PER CURIAM:

Michelle Valencia and Greg Singleton appeal wire fraud convictions arising from alleged efforts to manipulate natural gas markets. Each defendant raises myriad issues on appeal, which we have considered carefully, along with an exhaustive review of the trial court's record. Confident that each defendant received a fair trial and that the convictions rest on solid evidence, we affirm.

I.

We start with a factual overview of this case before delving into the particulars of the issues on appeal. We first describe the nature of defendants' job duties and give a sketch of the industry's relevant practices. We then describe the course of proceedings brought against the defendants and the salient details of their four-week trial, which took place in July and August of 2006. After documenting the facts and procedure, we consider the issues raised on appeal.

A.

The acts relevant to this appeal occurred in 2000 and 2001. During that time, Valencia was employed by Dynegy Marketing and Trade ("Dynegy") in Houston, Texas, as a natural gas trader on Dynegy's "West Desk." Singleton was employed by El Paso Corporation ("El Paso") as a natural gas trader for El Paso's Merchant Energy segment in Houston. Natural gas is transported to consumers throughout North America via a network of pipelines. Natural gas produced in one region is interchangeable with gas produced elsewhere; the significant difference among regions is the cost of transport. Contracts for future delivery are traded on the New York Mercantile Exchange, or NYMEX. The most basic type of natural gas trade is a "physical" trade. A physical trade calls for delivery of a set volume of gas to the buyer at a particular delivery location. A "baseload" trade is a kind of physical trade. It calls for delivery of natural gas each day for an entire month. Most baseload trades are negotiated during a period at the end of the preceding month called "bidweek." The most common unit of volume is one million British thermal units ("MMBtu"). Traders often buy or sell tens of thousands of MMBtu in a given transaction.

The price of a trade can be set if traders agree upon a dollar amount at the time of trade; this is called a "fixed" price. However, traders can also opt to use prices which will be set in the future, called "index prices." Commonly, traders use a price published either daily or monthly in a privately owned newsletter. These index prices also affect other natural gas transactions, such as swaps, where two traders agree to buy the same volume of gas from each other at the same time, but at different prices. In essence, swaps are financial transactions in which traders bet on, or hedge against, changes to an index price. Index prices also affect long-term supply contracts tied to index prices, options contracts, royalty payments, "tariffs" charged by pipelines, and futures contracts.

The index prices published in two newsletters are relevant to this case: Inside FERC Gas Market Report (" Inside FERC") and Natural Gas Intelligence (" NGI"). Each publication is privately owned and is not affiliated with any state or federal governmental entity. Inside FERC and NGI independently determine and publish index prices at the beginning of the month for natural gas delivered at dozens of different "hubs" across the country. The publications gather monthly price data through surveys of natural gas traders. Inside FERC provides a Microsoft-Excel form for making reports, and instructs traders: "Only report FIXED-PRICE, BASELOAD DEALS negotiated during bidweek." Traders must indicate the delivery points, prices, volumes, and dates of each trade. At the time of the acts alleged in this case, the publications requested, but did not require, identification of the other contracting party, or "counterparty." After receiving bidweek trades from market participants, each publication publishes indices which purport to represent the price of natural gas at delivery points across the country. It was the policy of both Dynegy and El Paso to require its natural gas traders to submit such information each month.

Both Valencia and Singleton bought and sold natural gas in order to fulfill long-term contracts, to utilize capacity, and ultimately, to bring profits to his respective employer. Each defendant was authorized to execute trades for physical delivery of gas throughout much of the western United States, as well as financially oriented trades based upon the same trading nodes. In addition to trading, Valencia and Singleton (along with other natural gas traders at their respective companies) were required to gather and submit bidweek trade information to Inside FERC and NGI. The government alleged that defendants submitted, or caused to be submitted, reports with false information to the publications in a scheme to manipulate the price of natural gas. Each defendant allegedly sought to raise the index price if the trader or his company had a net long position, i.e., had excess gas to sell, or lower the index price if he had a net short position, i.e., needed to purchase additional gas to meet contractual obligations. Valencia's and Singleton's alleged misrepresentations included reporting trades which never occurred, misstating the price or volume of real trades, and omitting real trades. By swaying gas indices one way or another at certain locations, Valencia and Singleton could allegedly boost their monthly performance and increase profits for their respective companies. Better performance would redound to the benefit of the trader in the form of promotions or higher year-end bonuses.

B.

Michelle Valencia was indicted on January 22, 2003. She was initially charged with three counts of false reporting under the Commodities Exchange Act ("CEA"), in violation of 7 U.S.C. § 13(a)(2), and four counts of wire fraud, in violation of 18 U.S.C. §§ 2 and 1343. Upon Valencia's pre-trial motion, the district court dismissed certain portions of the indictment charging Valencia with delivering or causing to be delivered false or misleading reports under the CEA. See United States v. Valencia, 2003 WL 23174749, at *19-21 (S.D.Tex. Aug. 25, 2003), vacated and modified upon reconsideration, 2003 WL 23675402, at *4-5 (S.D.Tex. Nov. 13, 2003). The court ultimately reasoned that the false reporting provision of the CEA was unconstitutionally overbroad because it did not contain a sufficient mens rea requirement. See 2003 WL 23675402, at *4-5. In an interlocutory appeal, a panel of this Court held that the statute could be construed so as to avoid constitutional infirmity, and reversed. United States v. Valencia, 394 F.3d 352, 355 (5th Cir. 2004). The Supreme Court denied Valencia's petition for writ of certiorari. Valencia v. United States, 544 U.S. 1034, 125 S.Ct. 2286, 161 L.Ed.2d 1062 (2005).

A second superseding indictment was filed as to Valencia on March 8, 2006. It alleged that she conspired with Singleton from July to September of the year 2000 to violate the CEA. Valencia was also alleged to have emailed over twenty reports to Inside FERC and NGI between December 30, 1999, and January 31, 2002, and also caused other employees to email such reports. The reports "affected and tended to affect index prices, and thereby, the price of natural gas." False information in the reports would affect the indices, increasing the profitability of Valencia's trades. Valencia was charged with one count of conspiracy to violate the false reporting provision of the CEA, thirteen counts of false reporting under the CEA, and nine counts of wire fraud.

Singleton was indicted on November 17, 2004. Like Valencia, Singleton was accused of having sent reports with false information to Inside FERC and NGI about trade volumes and prices between July and September of 2000. In a superseding indictment filed March 8, 2006, Singleton was charged with one count of conspiracy to violate the false reporting provision of the CEA, five counts of false reporting under the CEA, three counts of wire fraud, and one count of obstruction of justice for impeding an investigation concerning the false trades. In light of the nexus of common acts alleged, the defendants and government agreed to a joint trial. After numerous continuances, trial was set for July 2006.

Pretrial discovery involved production and analysis of vast amounts of data. Thousands of hours of telephone calls and other voice recordings, emails from scores of individuals, reams of paper records, and multiple electronic databases comprised the universe of potentially relevant material obtained from Dynegy, El Paso, the corporate parents of Inside FERC and NGI, governmental agencies, and other players or stakeholders in the natural gas markets. Before trial commenced, and during the trial itself, the parties often disagreed as to whether the government had turned over all relevant and discoverable information. In addition, the government consulted with and retained witnesses to analyze the potential effects of the acts alleged in the indictments on Dynegy's and El Paso's respective profitability.

Prior to trial, defendants moved to limit or exclude the testimony of two government witnesses. Defendants contended that the witnesses were offered to present expert testimony, but did not meet the strictures of admissibility. The first witness, Glenn Labhart, challenged by Valencia only, was the chief risk officer at Dynegy during the time period of the acts alleged in the indictment. The government retained Labhart to analyze Dynegy's monthly positions at the time of the acts alleged. Labhart was tasked with determining whether trade reports submitted to Inside FERC and NGI contained true or false information, whether Dynegy's monthly positions at certain trading points were long or short, and whether changes in the indices published in Inside FERC and NGI would have affected Dynegy's profits. Defendants also moved to limit or exclude the testimony of Matthew O'Loughlin. The government retained O'Loughlin in order to determine whether Valencia's and Singleton's false reports could have, or did, affect the monthly indices published by Inside FERC and NGI. The district court initially reserved ruling on the admissibility challenges. After a hearing, the court held that Labhart was a corporate fact witness presenting analysis which was part of his job duties while he was employed at Dynegy. Consequently, the court allowed him to testify. The court also conducted a hearing to determine whether O'Loughlin's opinion testimony was admissible. The court concluded that his testimony met the admissibility requirements of the Federal Rules of Evidence.

C.

Trial was held from July 10 to August 4, 2006. Over the course of trial, the government presented twenty-one witnesses. The witnesses included natural gas traders from Dynegy and El Paso at the time of the acts alleged, corporate custodians who testified about the compensation paid to Valencia and Singleton, corporate representatives who presented company records from both Dynegy and El Paso, the editors of Inside FERC and NGI, and the government's expert, Matthew O'Loughlin. In addition to live testimony, the jury heard dozens of phone calls and saw dozens of emails. The jury viewed demonstrative exhibits, as well as exemplars of trade tickets from natural gas transactions, and the monthly price reports sent by Valencia and Singleton to Inside FERC and NGI. Upon completion of the government's case-in-chief, defendants did not call witnesses or present evidence, but rather, moved for judgment of acquittal on all counts and rested. The court granted Singleton's motion as to count six (false reporting under the CEA by aiding and abetting) and count ten (obstruction of justice). The court denied the motions as to all other counts.

