
505 F.3d 578 (2007)
UNITED STATES of America, Plaintiff-Appellee,
v.
Kelley L. HUGHES (06-3024); Kevin L. Stacy (06-3025), Defendants-Appellants.
Nos. 06-3024, 06-3025.
United States Court of Appeals, Sixth Circuit.
Argued: July 26, 2007.
Decided and Filed: October 26, 2007.
*579 *580 *581 ARGUED: Chad A. Readler, Jones Day, Columbus, Ohio, W. Joseph Edwards, Law Office of W. Joseph Edwards, Columbus, Ohio, for Appellants. J. Michael Marous, Assistant United States Attorney, Columbus, Ohio, for Appellee. ON BRIEF: Chad A. Readler, Jones Day, Columbus, Ohio, W. Joseph Edwards, Law Office of W. Joseph Edwards, Columbus, Ohio, for Appellants. J. Michael Marous, Assistant United States Attorney, Columbus, Ohio, for Appellee.
Before: DAUGHTREY and GILMAN, Circuit Judges; ADAMS, District Judge.[*]

OPINION
JOHN R. ADAMS, District Judge.
Appellants Kelly Hughes and Kevin Stacy challenge their convictions and alternatively the district court's denial of their motion for judgment of acquittal and/or motion for a new trial. This appeal, originally filed by Hughes, was consolidated with case number 06-3025, filed by Stacy. There are nine issues raised on appeal, two by Hughes, seven by Stacy. Both Appellants allege that there was a variance between the indictment and the evidence *582 presented at trial on the conspiracy counts. They further argue that the variance resulted in substantial prejudice, thus necessitating a new trial. Stacy makes the following additional arguments: that the jury verdict was against the manifest weight of the evidence and was not supported by sufficient evidence with respect to the charge of making false statements and the charges of conspiracy to engage in insider trading and to obstruct justice; that the jury relied on material outside the record in its deliberations; that defense counsel failed to present an adequate defense at trial due to the district court's exclusion of certain expert testimony; that the jury instructions were inadequate; and that prejudice resulted from the cumulative effect of errors made by the district court. For the reasons stated below, we AFFIRM the judgment of the district court, although based upon a slightly different analysis from the one used by that court.

FACTUAL AND PROCEDURAL BACKGROUND
I. Kellogg buyout and suspicious stock purchases[1]
This case involves insider trading based on the purchase, by the Kellogg Company ("Kellogg"), of Worthington Foods ("WF"), a small meatless foods company located in Columbus, Ohio. From July 1999, when negotiations began with Kellogg, to October 1999, when the buyout occurred, Appellants' co-defendant Roger Blackwell ("Blackwell") sat on the Board of Directors at WF. As a member of the Board, Blackwell was privy to non-public information about the Kellogg purchase and the price per share negotiated by the parties.
From June to September 1999, many of Blackwell's friends and family bought WF stock. The largest purchaser was Justin Voss ("Voss"), a friend and business associate to Blackwell, who bought 38,000 shares. The second largest purchaser was Jack Kahl ("Kahl"), a friend and business associate to Blackwell, who bought 15,000 shares. Hughes, longtime employee and friend to Blackwell, and her husband, Stacy, were the fifth largest purchasers with 10,286 combined shares. Arnold Jack ("Jack"), longtime friend to Blackwell, and Black Jack Enterprises, co-owned by Jack and Blackwell, purchased 5,500 shares. Dale Blackwell, Blackwell's father, purchased 3,000 shares. Gertrude and Alfred Stephans ("the Stephanses"), parents of Blackwell's wife Kristina Stephans-Blackwell ("Stephans-Blackwell"),[2] purchased 1,800 shares. Finally, Blackwell's son, Christian Blackwell, purchased 350 shares of WF stock.
II. Appellants' relationship to Blackwell and other co-defendants
Blackwell is a former professor at the Ohio State University's School of Business. Hughes first worked for him while she was still in college, grading papers for the professor. She started her professional career working for him at Roger Blackwell Associates, Inc.[3] ("RBA") in 1990 as an administrative assistant. Eventually she became the Director of Marketing, and was responsible for all of RBA's finances, accounting and marketing.
*583 Most years, Hughes met with Blackwell during the month of August to discuss the company's finances. On August 31, 1999, Hughes had such a meeting with Blackwell, just days before her various purchases of WF stock.
Hughes had a close relationship with Blackwell. She was said to be devoted to him. Blackwell also valued Hughes as a friend, stating at his birthday party in 2003 that he "could not have gotten through all this[4] without her help and support." He also added that he valued her husband Stacy's support.
In addition to her duties at RBA, Hughes was responsible for the investment decisions for the Roger Blackwell Pension Plan ("Pension Plan"). Her first investment for the Pension Plan was in September of 1999 with the purchase of WF stock. At that time, only two employees of RBA, Stephans-Blackwell and Hughes, were vested in the Pension Plan.
Stacy, a surveyor by trade, married Hughes in 1990. He also had a close relationship with Blackwell and was an usher at Blackwell's wedding. Stacy and Hughes socialized many times with Blackwell and his wife, along with Blackwell's friends and family. Stacy even thought of Jack, one of Blackwell's closest friends, as his attorney.
III. Blackwell's direct connection to the conspiracy
Sometime in late July or early August of 1999, Blackwell discussed the possible buyout of WF with his then wife, Stephans-Blackwell. The two discussed the propriety of purchasing stock in WF at that time. Blackwell told his wife that they could not purchase WF stock at that time, but when she asked whether her parents would be able to purchase WF stock, Blackwell responded that he didn't think that would be a problem.
Stephans-Blackwell contacted her mother, Gertrude Stephans, and encouraged her to purchase WF stock. Blackwell was aware of this conversation. Additionally, Blackwell and Stephans-Blackwell provided her parents with $20,000 to purchase WF stock.
Blackwell also informed his friend and business associate, Kahl, of the imminent purchase of WF by Kellogg and the substantial rise in WF stock that would result. Blackwell told Kahl that, if he were to decide to purchase WF stock, he should "go in thinly" and not buy too much because it was a "thinly traded" stock.[5]
Kahl served on the Board of Directors for another company, Henkle, along with Blackwell. In January 2003, Kahl and other members of the Henkle Board received a letter from Blackwell stating:
I also want to clarify a few facts of this situation. First and foremost, I never discussed the merger of Kellogg and Worthington Foods with anyone nor disclosed anything that can be remotely considered insider information. That is *584 my policy at all times for all companies with which I am affiliated.
