples from the sperm fractions, the defendant could not be excluded as the contributor. A frequency calculation based on tests of the sample from the anorectal swab indicated that one in 450,000 African-Americans would be expected to share this DNA profile. Frequency calculations based on tests of the sample from the victim's shirt and the sock indicated that one in 8.4 million African-Americans would be expected to have the same DNA profile. The samples of DNA from the victim's shirt and the sock were determined to have come from a single source, male, and not a mixture of DNA from two or more persons. Frequency calculations based on the second series of tests performed on the sperm fractions from the first victim's shirt and the sock indicated a "match" to the defendant's DNA profile: one in 64 quadrillion (64 × 10¹⁵) African-Americans would be expected to have the same DNA profile as the sperm fraction.

b. Victim no. 2. The second victim had a drug and alcohol problem and would prostitute herself to obtain money to purchase drugs. She frequented a crack house at 65 Maynard Street, Springfield, where crack cocaine could be obtained for cash, goods, or sex. Sexual favors were often exchanged on the premises for cocaine. The defendant was a "regular" at 65 Maynard Street, where he obtained both drugs and sex.

At approximately 9 A.M. on February 2, 1998, the victim's partially naked body was discovered in an alley near a post office on State Street in Springfield, a few blocks away from Maynard Street. Her arms were extended over her head by her sweatshirt, which had been pulled up over her head. Fecal matter was found on a gray sweatshirt and two socks. A ballpoint pen was found under her body. The pen was a promotional item similar to one the defendant had obtained at a local auto body shop one week earlier.

An autopsy disclosed that death was caused by manual strangulation. Cocaine metabolites were present in the victim's blood. Her anus was dilated, consistent with penetration. The pathologist concluded that the victim had been dead one to two days when the autopsy was performed on February 3.

The first series of DNA tests on sperm fractions extracted from sperm cells found on a sock, a sweatshirt, and an undershirt taken from or near the victim's body could not exclude the defendant as the contributor. A frequency calculation based on the second series of tests on the sperm fractions from the sock and the undershirt indicated a match to the defendant's DNA profile: one in 64 quadrillion African-Americans would be expected to have the same DNA profile as that found in the sperm fraction. The DNA in the samples from the sock and the undershirt were determined to have come from a single source, male, and not a mixture.

c. Victim no. 3. She also was a regular at 65 Maynard Street, and prostituted herself to support her drug habit. She and the defendant arrived there together between 9 P.M. and 9:30 P.M. on February 10, 1998. The defendant was trying to exchange compact discs for drugs. They left, but returned sometime between 11 P.M. and 11:30 P.M., then left again.

The victim's son discovered her body on the evening of February 11, 1998, in the bedroom of her apartment at Lionel Benoit Road, Springfield, a short distance from Maynard Street. There was no evidence of forced entry. The victim's clothing had been pushed up around her head. The defendant's palm print was identified on a ceramic coffee mug on a dresser in the bedroom.

An autopsy revealed evidence of strangulation, but death was caused by asphyxiation from a sock that was lodged in the victim's airway. Cocaine metabolites were present in her blood. Her anus was dilated, consistent with penetration. The pathologist estimated time of death at between 10 P.M. on February 10, 1998, and 4 A.M. on February 11.

The first series of DNA tests could not exclude the defendant as a contributor of sperm fractions extracted from sperm cells on vaginal and anorectal swabs taken at the autopsy, and on sweatpants, a pillowcase, and a sanitary pad collected in the bedroom. A frequency calculation indicated that one in 490 African-Americans would be expected to share the DNA profile of the sperm fractions from the vaginal swab. The second series of tests produced similar results, but frequency calculations were not performed because the samples comprised a mixture of DNA consistent with the profiles of both the victim and the defendant, and a primary contributor could not be identified.

d. Victim no. 4. The fourth victim's teenage daughter filed a missing person report with the Springfield police on February 19, 1998, after her mother failed to return home the previous night. The teenager had walked around the apartment complex and discovered her mother's car, but it was not parked in the customary place. It had been ransacked, and the keys were on the floor.

At approximately 11 P.M. on February 18, 1998, the defendant, the victim, and a second woman arrived at the Springfield home of a third woman, where they all smoked crack cocaine. Not satisfied, they set out to find more. They tried unsuccessfully to sell the victim's gold Mickey Mouse earrings. At one point they drove to Holyoke in the victim's car. After they returned to Springfield, the victim drove off with the defendant sitting in the front passenger seat of her car. She was never again seen alive.

On February 20, 1998, Springfield police began an investigation of the disappearance of the fourth victim. They interviewed the two women who were with her on the evening of February 18, then tried to contact the defendant. On February 27, the defendant went to the Springfield police station where he related the events of the evening of February 18 to detectives. In contrast with the account given earlier by one of the women, the defendant said he did not drive away with the victim after they returned from Holyoke, and he said he never sat in the front seat of her car. He told detectives that, when they returned from Holyoke, they all parted ways and he walked to a bar, the Yellow Jacket, on State Street. There, he had a drink with a man who once worked with him. The man later drove the defendant to his girl friend's home. The bartender denied that the defendant had been at the Yellow Jacket that night, and the former coemployee denied that they were together that night.

While the defendant was in the police station on February 27, 1998, his girl friend arrived at the request of police to answer some questions about the defendant. At one point she was taken to the same room as the defendant to try to resolve a discrepancy in the time each said the defendant had arrived at the girl friend's home on February 19. The defendant tried to convince his girl friend that he arrived at 12:30 A.M. She said she was not sure about the time, but she later telephoned police to confirm that he did not arrive until 6 A.M. A friend of the defendant testified that the defendant appeared at his home at approximately 1 A.M. on February 19 asking to borrow fifteen dollars.

Also while the defendant was at the police station on February 27, 1998, he viewed the victim's car, including a blood stain in the front passenger seat. Arrangements had been made for a chemist to examine the car. Detectives asked the defendant if he would be willing to give palm and fingerprint samples, as well as a blood sample. He agreed. After his prints were taken, police drove the defendant to Baystate Medical Center where he signed a consent form. Four vials of his blood were then drawn.