On August 1, 2006, the court delivered its instructions to the jury, and counsel presented closing arguments. The jury then deliberated for several days. In a note to the court, on August 2, 2006, the jury requested further definitions of the terms "price" and "index." The court referred the jury back to the evidence in the case and the court's instructions. The jury then requested a dictionary. The court responded: "No, sorry." Near the end of the day on August 2, the jury stated in a note: "We are currently deadlocked over definitions of price versus index as it applies to the fourth element of the false reporting charges. Can we please get further clarification as to their definitions or are they synonymous?" The jury also requested the trial transcripts for two witnesses. The court again instructed the jury to consider the court's instructions, the jury's own recollection of the testimony, and the trial exhibits.

Around noon the next day, August 3, 2006, the jury delivered the court another note. It read: "We are deadlocked starting with the first counts and all others on the same points as we were all day yesterday as well. We do not agree on how the evidence fits the definitions of false reporting and conspiracy and neither side is willing to change." The court brought the jury into the courtroom, explained that it was unclear what the jury was struggling with, and asked the jury to attempt to resume deliberating after a lunch break. A few hours later, the jury sent another note stating: "Following your instructions we took a break for lunch in order to clear our heads and give the case a fresh look. After returning, we carefully reread your instructions several times and we are still hung on the conspiracy count, which we believe to be the foundation of the rest of the charges. Specifically for the purpose of this case, we cannot come to an agreement on the price of a commodity in relation to the index price. Without guidance from the court, we remain hopelessly deadlocked." In response, the court brought the jury into the courtroom. The court provided more overview, and instructed the jury to "consider each charge in each indictment separately and consider the elements as to each count separately and in light of the rest of the charge." The court asked the jury to continue deliberating.

At 5 p.m. on August 3, the jury sent another note to the court indicating that it had reached a verdict on three of the eight counts against Singleton, and nine of the twenty-three counts against Valencia. Namely, the note indicated that the jury had reached verdicts for the wire fraud counts. However, the jury remained "hopelessly deadlocked" over the remaining counts, i.e., the conspiracy and CEA counts. The court asked the jury to adjourn for the evening, continue thinking about the case, and return in the morning.

The next morning, August 4, 2006, the jury returned and indicated that it wished to continue deliberating. The jury then reached a verdict as to three of the CEA counts against Valencia and two of the CEA counts against Singleton. At that point, the court took the jury's partial verdict. The jury found Valencia guilty of seven counts of wire fraud, not guilty of two counts of wire fraud, and not guilty of three counts of false reporting under the CEA. The jury did not reach a verdict as to the remaining ten counts of false reporting or the one count of conspiracy. The counts upon which the jury was deadlocked were dismissed at the government's motion. The jury found Singleton guilty of one count of wire fraud, not guilty of two counts of wire fraud, and not guilty of two counts of false reporting under the CEA. The jury did not reach a verdict as to the remaining two counts of false reporting or the one count of conspiracy; these were also dismissed at the government's motion. The district court later denied Valencia's and Singleton's post-verdict motions for judgment of acquittal and motions for a new trial.

Valencia now argues: (1) that the government committed misconduct by reading an incriminating letter aloud during opening statements and referring to it again in closing argument, requiring retrial; (2) that the district court abused its discretion in allowing Glenn Labhart to testify about the effects of Valencia's actions on Dynegy's bottom line, and that the government's at-trial disclosure of a fee agreement with Labhart requires a new trial; (3) that the district court abused its discretion in allowing Matthew O'Loughlin to testify about the effects of false reports on Inside FERC's and NGI's indices; (4) that cumulative evidentiary errors at trial mandate retrial; (5) that the evidence is insufficient to sustain Valencia's convictions; and (6) that the district court erred when sentencing Valencia. Singleton joins Valencia's challenges to the admissibility of O'Loughlin's testimony, and challenges the sufficiency of the evidence to support the count of wire fraud of which he was convicted. We provide more details herein as they are germane to each issue raised on appeal.

II.

We first consider whether the government's reading of a whistle-blower letter by Jeffrey Hornback, a former Dynegy employee, during opening statements was reversible plain error. After detailing the circumstances in which the government used the letter, as well as Hornback's in-court testimony, we evaluate whether Valencia is entitled to a new trial on this basis. We conclude that no reversible error occurred.

A.

Following voir dire, the government represented to the court that it would not use any trial exhibits during its opening statement. For the most part, the government's hour-long opening statement complied with this representation. The prosecutor, Mr. Lewis, gave the jury an overview of natural gas markets, the acts alleged, and the means by which the government intended to prove the elements of the crimes charged. However, as Mr. Lewis delved into greater detail about the specific acts, he showed the jury an excerpt from Government Exhibit 14, a price report sent by Singleton to NGI on July 31, 2000. Mr. Lewis also described what certain key phone calls would reveal, identifying the conversations by date and exhibit number. Defense counsel did not object to the use of or reference to these trial exhibits.

With approximately fifteen minutes left in his allotted time for an opening statement, Mr. Lewis told the jury that the government would call Jeffrey Hornback as a witness. Hornback was a natural gas trader at Dynegy and a coworker of Valencia's on the West Desk. Mr. Lewis asserted that in July of 2000, Hornback sent a price report which included fake trades and omitted real trades. Hornback allegedly did so at the direction of Valencia and a supervisor at Dynegy. Mr. Lewis told the jury: "I'm going to show you evidence that employees at Dynegy and El Paso had choices as to what to do when they were confronted with opportunities to do something wrong and what I'm going to do is read from an exhibit that will, I believe, come into evidence and it's Exhibit 668." The document in question is a letter which Hornback apparently wrote in March of 2001, in response to an internal requirement that employees certify that they were unaware of violations of Dynegy's Code of Business Conduct. Defense counsel did not object, and the following was read to the jury:

I am writing this report to expand on the reasons I checked that I was aware of violations regarding financial integrity according to Dynegy's Code of Business Conduct. Let me make clear from the start that these violations are not in any way related to Dynegy's accounting practices or reporting, but, in my opinion, they are no less a violation of integrity than a misstatement of earnings would be.

The first and most widespread problem I have observed is the blatant fabrication of numbers reported to the publications that report monthly and daily gas price indices. These publications, such as Inside FERC, NGI, and Gas Daily, rely on the honesty and integrity of companies such as Dynegy to report any and all fixed-price physical transactions that actually occur. If transactions are actually reported, then the resulting index should always be a fair representation of the value of gas at a particular point, at a particular time.

However, at Dynegy I have observed month after month and day after day blatant lies being reported by Dynegy to intentionally attempt to skew the index in favor of Dynegy's position. I have seen members of top management in the organization instruct traders in their group that this is how the game is played and it is how Dynegy must behave in order to compete. I have even seen a top manager require that he be given a chance to review the numbers before they were sent to the publications so he could be sure they were where he wanted them to be, regardless of which prices were actually transacted.

To me, it has been a huge ethical dilemma to see volumes reported many, many times greater than would have even been possible to trade at a given point and to see prices reported well-outside the range of what actually traded.

I know almost every trader and manager at Dynegy condones this practice and I know that none of them mean any harm. They are only trying to enhance the profitability of their positions. Nonetheless, this practice is, in fact, harmful to many outside the company.

For example, the posted index prices have far-reaching impacts, affecting everything from the price utilities pay for their gas and ultimately the price that rate payers pay, to the price that royalty owners receive for their gas at the well-head.

In fact, one of my previous employers was so sensitive to the impact that index price reporting had on royalty payments, that the company's traders were forbidden to report to the publications for fear of class-action lawsuits.

Obviously, I am not a legal expert, but I feel very strongly that lying on price reports is not only dishonest and unethical, but could expose Dynegy to huge legal and public relations risks. It is, after all, a blatant form of price manipulation. Imagine how the public and press in California would react, for example, if they found out that Dynegy's traders were intentionally manipulating prices higher in their state. Consequently, I feel very adamant that Dynegy should enact a policy regarding honest reporting to the index publications to remove this risk.

By coming forward with these complaints, I am not trying to get any individuals into trouble or cause any dissension in the company. In fact, I would not have come forward at all had it not been for the code of conduct statements we were forced to sign. I just could not with a clean conscience sign that I was unaware of violations because of the reasons outlined above.

I sincerely hope this will remain anonymous and that no individuals will be reprimanded. At the same time, it is my true desire that these questionable practices should end.

Mr. Lewis then told the jury "you will hear that people had choices." After a few more sentences, he concluded his opening statement.

Mr. Flood, Valencia's counsel, then began his opening statement. He said: "I'm not going to talk about the letter of Mr. Hornback, who's going to be a government witness, because I think the evidence will show that that letter was an attempt to try to deflect from his own responsibility in what he had done and it has nothing to do with the evidence against Ms. Valencia." Mr. Flood then moved on to other matters.