Kahl questioned the truthfulness of the letter because he had personally received inside information about the WF/Kellogg buyout from Blackwell. Kahl understood the letter to express Blackwell's position in the investigation rather than a true and accurate version of the events surrounding the buyout.
IV. Investigations
In December 1999, the National Association of Securities Dealers ("NASD") launched an investigation into the purchases of WF stock, based on increased activity during the months directly preceding the buyout. NASD sent out a questionnaire requesting information on the relationships among purchasers of stock during the relevant period and those persons who knew about the buyout. Instead of providing the information requested in the questionnaire, Blackwell responded by stating: "I supplied no information about the transaction to any of the people on the list or any one [sic] else. When asked by anyone about [ ] Worthington Foods during this period, I replied that I was not able to comment about the company's stock, as is my standard practice." Blackwell eventually did disclose to NASD his relationships to his father, his son, Jack, Black Jack Enterprises, Hughes and Stacy, but continued to deny providing information to them about the buyout. It is important to note that in a three-day time frame, during the same month the NASD investigation began, Blackwell's telephone calling card was used from New York to call most of the people he was connected to who had purchased WF stock during the period under investigation, including Dale Blackwell, the Stephanses, Hughes and Stacy, Jack, and Voss.
In 2000, the Securities and Exchange Commission ("SEC") subpoenaed Blackwell to testify and to provide various documents regarding the matter. On January 9, 2001, Blackwell testified before the SEC that he did not disclose the information about the Kellogg buyout to anyone. He also responded to a later SEC inquiry, restating his position of non-disclosure. In February of 2002, Stephans-Blackwell, along with her parents, testified before the SEC, stating that they had not received insider information from Blackwell about the buyout. According to Stephans-Blackwell, she and her husband has previously agreed that no one would disclose that Blackwell had provided them with insider information about the buyout.
In November 2000, Hughes and Stacy both provided sworn testimony before the SEC. Both testified that they did not receive inside information about the purchase of WF and did not base their purchases of WF stock on any such information. On May 15, 2002, Hughes again testified before the SEC. She stated that she never received money from Blackwell to help pay for WF stock and that the $30,000 check she received from RBA, discussed below, was not used to purchase WF stock.
V. Hughes and Stacy's stock purchases
From April 1997 through August 1999, Stacy was a fairly conservative investor in his IRA accounts. His transactions involved the purchase of 200 shares in Checkpoint Systems, Inc. in April 1997, the purchase of 100 shares in Karrington Health, Inc. in September 1997, the sale and repurchase of those same stocks in December 1998, the sale of the Karrington stock in January 1999, and the purchase of 155 shares of Worthington Industries, Inc. in January 1999.
In September 1999, Stacy's purchases dramatically increased. Stacy purchased *585 180 shares of WF on September 7, 172 shares of WF on September 23, and 100, 500, and 1,500 shares of WF on September 29. The announcement of the Kellogg buyout occurred on October 1, 1999. From October 4-7, 1999, Stacy sold all the WF stock he had purchased in the previous month, making a substantial profit.
Hughes considered herself to be a knowledgeable investor based on her research and her attempt to educate herself on the equities markets in general. She handled the family finances and investments. Hughes was also a conservative investor prior to September 1999. From January 1997 through August 1999, Hughes bought and sold only four stocks: Checkpoint Systems  purchasing 100 shares in January 1997, 250 shares in April 1997, and selling 250 shares in December 1998; Karrington Health  purchasing 100 shares in August 1997 and selling 100 shares and then buying them back in December 1998 and reselling in January 1999; Wild Oats  purchasing 80 shares in July 1998; and Airnet Systems  making separate purchases of 100 shares and 185 shares in January 1999.
In activity that was inconsistent with her prior behavior in the stock market, in September 1999, Hughes began to sell shares of other stock in order to obtain WF Stock. On September 23, 1999, Hughes sold 136 shares of Airnet Systems and 80 shares of Wild Oats, and purchased 354 shares of WF. The next day she sold 285 shares of Airnet and 250 shares of Checkpoint Systems and purchased 382 shares of WF. On October 7, 1999, Hughes sold all of her WF stock, making a substantial profit.
In the couple's joint account, Hughes appeared to be moderately aggressive in her stock purchases.[6] From August through September 1998, the couple bought 320 shares of Airnet Systems, 100 shares of Huntington Bancshares, 200 shares of Worthington Industries and 200 shares of Banc Stk Group. In January and February 1999, they sold off their 247 shares of previously purchased Max & Ermas' stock, and purchased 100 shares of Consolidated Stores, 300 shares of Airnet Systems and 250 shares of WF. In July 1999, they purchased 10 more shares of Huntingon.
In September 1999, purchases in the joint account also drastically changed. On September 27, 1999, the couple purchased 6,250 shares of WF stock and an additional 100 shares two days later. By October 7, 1999, the couple had sold all of their WF stock for a substantial profit.
The profits received by Hughes and Stacy for the sale of the WF stock were as follows: $78,502.45 for the couple's joint account; $23,908.88 for Stacy's accounts; and $7,598.89 for Hughes's accounts. The couple's shared profits totaled $110,010.22. The Pension Plan, managed by Hughes, also profited by $57,000.
Between the two of them, Stacy and Hughes used seven different brokerage accounts to buy WF stock. Hughes had two personal accounts, one with Advest and one with Eisner. Stacy had three accounts, one with Advest, one with Eisner, and one with Charles Schwab. The two also had two joint accounts, one with Eisner and one with UBS. Additionally, the Charles Schwab account and the UBS account were both opened on September 29, 1999, and were used solely to purchase WF stock.
One of the Government's witnesses, Jody Stewart, testified about Hughes's and Stacy's checking account. Stewart stated that on September 23, 1999, there was a *586 deposit of $45,000.00 and on September 27, 1999, there was a deposit of $30,000.00. The later deposit was solely comprised of a check issued to Hughes by RBA that contained the notation "loan" and was signed by Blackwell. On September 27, 1999, Hughes made out a check to National Financial for $74,939.88. Two days later, she made out a check to Charles Schwab for $28,794.00.