On March 11, 1998, the fourth victim's naked and frozen body was found in some bushes adjacent to a building on East Columbus Avenue in Springfield. Her left hand had been positioned on her abdomen and her arm was propped up by branches that had been cut. Autopsy revealed that death was caused by manual strangulation. Time of death could not be determined because the body was frozen. Her blood contained metabolites of cocaine. Her anus was dilated, suggesting penetration.

On March 16, 1998, police went to a pawn shop in Springfield where they learned that the defendant had pawned the fourth victim's Mickey Mouse earrings on February 20 for twelve dollars. He previously had told police on February 27 that, after the victim dropped him off on their return from Holyoke, he suggested that she try selling the earrings at 65 Maynard Street.

On April 6, 1998, detectives told the defendant that the tests done on the blood on the front seat of the victim's car indicated that it was not his. They also told him that his attorney had contacted them, and that they were not going to talk to him about the fourth victim. Accordingly, they did not ask him any questions about the fourth victim. The defendant, however, asked about their progress in the investigations of the deaths of the other women, which had become a pressing public concern. He said he wished he could help them, but said that he did not know any of the other victims.

The first series of DNA tests on biological evidence collected at the autopsy and from the fourth victim's car could not exclude the defendant as the contributor of sperm fractions extracted from sperm cells recovered from the armrest and the front seat of the victim's car, or from a swab of the victim's buttocks and an anorectal swab taken at autopsy. A frequency calculation based on the tests indicated that one in every 450,000 African-Americans would be expected to have the same DNA profile as a sperm fraction from the front seat. That is, 99.9998% of African-Americans could be excluded as contributors. The second series of tests produced similar results. A frequency calculation indicated that one in every 709,000 African-Americans could be expected to have the same DNA profile as a sperm fraction from the armrest. Greater precision could not be obtained because the sample was too small to permit full testing at the thirteen core STR loci; testing was performed at six STR loci.

2. Suppression issues. a. Voluntariness of consent to search. The defendant argues that the motion to suppress evidence derived from his blood samples should have been allowed because his consent to a search and seizure of his blood was obtained by trickery, and therefore was not voluntary. Specifically, the defendant contends that detectives affirmatively misled him to believe that the test results of his blood sample would be used only to compare with tests done on blood found inside the fourth victim's car, and not for any other purpose.

When the Commonwealth relies on consent to justify the lawfulness of a search, it must prove that consent was given freely and voluntarily. See Bumper v. North Carolina, 391 U.S. 543, 548-549 (1968). Voluntariness of consent "is a question of fact to be determined from the totality of all the circumstances." Schneckloth v. Bustamonte, 412 U.S. 218, 227 (1973).

The defendant focuses on one factor that has been identified under the "totality of all the circumstances" test, namely, deception as to purpose. See 3 W.R. LaFave, Search and Seizure, § 8.2(n), at 706-713 (3d ed. 1996 Supp. 2004). He relies on Beasley v. State, 204 Ga. App. 214, 216 (1992) (consent to give urine sample not voluntary where defendant charged with possession of cocaine based on urine test results obtained after police officer represented that "test results would be used only 'for determining bond'" [emphasis added]), and Graves v. Beto, 424 F.2d 524, 525 (5th Cir.), cert. denied, 400 U.S. 960 (1970) (consent to take blood sample limited where defendant convicted of rape based on blood test results obtained after police officer represented that blood sample "would be used only to determine the alcoholic content of his blood" [emphasis added]). In those cases, consent was negated or limited by a defendant's reliance on the express representation of a police officer that the particular object to be seized would be used solely for a particular purpose, and therefore a warrant was required for a search and seizure for purposes beyond the self limiting representation of police. See 3 W.R. LaFave, supra at § 8.2(n), at 708. Here, police made no express representation that the test results of the defendant's blood would be used solely for comparison to the tests on blood found in the fourth victim's car, and the judge so found. Police are under no obligation to inform a defendant of all purposes for which his blood tests might be used. See Commonwealth v. Sanna, 424 Mass. 92, 98 (1997); Pace v. State, 271 Ga. 829, 832 (1999).

The defendant argues that the judge's "conclusion that the defendant's blood was sought in connection with [the fourth victim's] disappearance only, is clearly erroneous." He contends that the judge could not have made this finding where the disinterested nurse who drew the defendant's blood on February 27, 1998, gave uncontradicted testimony that a police officer told her that the defendant's blood was being taken in connection with several recent murders, "plural," not just the disappearance of the fourth victim, whose body had not been found as of February 27.

The defendant's argument is based on testimony taken out of context. The nurse's testimony, in context, indicates that she had been told that there were others whom police wanted to question concerning murders and from whom blood samples would be sought. She did not testify that police told her the defendant was a suspect in the murders. In fact there were suspects who had been questioned in connection with the murders, and as of February 27, 1998, the defendant was not one of them. Blood samples had been taken from eight individuals, and the defendant was neither the first nor the last of them. The judge found that "on February 27 the defendant was not yet a suspect in any other unsolved homicide." The judge's finding that the defendant's blood was sought only in connection with the fourth victim was not clearly erroneous. But, even if the police were interested in using the defendant's blood test results in other cases and had not fully disclosed their intentions, such an omission would not invalidate the defendant's consent. Commonwealth v. Sanna, supra; Pace v. State, supra.

The defendant himself placed no condition or limitation, express or implied, on his consent, and the judge so found. See Commonwealth v. Cantalupo, 380 Mass. 173, 178 (1980). Indeed, the consent form signed by the defendant at the hospital stated:

"After being advised of the rights outlined above including my absolute right to refuse to consent to the taking of a sample of my blood, [I] do hereby authorize, [the nurse] to take a sample of my blood and to deliver said sample to [p]olice [o]fficers of the [c]ity of Springfield, Massachusetts for whatever purpose said [p]olice [o]fficers desire in connection with a police investigation" (emphasis added).