Hornback testified at trial for nearly two full days. His live testimony described the day-to-day practices of natural gas trading at Dynegy's West Desk. Through Hornback, the government introduced myriad emails, phone calls, and other documents. The tone and substance of Hornback's testimony in many ways mirrored the accusations which the prosecutor read to the jury during opening statements. For instance, Hornback at one point stated that "as far as I can recall, we had never really accurately reported prices." Rather, when Hornback first joined the West Desk, traders would list what they considered representative trades, in "an honest attempt to just report what we did see in the market." However, after Dynegy's West Desk came under the supervision of a hard-knuckled vice president named Steve Barron some time in late 1999 or early 2000, Dynegy's reporting practices "became, in my view, an intentional attempt to skew the index one way or the other to benefit our positions." Hornback said that there were times where Barron "specifically told me to go back and change numbers that I had previously planned to submit." The change in reporting practices was apparently largely driven by a sense that Enron was dominating natural gas markets and manipulating prices, and that other players in the market had to push back in order to stay afloat.

Hornback often interacted with Valencia. He stated that in or around July of 2000, Dynegy began reporting trades entered into by West Coast LLC ("West Coast"). West Coast was a partly owned joint venture of Dynegy and NRG Energy, Inc. Hornback testified that Valencia had explained to him that "we were going to use [West Coast] to report offsetting transactions to the transactions that we submitted at Dynegy to give them basically more validity as far as the publication was concerned." In other words, West Coast would report trades with Dynegy at price levels favorable to Dynegy's market positions in an effort to show that greater volumes of gas were being bought and sold at a given location. Hornback explained that publications such as Inside FERC and NGI might become suspicious if Dynegy alone reported transactions involving atypically large volumes of gas, "but if you had two independent parties that both had the same data, then the publication would be more likely to believe it as being real."

Recorded phone calls corroborated the assertion that Valencia participated in fabricating trades, inflating the volume of real trades, or omitting real trades to favor Dynegy's positions. With Hornback on the stand, the government played a phone call from July 31, 2000, between Valencia and Michael Stewart, an employee of West Coast. In the call, Valencia tells Stewart to fax West Coast's price reports to the publications. Valencia states at one point that she wants Stewart to fax the data from a separate machine ( i.e., a machine in West Coast's office, not Dynegy's office). Valencia states: "I'm — just worried, I just want it to look as legitimate as possible." In another phone call between Stewart and Hornback, also from July 31, 2000, Stewart indicates that "the guy from Inside FERC called me about these monthly . . . numbers that Michelle emailed him." Hornback says to Stewart: "these numbers are . . . in the legitimate range, so I'm not sure why he would be questioning them." When asked to explain this statement, Hornback testified that "there must have been real trades that happened in the ranges of prices that we reported. . . . It seems that they were not real trades done at Dynegy though."

Additionally, the government introduced a physical trade ticket documenting a fixed-price, baseload deal entered into on July 28, 2000, for delivery for the next month. Although Hornback filled out the information on the ticket, he did so on behalf of Valencia, who had entered into a contract for Dynegy to buy 10,000 MMBtu of natural gas from El Paso at a price of $4.60 per MMBtu at a delivery point called SoCal Ehrenberg. Hornback agreed that this trade met the criteria for reporting to Inside FERC and NGI, but noted that the trade was not listed on Dynegy's monthly report sent on July 31, 2000, from Hornback to Kelly Doolan, the chief editor of Inside FERC.

With Hornback still on the stand, the government demonstrated that several other trades meeting Inside FERC's and NGI's reporting criteria were not in a report sent by Valencia on November 30, 2000. The government played a phone call from November 30, 2000, between Valencia and Rick Anderssen, a natural gas trader at Pan Canadian Energy Services. Anderssen was Valencia's boyfriend at the time of the call; he is currently Valencia's husband. In the call, Valencia tells Anderssen that she "already turned in my indexes and I couldn't get hold of you because the deadline was one o'clock." She then tells Anderssen: "I turned in twenty thousand at fifteen fifty." In other words, Valencia reported trading 20,000 MMBtu at a price of $15.50. Anderssen states: "I'll put it down right here. Twenty at fifteen fifty. Okay." Later, Valencia begins to describe what she reported for a delivery point in Northern California called Malin. Anderssen asks: "Flat to SoCal?" Valencia responds: "F*** no, I want Malin low!" Anderssen then asks: "I mean you — you think Malin will be low, yeah. I do too." Valencia answers: "No. I'm only reporting it low." Hornback testified that "flat to SoCal" meant the index price at Malin would be the same as the index price at SoCal, or Southern California.

Hornback testified that at some time during the summer of 2000, he overheard Valencia apparently "speaking to a trader at another company discussing what prices to report or not report for the first of month index report." He stated that it caused him to become concerned:

I had been feeling personally conflicted from an ethical standpoint, you know, ever since it started to become more of what I felt was a dishonest approach to reporting the indices. And that particular conversation I remember, you know, it made an impression because it kind of scared me a bit, because I remembered enough from business law classes in college to know that you shouldn't be talking to somebody at another company about things like that.

Due to such concerns, some time after overhearing Valencia's conversation, Hornback "decided not to participate anymore with the reporting."

I decided that I would report to Michelle what the actual trades were that I had transacted and then from there I said that I did not want to know anything else about it. Basically I didn't want to see what was sent in, I didn't want to be a part of any conversations about it, kind of just tried to distance myself from it.

Hornback indicated that Valencia appeared to understand and respect this decision. Hornback did not tell Barron about his decision to distance himself from the monthly reports.

After Hornback had been on the stand for several hours, the government began a line of questioning concerning Dynegy's Code of Business Conduct. Specifically, in early 2001, employees were required to certify that they knew of no ethical violations. Hornback stated that he could not make such a certification in good conscience given what he had witnessed. Mr. Flood, Valencia's counsel, then requested a sidebar and stated: "During the opening statement I unfortunately did not object, I guess. . . . Mr. Lewis read from a letter that I think is rank hearsay." The court noted that the government had not offered Hornback's letter into evidence, and reasoned that the fact that he wrote the letter was not hearsay. However, the court continued that if the government wished to introduce the letter to prove its contents, i.e., that Hornback had witnessed lies being reported to the publications in order to skew indices in Dynegy's favor, there would be a hearsay question. Singleton's counsel, Mr. Nugent, also asserted that the letter was "rank hearsay." The government disagreed. The court stated that it would not admit the letter at that time, but would consider admitting portions of it if the need arose.

Hornback testified that after he declined to certify that he knew of no ethical violations, he was contacted by the office of Eric Bruce, Dynegy's top compliance officer. Hornback met with Bruce, and Bruce requested that Hornback describe his observations in a written memorandum. He did so, but did not sign or date the document, and left it under Bruce's door around the beginning of March, 2001. When asked why he did this, Hornback said: "I didn't want it to appear that I was trying to betray anybody on the floor, that I was trying to get anybody in trouble or anything like that." Hornback stated that Exhibit 668 was a copy of the letter. The government offered the letter into evidence; the court stated: "I'm not going to receive it right now." Some time later, Hornback stated that the contents of the writing summarized what he had discussed with Bruce concerning misreporting of trades and pressure from managers to engage in misconduct. When Hornback could not recall the exact contents of the document, the government requested that Hornback be allowed to read from it. The court conducted a sidebar at which it ruled that the government did not need the document to show that Valencia, like Hornback, had a choice as to whether to misreport trade information. Hornback also testified that he received a very bad performance review from Steve Barron in early 2001. Hornback quit his job at Dynegy in mid-March of 2001, about two weeks after delivering the memorandum to Bruce.

On re-direct examination, Mr. Lewis returned Hornback to his interactions with Valencia and other traders on the West Desk regarding the reporting of trades. Hornback testified that, on one particular occasion, he had objected to efforts to "skew" indices lower because this could reduce royalty payments to mineral rights owners, some of whom could be "little old ladies" who depended on the royalties for their income. Hornback could not recall whether Valencia reacted in any way to this statement.

Trial proceeded for several more weeks with no references to Hornback's letter. In the first minutes of his closing argument, Mr. Lewis stated: "[T]hese index prices were not some sort of theoretical concept floating out there in somewhere other than the real world. I mean, people really bought gas at these prices. And those little old ladies that Jeff Hornback described, they're real." Mr. Lewis beseeched the jury to be mindful of Hornback's testimony. These were the only direct references to Hornback made during Mr. Lewis's closing. Thereafter, Mr. Lewis replayed many telephone calls and displayed for the jury key documents which had been entered into evidence. He insisted that greed motivated the defendants, and not a blind need to follow orders or a desire to act as a counterweight to Enron's perceived dominance of the markets.

During his closing argument, Mr. Flood repeatedly mentioned Hornback's in-court testimony to show that Valencia was but one of many contributors to the monthly price reports, and thus that others were equally or more culpable for misrepresentations in the reports. He then stated that he would remind the jury of certain things Mr. Lewis said during opening statements:

"Top management instructed us to report this way. That is how the game is played." Mr. Lewis said that to you in opening statement. "Top management changed the numbers on the report. I know that every trader and manager at Dynegy condones this practice and I know none of them mean any harm." Those were his words in opening statement. "I know none of them mean any harm."

Mr. Lewis objected that this line of argument "misstates the opening statement." The court instructed the jury that "it's up to you to recall Mr. Lewis's opening." Mr. Flood then moved on, and thereafter made only fleeting references to Hornback as a participant in the reporting of false trades.

