KAFKER, J.

Irrelevant deoxyribonucleic acid (DNA) statistics and improper appeals to juror emotions by the prosecutor needlessly complicated the convictions of the defendant, Michael Grinkley, for indecent assault and battery on a child. Nonetheless, overwhelming evidence of guilt — the defendant was found in bed with his pants down, on top of one young victim and beside the other — and a discerning jury, who acquitted the defendant on rape and other charges, rendered the errors harmless. We therefore affirm.

This case arises from several incidents of sexual assault involving the defendant and two minors, his great-nieces. The defendant was indicted on four counts of natural sexual intercourse with a child under the age of sixteen and seven counts of indecent assault and battery of a child under the age of fourteen. After a jury trial in December, 2005, he was found guilty of four lesser included offenses of indecent assault and battery of a child under the age of fourteen on the indictments charging statutory rape, but he was found not guilty on the three remaining, separately charged indecent assault and battery counts. On appeal, the defendant argues that his convictions should be reversed because (1) the judge improperly allowed an expert witness to testify to an irrelevant and misleading DNA statistic and (2) the prosecutor's closing argument was improper because it appealed to juror sympathies and misrepresented the defense argument.

1. Background. On February 10 or 11, 2003, the older victim, then eight years of age, and her sister, the younger victim, then six years of age, were being cared for by their aunt Natalie at home in the Dorchester section of Boston. Also residing in the home were the victims' grandmother Celeste, their grandfather Chester, their five year old brother, and two adult cousins, Tosheiya and Lakeisha (the victims' first cousins, once removed). The defendant was the victims' great-uncle (their grandmother Celeste's brother), who lived in the third-floor apartment of the home. Because Celeste, the primary caregiver, was away on the evening of February 10 or 11, Natalie put the girls and their brother to bed together in the girls' room at approximately 9:00 P.M. The older victim was wearing a long T-shirt and under-wear. The younger victim was wearing pajama pants with a small rip near the crotch area. After tucking them in. Natalie went up to the defendant's room to smoke a cigarette with several family members and friends, including her cousins. Although some of the men had been consuming alcohol, testimony indicated that the women were not drinking that evening.

At some point thereafter, the defendant went downstairs into the room where the children were sleeping together in the same bed. The older victim testified that the defendant climbed on the bed, "unzipped his fly and pulled out his private spot" (later described by the child as his penis). He moved the covers down to her knees, and climbed on top of her. The defendant then slid her underwear down, at which point "[h]is private spot went up to mines." She felt the defendant's penis "moving around" on top of her vagina. After a period of time, the defendant climbed off her and moved onto her sister, the younger victim. The older victim felt a part of the bed going down and his leg next to her. She then went back to sleep.

The younger victim, who was lying on her stomach, testified that the defendant lay on top of her. The defendant put his "private" through the rip in her pajama pants onto her "private." She then felt something wet and slimy "inside [her] private." She described something "hard" "going against [her] legs." The younger victim also asserted that the defendant touched both her sister and her "in a place that we don't want to be touched in."

Fifteen to twenty minutes later, several of the adults realized that the defendant was not with them in his apartment and went downstairs to look for him. They opened the door to the girls' darkened bedroom and found the defendant, still on top of the younger victim. The defendant's pants were at least partially undone and Natalie stated that his buttocks were exposed. Natalie was also able to see the younger victim's bare legs underneath the defendant, while Tosheiya testified that she could see the younger girl's exposed buttocks. Natalie asked the defendant what he was doing, prompting him to respond brazenly: "Ha-ha. You know how that goes." Lakeisha similarly stated that when discovered, the defendant said "Ha-ha-ha. You know how we do." Tosheiya asserted that she heard the defendant respond "`Don't act like you don't know how it goes down in this family,' or something to that effect."

His casual remarks, along with the compromising situation in which he was found, caused several of the family members to kick, punch, and nearly stab the defendant with a hastily grabbed knife while removing him from the bedroom.

After the defendant had been separated from the girls and taken to another room, Natalie spoke with each child individually. When the police arrived later that evening, the two victims were taken to the Boston Medical Center for the administration of a sexual assault evidence collection kit. The resulting forensic samples were forwarded to the Boston police crime laboratory (laboratory) for processing and DNA testing.