The judge found, with ample record support, that the defendant, who was not in custody, was advised that he had a right to refuse to consent to the taking of a blood sample, that his consent was not the product of coercion or deceit, and that police had been courteous, honest, and forthright in their treatment of the defendant. The judge also found, with record support, that the defendant "stated repeatedly that he was willing to help the police in any way he could." There was no error in the determination that, in the totality of all the circumstances, the defendant's consent was given freely and voluntarily.

b. Scope of consent. Closely related to the question of voluntariness of consent to search is the issue of the scope of consent to search. The defendant contends that the scope of his consent was limited by what police told him, namely, that they wanted to test his blood and compare the results with testing done on blood found in the fourth victim's car.

"The standard for measuring the scope of a suspect's consent under the Fourth Amendment [to the United States Constitution] is that of 'objective' reasonableness — what would the typical reasonable person have understood by the exchange between the officer and the suspect?" Florida v. Jimeno, 500 U.S. 248, 251 (1991). The scope of consent is ordinarily understood to refer to the scope of the search, and is defined by the areas, objects or things for which consent is given. Id.

Here, a reasonable person likely would have concluded that police were seeking the defendant's blood tests results, including his DNA profile. The object of the intended search was a sample of the defendant's blood and the identifying information that could be obtained by DNA testing of the sample. The testing actually done on the defendant's blood sample was no more intense or intrusive of his privacy interests than what was expressly sought. The scope of the search, blood tests, was confined to what a reasonable person would have understood from the request by police. Florida v. Jimeno, supra.

Contrary to the defendant's claim, the scope or object of the intended search was not the investigation into the disappearance of the fourth victim, which police had mentioned. Their statement about that investigation may have been relevant to the voluntariness of the defendant's consent, but in this case it did not define the scope of his consent. "The touchstone of the Fourth Amendment is reasonableness." Florida v. Jimeno, supra at 250. "It would not be reasonable to require law enforcement personnel to obtain additional consent or another search warrant every time a validly obtained DNA profile is used for comparison in another investigation." Pace v. State, supra at 832. "[L]ike a fingerprint, DNA remains the same no matter how many times blood is drawn and tested and a DNA profile can be used to inculpate or exculpate suspects in other investigations without additional invasive procedures." Id.

Although it is a suspect's right to limit the scope of a search to which he consents, Florida v. Jimeno, supra at 252, the defendant did not avail himself of that right. The judge properly concluded that the scope of the search for and of the defendant's blood was not limited in the manner argued by the defendant.

c. Statement of April 6, 1998. The defendant argues that the motion to suppress his statement of April 6, 1998, should have been allowed because police deliberately circumvented his attorney's instruction not to speak to him about the case involving the disappearance of the fourth victim. The judge found that counsel telephoned police on March 3 and directed them to have no contact with the defendant concerning their investigation into the disappearance of the fourth victim. Thereafter, police learned that the defendant had been excluded as a contributor of the blood found in the fourth victim's car. On April 6 they went to his place of employment and told him of the test results without first telling counsel. They also told the defendant that they knew he had an attorney in the particular matter, and would not question him about that case. The defendant spontaneously asked the officers about their investigation into the other murders. They showed him photographs of the other victims. He said he wished he could help them, but he did not know the other victims.

The defendant accepts the judge's factual findings and concedes that police were free to contact him about investigations other than the one involving the fourth victim. See Commonwealth v. Rainwater, 425 Mass. 540, 557 (1997), cert. denied, 522 U.S. 1095 (1998). What they could not do, he argues, is contact him about the case involving the fourth victim after having been forbidden by counsel to do so. The defendant contends that his statement of April 6, 1998, obtained in derogation of his counsel's demand, violates the core principles underlying the Fifth and Sixth Amendments to the United States Constitution and art. 12 of the Massachusetts Declaration of Rights, and should have been suppressed.

The defendant concedes that Edwards v. Arizona, 451 U.S. 477 (1981), which requires suspension of interrogation after a defendant asserts his Fifth Amendment right to counsel, applies only in cases involving custodial interrogation. See McNeil v. Wisconsin, 501 U.S. 171, 178 (1991). Here, the defendant was not in custody on April 6, 1998, when he made his statement. The defendant also concedes that the Sixth Amendment right to counsel had not attached as of April 6 because no criminal proceeding had been instituted against him. Id. at 175-178. In the circumstances, art. 12 does not create any more expansive or additional right than the Fifth and Sixth Amendments. See Commonwealth v. Beland, 436 Mass. 273, 288 (2002). The judge correctly concluded, "[t]he right which the defendant claims the police violated simply does not exist."

The Supreme Court has "never held that a person can invoke his Miranda rights anticipatorily, in a context other than 'custodial interrogation.'" McNeil v. Wisconsin, supra at 182 n. 3. To rule otherwise, as the defendant would have us do, would "seriously impede effective law enforcement." Id. at 180. We previously have declined invitations to hold that art. 12 provides greater protection than the Fifth and Sixth Amendments in this area, concluding that there is "no sufficient reason to depart from the balance the Supreme Court has struck between the interests of defendants and interests of the public in Maine v. Moulton, [ 474 U.S. 159, 180 (1985)], and McNeil v. Wisconsin, [ supra]." Commonwealth v. Rainwater, supra at 554-555. See Commonwealth v. Beland, supra. This case presents nothing that persuades us to change that view.

Primary responsibility for exercising the protections guaranteed by Miranda v. Arizona, 384 U.S. 436 (1966), and art. 12 rests with suspects. See Commonwealth v. Beland, supra. That is precisely what occurred here. The defendant understood his rights and he understood how to exercise them. When he was invited to the police station to give a statement, the defendant declined, saying he understood his rights and would take the advice of his attorney and not give any further statements. The judge found that the defendant's inquiries into progress of the investigations of the other cases were not prompted by the news of the blood test results, but were spontaneous and voluntary questions by the defendant about the other cases. The motion to suppress the statement was properly denied.