Singleton's counsel, Mr. Nugent, also made references to Hornback's testimony, such as representing that Hornback had said: "We never sent in accurate numbers. We sent in representational numbers that were accurate market information, but they weren't based on real trades." Mr. Nugent later said: "But Jeff Hornback finally talks to the vice president of Dynegy, I think his name was Eric Bruce, and he talks to him for an hour and [a] half and he writes that letter and says, `top management is pressuring us at Dynegy.'"

During rebuttal, Ms. Beek, co-counsel for the government, stated:

Mr. Flood attributed comments to Mr. Lewis and so did Mr. Nugent that were not Mr. Lewis's. For example, Mr. Lewis read you in the opening a whistleblower letter from Mr. Hornback, where Mr. Hornback says, "I am writing this report to expand on the reasons I checked that I was aware of violations regarding financial integrity according to Dynegy's code of business conduct." And then he goes on to say, "However, at Dynegy I have observed month after month, day after day blatant lies being reported by Dynegy to intentionally attempt to skew the index in favor of Dynegy's position. I have seen members of top management in the organization instruct traders in their group that this is how the game is played and it's how Dynegy must behave in order to compete." Those were from Mr. Hornback's whistle-blower letter. They were not comments of Mr. Lewis.

Defense counsel did not object to Ms. Beek's statements.

B.

Valencia says the government should not have read Hornback's letter during the opening statement or read portions of it during the closing argument. Valencia says the letter was inculpatory hearsay, and was prejudicial because it: (1) described the breadth of the false reporting; (2) divulged the intent and identities of the individuals who made false reports; (3) established the materiality of the false reports; and (4) suggested what the consequences would be. In sum, the letter "established every element of the government's case and was the single most damaging piece of information the jury heard." Valencia insists that the government should have known that the letter might be inadmissible, and thus should not have mentioned it until it was admitted. Valencia says the letter was not cumulative of Hornback's in-court testimony, as he could not recall all of its contents. The jurors' difficulty in reaching a verdict, and the split verdict it rendered, purportedly militate in favor of a new trial. Valencia also argues that the government improperly bolstered its own credibility and the credibility of the Hornback letter by stating that it was the prosecutor's job to tell the jury the truth.

The government stated at oral argument that Valencia's failure to object to use of the letter in advance of trial means she forfeited any challenge to its admissibility. In its brief, the government does not actually argue that it was proper to read Hornback's letter during the opening statement, but rather, insists that the contents of the letter were never explicitly deemed inadmissible hearsay. Moreover, the government says references to the letter during closing arguments were invited by defense counsel. The purpose of mentioning the letter during closing was, ostensibly, to correct misattributions. The government says there is no prejudice because the district court instructed the jury that lawyers' statements and arguments are not evidence. More importantly, the government says that evidence of Valencia's and Singleton's guilt was "overwhelming." The government insists that Hornback's testimony conveyed much of what his letter said, and that Valencia's own emails and phone calls provide ample evidence of guilt.

C.

Valencia concedes that she failed to timely object to references to Hornback's letter. We therefore review for plain error. See FED.R.CRIM.P. 52(b); United States v. Mares, 402 F.3d 511, 520 (5th Cir. 2005). Valencia must show "(1) error, (2) that is plain, and (3) that affects substantial rights." Mares, 402 F.3d at 520 (quotation omitted). If these conditions are present, we may exercise our discretion to correct the error if it "seriously affects the fairness, integrity, or public reputation of judicial proceedings." Id.

"This court applies a two-step analysis when reviewing claims of prosecutorial misconduct. The court must first decide whether the prosecutor made an improper remark. The court evaluates the remark in light of the context in which it is made." United States v. Morganfield, 501 F.3d 453, 467 (5th Cir. 2007), cert. denied, ___ U.S. ___, 128 S.Ct. 2500, 171 L.Ed.2d 790 (2008) (citations omitted). To determine whether improper statements require a new trial, we ask whether the prosecutor's remarks "prejudiced the defendant's substantive rights," and "cast[] serious doubt on the correctness of the jury's verdict." Id. (quotations and citations omitted). We must weigh: "(1) the magnitude of the prejudicial effect of the prosecutor's remarks, (2) the efficacy of any cautionary instruction by the judge, and (3) the strength of the evidence supporting conviction." Mares, 402 F.3d at 515-16 (quotation omitted). We also presume that a jury can and will follow an instruction that attorneys' statements are not evidence, "unless there is an overwhelming probability that the jury will be unable to follow the instruction and there is a strong probability that the effect is devastating." Morganfield, 501 F.3d at 468 (citations and internal quotations omitted). As for "invited error:"

The doctrine of invited error provides that when injection of inadmissible evidence is attributable to the actions of the defense, the defense cannot later object to such "invited error." Under this doctrine, a defendant cannot complain on appeal of alleged errors which he invited or induced, especially where the defendant may not have been prejudiced by the error. We will not reverse on the basis of invited error, absent manifest injustice.

United States v. Green, 272 F.3d 748, 754 (5th Cir. 2001) (citations and internal quotations omitted).

In United States v. Flores-Chapa, 48 F.3d 156, 161 (5th Cir. 1995), we found that prejudicial prosecutorial comments constituted reversible plain error. At trial, the district court ruled that hearsay testimony connecting the defendant to a drug conspiracy was inadmissible. Id. at 159. However, the government referred to the hearsay when examining the next witness and again during closing argument. Id. at 159 n. 6. The court held that the error was plain, and that the statement was prejudicial "in the context of the entire trial," in light of the "paucity of evidence" that the defendant was involved in the conspiracy. Id. at 160-61; see also United States v. Novak, 918 F.2d 107, 109 (10th Cir. 1990) (noting that a prosecutor may not "refer to evidence of questionable admissibility" during opening statements). In Novak, the government made two assertions during opening statements which were never substantiated: (1) that a citizen reported to police that the defendant sold cocaine from his home; and (2) that cocaine seized in the defendant's home was very pure — "dealer" grade as opposed to "user" grade. 918 F.2d at 108, 110. The government's case was "completely circumstantial" and relied heavily on inferences drawn from these statements to prove the defendant's intent. Id. at 110-11. Therefore, the prosecutor's improper remarks and conduct at trial prejudiced the defendant in a way that was incurable by the court's admonition that the opening statements were not evidence. Id. Valencia relies primarily on the authority of Flores-Chapa and Novak in asserting that the government's use of Hornback's letter was plain error which prejudiced her substantial rights.

D.

We may quickly dispense with the government's assertion that Valencia forfeited the opportunity to challenge the letter's admissibility by failing to object to its use before trial. The district court did set a pre-trial deadline for objections to the authenticity of trial exhibits. However, it is abundantly clear, based on our review of the record, that the lack of a pre-trial objection did not preclude all challenges to the admissibility of an exhibit, as the government suggests. The district court regularly entertained objections based on relevance, lack of foundation, the hearsay rule, or undue prejudice. The government represented in its brief that the court never deemed the letter hearsay. However, prior to jury instructions, the government acknowledged that Hornback's letter was never admitted into evidence due to defendants' hearsay objections. We find the government's insouciance regarding the record and the district court's orders unsettling.

We conclude that Mr. Lewis's decision to read Hornback's letter verbatim in his opening statement was improper. See Morganfield, 501 F.3d at 467. There was a very good chance that the court would deem the letter inadmissible hearsay, and the court apparently sustained defendants' objection to admission of the letter for this reason. The government says that the court did not explicitly deem the letter hearsay, or alternatively, that the letter could have been admitted under the business record exception to the hearsay rule. These post-hoc justifications are a thin reed. Mr. Lewis could have simply paraphrased Hornback's anticipated testimony, avoiding the inflammatory language of the letter. We admonish the government that its cavalier approach to the district court's rulings has made our task of reviewing the fairness of the trial and the soundness of the verdict considerably more difficult.

The decision to read Hornback's letter in its entirety was "error." See Mares, 402 F.3d at 520. However, even assuming the error was "plain," see id., any prejudice to Valencia was minor. The letter does not mention Valencia by name. Moreover, over the course of two days, Hornback's testimony directly implicated Valencia in a scheme to misstate Dynegy's trades in the monthly reports to Inside FERC and NGI. The government presented copious evidence of false trade data sent either by Valencia or at her direction. Recordings of Valencia's phone calls provided ample evidence from which the jury could conclude that she intended to manipulate the trade reports in order to skew the indices in favor of her trading positions, or those of Dynegy as a whole. A thorough review of the record refutes Valencia's contention that Hornback's trial testimony was "vague and ineffective." We cannot agree with Valencia's assertion that Hornback's letter "was the single most damaging piece of information the jury heard." The extensive, incriminating in-court testimony provided by Hornback and others, in conjunction with inculpatory, properly admitted exhibits, heavily dampened the magnitude of whatever prejudicial effect Hornback's whistle-blower letter had upon the jury. See Morganfield, 501 F.3d at 467; Mares, 402 F.3d at 515-16. The judge did not give a cautionary instruction to the jury to disregard the contents of Hornback's letter, but Valencia failed to ask for one. Finally, the strength of the evidence supporting the wire fraud convictions is strong independent of the assertions in Hornback's letter. See Morganfield, 501 F.3d at 467. Therefore, the improper remarks are not so prejudicial that a new trial must be held. See id.; Mares, 402 F.3d at 515-16.