Both girls testified that the defendant had inappropriately touched each of them on a prior occasion. At trial, the older victim testified that at some point in the past she, her younger sister, and their brother were watching a movie in the defendant's apartment when the defendant licked her vagina. She could not remember if her clothes were on or off. She also testified that on the same occasion, the defendant "got [her] hand and made [her] touch his penis." He was wearing clothes at the time. The incident was interrupted when their grandmother said there was a telephone call for the defendant.

The younger victim similarly testified to an occurrence in the defendant's apartment on the third floor where the defendant looked at pornographic magazines (described by the girl as "magazines with nasty things in it") with all three children. He then made the girls pull down their pants and get in his bed, where he proceeded to lick their "private spot." The younger victim stated that he then pulled the covers over them, made his private spot touch hers, and moved up and down repeatedly. Additional facts regarding the assaults and the follow-up investigation will be set forth as necessary to address the particular legal issues raised.

2. DNA evidence. On the night of the February 10 or 11, 2003, incident, a number of biological samples were taken from the victims as part of the sexual assault kit examination. No semen was found in any of the testing. The samples, which contained amylase, an enzyme contained in biological fluids such as saliva, were submitted to the laboratory for DNA extraction, amplification, and analysis in accordance with its applicable protocols. The laboratory performed polymerase chain reaction (PCR) testing and produced a profile of the DNA found in the samples. At the time, the laboratory examined sixteen different loci on the DNA molecules found in the samples to compile the DNA profile or profiles present in each sample. By means of comparisons to reference DNA samples of the victims and the defendant, the laboratory attempted to determine whether the particular known individual's alleles were the same as those present at the loci in the DNA samples tested.

Of the samples tested in this case, one — the perianal swab from the older victim — showed results indicating the presence of DNA not belonging to the victim in addition to the victim's own DNA and thus was the only sample of any possible relevance in linking the defendant to the crimes charged via a DNA match. The profile found in that sample was an "inclusion" for the older victim; her alleles were the same as those identified at every location tested. The defendant was neither included nor excluded as a contributor of the additional DNA found in the sample. Five of his alleles were possible matches to alleles found in the sample that could not have been contributed by the older victim. Twelve alleles of the defendant were not found in the sample. No allele was detected that could not have been contributed by either the older victim or the defendant. The older victim's alleles were also darker, thereby indicating that she was the primary contributor to the sample.

None of the testing methodology so far described is at issue on appeal. At trial, however, the Commonwealth's expert witness, Julie Lynch, testified to a statistic purporting to represent the proportion of different racial populations that could be possible contributors of their entire DNA to the profile identified in the perianal sample. Despite the defendant's repeated pretrial and trial objections and requests for a Daubert-Lanigan hearing see Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579 (1993); Commonwealth v. Lanigan, 419 Mass. 15 (1994), regarding the statistical methodology, all of which were denied, the prosecution was permitted to offer Lynch's statistical opinion that only approximately "one in three hundred and twenty billion African-Americans . . . are included as being possible contributors to the mixture detected on that sample." Lynch explained that that meant "basically a fraction of one percent of the population . . . could be expected to have contributed their entire DNA profile to [that] particular mixed piece of evidence." On cross-examination, Lynch acknowledged that the defendant was not included within the one in three hundred and twenty billion African-Americans who could be considered a contributor, as his entire profile was not detected in the mixture. In fact, the only identifiable complete DNA profile in the mixture was that of the older victim. The statistics were therefore irrelevant and potentially confusing and misleading.

3. Admissibility of the DNA evidence and statistics. As explained above, the DNA evidence here was for a mixed sample. The victim was apparently the primary contributor, and the defendant was found to be a possible contributor in each of the five loci where DNA from an additional contributor was detected. The question we confront is what type of statistical evidence is appropriate in this context. What the Commonwealth submitted here, ostensibly product rule statistics for a mixed sample without an identified primary contributor, was not permissible for reasons that should have been obvious.