Timothy Salzer, one of Hughes's and Stacy's brokers, also testified for the Government. Based on the couple's recent purchase of 6,000  plus shares of WF, Salzer requested new financial information on them because he did not believe it was a suitable investment for them due to their limited liquidity. The liquid net worth stated by Hughes and Stacy in 1998 was between $50,000 and $100,000. In September 1999, that number changed to between $100,000 and $500,000. When Salzer learned of the Kellogg buyout, he was surprised because of the size of the couple's recent purchases of WF stock and the proximity to the buyout. Because the couple virtually doubled their money in eight days, Salzer concluded that they either had great luck or had inside information.
VI. Procedural History
On August 26, 2004, a grand jury indicted Hughes, Stacy, Jack, Voss, and Black Jack Enterprises on the following charges: insider trading in violation of 15 U.S.C. §§ 78j and 78ff, conspiracy to commit insider trading in violation of 18 U.S.C. § 371, obstruction of agency proceedings in violation of 18 U.S.C. § 1505, conspiracy to obstruct agency proceedings in violation of 18 U.S.C. § 371, and making false statements in violation of 18 U.S.C. § 1001. Blackwell, Hughes and Stacy were convicted of most of the charges against them. Voss, Jack, and Black Jack Enterprises were acquitted on all counts.
Hughes was indicted on the following charges: Count 1  conspiracy to engage in insider trading; Counts 2 through 15  manipulative and deceptive devices (insider trading); Count 23  conspiracy to obstruct justice; Count 25  obstruction of an agency proceeding; Counts 26 and 27  false statements; and, Counts 28 through 31  obstruction of an agency proceeding. Hughes was acquitted on Counts 29 and 31. She was convicted of the remaining counts and was sentenced to thirty-three months' incarceration and three years of supervised release on Counts 1, 2-15, 23, 25-28, and 30, to be served concurrently. She was also ordered to pay a fine of $53,443.00 and special assessments totaling $1,300.00.
Stacy was indicted on the following charges: Count 1  conspiracy to engage in insider trading; Counts 10 through 15  manipulative and deceptive devices (insider trading); Count 23  conspiracy to obstruct justice; Counts 31, 32, and 34  obstruction of an agency proceeding; and Counts 33, 35, and 36  making false statements. Stacy was acquitted on Count 31. He was convicted of the remaining counts and was sentenced to twenty-seven months' incarceration and three years' supervised release on Counts 1, 10-15, 23, and 32-36, to be served concurrently. He was also ordered to pay a fine of $53,443.00 and special assessments totaling $1,300.00.

DISCUSSION
I. Variance
Both Appellants allege that a variance occurred because the evidence presented at trial demonstrated multiple conspiracies, whereas the indictment alleged single conspiracies. Appellants also assert that they suffered substantial prejudice as a result thereof. The Government disputes *587 this claim. The district court, in its denial of the motion for judgment of acquittal and/or a new trial, found that no variance had occurred and that the jury instructions properly prevented any prejudice due to spillover.
A. Standard of Review
We perform a de novo review when determining whether there has been a variance at trial from the indictment. United States v. Smith, 320 F.3d 647, 656 (6th Cir.), cert. denied, 538 U.S. 1023, 123 S.Ct. 1954, 155 L.Ed.2d 868 (2003). "A variance occurs when the charging terms [of the indictment] are unchanged, but the evidence at trial proves facts materially different from those alleged in the indictment." United States v. Chilingirian, 280 F.3d 704, 711 (6th Cir.2002) (quotations omitted). "[I]f an indictment alleges one conspiracy, but the evidence can reasonably be construed only as supporting a finding of multiple conspiracies, the resulting variance between the indictment and the proof is reversible error if the appellant can show that he was prejudiced thereby." United States v. Warner, 690 F.2d 545, 548 (6th Cir.1982) (citing Kotteakos v. United States, 328 U.S. 750, 66 S.Ct. 1239, 90 L.Ed. 1557 (1946)). Therefore, to obtain the reversal of a criminal conviction due to a variance, an appellant must demonstrate (1) that a variance occurred and (2) that it affected a substantial right. United States v. Manning, 142 F.3d 336, 339 (6th Cir.1998).
We first look to whether the evidence can "reasonably be construed only as supporting a finding of multiple conspiracies" rather than the single conspiracy alleged in the indictment. Warner, 690 F.2d at 548. "Whether single or multiple conspiracies have been shown is usually a question of fact to be resolved by the jury . . . and [is] to be considered on appeal in the light most favorable to the government." United States v. Schultz, 855 F.2d 1217, 1222 (6th Cir.1988) (internal quotation marks omitted). "The principal considerations in determining the number of conspiracies are the existence of a common goal, the nature of the scheme, and the overlapping of the participants in various dealings." Smith, 320 F.3d at 652.
To demonstrate substantial prejudice, the appellant must show that the variance prejudiced his ability to defend himself or prejudiced the overall fairness of the trial. Manning, 142 F.3d at 339. A reviewing court looks to whether there is a danger that the appellant was convicted based on evidence of a conspiracy in which the appellant did not participate (guilt transference). United States v. Blackwell, 459 F.3d 739, 762 (6th Cir.2006). However, "[t]he concept of variance is designed to prevent the prosecution from convicting the defendant of a different offense, not a lesser variation on the charged offense." United States v. Solorio, 337 F.3d 580, 590 (6th Cir.2003).
Even if an appellant can demonstrate that a variance resulted in guilt transference, typically any danger of prejudice can be cured with a cautionary instruction to the jury that if it finds multiple conspiracies, it cannot use evidence relating to one conspiracy in determining another conspiracy. Blackwell, 459 F.3d at 762. However, as the Blackwell court noted, "the more evidence presented at trial that is unrelated to the defendant's conduct, or a conspiracy in which the defendant took part, the less likely instructions are to cure the danger of guilt transference." Id. If there is a significant amount of evidence produced at trial that was unrelated to the appellant, we must look to the following factors to determine if the variance was prejudicial: (1) "the number of conspiracies the evidence establishes[;]" *588 (2) "the number of non-conspiratorial co-defendants tried with defendant[;]" and (3) "the size of the conspiracy alleged in the indictment[.]" Id.