Valencia relies on Novak and Flores-Chapa, but these cases lend scant support to her position. In Novak, the government never presented evidence that it had promised in opening statement; references to this "evidence" at trial nevertheless played a key role in proving the defendant's intent. 918 F.2d at 110. Here, Hornback's in-court testimony restated and significantly expanded upon the assertions of his letter, and for the first time, implicated Valencia personally. This is a far cry from Flores-Chapa, where it was solely inadmissible hearsay which connected the defendant to a drug conspiracy. 48 F.3d at 160-61. What is more, in that case, even after the district court ruled that the evidence was inadmissible, the prosecutor made several references to it. In this case, the impact of Hornback's letter was greatly outweighed by his live testimony connecting Valencia to the scheme. The district court also never forbade the government from making reference to the letter.

In contrast to opening statements, the government's references to Hornback's letter during closing arguments were invited. See Green, 272 F.3d at 754. Mr. Lewis did not mention the letter during closing argument. Defense counsel for both Valencia and Singleton made references to the letter. On rebuttal, Ms. Beek clarified who said what. Ms. Beek showed that certain statements came from the letter, as opposed to Mr. Lewis. Moreover, the court instructed the jury that it was "up to you to recall Mr. Lewis's opening." The district court later instructed the jury that the lawyers' arguments and statements were not evidence. Valencia shows no reason to disregard our normal presumption that the jury is capable of following the court's instructions. See Morganfield, 501 F.3d at 468. Having brought up the substance of the accusations contained in Hornback's letter in closing, Valencia cannot contend she was prejudiced because the government sought to set the record straight. Given these circumstances and the weight of the evidence as a whole, we cannot conclude that Ms. Beek's reference to Hornback's letter caused "manifest injustice." See Green, 272 F.3d at 754.

Valencia also complains that the government vouched for its own credibility and that of Hornback's letter by stating during closing argument that it was the prosecutor's job to tell the jury the truth. We do not condone such statements, but it is implausible to say that this oblique remark, to which Valencia did not object and which did not directly bolster the credibility of Hornback's letter, caused devastating prejudice. Cf. United States v. Gracia, 522 F.3d 597, 601-02 (5th Cir. 2008). The jury's lengthy deliberations lend support to Valencia's argument that the case was close and the jury struggled to reach a verdict. However, the jury was tasked with recalling several weeks worth of evidence and deliberating upon more than thirty distinct counts between Valencia and Singleton. Also, it is evident from the jury's notes that it mainly struggled to understand the conspiracy and CEA counts, which we do not review in this appeal. Given the jury's difficult assignment, and the panoply of evidence supporting guilty verdicts on the wire fraud counts, it would be far-fetched to conclude that the mere length of the jury's deliberations "casts serious doubt on the correctness of the jury's verdict," rising to the level of plain error. See Morganfield, 501 F.3d at 467 (quotation omitted); cf. United States v. Fields, 483 F.3d 313, 379 (5th Cir. 2007) (Benavides, J., dissenting) (noting that courts "have been unwilling to find error harmless where the record . . . affirmatively shows that the jurors struggled with their verdict") (emphasis added).

We admonish the government that it was improper to read Hornback's letter during opening statement, and that its attempts to justify the error before this Court are completely unpersuasive. Nevertheless, considering the entirety of Hornback's testimony in the context of trial, we conclude that references to the letter, individually or cumulatively, are not reversible plain error. See Mares, 402 F.3d at 515-16.

III.

Valencia next raises issues concerning the testimony of Glenn Labhart. Valencia makes three distinct arguments: (1) that Labhart was an expert witness, implicating the strictures of prior disclosure under Federal Rule of Criminal Procedure 16(a)(1)(G), as well as foundation and reliability requirements under Federal Rule of Evidence 702; (2) that Labhart provided summary testimony which did not comply with Federal Rule of Evidence 1006; and (3) that the government's at-trial disclosure of a fee agreement with Labhart was a material violation of Valencia's rights under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). Valencia contends that each error is sufficient to require a new trial. We first recount the circumstances and details of Labhart's testimony, and then consider each argument in turn. None is meritorious.

A.

Glenn Labhart was the chief risk officer of Dynegy from 1997 to 2004. His duties included monitoring Dynegy's trading operations, and specifically, enforcing the "trading limits and risk tolerances" set by higher-ups at Dynegy. Labhart assisted Dynegy in responding to the government's requests that Dynegy determine whether trades reported by Dynegy and West Coast in 2000 and 2001 were real or fictitious ("true-false analysis"). He also determined whether Dynegy had net long or short positions at various trading points ("long-short analysis"). Finally, he analyzed how Dynegy stood to benefit through changes in the indices published by Inside FERC and NGI during that time period ("penny-up penny-down analysis"). Before trial, Valencia moved to exclude Labhart's proposed testimony on the grounds that it was expert in nature, and failed to meet the requirements of Federal Rule of Evidence 702 and Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993). Valencia also averred that the government had not provided a report detailing the nature of Labhart's conclusions, and that the government failed to turn over databases containing the universe of trade records upon which Labhart based his analysis. The district court initially reserved ruling on this motion. On the thirteenth day of trial, July 26, 2006, before Labhart had been called to the stand, the court conducted a hearing outside of the jury's presence to evaluate these matters.

At the hearing, Labhart said he did not complete the true-false, long-short, or penny-up penny-down analyses relevant to this case by the time he left Dynegy in 2004. He was subpoenaed by the government in March of 2006, several months in advance of trial, to pick up where he left off in 2004 and complete this work. Labhart was familiar with the trade reports sent to Inside FERC and NGI from his time as chief risk officer. He stated that for the true-false analysis, trades were reviewed from a system called AREV. AREV was generally used to track physical positions for buying, selling, storing, and transporting natural gas. For the long-short and penny-up penny-down analyses, Labhart mainly used a risk management system at Dynegy called Abacus. Dynegy tracked both physical and financial transactions via Abacus.

If Labhart needed additional portfolios or records from within the Abacus system, he requested the information from Dynegy, which provided it to him. A Dynegy computer programmer named Wanda Chovanec assisted Labhart in combing through the databases. Labhart stated that, just days before he was called to testify, he prepared a final summary report of his analysis ("the summary report"). The summary report contained positions for each relevant month and location at which prices were reported, both physical and financial, as contained in Abacus and AREV. Labhart did not show the summary report to the government until the day before the July 26 hearing.

After Labhart stated the foregoing, Valencia's counsel re-urged the motion to exclude Labhart's testimony on the basis that it was expert testimony, and that Labhart had not been shown to be qualified to render such opinions. The court overruled the motion as to the true-false and long-short analyses, stating: "He's saying he was reviewing the records and doing exactly what he did while he was in the job. He had access to the database as part of his work and he was the risk manager." The court reasoned that such analysis was "clearly what he did during the day at work. And the fact that he's had to reconstruct it later is immaterial to the ruling." However, the court reserved ruling on whether Labhart was qualified, based on his job duties while he was at Dynegy, to relate the penny-up penny-down analysis to the jury. The court ordered the government to lay a foundation before seeking to elicit such analysis.

The government noted that it had provided Labhart with enlargements of the monthly price reports sent to Inside FERC and NGI containing the allegedly false reports. The government instructed Labhart to mark a blue star next to each reported trade which matched an actual fixed-price trade. If the volume was inaccurate, Labhart was told to write the correct volume in red ink next to the reported trade. The government also told Labhart to list omitted reports on the report in purple. Finally, Labhart was to include the results of his penny-up penny-down analysis, i.e., the amount Dynegy stood to gain or lose from movements in a particular index price. The court instructed the government to keep the penny-up penny-down figures covered unless and until such analysis was deemed admissible. Defense counsel did not object to the use of the marked-up reports in this manner.

With the jury present, Labhart stated that the index prices published by Inside FERC and NGI affected Dynegy's risk positions and profits. Consequently, Dynegy vigilantly monitored its positions for locations and contracts affected by the published indices. As part of his job as risk manager, Labhart could calculate daily how any given trade by the West Desk, or an individual trader on the Desk, could affect Dynegy's profits. Labhart also ran profit and loss calculations each month when the index prices were released. Labhart stated that he used the same methodology when preparing his penny-up penny-down analysis in anticipation of his testimony in this case.

Outside of the presence of the jury, the court ruled that Labhart's penny-up penny-down analysis was based upon his experience and duties as the chief risk officer of Dynegy. The court deemed this "complex lay opinion," which was not expert testimony and therefore was not subject to Federal Rule of Evidence 702. In the alternative, the court ruled that if Labhart's opinions were expert in nature, such met the requirements of Rule 702. Counsel for Valencia asserted that Labhart was a "summary witness," that no report of Labhart's opinions was given to defense counsel, and moreover, "we have not been able to have his methodology and/or his opinions independently assessed by our own expert to determine if they're, in fact, valid." The court noted that defense counsel had not received Labhart's recent summary report until the day before Labhart testified (Labhart put together the report only days before testifying), but "you have had the underlying data and could have done this comparison with the databases that were provided several weeks ago."