The product rule computes the probability (also called a frequency profile) that the DNA found in a given biological sample matches a randomly selected individual in the general population. Commonwealth v. Lanigan, 419 Mass. at 20, 21. Commonwealth v. Gaynor, 443 Mass. 245, 268 (2005). The product rule has been held to be a scientifically valid method for calculating frequency profiles involving single source (i.e., nonmixed) DNA samples. See Commonwealth v. Rosier, 425 Mass. 807, 815-817 (1997). Statistics derived from the product rule have also recently been held valid in mixed sample contexts "where the [DNA] 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." Gaynor, supra at 268. In the instant case, the sample was mixed. The defendant was not identified as a primary contributor and therefore the sample was not treated as originating from a single source. Moreover, if a primary contributor could have been identified, it would apparently have been the older victim. To apply the product rule to a mixed sample without a primary contributor having been identified as a single source makes absolutely no sense. See id. at 250 (frequency calculations using product rule computations were not performed on some samples because "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").

The prosecutor argued to the judge at trial that the statistical evidence here was justified by Commonwealth v. McNickles, 434 Mass. 839 (2001), contending that it was "`exactly the same statistic." The McNickles case, however, concerned the use of likelihood ratios, not product rules. See, e.g., id. at 845-848; Commonwealth v. Gaynor, 443 Mass. at 268. The likelihood ratio "compares the probability that the defendant was a contributor to the sample with the probability that he was not a contributor to the sample." Commonwealth v. McNickles, 434 Mass. at 847. It is appropriately used for statistically evaluating the "test results of mixed samples when the primary and secondary contributors cannot be distinguished." Commonwealth v. Gaynor, supra. It is undisputed that the laboratory here, however, lacked the capacity to compute likelihood ratios. There was no attempt to calculate such ratios in this case.

In Commonwealth v. McNickles, 434 Mass. at 851-854, the Commonwealth also introduced evidence that the defendant's alleles matched the alleles identified at one location of a PCR test of a mixed sample. The rest of the locations examined were not readable. The Commonwealth also provided statistics regarding the frequency of those alleles at that location, which was one in 1,000 for Caucasians and one in 100 for African-Americans. The court stated: "That a scientific test provides descriptive information short of a definitive identification does not make it unreliable, irrelevant or otherwise inadmissible." Id. at 854. The court also stressed that "[a]t no time did the Commonwealth exaggerate the significance of having the same type at the single . . . site or otherwise suggest that the PCR-based testing itself identified the defendant as the perpetrator." Id. at 854-855. No such limited DNA evidence or qualified statistics were, however, introduced here.

In sum, the statistics offered were not authorized by Commonwealth v. McNickles, supra, or the other DNA cases. Rather, the statistics were meaningless and improper. The jury would, however, have wondered why they were being presented. The prosecutor appeared to be arguing that the expert's testimony was that the "genetic profile [of] what is in the perianal evidence is . . . very, very, very rare . . . [and] is highly inculpatory," because the chances for "someone else to have contributed the DNA . . . are one in a cabillion." Essentially, the prosecutor seemed to be contending that, because the DNA profile of the perianal sample was exceptionally rare, and the defendant met part of the profile, it was highly likely that the DNA was the defendant's. This was of course inaccurate and misleading. The DNA profile reflected a mixed sample, not one person's. The likelihood that the defendant contributed to the mixed sample analyzed was not in any way calculated. The judge therefore abused her discretion in allowing the Commonwealth to offer statistics based on a product rule calculation over the repeated objection of the defendant. The question then becomes whether the error was harmless.

Given the overwhelming evidence of the sexual assaults and the defendant's acquittal on the rape charges, we conclude that the defendant was not prejudiced by the Commonwealth's improper use of the product rule statistics. The defendant was found in bed on February 10 or 11 with several young children. His pants and zipper were down while he lay on top of the younger victim, whose pajama bottoms had also been pulled down. He also made inculpatory remarks when he was discovered. The girls described the different sexual assaults in some detail. Although the DNA evidence taken from the perianal sample arguably added proof not otherwise present of a rape of the older victim, it was totally unnecessary to the mountain of proof provided to establish the sexual assaults by the defendant.