3. Change of venue. The defendant filed a motion, based on massive and prejudicial pretrial publicity, for a change of venue from Hampden County to Suffolk County. The judge agreed that a change of venue was warranted, but he ordered the case transferred to Berkshire County for trial. The defendant claims that the transfer of the case to Berkshire County deprived him of a jury drawn from a fair cross section of the community, in violation of art. 12. Commonwealth v. Soares, 377 Mass. 461, cert. denied, 444 U.S. 881 (1979). Specifically, the defendant contends that, in the circumstances, the fair cross section standard requires a pool of potential jurors representative of the Hampden County population, and the only county having as many potential minority jurors as Hampden County is Suffolk County.

When a defendant moves for a change of venue, he waives his right to be tried by a jury drawn from a pool that is representative of the venue where the crime occurred. Commonwealth v. Rankins, 429 Mass. 470, 476 (1999). He has no State constitutional right to have his case transferred to a county having a minority population at least as great as the county where the crime took place. Id. The defendant does not argue that the trial jury were not drawn from a fair cross section of people in Berkshire County.

The defendant contends that the Rankins case cannot be reconciled with Smith v. Commonwealth, 420 Mass. 291 (1995), where this court held that a defendant whose case was transferred to Middlesex County from Suffolk County was entitled to a jury drawn from a pool that was representative of a fair cross section of Suffolk County. There the defendant had opposed the change of venue, which had been ordered for the administrative convenience of the District Court Department. The defendant in the Smith case, unlike the defendants here and in the Rankins case, had not sought a change of venue, and therefore he did not waive his right to be tried by a jury drawn from a fair cross section of the community where the crime occurred.

A judge has substantial discretion to transfer a case to another county based on pretrial publicity. Commonwealth v. James, 424 Mass. 770, 775 (1997). A judge may consider race in making his decision. Commonwealth v. Rankins, supra. Contrary to the defendant's contention, the experienced trial judge appropriately considered racial issues, including the fact that the defendant, all the victims, and several Commonwealth witnesses are African-Americans. He noted that there was "no allegation that race was a motivating factor in any of these crimes, and there appears to be no reason to suppose that the case is racially charged." He called on his "experience with African-American defendants in the Berkshire Superior Court," concluding that the defendant would receive a fair trial in Berkshire County, and that there was no "compelling reason to inconvenience the victims' families or to add to the Commonwealth's burden and expense by transferring this case to Suffolk County." The defendant has failed to show an abuse of discretion in the transfer of his case to Berkshire County.

4. Joinder. The Commonwealth moved, pursuant to Mass. R. Crim. P. 9 (a) (1), (3), for joinder of the indictments for trial on the ground that the offenses were related. The judge found that joinder would serve the interests of judicial economy, as many of the same witnesses were involved in all or most of the cases and would have to testify at four trials unless the cases were joined. See Commonwealth v. Hoppin, 387 Mass. 25, 32 (1982). He also found that the four cases were "related" under the rule, reasoning that the evidence supported a finding of "a common plan by the defendant to prey upon similarly situated cocaine-addicted women for sexual gratification" and that there exists "a sufficient temporal and schematic nexus to make joinder proper." He ordered the cases joined. The defendant asserts that the ruling is not supported by evidence of a common scheme or plan and is therefore erroneous.

A decision to join offenses for trial is committed to the sound discretion of the trial judge. See Commonwealth v. Montanez, 410 Mass. 290, 303 (1991). The defendant bears the burden of demonstrating that the offenses were unrelated, and that prejudice from joinder was so compelling that it prevented him from obtaining a fair trial. Commonwealth v. Wilson, 427 Mass. 336, 346 (1998). Offenses are related, for purposes of joinder, "if they are based on the same criminal conduct or episode or arise out of a course of criminal conduct or series of criminal episodes connected together or constituting parts of a single scheme or plan." Mass. R. Crim. P. 9 (a) (1). Where offenses are related as defined in rule 9 (a) (1), "[t]he trial judge shall join the charges for trial unless he determines that joinder is not in the best interests of justice." Mass. R. Crim. P. 9 (a) (3). Factors that may be considered in determining whether offenses are related for purposes of joinder include factual similarities, see Commonwealth v. Ferraro, 424 Mass. 87 (1997), and closeness of time and space, see Commonwealth v. Delaney, 425 Mass. 587, 594 (1997), cert. denied, 522 U.S. 1058 (1998). "[O]ffenses are related if the evidence 'in its totality shows a common scheme and pattern of operation that tends to prove all the indictments.'" Id., quoting Commonwealth v. Feijoo, 419 Mass. 486, 494-495 (1995). The appropriateness of joinder often turns on whether evidence of the other crimes would be admissible in a separate trial on each indictment. See Commonwealth v. Gallison, 383 Mass. 659, 672 (1981). Evidence of other criminal conduct is not admissible to prove the propensity of the defendant to commit the indicted offense. Id. For purposes of joinder, however, it may be used to show a common scheme or pattern of operation. Commonwealth v. Feijoo, supra.

The factual similarities of the cases are strong. Each victim had a history of cocaine use and had taken cocaine a short time before death. Each victim had been sexually assaulted, including anally, then strangled. There was evidence that the defendant was a heavy cocaine user, and that he liked having intercourse when he smoked crack cocaine. There was evidence that about this time the defendant spoke of having anal intercourse with women. Each victim was found in a state of total or partial nudity, and forensic evidence sharing the defendant's DNA profile was recovered in each case from the victim's body or on items in the immediate vicinity where her body was found. The evidence in each case suggests that death occurred at approximately the same time as sexual intercourse with the defendant. The crimes were sufficiently close in time (three and one-half months) and space (a small geographic area of Springfield in neighborhoods known for prostitution and crack cocaine that were frequented by the defendant and the victims) to make joinder appropriate.

The defendant frequented places where he could meet cocaine-addicted women willing to participate in sexual activity in exchange for cocaine, he indulged his own desires for cocaine and anal intercourse with each of the women, then he strangled them. The similarity between the victims, the manner and circumstances in which each had been engaged by the defendant at about the time of her death, and the manner in which death was caused, support the judge's findings of common scheme and relatedness of offenses. See Commonwealth v. Mamay, 407 Mass. 412, 416-417 (1990).