B. Conspiracy to Engage in Insider Trading
Hughes and Stacy claim that the evidence at trial demonstrated somewhere between ten to twenty different conspiracies. This is based not only on the evidence presented about the other codefendants, but also the evidence presented regarding the unnamed conspirators. Because Voss, Jack, and Black Jack Enterprises were acquitted of the crimes charged in the indictment, they will not be included in the discussion of whether a single conspiracy existed. The question becomes whether there was a single conspiracy among the remaining alleged conspirators: Blackwell, Stephans-Blackwell (unindicted conspirator), the Stephanses (unindicted conspirators), Hughes and Stacy, Kahl (unindicted conspirator), and Dale and Christian Blackwell (unindicted conspirators).
The Government's theory at trial and on appeal is that the goal of the conspiracy was to enrich Blackwell's inner circle of friends and family. In support of that theory, it has attempted to demonstrate that all co-defendants and unindicted conspirators knew or should have known that they were involved in a single conspiracy because they were all close friends and had socialized together often. However, "mere association with conspirators is not enough to establish participation in a conspiracy." United States v. Pearce, 912 F.2d 159, 162 (6th Cir.1990). Moreover, it is not sufficient simply to demonstrate that the alleged conspirators knew each other and committed the same or associated crimes. The essence of a conspiracy is the agreement to commit the offense and not the commission of the substantive offense. United States v. Fife, 573 F.2d 369, 373 (6th Cir.1976); United States v. Sutton, 642 F.2d 1001, 1020 (6th Cir.1980).
The Government did not present evidence that Dale Blackwell, the Stephanses, or Voss socialized with Appellants. Although the Government did present evidence of social connections among other conspirators, it did not demonstrate that those individuals discussed the stock market or ways to increase their own wealth. It also failed to present evidence, either direct or circumstantial, that there was any sort of agreement amongst all those named in Count 1 of the indictment.
Despite viewing the evidence in the light most favorable to the Government and making all reasonable inferences in its favor, we find that there was no basis for a reasonable juror to conclude that Appellants knew or should have known that Blackwell had provided inside information to anyone else. To the contrary, it was more likely that they believed that Blackwell would not disseminate the information widely. The evidence established that Blackwell knew WF was a thinly traded stock, and that Hughes considered herself to be knowledgeable in the stock market. Blackwell and Hughes would both have known that the purpose of the charged scheme would have been defeated if many people were involved. The marked increase in WF stock purchases caused by widespread participation would drive up the market price of the stock before all of the intended beneficiaries of the information could buy the volume they hoped to buy, thereby diminishing the benefit to Blackwell's friends and family.
Based upon the above, we disagree with the district court and find that the Government presented no evidence of a common goal amongst the persons listed as conspirators *589 in the indictment. Furthermore, although there was a single conspiracy alleged in Count 1 of the indictment, we find that the evidence presented at trial demonstrated multiple unrelated conspiracies, resulting in a variance.
However, even though a variance occurred, we find that Appellants did not suffer substantial prejudice so as to justify a new trial. The Supreme Court concluded in Berger v. United States, that "the fact that the proof disclosed two conspiracies instead of one, each within the words of the indictment, cannot prejudice [a defendant's] defense of former acquittal of the one or former conviction of the other, if he should again be prosecuted." 295 U.S. 78, 55 S.Ct. 629, 79 L.Ed. 1314 (1935), overruled on other grounds by Stirone v. United States, 361 U.S. 212, 80 S.Ct. 270, 4 L.Ed.2d 252 (1960). The Supreme Court further held that a variance should never be regarded as material "where the allegation and proof substantially correspond, or where the variance was not of a character which could have misled the defendant at trial" and where it did not "deprive the accused of his right to be protected against another prosecution for the same offense." Id. at 83, 55 S.Ct. 629.
In this case, the conspiracy involving Stacy and Hughes was contained within the indictment. Those claims substantially corresponded to the evidence presented at trial. Therefore, neither Stacy nor Hughes would have been misled by the indictment. The indictment also put them on notice about what evidence would be produced against them at trial. Finally, the variance between what was charged in the indictment and the evidence produced at trial does not deprive either Appellant of his or her right to be protected against another prosecution for the same offense in the future.
Appellants argue that we should apply the holding in Kotteakos, supra, and find that the variance was fatal. In Kotteakos, an overarching conspiracy was alleged in the indictment but the proof at trial demonstrated multiple conspiracies. 328 U.S. at 755, 66 S.Ct. 1239. The key defendant had experience obtaining loans under the National Housing Act. Id. at 753, 66 S.Ct. 1239. He used that knowledge to assist other co-defendants in obtaining loans under the Act for various fraudulent reasons. Id. at 753-54, 66 S.Ct. 1239. Each loan was unique unto itself. Id. Additionally, none of the loans went through at the same time and each was obtained for a different purpose. Id. The Supreme Court found that an impermissible variance occurred between the indictment and the evidence at trial, and that the variance resulted in substantial prejudice to the defendants. Id. at 775-76, 66 S.Ct. 1239.
This case, however, is factually distinguishable from Kotteakos because each conspiracy involved the transmission of inside information. That information was then used to purchase WF stock at a low price. The purchasers made substantial profits when the stock was sold after the announcement of the Kellogg buyout. The facts in this case are more similar to those in Berger.
In Berger, the evidence at trial demonstrated two different conspiracies to pass counterfeit money. 295 U.S. at 80, 55 S.Ct. 629. However, the facts of each of the alleged conspiracies were contained in the indictment. Id. The Supreme Court found that although there was a variance between what was charged and the evidence presented, it was not material and did not affect the defendants' substantive rights. Id. at 83-84, 55 S.Ct. 629. In the case at bar, there were three separate conspiracies to engage in insider trading. Although the Government did not prove *590 that there was one common goal amongst the conspirators, it certainly would not have shocked Appellants at trial to be confronted by evidence involving the conspiracy among Blackwell, Hughes and Stacy. Therefore, the variance was immaterial.