4. Closing argument. The defendant claims that the prosecutor's closing argument compounded the error regarding the DNA statistics and contained improper emotional appeals and misrepresentations of defense positions. To determine whether prejudice resulted from the error, "we consider (!) whether the defendant seasonably objected; (2) whether the error was limited to collateral issues or went to the heart of the case; (3) what specific or general instructions the judge gave to the jury that may have mitigated the mistake; and (4) whether the error, in the circumstances, possibly made a difference in the jury's conclusions." Commonwealth v. Kater, 432 Mass. 404, 422-423 (2000). In making this determination, we consider "the entire record, including the balance of the prosecutor's argument." Id. at 423, quoting from Commonwealth v. Kozec, 399 Mass. 514, 523 (1987).

We begin with the prosecutor's suggestion that "a mixture of DNA [was] found in [the older victim's] perianal area . . . [and only] [o]ne in three hundred and twenty billion [have that DNA]." As explained above, the statistical evidence was irrelevant and potentially misleading. Defense counsel's objection was, however, sustained. Furthermore, defense counsel herself had made a similarly misleading reference to the statistics in her closing, arguing "whoever contributed to that swab . . . was one in a billion, billion," and that person was not the defendant. The proper response by the prosecutor, however, should have been to object to the defendant's abuse of the statistics or to answer and correct the prejudicial irrelevancy, not combat it with further misleading information. See Commonwealth v. Kozec, 399 Mass. at 519 n. 9. See also Commonwealth v. Bradshaw, 385 Mass. 244, 277 (1982) ("We have on more than one occasion requested that prosecutors seek redress from the trial judge rather than resort to retaliatory reply to an improper defense argument").

Regardless, as explained above, it is not clear what significance the reference to the DNA statistics in the closing argument has on appeal. The defendant was acquitted on the rape charges, which the DNA evidence was offered to prove. There was overwhelming evidence of sexual assaults, so we cannot discern the value, if any, the DNA evidence added in terms of proving sexual assault. We therefore deem the prosecutor's reference to the figure in closing harmless.

Overreaching by the prosecutor, both on her own initiative and in response to defense tactics, was not, however, confined to the DNA evidence. In Commonwealth v. Deloney, 59 Mass. App. Ct. 47, 53 (2003), we stated that this very same prosecutor's "references to `how hard' it must have been for [the victims and parents] to testify . . . crept perilously close to the line separating a proper discussion of the evidence from an improper appeal to jury sympathy." See ibid. Despite that warning, the prosecutor adopted the same approach here, stating: "Don't under-estimate for one second how hard it was for those little girls to come into this courtroom, in front of this court, in front of all of you strangers, get up on the witness stand, and talk about the private intimate details of their victimization at the hands of this child rapist." The prosecutor stated further: "Think about literally how hard it was for [the younger victim] to talk. . . . The shame she felt. She couldn't talk about it. She needed to write it down and then read it."

We recognize that the line between improper appeals to juror emotions and proper explanations of difficulties witnesses may have had in testifying is not always clear, particularly with regard to young children in sexual assault cases. The younger child in this case did in fact have difficulty recounting the details of the incidents, requiring her to write down what happened to her. In isolation, the prosecutor's comments would be less troubling. In context, they are part of a coherent rhetorical appeal to juror emotions.

The prosecutor also stated: "Think about what it must have been like for [the older victim] that night after he had gotten on top of her. . . . Think about what life was like for [the younger victim]. She sees it done to her big sister, and her sister doesn't do anything about it." Although the actions described were arguably grounded in the evidence, it is not clear what purpose was served by the statements that the jurors were to "think about what it must have been like" for the victims except to have them place themselves in the children's position. "The invitation to the jury to put themselves in the position of the victim is usually improper." Commonwealth v. Jordan, 49 Mass. App. Ct. 802. 816 (2000). See Commonwealth v. Harris, 11 Mass. App. Ct. 165, 176 n. 9 (1981) (remarks such as "[t]hink of what [this eleven year old girl] went through by giving this story. Do you think she wanted to put herself through this? Please don't let her down" were "improper attempt[s] to arouse juror sympathy").