With the jury present once more, Labhart first related his true-false analysis and long-short analysis. He noted that many of the trades reported by Valencia did not exist in the AREV database, that some volumes appeared to have been reported inaccurately, and that some real trades meeting the reporting criteria were not included. In other words, many of the reports sent by Valencia or at her direction contained fictitious or inaccurately reported trades, and omitted real trades. Labhart also testified that, based on his review of Dynegy's databases, Valencia often entered into transactions called "swing swaps" and "basis swaps," which used index prices published by Inside FERC and NGI. Labhart noted that Valencia often kept "open" positions, meaning her monthly natural gas portfolio was either long or short. As a result, changes in the published indices would have a direct effect on the performance of Valencia's portfolio.

Labhart then presented his long-short and penny-up penny-down conclusions to the jury. For each report, he indicated how much money Dynegy would have gained or lost from a one-cent movement in the indices for each trading point, as well as what Dynegy stood to gain from changes in the indices' basis figures. Labhart's true-false and penny-up penny-down analyses were depicted on the blow-ups of Dynegy's monthly reports, which had already been admitted into evidence. The jury was thus able to see Labhart's conclusions superimposed on the reports. Defense counsel did not object to the use of reports in this fashion, although counsel reiterated objections based upon Labhart's methodology. The true-false, long-short, and penny-up penny-down analysis took much of the afternoon of July 26; at the conclusion of this testimony, the government passed the witness.

On the morning of July 27, the government recalled Labhart in order to clarify what the markings on the enlarged reports were intended to mean. The government then pursued a new line of questioning concerning the time Labhart had spent preparing his analysis. Labhart said that after being subpoenaed, he spent about 350 hours of his time, for which he had been paid $36,500 by the government and about $20,000 by Dynegy. He anticipated billing for additional work, for a total of roughly $50,000 from the government. He stated that he was not being paid for his in-court testimony. Counsel for Valencia and Singleton cross-examined Labhart on the details of his compensation. Counsel protested that compensation details had never been disclosed, whereupon the court ordered the government to provide such information to defendants. That evening, the government emailed documents including a form number "OBD-47" for Labhart, which bears the caption "Expert Witness Statement of Work." Under the heading "Purpose and Objective of Expert Witness Services," the following items are checked:

(1) To support litigation/prosecution theory or conclusion

(2) To determine loss dollar value

(3) To provide technical explanations of fact

(5) Other: To determine (1) accuracy of reported gas trades (2) benefit to company from changes in index price movement.

Item (4), "To offer professional opinion," is not checked. The following items are checked under the heading "Preparation and Scope of Work Requested:"

(1) Review case files, records or evidence

(3) Create and produce technical charts, graphs or accountings

(4) Evaluate and compare technical facts

(6) Validate hypothesis

(11) Oral presentation to Counsel

(12) Prepare and present expert reports, graphics, models, exhibits and demonstrations for trial preparation and for trial.

B.

We agree with the district court's conclusion that Glenn Labhart was a lay witness, and not an expert witness. Labhart's testimony related to his former job duties as risk officer at Dynegy. He recreated much of the analysis he regularly performed when evaluating risk tolerances. He was properly characterized as a fact or lay opinion witness. See FED.R.EVID. 701 ; Nat'l Hispanic Circus, Inc. v. Rex Trucking, Inc., 414 F.3d 546, 551-52 (5th Cir. 2005) ("Rule 701 does not exclude testimony by corporate officers or business owners on matters that relate to their business affairs, such as industry practices and pricing."). Labhart apparently analyzed a great deal of data first provided to him after he left Dynegy's employ. However, this does not change the character of his testimony. Valencia insinuates that the timing of Labhart's departure from Dynegy — after the acts complained of but before trial — means his testimony must be considered expert in nature. We cannot agree. Because Labhart's knowledge and analysis were derived from duties he held at Dynegy, his opinions were admissible as testimony based upon personal knowledge and experience gained while employed by Dynegy. See FED.R.EVID. 701. He engaged in precisely the kind of analysis he regularly performed as chief risk officer; the fact that he drew particular opinions and projection for the purposes of this case does not make him an "expert" within the meaning of Federal Rule of Evidence 702. See Tex. AM Research Found, v. Magna Transp., Inc., 338 F.3d 394, 403 (5th Cir. 2003).

C.

We now turn to Valencia's argument that Labhart gave inadmissible summary testimony. See FED.R.EVID. 1006. We review evidentiary rulings for an abuse of discretion, and in the event of error, we will affirm provided the error is harmless. See United States v. Bishop, 264 F.3d 535, 546 (5th Cir. 2001). While "[r]eview of evidentiary rulings is heightened in a criminal case," United States v. Gutierrez-Farias, 294 F.3d 657, 662 (5th Cir. 2002), to obtain reversal, the appellant "must demonstrate that the district court's ruling caused [her] substantial prejudice." Bishop, 264 F.3d at 546. Here, we may only review for plain error because Valencia failed to object to Labhart's use of the enlarged reports to convey his analysis. See FED.R.CRIM.P. 52(b).

We see no merit in Valencia's contention that Labhart gave inadmissible summary testimony. Labhart's testimony, and that of programmer Wanda Chovanec, indicate that Labhart analyzed all relevant trades contained in Dynegy's vast databases. The government stated at trial that it had provided defense counsel with these records approximately one month before trial began. Valencia did not dispute that the records had been provided, nor that the databases were available for her inspection. Valencia provides no support for the statement in her appellate brief that the government "provided the defense with disks containing various data splices that had been `cherry-picked' from those databases." At trial, Labhart provided visual depictions of his true-false, long-short, and penny-up penny-down analyses, which he was qualified to perform based upon his previous experience as chief risk officer. He wrote these findings on enlarged copies of the reports sent to Inside FERC and NGI, which had been admitted into evidence. In so doing, Labhart was able to compare and contrast the false price reports with the financial reality at Dynegy for each given month. This would assist the jury in understanding how voluminous records boiled down to the potential for profit or loss — a means of showing Valencia's motive for falsifying trade reports. See United States v. Jennings, 724 F.2d 436, 442 (5th Cir. 1984) (citation omitted) (recognizing that summaries are useful to demonstrate the contents of accounts or other business transactions).

Valencia complains that the government failed to introduce Dynegy's data-bases into evidence or show that the data-bases were business records subject to a hearsay exception. Valencia cites our decision in Bishop, where we noted that "Rule 1006 allows admission of summaries when (1) the evidence previously admitted is voluminous, and (2) review by the jury would be inconvenient." 264 F.3d at 547. Bishop is different because it concerned summaries of live testimony and exhibits presented in court, see id., not voluminous records which "cannot conveniently be examined in court." See FED.R.EVID. 1006; United States v. Nguyen, 504 F.3d 561, 571-72 (5th Cir. 2007) (citation omitted) (recognizing that Rule 1006 does not concern summaries of trial exhibits or trial testimony). Therefore, despite its broad wording, Bishop did not mandate that the databases be admitted into evidence before Labhart could summarize their contents, as this would contravene the plain language and purposes of Rule 1006. Additionally, Labhart and Chovanec testified that Dynegy maintained its databases in the course of regularly conducted business activities. Therefore, the district court did not err, much less abuse its discretion, in overruling Valencia's hearsay-based objections to Labhart's conclusions. See FED.R.EVID. 803(6).

Valencia was not sandbagged by Labhart's testimony or reliance on records from Dynegy's AREV and Abacus databases. Counsel cross-examined Labhart on the thoroughness and relevance of his calculations, revealing familiarity with Dynegy's use of databases to track its transactions and open positions. See Jennings, 724 F.2d at 442; Harris v. United States, 356 F.2d 582, 585 (5th Cir. 1966). In light of the nebulous objections raised by Valencia in the trial court — indeed, given her failure to cite Rule 1006 until appeal — we would be justified in concluding that an objection based on Rule 1006 was not properly raised, and therefore, is forfeited. However, following our independent review of the record, we are confident that Labhart's means of presenting his conclusions to the jury was proper. Valencia has not shown an abuse of discretion, much less "substantial prejudice" or plain error mandating a new trial. See Bishop, 264 F.3d 535, 546.

D.

We likewise hold that the government's failure to disclose its fee agreement with Labhart was not reversible error. Under the Brady rule, the government must give the defense impeachment evidence and exculpatory evidence. See United States v. Johnson, 872 F.2d 612, 619 (5th Cir. 1989). We review de novo whether the government violated the Brady rule. United States v. Fernandez, 559 F.3d 303, 319 (5th Cir. 2009). As the proponent of a new trial, Valencia must prove that: "(1) the prosecution did not disclose evidence; (2) the evidence was favorable to the defense; and (3) the evidence was material." United States v. Infante, 404 F.3d 376, 386 (5th Cir. 2005). Evidence is material if there is a reasonable probability that the outcome of the trial would have been different had such evidence been revealed to the defense. Id. "The question is not whether the defendant would more likely than not have received a different verdict with the evidence, but whether in its absence he received a fair trial, understood as a trial resulting in a verdict worthy of confidence." United States v. Brown, 303 F.3d 582, 593 (5th Cir. 2002) (quoting Kyles v. Whitley, 514 U.S. 419, 434, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995)). Moreover, the harmless error rule applies to Brady violations. Id. at 597. In United States v. Cervantes-Pacheco, 826 F.2d 310, 315-16 (5th Cir. 1987) (en banc), we reasoned that when the government has a fee arrangement with a testifying informant, it must make a "complete and timely disclosure" of this to defense. This gives the defendant "an adequate opportunity to cross-examine the informant and government agents," especially if the informant's compensation depends in part on a conviction. See id. at 316.