Additionally, it is noted that this Court held in Blackwell that the jury instructions given by the district court properly protected the defendants from any guilt transference. 459 F.3d at 763. Hughes argues that this holding should not be binding because the bulk of the evidence presented at trial was unrelated to her. However, upon review of the transcript, we find this argument to be without merit. The question we must consider is not whether there was substantial evidence unrelated to Hughes specifically, but whether there was substantial evidence unrelated to the conspiracy in which she allegedly participated in. Therefore, evidence concerning Blackwell's actions, which comprises approximately one-third of the transcript, is certainly relevant as he is a conspirator in the Blackwell, Hughes, and Stacy conspiracy. In addition, Hughes significantly downplays the amount of evidence and testimony presented against her at trial. Nearly every witness in the Government's case-in-chief testified about Hughes. There were at least fifty exhibits presented by the Government that involved Hughes and/or her husband Stacy. This fact alone weighs in favor of finding, as the Court did in Blackwell, that the jury instructions properly protected the Appellants from guilt transference.
Even if we were to find a majority of the evidence was unrelated to Hughes, the factors to consider to determine if prejudice existed do not weigh in her favor. These include: (1) the number of conspiracies the evidence demonstrated; (2) the number of non-conspirator co-defendants tried along with the defendant; and (3) the size of the conspiracy alleged in the indictment. Blackwell, 459 F.3d at 762. Although Appellants argue that the evidence demonstrated between ten and twenty different conspiracies, clearly the jury disagreed, acquitting Voss, Jack, and Black Jack. If we were to consider the three separate conspiracies discussed previously, it is unlikely that the jury was confused and thus transferred guilt from one defendant to the other when such a small number of alleged conspiracies was involved.
The second factor requires consideration of the number of co-defendants tried along with Appellants. In this case, six people were tried together, three of whom were part of the Blackwell, Hughes and Stacy conspiracy. This is not a case where the sheer number of co-defendants resulted in a likelihood of juror confusion or guilt transference.
Finally, we look to the size of the conspiracy alleged in the indictment. This factor appears to be the only one that weighs in Appellants' favor. There are eleven alleged conspirators, six named in the indictment along with five unindicted conspirators. Because there was a significant amount of testimony and evidence presented against Hughes and Stacy at trial and because a majority of the above factors do not demonstrate that there was a likelihood of guilt transference, we cannot find that adequate jury instructions would not have prevented prejudice against Appellants due to a variance.
As previously stated, this Court determined in Blackwell that the jury instructions given by the district court adequately protected the defendants from substantial prejudice due to guilt transference. 459 F.3d at 762-63. This holding was based on the finding that "[t]he trial court instructed the jury that it could not consider evidence of one defendant's actions, . . . *591 against another defendant, . . . unless the jury first found the defendants to be co-conspirators[.] . . . We presume the jury followed these instructions." Id. at 763.
Because the Blackwell court decided the exact issue regarding the jury instructions, and its conclusion was not based on a determination of the weight of evidence against any one of the defendants, the holding becomes the "law of the case" and "may not be revisited unless `substantially new evidence has been introduced, . . . there has been an intervening change of law, or . . . the first decision was clearly erroneous and enforcement of its command would work substantial injustice.'" United States v. Corrado, 227 F.3d 528, 533 (6th Cir.2000) (quoting Miles v. Kohli & Kaliher Assocs., Ltd., 917 F.2d 235, 241 n. 7 (6th Cir.1990)); see also United States v. Wilson, No. 91-1510, 1992 WL 179240, at *8 (6th Cir. July 28, 1992) (ruling that a panel's holdings in the appeals of three co-defendants became the law of the case as to the identical issues raised in the subsequent appeal of a fourth co-defendant); Heathcoat v. Potts, 905 F.2d 367, 370 (11th Cir.1990) ("[F]indings of fact and conclusions of law by an appellate court are generally binding in all subsequent proceedings in the same case in the trial court or on a later appeal." (citation omitted)). Appellants have failed to produce new evidence, argue an intervening change of law, or demonstrate that the previous decision was clearly erroneous and enforcement of its holding would result in substantial injustice. Therefore, we find that Appellants did not suffer substantial prejudice because the jury instructions adequately protected them from guilt transference.
C. Conspiracy to Obstruct Justice
Unlike the conspiracy charge discussed above, in viewing the evidence in a light most favorable to the Government, it would be reasonable for a juror to infer that all of the named conspirators in Count 23 had the agreed common goal of concealing the fact that Blackwell had provided them with inside information. There is also evidence that each conspirator knew or had reason to know of the participation of the others in the conspiracy and that it was necessary for each to provide false and misleading testimony before the SEC to conceal his or her actions.
During December 1999, as the NASD launched its investigation, Blackwell received a list of suspicious purchasers from NASD requesting that he identify the persons he knew and their relationship to him. On that list were Jack, Kahl, Hughes, Hiser, Juanita Hiser, Stacy, Voss, the Stephanses, and Dale and Christian Blackwell. Blackwell's phone records show that within a three-day period that month, he used his telephone calling card from New York to make calls to Dale Blackwell, the Stephanses, Hughes and Stacy, Jack, and Voss.
Kahl testified that he received a letter from Blackwell providing a blanket denial of the allegations that he provided insider information. He understood that the letter was Blackwell's position in the investigation and not the truth.
Blackwell kept track of who was being subpoenaed. He was aware that Voss had given testimony before the SEC. When Hiser received her subpoena from the SEC, she immediately contacted Stacy. Stacy informed her about both his and Hughes's subpoenas. Stacy also informed Jack about the subpoena.
On November 8, 2000, the contact information contained in the RBA computer for Voss, Stephanses, Kahl, and Stephans-Blackwell's cousin, Sigi MacFetridge, was deleted. Stephans-Blackwell testified that she and Blackwell had an agreement that both she and her parents would testify *592 before the SEC that they did not receive confidential information from him regarding the Kellogg buyout. The evidence also demonstrated that all persons who testified before the SEC essentially made the same statement, which was that they did not receive inside information from Blackwell and that their stock purchases were not based on any inside information. It was reasonable for a juror to conclude that there was an agreement amongst all the alleged conspirators to hide the fact that they had received inside information from Blackwell. Viewing the evidence in a light most favorable to the Government, we find that the evidence at trial demonstrated a single conspiracy to obstruct justice, just as was alleged in the indictment. Thus, no variance occurred. It is, therefore, unnecessary to discuss whether Appellants suffered substantial prejudice.