Contrary to the defendant's claim, the differences in the cases do not render joinder inappropriate. There is no requirement that the circumstances of the cases be identical. Although the judge found that the similarities of the cases were not sufficiently strong to establish a signature mark to prove the identity of the killer, Commonwealth v. Brusgulis, 406 Mass. 501, 505 (1990), the circumstances of the cases were not offered for that purpose. Here, powerful DNA and other evidence linked the defendant to the victims at the times of death. The judge correctly concluded that the circumstances of the cases were sufficiently similar to show that the defendant was acting pursuant to a common scheme, and were relevant to questions of his intent and motive. See Commonwealth v. Maimoni, 41 Mass. App. Ct. 321, 327-328 (1996) (prior bad acts properly admitted to prove defendant's pattern of conduct and his mental state at time of murder as one of "sexual aggressiveness").

With respect to the question of potential prejudice to the defendant, the judge correctly noted that considerations of judicial economy must yield to the defendant's right to a fair trial. See Commonwealth v. Sylvester, 388 Mass. 749, 758 (1983). He considered whether joinder would foreclose or substantially impair the defendant's presentation of a particular defense. See Commonwealth v. Gallison, supra at 672. He also considered whether the defendant might wish to testify on his own behalf on one of the offenses, but not another, thus being forced to choose the unwanted alternative of testifying as to both or testifying as to neither. See Commonwealth v. Williams, 18 Mass. App. Ct. 945, 947 (1984).

The judge noted that the defendant had not filed a notice of alibi or otherwise indicated that he may have a defense in one case but not in another. Eight months later, however, the defendant served a notice of alibi as to the case of the second victim only, together with a motion for relief from prejudicial joinder. The motion was denied. The defendant's notice of alibi identified two witnesses, and stated that he was not in the vicinity of the second victim's murder on February 2, 1999. Counsel filed an affidavit indicating that the defendant also might testify. No other specifics were offered. The pathologist testified that the the victim had been dead for one to two days when the autopsy had been performed on February 3. Therefore the defendant's whereabouts on February 2, 1999, may not have been particularly exculpatory. Because the judge had not been given any particulars about the testimony that might have been proffered through the defendant or the named alibi witnesses, the defendant failed to show how he would have been prejudiced. It is not enough simply to assert the loss of a defense. See Commonwealth v. Wilson, supra at 347; Commonwealth v. Williams, supra. We note that the two witnesses through whom the alibi presumably could have been presented at trial without the defendant's testimony did not testify.

The judge also considered whether proof of the defendant's guilt as to one offense might be used improperly to convict him of a second offense. He noted that much of the evidence in each case would be admissible in the other cases not only on the question of common scheme, but also as to the issues of the defendant's premeditation, his intent to kill, and his motive, and correctly concluded that there was little chance of "seepage . . . of evidence not otherwise admissible." Commonwealth v. Gallison, supra at 672, quoting Commonwealth v. Beneficial Fin. Co., 360 Mass. 188, 221 (1971), cert. denied sub nom. Farrell v. Massachusetts, 407 U.S. 910 (1972), and Beneficial Fin. Co. v. Massachusetts, 407 U.S. 914 (1972). The defendant has failed to show the existence of any prejudice, much less prejudice that is so compelling that he was denied a fair trial.

Finally, the judge's instructions to the jury that they were to consider each of the indictments separately and that they could only consider evidence of the other crimes solely on the issues of the defendant's intent, motive, state of mind, or plan were adequate to offset any possible prejudice from the joinder. See Commonwealth v. Wilson, supra at 347. The defendant has failed to show that joinder was improper and that it was prejudicial.

5. Admissibility of DNA evidence. The defendant filed a pretrial motion in which he requested a " Daubert-Lanigan hearing" with respect to the admissibility of DNA evidence the Commonwealth was expected to offer in all four cases. See Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579 (1993); Commonwealth v. Lanigan, 419 Mass. 15 (1994). Hearings were conducted as to both the first and second series of tests, and the defendant now appeals from rulings that (1) Cellmark's methodology in dealing with mixtures and technical artifacts is generally accepted within the scientific community; (2) Cellmark both performed its tests properly and reported the results accurately; (3) Cellmark's election not to stay within conservative recommendations of test kit manufacturers regarding the minimum quantity of DNA tested did not invalidate the studies; (4) the use of the product rule to make frequency calculations for identifiable primary contributors in a mixed DNA sample is acceptable; and (5) the database used by Cellmark to make frequency calculations is adequate and common within the field.

a. Mixtures and artifacts. The defendant first claims that the evidence does not support the judge's conclusion that Cellmark can reliably distinguish technical artifacts from true alleles, or primary from secondary contributors in mixed samples of DNA. He does not challenge the scientific validity of PCR testing. We previously have held that PCR-based DNA analysis, both generally and at the DQ Alpha locus, the PM loci, and the D1S80 locus, is a scientifically valid methodology for developing DNA profile evidence. See Commonwealth v. Vao Sok, 425 Mass. 787, 799, 801-802 (1997). We reached the same conclusion with respect to PCR testing at STR loci identified as CSF1PO, TPOX, and TH01. Commonwealth v. Rosier, 425 Mass. 807, 812-813 (1997). The defendant does not challenge the scientific validity of PCR testing at the other STR loci designated under CODIS. His challenge focuses on the second, or reliability prong of the Daubert-Lanigan inquiry.

A determination of the reliability of the testing process entails a fact-based inquiry, including questions of credibility. See Commonwealth v. Vao Sok, supra at 797, 798. The analysis calls on a judge to determine whether testing was properly performed, Commonwealth v. McNickles, 434 Mass. 839, 850 (2001), and whether an expert's conclusions based on clinical experience and observations were sufficiently reliable. Canavan's Case, 432 Mass. 304, 313 (2000). The judge's decision under the reliability prong is reviewed under the abuse of discretion standard. Id. at 312.

Testimony indicated that it is not unusual to find mixtures of DNA (a sample containing DNA from two or more persons) in cases of sexual assault. The presence of a mixture of DNA can create difficulties in interpreting test results. For example, where DNA from two contributors is present in a mixture in relatively equal amounts, the dots or bands (depending on the particular test) produced by the two samples during the testing process will display comparable intensity (darkness). In such cases the Cellmark analyst will report the existence of a combination without attempting to interpret the results.