The government should have revealed in advance of trial that it had a fee agreement with Labhart. Such could have been used to impeach Labhart's motive to tell the truth. However, we conclude that if such failure constituted a Brady violation, it was immaterial and harmless. The jury heard that Labhart was receiving tens of thousands of dollars from the government and Dynegy in exchange for over 350 hours of analysis in this case. He also stated that he was subpoenaed to complete work he had not completed while he was employed by Dynegy. Labhart was not testifying as an accomplice to a crime seeking a reduced sentence or a "hired gun" in search of a bounty. Rather, Labhart was the most qualified person to perform the true-false, long-short, and penny-up penny-down analyses. There is no indication that Labhart anticipated receiving a bonus in the event of a guilty verdict. The facts and circumstances of Labhart's unique role in this case do not implicate the policy concerns we have expressed about undisclosed payments to informants testifying in criminal cases. Cf. Cervantes-Pacheco, 826 F.2d at 315-16, overruling Williamson v. United States, 311 F.2d 441, 444 (5th Cir. 1962) ("[W]e cannot sanction a contingent fee agreement to produce evidence against particular named defendants as to crimes not yet committed.").

Valencia asserts that the use of a form commonly used for expert witnesses shows the government's duplicity in hiding the nature of Labhart's "expert" testimony. We cannot agree that the government's use of a particular form to hire Labhart transforms his findings into expert testimony. We reiterate that Labhart performed analysis which was part and parcel of his daily duties while he was chief risk officer of Dynegy. Valencia also suggests that the boxes checked on the contract indicate that Labhart would allow his conclusions to be swayed by a "contractual obligation to provide inculpatory, expert testimony at trial." It is true that on the OBD-47 form, the box "support litigation/prosecution theory or conclusion" was checked. However, the government had always represented that Labhart's testimony and conclusions would support its theory of the case. Ultimately, it is the duty of the jury to determine whether Labhart was a credible witness despite his compensation arrangements. See Cervantes-Pacheco, 826 F.2d at 316 ("[W]e hold that the credibility of the compensated witness, like that of the witness promised a reduced sentence, is for a properly instructed jury to determine.").

Valencia says advance disclosure of the fee contract would have allowed her to more effectively impeach Labhart regarding his pecuniary motives, because counsel was focused on numerous other matters in this complex trial. Even if this is true, Valencia fails to show that the error was prejudicial — that inability to better impeach Labhart rendered her trial unfair. Even without the written contract, counsel for both Valencia and Singleton thoroughly cross-examined Labhart regarding his fee agreement and insinuated that the payments biased him towards the government's position. See United States v. McKinney, 758 F.2d 1036, 1050 (5th Cir. 1985) ("If the defendant received the material in time to put it to effective use at trial, his conviction should not be reversed simply because it was not disclosed as early as it might have and, indeed, should have been."). We do not condone the government's failure to reveal in advance its compensation agreement with Labhart. However, this misstep does not undermine confidence in the jury's verdict. See Infante, 404 F.3d at 386; Brown, 303 F.3d at 593, 597. Valencia is not entitled to a new trial on this basis.

IV.

The next issue on appeal, urged by both Valencia and Singleton, concerns the testimony of the government's expert, Matthew O'Loughlin. Defendants contend that the district court abused its discretion in admitting O'Loughlin's testimony, which fell short of the admissibility requirements of expert opinions. We disagree.

A.

The government hired O'Loughlin to evaluate whether defendants' false reports affected or tended to affect the natural gas indices published by Inside FERC and NGI. On July 5, 2006, days before trial began, defendants requested a hearing regarding expert testimony; on July 8, 2006, defendants moved to exclude O'Loughlin's testimony as unreliable under Federal Rule of Evidence 702 and Daubert. At a pretrial hearing, the court stated that the motion to exclude was untimely, but indicated that it would entertain the objection pursuant to its gatekeeping duty. Several days into trial, on July 14, 2006, the district court held a conference outside of the presence of the jury, at which it considered the admissibility of O'Loughlin's testimony. O'Loughlin was not present, but the court considered the report he had prepared pursuant to Federal Rule of Criminal Procedure 16, as well as counsel's arguments. The court explained that the purpose of the conference was to determine whether O'Loughlin's presence at a Daubert hearing would be necessary, or whether the court could make admissibility determinations based on information already available.

It was undisputed that indices published by Inside FERC and NGI were commonly used by participants in natural gas markets to price transactions. However, defendants contested whether false reports could have an effect on the published indices. Legally, this was relevant to the materiality of the reports for both the CEA and the wire fraud counts. At first blush, the relationship between price reports and indices would seem elementary: the editors of both Inside FERC and NGI testified that the reports were their starting point when determining index prices. However, neither editor could say that a given report could, or actually did, change the index price. Specifically, Kelly Doolan, chief editor of Inside FERC, testified that he examined the volume-weighted averages for bidweek trades. However, he said he exercised his judgment in determining whether to include all reported trades in the averages. Doolan frequently discarded what he considered outlier data, and said it "would be speculation" to conclude that a particular report had changed, or could change, the indices. He might also decide to alter an index price based on information not contained in the price reports. Mark Curran, chief editor of NGI, examined the trade reports and used editorial discretion in determining whether to include all trades in the volume-weighted average. However, Curran testified that, after deciding which figures to include or exclude, he typically published the final volume-weighted averages as NGI's index prices.

In light of the assertions of Doolan and Curran regarding their editorial discretion in creating the monthly indices, the government could not simply ask the jury to infer or assume that a given false price report had changed, or had the tendency or ability to change, a given index price. The government retained O'Loughlin in order to probe the relationship of the price reports to the indices. O'Loughlin was provided the trade reports submitted to Inside FERC. He calculated volume-weighted averages for all reported trades and delivery nodes, and compared these figures to Inside FERCs published index prices. In this manner he could test how closely the indices tracked the raw data. For NGI, the monthly reports were not available. However, O'Loughlin was able to obtain the final worksheets used by chief editor Mark Curran to calculate the publication's volume-weighted averages, which ultimately became the index prices. For both Inside FERC and NGI, O'Loughlin found a strong correlation between the volume-weighted averages of the trades submitted and the published index prices. O'Loughlin then examined the months when Valencia or Singleton allegedly submitted false reports. He removed the false trades in an effort to see whether the trades had the potential to change the volume-weighted averages. He found that they did.

At the July 14 conference, the court acknowledged that O'Loughlin's methodology was "not one that he purports others have used," and that he was "largely summarizing factual material and then doing an arithmetic calculation." However, the court also reasoned that this issue was res nova, as "there has been no need in the world for anyone to do the work that O'Loughlin has done." The upshot of O'Loughlin's analysis was, as the district court stated, an inference that Inside FERCs and NGI's published indices closely tracked the volume-weighted averages of the trades submitted to the publications. Thus, the editorial judgment of the editors was minimal. Defense counsel objected that O'Loughlin would essentially be asking the jury to disregard the editors' testimony. Moreover, Doolan and Curran exercised discretion in deciding which trades to include or exclude from the volume-weighted averages. Therefore, defendants asserted that O'Loughlin failed to account for a critical step in the process of creating the indices. The court ruled that this did not undermine O'Loughlin's methodology, but instead challenged an assumption he made in reaching his conclusions. The court reasoned that defendants' critiques affected the weight of the testimony, not its admissibility, and were fodder for cross-examination. Consequently, the court held that the government had met its burden of proving under Daubert that the expert's methodology and opinions were reliable, and were based on sufficient data.

O'Loughlin testified near the end of trial, from July 27 to 28, 2006. He stated that, in order to analyze Inside FERCs practices, he examined the trade reports for a twenty-eight month period from January 2000 through April 2002. These reports were used to create approximately 1,600 index prices for various trading points. He used the reports to create a volume-weighted average for each point. O'Loughlin found that for the vast majority of the data points, approximately eighty percent, the volume-weighted average was the same as the published index price. The data points were depicted on a graph and displayed to the jury. Based on these observations, O'Loughlin opined that the volume-weighted average "has a strong relationship with the index price," and that a change in the volume-weighted average was "very likely to lead to a different index price." In light of the frequency with which the volume-weighted average and the index price were very close or identical, he inferred that the volume-weighted average was "an explanatory variable for the index price." He further explained that, in light of the data, it appeared that there was a relationship between the volume-weighted averages and the index prices, "and the relationship is showing that the change in the volume-weighted average appears to be one for one with the change in the index price."

O'Loughlin reached a similar conclusion for NGI. Although he lacked the bidweek data submitted to NGI, O'Loughlin read from NGI's methodology statement, which stated: "The prices that appear in NGI's gas price index represent the volumetric weighted average of negotiated fixed price transactions that occur during a particular time period. . . ." As O'Loughlin explained, the methodology statement acknowledged that outlier data would be discarded, and that the volume-weighted average of the remaining data set would be published as the index price. On this basis, he concluded that the volume-weighted average was the most important component of NGI's indices.