II. Sufficiency of the Evidence and Manifest Weight
Stacy argues that the evidence at trial was insufficient to convict him and that his convictions were therefore against the manifest weight of the evidence for Count 1  conspiracy to insider trade, Count 23  conspiracy to obstruct justice, and Counts 33, 35 and 36  making false statements. Although Stacy's seventh stated issue raises a general claim of insufficiency against his convictions on all counts, he makes only argument on the specific counts listed above. Therefore, any other sufficiency challenges are waived. See United States v. Layne, 192 F.3d 556, 566 (6th Cir.1999).
The district court, in its denial of the motion for judgment of acquittal pursuant to Rule 29 and motion for a new trial pursuant to Rule 33 of the Federal Rules of Criminal Procedure, found that sufficient evidence was presented for a rational trier of fact to find that all the elements of the offenses were proved beyond a reasonable doubt and that the evidence did not preponderate against the verdicts of guilt arrived at by the jury. Therefore, the court found that the convictions were not against the manifest weight of the evidence.
A. Standard of Review
When considering a challenge to the sufficiency of evidence to sustain a conviction on direct appeal, the relevant question is whether "any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt." Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979). The appellate court must view all evidence and resolve all reasonable inferences in favor of the government. Id; see also United States v. Searan, 259 F.3d 434, 441 (6th Cir.2001). Because the issue is one of legal sufficiency, the court neither "independently weighs the evidence, nor judges the credibility of witnesses who testified at trial." United States v. Talley, 164 F.3d 989, 996 (6th Cir.1999). An appellate court cannot substitute its judgment for that of the jury. United States v. Hilliard, 11 F.3d 618, 620 (6th Cir.1993). Finally, it must be remembered that "circumstantial evidence alone can sustain a guilty verdict and . . . [such] evidence need not remove every reasonable hypothesis except that of guilt." United States v. Stone, 748 F.2d 361, 362 (6th Cir.1984). This standard is a great obstacle to overcome, United States v. Winkle, 477 F.3d 407, 413 (6th Cir.2007), and presents the appellant in a criminal case with a very heavy burden. United States v. Jackson, 473 F.3d 660, 669 (6th Cir.2007).
A motion for a new trial under Rule 33 of the Federal Rules of Criminal Procedure may be premised upon the argument that the jury's verdict was against the manifest weight of the evidence. Generally, *593 such motions are granted only "in the extraordinary circumstance where the evidence preponderates heavily against the verdict." United States v. Turner, 490 F.Supp. 583, 593 (E.D.Mich.1979), aff'd, 633 F.2d 219 (6th Cir.1980). A district judge, in considering the weight of the evidence for purposes of adjudicating a motion for new trial, may act as a thirteenth juror, assessing the credibility of witnesses and the weight of the evidence. United States v. Lutz, 154 F.3d 581, 589 (6th Cir.1998). Our review of the trial court's ruling is limited to determining whether it was a clear and manifest abuse of discretion. United States v. Hernandez, 227 F.3d 686, 695 (6th Cir.2000).
B. Conspiracy Counts
Under 18 U.S.C. § 371, it is illegal for "two or more persons [to] conspire either to commit any offense against the United States, or to defraud the United States, or any agency thereof in any manner or for any purpose." To prove conspiracy, the government must establish: (1) the existence of an agreement to violate the law; (2) knowledge and intent to join the conspiracy; and (3) an overt act constituting actual participation in the conspiracy. See, e.g., United States v. Martinez, 430 F.3d 317, 330 (6th Cir.2005); United States v. Jamieson, 427 F.3d 394, 402 (6th Cir.2005). "A tacit or material understanding among the parties to a conspiracy is sufficient to establish the agreement [, and] conspiracy may be inferred from circumstantial evidence which may reasonably be interpreted as participation in a common plan." United States v. Walls, 293 F.3d 959, 967 (6th Cir.2002) (internal citations omitted).
1. Insider Trading
We find that in its denial of the motion for judgment of acquittal/motion for a new trial, the district court provided an appropriate listing of the elements necessary for the Government to prove beyond a reasonable doubt the offense of insider trading:
(1) that the defendant, in connection with the purchase or sale of [WF] stock, employed manipulative and deceptive devices and contrivances by employing a device, scheme or artifice to defraud, or engaging in acts, practices or courses of business which operated or would operate as a fraud and deceit upon other persons;
(2) that the defendant acted knowingly and wilfully and with the intent to defraud; and (3) that the defendant knowingly used or caused to be used means or instrumentalities of interstate commerce or the facilities of a national securities exchange in connection with the purchase or sale of securities in furtherance of the fraudulent conduct.
J.A. at 1194. Additionally the Government was also required to prove that Hughes and Stacy, as tippees: (1) received material, confidential, and non-public information from Blackwell knowing that he was the source; (2) knew or should have known that Blackwell breached his fiduciary duty; and (3) knowingly and wilfully purchased or caused to be purchased shares of WF stock based on that information. Dirks v. SEC, 463 U.S. 646, 660, 103 S.Ct. 3255, 77 L.Ed.2d 911 (1983).
Stacy argues that the Government presented nothing but speculation and conjecture in support of its theory that Blackwell provided inside information to Hughes and/or himself. He points out that the Government presented nothing but circumstantial evidence and he, along with Blackwell and Hughes, took the witness stand and denied engaging in insider trading. Stacy also points to the reasonable explanations testified to at trial for both his and *594 his wife's purchases of WF stock. However, "[t]he government may meet its burden through circumstantial evidence alone, and such evidence need not exclude every possible hypothesis except that of guilt." United States v. Jackson, 55 F.3d 1219, 1225 (6th Cir.1995) (citing United States v. Peters, 15 F.3d 540, 544 (6th Cir.), cert. denied, 513 U.S. 883, 115 S.Ct. 219, 130 L.Ed.2d 146 (1994)). Additionally, it is well settled that when a defendant "offer[s] an innocent explanation for the incriminating facts proved by the government, the jury is free to disbelieve [it]." United States v. Schreane, 331 F.3d 548, 562 (6th Cir.2003).