Similar interpretive challenges arise from the presence of technical artifacts: mistakes in the PCR amplification process that replicates a defined segment of DNA through the use of enzymes. Artifacts typically give faint readings. When a test result suggests either an artifact or a secondary contributor and the difference in intensity between readings is slight, the Cellmark analyst will not attempt to distinguish them but will report the result as not interpretable.

Where DNA from two contributors is present in a mixture in unequal amounts, readings produced by the larger sample will be darker than those of the smaller sample, roughly in direct proportion to the difference between the amounts of the samples. Where the differences are such that intensity readings from the greater sample are dark and those from the lesser sample are faint, the Cellmark analyst may conclude that the darker reading is produced by a primary contributor and report her conclusion in the same way she reports a single source sample, conformably with the recommendation of the National Resource Council (NRC), The Evaluation of Forensic DNA Evidence (1996) at 129. We have treated the reports of the NRC as authoritative works for purposes of determining generally accepted standards within the scientific community for (a) the validity of the underlying scientific theory, or (b) the reliability of the underlying process for developing forensic DNA evidence. See Commonwealth v. Rosier, supra at 815; Commonwealth v. Vao Sok, supra at 801.

Here, only test results based on a single source of DNA, or, where there appeared to be a mixture, only test results that yielded strong evidence (dark bands or dots) of a primary contributor were used. Consequently, a weak reading of a secondary contributor or technical artifact had no effect on Cellmark's ability to report the result of the primary contributor and calculate a statistic of the probability of a random match to the defendant's DNA profile. The judge's findings that Cellmark's methodology in reporting tests of a mixed sample with an identifiable primary contributor in the same way it reports tests of a single source sample conforms to the recommendation of the NRC, and that Cellmark's methodology in dealing with the presence of mixtures or technical artifacts is generally accepted within the scientific community, were made with record support and well within his discretion.

b. Conduct of tests. The defendant suggests other known conditions that could affect the accuracy and reliability of test results, including contamination, loci dropout, allele dropout, differential amplification, stochastic effect, spikes, and peak imbalances. We need not engage in a lengthy discussion about these conditions. For purposes of this appeal, it is sufficient to note that Cellmark analysts considered each of these issues and factored them into their test results. The judge accepted the testimony of Dr. Robin Cotton, Cellmark's forensic laboratory director, with respect to the many controls and safeguards that Cellmark uses to adjust for these challenges, including threshold control dots to detect minimum sample size at the DQ Alpha locus, the use of two different manufacturers' test kits that test DNA at four of the same loci for CODIS testing, controls to ensure against contamination at the PM and D1S80 loci, and others. The judge found that Cellmark follows the standards adopted by the DNA advisory board, a group of individuals authorized by Congress to advise the Federal Bureau of Investigation on DNA testing, and the guidelines published by the Technical Working Group for DNA Analysis Methods (TWGDAM) in 1995, except in cases where those guidelines have been superseded by the DNA advisory board standards. The judge did not abuse his discretion in ruling that the test results were sufficiently reliable to be put before the jury and that the questions raised by the defendant were more appropriately addressed to the weight of the evidence. See Commonwealth v. McNickles, 434 Mass. 839, 850-854 (2001).

c. Manufacturer's recommendations. There is no merit to the defendant's contention that Cellmark's failure to comply with the minimum standards for DNA sample sizes set by the test kit manufacturers invalidated the test results. The user's manuals for the Profiler Plus and Cofiler kits, both manufactured by PerkinElmer, Inc., recommended that they not be used with less than one to 2.5 nanograms (one one-billionth of a gram) of questioned DNA, but many of the tests were conducted with less than the recommended amounts. The judge found that the manufacturers' recommendations were just that, recommendations. They were intended to ensure optimal results. He found, with record support, that Cellmark had conducted validation studies that supported the reliability of testing based on amounts smaller than recommended by the manufacturers, amounts as small as one-half nanogram. Cellmark had also conducted validation studies indicating that analysts could reliably interpret computerized test results (the second series of tests performed under CODIS) based on readings as low as forty relative flourescent units (RFUs) rather than the conservative level of 150 RFUs recommended by the manufacturers, and the judge found that Cellmark's readings were reliable. There was no abuse of discretion.

The defendant's expert, Dr. Donald E. Riley, ultimately conceded Cellmark's conclusions (that test results matched the defendant's profile) were supported by the data and that the defendant's and the victims' identified alleles matched those identified by Cellmark's testing. The judge observed that "[a]ll that Dr. Riley can really say is that there is a 'potential' for this system of analysis to miss alleles and distort results. I recognize that the potential for error exists in any scientific testing, but I am satisfied that Cellmark has done all that is reasonably possible to eliminate that potential" (emphasis in original). He correctly noted that the issues raised by the defendant went to the weight of the evidence, not its admissibility.

d. Use of the product rule. The defendant argues that the judge erred by accepting the proposition that if an analyst can distinguish between a primary contributor and a secondary contributor in a mixed DNA sample, the analyst may properly treat the primary contributor as a single source for statistical purposes. His argument, essentially, is that the "product rule," which has been held to be a scientifically acceptable method for calculating frequency profiles based on results of PCR testing of single source samples, see Commonwealth v. Rosier, supra at 816-817, produces unreliable results in mixed samples; and that a "likelihood ratio," which has been held to be a scientifically acceptable method of calculating frequency profiles based on results of testing of mixed samples, should have been used. See Commonwealth v. McNickles, supra at 845-848.