Continuing in the same vein, the prosecutor responded inappropriately to an implausible argument by the defense that the victims' statements were motivated by the rape, examination. In elaborating on her argument that the victims' statements were all a product of the suggestibility of children exposed to the hysterical reactions of the adults that night, defense counsel stated: "Well the suggestibility to the children did not end there. They were whisked in the middle of the night in an ambulance to the hospital where the two girls underwent a horrific physical exam. What child would think nothing had happened to them but yet they had to go through that exam? And we heard how painful it was. [The older victim] had to be given a sedative."

In response, the prosecutor argued: "[A]sk yourself what possible motive do those girls have to come into this courtroom? Why? So that eight or so hours after they tell their aunt what happened they get to have a rape kit done? That's the motive? And think about what [emergency room nurse] Dianne Courtney said that involves. Swabbing of the vagina, the anus, the perianal area. They were six and eight years old when they laid there in the frog position in the emergency room that night."

There was no reason for either the prosecution or the defense to discuss or dwell on the pain and difficulty of the rape examination. Obviously the rape examination was the product of the sexual assault claim, not vice versa. Therefore, whatever conceivable relevance there was to a description of the difficulty and pain of a rape examination was outweighed by its prejudicial effect. See Mass. G. Evid. § 403 (2009-2010). Although goaded by the defense, the prosecution capitalized on the opportunity by drawing out its details as part of an over-all emotional appeal to the jury. As explained above, the prosecutor should have objected to the defendant's argument or explained its implausibility without trying to exploit it.

In sum, the overwrought and inflammatory closing argument of the assistant district attorney, which ignored a prior warning from this court about many of the same unacceptable points of argument, was prosecutorial error. In examining the total effect of the different emotional appeals, we begin with the recognition that the defendant made a timely objection that the "whole closing was an appeal to sympathy. What it must have felt like, what it was like for her." The judge denied a motion for a mistrial and gave no specific curative instructions. The judge did give standard instructions to the effect that arguments are not evidence, that the jury's "verdict must be based solely on the evidence," and that it "would be improper for [the jury] to consider any personal feelings or bias or prejudice or sympathy in deciding whether [they] should return a verdict of guilty or not guilty."

We must determine whether these improper appeals to emotions, in these circumstances, possibly made a difference in the jury's conclusions. In the instant case there was overwhelming evidence of sexual assaults — particularly on the night of February 10 or 11 when the defendant was found in bed with the two victims, on top of one of them, with his pants unzipped and her bare bottom exposed — and a close question whether there was penetration. The jury, apparently able to distinguish between, on one hand, argument and evidence and, on the other, excessive contentions by both parties, convicted the defendant on the sexual assaults and acquitted him on the rape charges. See Commonwealth v. Bradshaw, 385 Mass. at 277 (where "most of the prosecutor's remarks were grounded in the evidence and the few extravagant remarks were responsive to equally extravagant defense tactics in final argument. . . . [t]he jury could be expected to take both arguments with a grain of salt"); Commonwealth v. Kozec, 399 Mass. at 517 (properly instructed jurors considered able to sort out excessive claims of both parties and separate argument from evidence where "they in part rejected the position of each side, returning a verdict of not guilty on one charge and guilty on the other").

In sum, the verdict was reflective of the jury's discernment and deeply rooted in the evidence, and it need not be overturned due to prosecutorial errors. Nonetheless, we are again left "to wonder why some prosecutors persist, by ill advised rhetoric and inflammatory remarks, in attempting to snatch defeat from the jaws of victory." Commonwealth v. Jones, 9 Mass. App. Ct. 103, 119 (1980), S.C, 382 Mass. 387 (1981). We understand that these are hotly contested cases in which the stakes and emotions could not be higher. We also understand that the defense was provocative. That being said, "the prosecutor, as a representative of the government, must hold himself [or herself] to a consistently high and proper standard." Commonwealth v. Kozec, 399 Mass. at 519. We do not consider that high and proper standard close to being met in the closing argument here.

Judgments affirmed.