At trial, the Government produced evidence that Blackwell found out about the proposed buyout in late July or early August 1999. This information was not public. It was only through Blackwell's position on the Board at WF that he gained this knowledge. On August 31, 1999, Hughes had a meeting with Blackwell at RBA. Hughes had a close relationship with Blackwell and had worked for him for a number of years. Hughes was a knowledgeable investor. Additionally, Stacy testified that it would be impossible for Hughes to have inside information and not tell him.
Beginning on September 7, 1999, and continuing until September 29, 1999, just days before the buyout, Stacy and Hughes, previously cautious investors, bought thousands of shares of WF. Additionally, the couple received a check from Blackwell for $30,000 which they promptly used to purchase WF stock. By October 7, 1999, Hughes and Stacy had sold all of their WF stock, obtaining $110,010.22 in profits. Based on these facts, we find that there was sufficient circumstantial evidence to demonstrate that Blackwell obtained non-public information about the buyout and conveyed that information to Hughes, who in turn shared that information with her husband, Stacy, and that the couple purchased WF stock based on that information. As a knowledgeable investor, Hughes would have known that Blackwell was breaching his duty to the company, its shareholders, and the investing public by sharing this information. The jury could also infer that Appellants' close relationship would mean that Stacy was also aware, through Hughes, that Blackwell's conduct was improper.
Stacy again asserts that each of the persons alleged to have conspired with Blackwell did not have a connection with the other alleged conspirators and thus had no common goal. This issue was discussed above and will not be discussed further.
Based on the above, we find that, viewing the evidence in a light most favorable to the Government, a rational trier of fact could have found beyond a reasonable doubt that the Government had proven the essential elements of conspiracy to engage in insider trading with respect to Stacy. Additionally, because the amount of evidence presented was substantial, the denial of Stacy's Rule 33 motion was not a clear and manifest abuse of discretion.
2. Obstruction of Justice
Stacy argues that the evidence presented against him at trial on this charge was insufficient to sustain a conviction and, thus, that his conviction was against the manifest weight of the evidence. He points out that there was no evidence that he entered into an agreement with Stephans-Blackwell or Blackwell to delete names from RBA's database, an act Stephans-Blackwell admitted to on the witness stand. He further argues that he did not trade on inside information, thus making his statements to the SEC accurate. Finally, Stacy claims that even *595 if it was found that he did make false statements, the Government presented no evidence that he conspired with anyone, including his wife, to obstruct justice.
As stated above, a conspiracy can be shown through circumstantial evidence, that "may reasonably be interpreted as participation in a common plan." Walls, 293 F.3d at 967. Additionally, even though there was no evidence that Stacy was involved in the deletion of names from the computer, that act was not the only overt act listed in the indictment. The indictment alleges that Stacy failed to provide copies of his brokerage account statements for the month of September 1999, when all the significant purchases of WF were made, and that he gave false statements to the SEC during his testimony, in an attempt to obstruct or to impede the SEC proceeding.
During trial, the Government produced documents detailing Appellants' stock transactions. Those documents demonstrated that most of the WF shares were purchased by the couple during the month of September. A reasonable jury could infer, based on the close relationship between Hughes and Stacy, that the two agreed not to produce the September 1999 statements for their November 2000 testimony before the SEC. The jury could infer that they did so in an attempt to conceal the number of shares they had purchased during that month, thereby concealing the fact that they were trading based on inside information obtained from Hughes's boss, Blackwell, who also sat on the Board of Directors for WF. Additionally, because the jury found that Stacy had committed insider trading, it was necessary for them to determine that his statements to the SEC were false.
Based on the above, we find that, viewing the evidence in a light most favorable to the Government, a rational trier of fact could have found beyond a reasonable doubt, that the Government had proven the essential elements of conspiracy to obstruct justice, and that Stacy was, therefore, guilty of the crime charged. Additionally, because the amount of evidence presented was substantial, we do not find that the district judge's decision to deny the Rule 33 motion was a clear and manifest abuse of discretion.
C. Making False Statements
Under 18 U.S.C. § 1001, it is unlawful for any person to (1) "knowingly and willfully" (2) make "any materially false, fictitious, or fraudulent statement or representation" (3) "in any matter within the jurisdiction of the executive, legislative, or judicial branch of the Government of the United States." On November 21, 2000, Stacy testified before the SEC and provided the following responses:
Q. Did you talk with anyone regarding your purchase of Worthington Foods in the Advest account [on September 1, 1999]?
A. No
. . . .
Q. Did you know about any possible business activities concerning Worthington Foods prior to the announcement on  in October of the deal with Kellogg?
A. No
. . . .
Q. Any business deals involving any business combination, or any business deals involving Worthington Foods prior to the news announcement in October[?]
A. No
. . . .
Q. So prior to the announcement in October of 1999 did you have any reason to believe that Worthington Foods was considering any business combinations, mergers, or acquisitions?

*596 A. No.
Q. Did you have any idea prior to the announcement in October of 1999 that there was any significant business developments at Worthington Foods?
A. No
. . . .
Q. Did Kelley [sic] [Hughes] ever tell you about anything she had heard at work regarding Worthington [Foods] in August of 1999?
A. No.
Q. Did Kelley [sic] [Hughes] ever tell you about anything she had heard at work regarding Worthington [Foods] in September of 1999?
A. No.
Stacy claims that the jury incorrectly found that he had made false statements to the SEC. He additionally contends that the verdicts on these counts were against the manifest weight of the evidence because the statements were only false if he actually did trade on inside information. The only proposition offered to support this argument is his insistence that he was "telling the truth." It is assumed that Stacy is referring to both his and Hughes's testimony offering allegedly reasonable explanations for their stock purchases. However, the jury was free to disbelieve Appellants' explanation. Schreane, 331 F.3d at 562.
As previously stated, there was sufficient evidence to support the jury's finding that Stacy conspired to and did commit insider trading. In finding that Stacy committed insider trading, the jury necessarily found his statements before the SEC  that he did not commit insider trading  to be false. Therefore, we conclude that the Government presented sufficient evidence to allow a reasonable juror to convict Stacy of making false statements as charged in Counts 33, 35, and 36.