Likelihood ratio analysis is appropriate for test results of mixed samples when the primary and secondary contributors cannot be distinguished. Id. at 846. It need not be applied when a primary contributor can be identified. Contrary to the defendant's view, the use of the product rule is scientifically acceptable where the analyst can distinguish between a primary and a secondary contributor in a mixed sample, and thereafter treat the primary contributor as a single source for statistical purposes. See NRC, Evaluation of Forensic DNA Evidence (1996), Executive Summary Recommendation 4.1 at 5; State v. Roman Nose, 667 N.W.2d 386, 398 (Minn. 2003). In any event, at trial, Dr. Christopher Basten, the Commonwealth's statistician in the field of population genetics, recalculated the profile frequencies using likelihood ratios and reached results comparable to those obtained under the product rule.

e. Cellmark's database. Finally, the defendant contends that the judge erred in ruling that Cellmark's database was adequate, that the absence of data on African-Americans from Springfield in Cellmark's database was not a matter of concern, and that the results of Cellmark's profile frequency calculations through use of the product rule were reliable and accurate.

The product rule refers to the product, or multiplication, of the frequencies (probabilities) with which each allele in a tested sample of DNA occurs in the population included in the database. The resulting number is the probability of someone's having the same characteristics as the sample tested. See Commonwealth v. Curnin, 409 Mass. 218, 224 (1991). The product rule is based on two assumptions about the nature of genetic variants in the population. Those assumptions are known as the condition of Hardy-Weinberg equilibrium, and the condition of linkage equilibrium. Hardy-Weinberg equilibrium is a condition that is achieved when there is no particular relationship between the occurrence of alleles within a single genetic marker. That is, when a person's parents meet randomly in the general population. Linkage equilibrium is a condition that is achieved when there is no particular relationship between genes. That is, when genes are inherited independently, and not linked to other genes. See id. at 225 n. 11; NRC, Evaluation of Forensic DNA Evidence (1996) at 90-91, 106-107.

Dr. Basten testified that he had conducted a study of fifty databases, including Cellmark's, to determine whether the frequencies of alleles at particular loci varied between databases or between racial groups within the databases for the population of the United States. He had presented the details and the results of his study at two scientific conventions. Dr. Basten concluded that there was consistency of allelic frequency between the various databases and within the various racial groups, but that allelic frequency varied between racial groups. He also concluded that the size of Cellmark's database, 103 persons, is adequate and common within the field, and that a database larger than Cellmark's would produce no significant difference in result. He explained that the reliability of profiling depends more on the number of alleles in the database than the number of persons in the database. Dr. Basten discounted the absence of data on any African-American from Springfield in Cellmark's database because his study of the fifty databases indicated that the African-American population is fairly homogeneous across the United States. He endorsed Cellmark's use of the product rule in making its profile frequency calculations, and verified Cellmark's test results through the use of the theta factor, a statistical adjustment recommended by the NRC when dealing with possible subgroups, or small isolated populations. See NRC, Evaluation of Forensic DNA Evidence (1996) at 29-30. Dr. Basten employed the more conservative of the two theta factors recommended by the NRC. Verification also was made through the use of "confidence intervals." See Commonwealth v. Rosier, supra at 814 n. 14.

The judge found that Dr. Basten's use of the conservative theta factor and confidence intervals were an "appropriate corrective measure" to account for any possible substructure with the African-American community in Springfield. He acted within his discretion in ruling that Cellmark's database was adequate and that the use of the product rule produced reliable results. See Commonwealth v. Rosier, supra at 813-814.

6. Hearsay testimony. The defendant objected to the admission of hearsay through police officers who testified that one of the three women who had smoked cocaine with the defendant on the evening of February 18, 1998, told police that she saw the fourth victim drive off with the defendant after they returned from Holyoke. See Commonwealth v. Silanskas, 433 Mass. 678, 693-694 (2001). The testimony properly was admitted, not for its truth, but to show the state of police knowledge that drew their attention to the defendant: there were significant discrepancies between the accounts given by the woman and the defendant as to the events of February 18, 1998. See Commonwealth v. Rosario, 430 Mass. 505, 508-509 (1999). In any event it was merely cumulative of the woman's properly admitted testimony, and therefore could not constitute reversible error. See Commonwealth v. Connor, 407 Mass. 663, 670 (1990).

The defendant also contends that a similar error occurred when a police officer testified in response to a question by the prosecutor on redirect examination that a crack cocaine dealer had given a written statement in which he said that the defendant "used to be a runner," or acted as a middle man for drug dealers, at 65 Maynard Street. The Commonwealth contends that the testimony was offered under the doctrine of verbal completeness to clarify the context of a portion of the same statement used by defense counsel in his cross-examination of the same police officer. Defense counsel had elicited testimony from the officer that the drug dealer told him that the fourth victim used to purchase crack cocaine at various locations. The portion of the statement that was admitted on redirect examination did not serve to clarify the context of the portion that was admitted during cross-examination or serve to correct any distortion that might have been caused by a fragmented version of events. It involved a different subject, and as such, was not admissible under the rule of completeness. See Commonwealth v. Watson, 377 Mass. 814, 825-831 (1979), S.C., 409 Mass. 110 (1991). As there was no objection, we review under the substantial likelihood of a miscarriage of justice standard. Commonwealth v. Vinnie, 428 Mass. 161, 163-164, cert. denied, 525 U.S. 1007 (1998). The defendant does not explain how he was prejudiced, and we perceive no harm — in this case. Whether the defendant may have been a middle man for drug dealers at 65 Maynard Street would probably not raise an eyebrow in light of the full range of illegal activity that openly took place at 65 Maynard Street, where the defendant was all too well known.

7. Comments on the defendant's right to remain silent. During cross-examination of the State police chemist, defense counsel established that some tangible evidence had not been sent to Cellmark and had not been analyzed by the State police. The prosecutor was permitted, on redirect examination of both the chemist and a Cellmark analyst, to elicit testimony that the evidence nevertheless had been preserved and was available for testing by either the Commonwealth or the defendant. This is a sensitive area. However, we are satisfied that the testimony did not constitute impermissible comment that the defendant failed to produce evidence or that he had any burden to test the evidence. See Commonwealth v. Conkey, 430 Mass. 139, 147 (1999), S.C., ante 60 (2004). The testimony was admissible to bolster the credibility of the witnesses by showing they did not try to hide anything. See Commonwealth v. Evans, 438 Mass. 142, 153-154 (2002), cert. denied, 538 U.S. 966 (2003). Moreover, the defendant did offer expert testimony as to some evidence, vitiating any effect the testimony may have had. Id. The judge thoroughly instructed the jury at the end of the trial that the defendant had no burden to produce evidence, and that the burden of proof never shifted to him. There was no error.

The defendant next contends that the prosecutor improperly elicited testimony from a police officer who, when he spoke to the defendant on April 6, 1998, mentioned that he knew the defendant had consulted an attorney. The officer had testified that he told the defendant they wanted to tell him about the blood test results in the case of the fourth victim, as promised, but that they knew he had consulted an attorney and therefore they would not question him about that case. He testified that the defendant thanked them, then engaged them in conversation about the other cases. The judge interrupted the witness in the middle of his answer to the question, "Did he say anything else?" By doing so, the judge effectively curtailed testimony that would have constituted improper comment on a defendant's exercise of his right to remain silent and averted reversible error. See Doyle v. Ohio, 426 U.S. 610, 619 (1976). The officer was about to describe the portion of the defendant's statement in which he declined, on advice of counsel, to accompany the officers to the police station to give a statement about the other murders. While it would have been preferable to avoid the testimony about the defendant having consulted an attorney (pursuant to a pretrial stipulation, for example), it did not, in context, amount to a comment on his exercise of the right to remain silent, see Commonwealth v. Waite, 422 Mass. 792, 798 (1996); Commonwealth v. Sazama, 339 Mass. 154, 157-158 (1959), and there was no suggestion that the defendant should be disparaged for having retained counsel. Moreover, it could not have been construed as a comment on the defendant's right to remain silent because he was not silent but instead spoke freely. There was no error.

8. Motion to limit cross-examination of the defendant. The defendant alleges error in the denial of his motion to limit the prosecutor's cross-examination of him to the case of the second victim, in whose case he had filed a notice of alibi defense. He argues that the denial of his motion deprived him of his right to present an alibi defense.

The issue is related to the question of joinder, and as discussed above in part 4, supra, the defendant was not deprived of the opportunity to present an alibi defense. He could have presented an alibi defense through two alibi witnesses without having to testify himself. The principle that the scope of cross-examination is within the sound discretion of the trial judge applies to criminal defendants who choose to testify on their own behalf. See Commonwealth v. Judge, 420 Mass. 433, 444-445 (1995). Thus, the defendant has failed to show that the denial of his motion was an abuse of discretion.

9. Prosecutor's closing argument. The defendant contends that the prosecutor's closing argument was improper in many respects, some of which were the subject of an objection. Where there was an objection, we review under the prejudicial error standard. See Commonwealth v. Wilson, 427 Mass. 336, 351 (1998). In the absence of an objection we review to determine whether the argument, if improper, created a substantial likelihood of a miscarriage of justice. See Commonwealth v. Allison, 434 Mass. 670, 686 (2001). In determining whether an argument was improper we examine the remarks "in the context of the entire argument, and in light of the judge's instructions to the jury and the evidence at trial." Commonwealth v. Viriyahiranpaiboon, 412 Mass. 224, 231 (1992).

The defendant grouped the perceived improprieties in four categories. In the first category are three instances where he claims the prosecutor appealed to the emotions of the jury, all without objection. The first instance occurred while the prosecutor was describing the defendant's false statements and omissions in his statement to police on February 27, 1998. The prosecutor referred to the twelve dollars the defendant received from pawning the fourth victim's earrings as "the value he put on [her] life." The comment was brief, isolated, and out of step with the strong and proper point he was in the process of developing. Considering the closing in its entirety and the judge's repeated instructions on the purpose of closing argument and the instruction that the jury should not be influenced by sympathy, there was no substantial likelihood of a miscarriage of justice. See Commonwealth v. Mitchell, 428 Mass. 852, 857 (1999). Contrast Commonwealth v. Worcester, 44 Mass. App. Ct. 258, 263-264 (1998) (repeated comments about value of victim's life in combination with exhortations to "do something about [the killing]," held prejudicial).

Second, the prosecutor's references to "the killing zone" were not appeals to sympathy, but, in the context of the entire argument, a shorthand expression for his argument that physical evidence of the defendant's involvement in the murders was found within the immediate area where each killing occurred or on the victim's body or personal belongings. The comments were somewhat awkward, but they were not improper.

In the third instance, the prosecutor referred to the killings as creating a "time of terror" in Springfield. The comment should not have been made, but it was not repeated or hard driving, and it was not a principal focus of what otherwise was a proper closing argument. Moreover, the absence of an objection from experienced defense counsel is some indication that the comment did not land a foul blow. See Commonwealth v. Kozec, 399 Mass. 514, 518 n. 8 (1987).

In the next category of alleged impropriety, the defendant claims that the prosecutor impermissibly commented on the defendant's failure to assist police. The defendant objected to this portion of the argument. The argument was proper comment on the defendant's motive for making false statements and omissions in his statement to police on February 27, 1998. See Commonwealth v. Morales, 440 Mass. 536, 551-552 (2003). Even if improper, there was no prejudice. The comment was brief, the judge's instructions informed the jury that closing argument is not evidence and that the defendant has no obligation to produce evidence, see Commonwealth v. Martino, 412 Mass. 267, 282-283 (1992), and the cases against the defendant were overwhelming. See Commonwealth v. Wilson, 427 Mass. 336, 353 (1998).

In the third category of alleged impropriety, the defendant claims that, on several occasions, the prosecutor asserted his personal belief as to the facts. See Commonwealth v. Wilson, supra at 352. There was no objection. We have reviewed the transcript and are satisfied that in each instance the prosecutor was drawing a permissible inference or was inviting the jury to draw a permissible inference. There was no error.

Finally, the defendant argues that the prosecutor misstated the evidence in two respects. We have reviewed the transcripts and are satisfied that the prosecutor was arguing permissible inferences that were grounded in the evidence.

10. Relief under G.L. c. 278, § 33E. We have reviewed the transcript, the record, and the briefs, and conclude that there is no reason to reduce the verdicts or grant a new trial pursuant to our power under G.L. c. 278, § 33E.

Judgments affirmed.