III. Material Outside the Record
Stacy claims that he is entitled to a new trial based on a juror's post-trial comments in a COLUMBUS MONTHLY article that he (the juror) and two other jurors saw Blackwell mouth the words "I hate you" to Stephans-Blackwell while she was on the stand. When this occurred at trial, the district court allowed counsel to cross-examine Blackwell about the incident. Blackwell denied the alleged conduct. The three jurors conveyed their observations to the other jurors. The juror stated in the article that when Blackwell denied mouthing the words, "his fate was virtually sealed from my perspective." Dennis Read, A Blackwell juror talks, COLUMBUS MONTHLY, Dec. 2005, at 10.
Stacy argues that because the jurors questioned Blackwell's credibility based on this incident, "it follows that the jury would also question Mr. Stacy's credibility." He alleges that his Fifth Amendment right to a verdict based solely on the evidence produced at trial was violated, but he cites no law in support of what appears to be a due process argument.
In Sheppard v. Maxwell, the Supreme Court stated that a jury's verdict must be based on evidence presented in open court and not on evidence from outside sources. 384 U.S. 333, 350-51, 86 S.Ct. 1507, 16 L.Ed.2d 600 (1966). However, in this instance, the COLUMBUS MONTHLY article is not alleged to have influenced the jury. It was Blackwell's own actions during the trial and his subsequent denial of those actions under oath that convinced one juror that Blackwell was not credible. Therefore, we cannot find that the verdict was based on evidence outside of the trial.
Additionally, Stacy fails to explain how this alleged "spillover" effect applied only to him and did not affect the credibility of the other co-defendants who were acquitted *597 of the same charges. As discussed above, there was sufficient evidence presented at trial that discredited Stacy's testimony that he did not receive inside information from Blackwell. Therefore, because there is no evidence that the jurors relied on information outside the record, Stacy is not entitled to a new trial on this basis.
IV. Sixth Amendment Right to Present an Adequate Defense
Stacy alleges that he was deprived of his Sixth Amendment right to present an adequate defense because the trial court prevented him and his co-defendants from presenting expert testimony on a "leakage" theory offered by Dr. Gregg Jarrell. This Court, in Blackwell, supra, discussed this very issue and determined that the district court was correct in excluding portions of Professor Jerrell's testimony regarding the "leakage" theory as irrelevant under Fed.R.Evid. 401 because the testimony did not make that theory "any more likely than tipping, and thus did not make Defendant's innocence more probable than it would be without the evidence." 459 F.3d at 753 (internal quotations omitted).
Because this Court previously addressed the exact issue, it is the "law of the case" and "may not be revisited" for the same reasons as previously discussed. Corrado, 227 F.3d at 533. Stacy failed to produce new evidence, argue an intervening change of law or demonstrate that the previous decision was clearly erroneous and enforcement of its holding would result in substantial injustice. Therefore, Stacy's fifth assignment of error is without merit.
V. Jury Instructions
Stacy alleges that the district court erroneously declined to instruct the jury that it could not convict him on circumstantial evidence alone if it determined that the circumstantial evidence supported either of two reasonable probabilities. However, upon review of the record, we find that Stacy failed to present any proposed jury instructions. Additionally, Stacy does not point to evidence in the record demonstrating that he made an objection to the Court's instructions. Failure to object to jury instructions at trial results in a forfeiture of one's right to raise the alleged error on appeal. We thus review the jury instructions for plain error. See Reynolds v. Green, 184 F.3d 589, 594 (6th Cir.1999). "Plain error is an obvious and prejudicial error that requires action by the reviewing court in the interests of justice." Id. (quotations omitted).
Stacy has failed to demonstrate that plain error has occurred. Additionally, the "two inferences" instruction Stacy refers to, which was proposed by Blackwell, was addressed by this Court in Blackwell, supra. The Blackwell court found the instruction would not have added any value to the jury charge and was itself erroneous and improper. 459 F.3d at 766. Because this exact issue was discussed by the Blackwell court, its findings become the "law of the case" and are therefore binding. Corrado, 227 F.3d at 533. Stacy's argument is without merit.
VI. Cumulative Effect of Errors
Stacy's final argument is that even if each of the alleged errors discussed above does not individually constitute reversible error, the cumulative effect of such errors rendered his trial fundamentally unfair in violation of due process. "Errors that might not be so prejudicial as to amount to a deprivation of due process when considered alone, may cumulatively produce a trial setting that is fundamentally unfair." Walker v. Engle, 703 F.2d 959, 963 (6th Cir.1983); see also United States *598 v. Hines, 398 F.3d 713, 719 (6th Cir.2005). As previously discussed, however, except for his argument that a variance occurred, all of Stacy's arguments lack merit. The sole finding of variance, without a correlative finding of substantial prejudice, is simply insufficient to establish a deprivation of Stacy's due process rights or to show that he did not receive a fair trial.

CONCLUSION
For the aforementioned reasons, we AFFIRM the judgment of the district court.
NOTES
[*]  The Honorable John R. Adams, United States District Judge for the Northern District of Ohio, sitting by designation.
[1]  All general facts not specific to Appellants Hughes and Stacy are derived from this Court's opinion in United States v. Blackwell, 459 F.3d 739 (6th Cir.2006).
[2]  Blackwell and Stephans-Blackwell divorced prior to the trial and Stephans-Blackwell was the main witness against Blackwell at trial.
[3]  In 1999, RBA had four employees  Blackwell, Stephans-Blackwell, Hughes, and Mary Hiser.
[4]  Blackwell was referring to the SEC investigation.
[5]  A stock is considered "thinly traded" when "transactions occur only infrequently and there are a limited number of interested buyers and sellers. Prices of thinly traded securities tend to be more volatile than those traded more actively because just a few trades can affect the market price substantially. It can also be difficult to sell shares of thinly traded securities, especially in a downturn, if there is no ready buyer. Shares of small and micro-cap companies are more likely to be thinly traded than those of mid- or large-cap stocks." http://www.thestreet.com/financial-dictionary/thinly-traded.html
[6]  As previously stated, Hughes handled all of the family investments.