In addition, O'Loughlin testified about whether the false trades submitted by defendants had the potential to change the volume-weighted averages compiled by Inside FERC and NGI. In the case of trades submitted by Valencia to Inside FERC and identified by Glenn Labhart as false, O'Loughlin found that removing the trades could change the volume-weighted averages, in some cases by several cents. O'Loughlin engaged in this analysis for the false trades submitted to NGI by both Valencia and Singleton, and concluded that these also had the potential or tendency to affect the volume-weighed averages. Finally, O'Loughlin drew the inference that because volume-weighted averages were the most important component of the indices, and because trade reports had the potential to affect volume-weighted averages, false trade reports had the tendency or capability to affect the final published index prices. In particular, O'Loughlin noted that, based upon his review of Curran's testimony, most trades submitted by Dynegy and El Paso would be included in the volume-weighted averages, and hence, the indices. His conclusion covered both Inside FERC and NGI: "[G]iven that there is this relationship that we've seen both for Inside FERC and NGI between changes in the volume-weighted average and changes in the index price, my sense is that if, in fact, the volume-weighted average is being affected by more than a penny, it is likely that there was some influence on the index price."

B.

Valencia and Singleton argue that the district court abused its discretion in admitting O'Loughlin's opinions. Defendants do not contest his credentials or qualifications, but rather, argue that his testimony was not relevant or reliable. Defendants fault the district court for failing to require that O'Loughlin be present for voir dire at a pre-trial Daubert hearing, and for ruling on the admissibility of his opinions mid-trial. Defendants say the court abdicated its gatekeeping role over expert testimony, and allowed spurious testimony to reach the trier of fact. Defendants criticize O'Loughlin's calculations as unreliable tweaks to volume-weighted averages, which disregarded Doolan's and Curran's testimony that their respective indices "were determined through subjective editorial judgments." As such, defendants characterize O'Loughlin's conclusions as impermissible attacks on the witnesses' credibility. Defendants say O'Loughlin's analysis was unreliable because he did not conduct multiple-regression analysis to account for potentially confounding variables affecting the index prices. On this basis, O'Loughlin allegedly posited an unsubstantiated causal link between the defendants' false reports and changes in the published indices. Defendants say the admission of the testimony prejudiced them because the aura of credibility conferred by O'Loughlin's expert status likely swayed the jury to find the materiality element of the crimes, which no other evidence in the record directly provided.

Moreover, Valencia contends that O'Loughlin wrongly conveyed hearsay directly to the jury. O'Loughlin evaluated so-called voice broker data generated by services such as the now-defunct Enron Online, which matched buyers and sellers of gas. After evaluating these records, O'Loughlin testified that Dynegy used Inside FERC's indices in California to price gas contracts. This contradicted Jeffrey Hornback's testimony that Dynegy used NGI's indices to price contracts in California. O'Loughlin allegedly failed to state in his expert report that he would consider voice broker data or present this conclusion. Singleton argues separately that O'Loughlin wrongly concluded that Singleton's July 31, 2000 email to NGI "likely" influenced NGI's Southern California Border index. However, Singleton says that the volume-weighted average for Curran's worksheets for that month remains the same whether Singleton's trades are included or excluded. Singleton says this erroneous statement indicates that O'Loughlin's opinions are unreliable.

The government says O'Loughlin was qualified to testify as an expert in the fields of economics and energy affairs with experience evaluating natural gas indices. Moreover, his analysis conforms to the requirements of relevance and analytical rigor required of an expert in economics or statistics. The government contends that multiple regression analysis was not necessary because O'Loughlin was trying to establish that, notwithstanding the editorial judgment exercised by the editors of Inside FERC and NGI, defendants' false reports had the potential or tendency to influence index prices. The government also states that defendants did not criticize the lack of regression analysis during voir dire of the witness or before the district court. Because success of a scheme is not an element of wire fraud, the government says it was proper to allow O'Loughlin to testify that it was possible or likely that the false reports were market-moving communications without eliminating all confounding or contributory factors.

C.

We review the admission or exclusion of expert testimony for an abuse of discretion. Kumho Tire Co. v. Carmichael, 526 U.S. 137, 152, 119 S.Ct. 1167, 143 L.Ed.2d 238 (1999). The district court's ruling will not be disturbed on appeal unless it is manifestly erroneous. United States v. Norris, 217 F.3d 262, 268 (5th Cir. 2000). If the court abuses its discretion, judgment will be affirmed under the harmless error doctrine unless the error affected a substantial right of the defendant. Id. (citations omitted); Watkins v. Telsmith, Inc., 121 F.3d 984, 988 (5th Cir. 1997) (noting that "[d]istrict courts enjoy wide latitude in determining the admissibility of expert testimony") (citations and internal quotations omitted).

Under Federal Rule of Evidence 702, district courts are assigned a gatekeeping role to determine the admissibility of expert testimony. Daubert, 509 U.S. at 592-93, 113 S.Ct. 2786. The court must find that the evidence is both relevant and reliable before it may be admitted. Id. To do so, the court must evaluate whether the reasoning and methodology underlying the testimony is valid and can be reliably applied to the facts of the case. Id. This requires more than a glance at the expert's credentials; the court must also ensure that the expert has reliably applied the methods in question. See Moore v. Ashland Chem. Inc., 151 F.3d 269, 276 (5th Cir. 1998) (en banc). Factors to consider when evaluating reliability include: (1) whether a theory or technique can be tested; (2) whether the theory or technique has been subjected to peer review and publication; (3) the known or potential rate of error; (4) the existence and maintenance of standards and controls; and (5) general acceptance of the theory in the scientific or expert community. Daubert, 509 U.S. at 593-95, 113 S.Ct. 2786. Nevertheless, the reliability inquiry is flexible, and the judge has discretion in determining which factors are most germane in light of the nature of the issue, the particular expertise, and the subject of the expert's testimony. Id. at 594-95, 113 S.Ct. 2786; Kumho Tire, 526 U.S. at 142, 119 S.Ct. 1167. Overall, the trial court must strive to ensure that the expert, "whether basing testimony on professional studies or personal experience, employs in the courtroom the same level of intellectual rigor that characterizes the practice of an expert in the relevant field." Kumho Tire, 526 U.S. at 152, 119 S.Ct. 1167.

The relevance and reliability of expert testimony turns upon its nature and the purpose for which its proponent offers it. See, e.g., Hodges v. Mack Trucks, Inc., 474 F.3d 188, 195 (5th Cir. 2006) ("Of course, whether a proposed expert should be permitted to testify is case, and fact, specific") (citing Kumho Tire, 526 U.S. at 150-51, 119 S.Ct. 1167). Expert statistical opinions are often offered to demonstrate a causative relationship between a dependent variable and an explanatory variable because the existence vel non of a causal link is a legally relevant fact of consequence. See, e.g., Munoz v. Orr, 200 F.3d 291, 300 (5th Cir. 2000) (in Title VII disparate impact claim, "the evidence will focus on the degree of statistical disparity between protected and non-protected workers in regards to employment or promotion"); Sheehan v. Daily Racing Form, Inc., 104 F.3d 940, 942 (7th Cir. 1997) (Posner, J.) (in age discrimination claim, "equating a simple statistical correlation to a causal relation . . . indicates a failure to exercise the degree of care that a statistician would use in his scientific work"); McClain v. Metabolife Int'l, Inc., 401 F.3d 1233, 1243 (11th Cir. 2005) (in claims for toxic torts or negligent prescription, evidence of correlation or temporal proximity cannot establish required causative nexus between defendant's act and plaintiff's ensuing injury). Logically and legally speaking, in such cases causal evidence is relevant to the jury. See Huss v. Gayden, 571 F.3d 442, 457 (5th Cir. 2009) (noting, in dictum, that evidence of injury by negligent prescription must be shown "to a reasonable degree of medical certainty") (citation omitted), petition for cert. filed, 78 U.S.L.W. 3447 (U.S. Jan. 8, 2010) (No. 09-842).

Evidence of mere correlation, even a strong correlation, is often spurious and misleading when masqueraded as causal evidence, because it does not adequately account for other contributory variables. See, e.g., id. at 459 ("Any scientist or statistician must acknowledge, however, that correlation is not causation."); Munoz, 200 F.3d at 301-02; Sheehan, 104 F.3d at 942 ("Completely ignored was the more than remote possibility that age was correlated with a legitimate job-related qualification, such as familiarity with computers."). However where evidence of correlation itself is potentially relevant and unlikely to mislead the jury, an expert who reliably discerns this relationship can present such conclusions to the jury. See Pirlott v. NLRB, 522 F.3d 423, 435, 436 (D.C. Cir. 2008) (reasoning that, in order to charge objecting nonmembers union dues, union can show "a positive correlation between wages and union density in the relevant market at issue"; expert witness presented such evidence) (internal quotation and citation omitted); United States v. W.R. Grace, 504 F.3d 745, 765 (9th Cir. 2007) (holding that "the fact that a study is associational — rather than an epidemiological study intended to show causation — does not bar it from being used to inform an expert's opinion about the dangers of asbestos releases"); United States v. White, 356 F.3d 865, 870 (8th Cir. 2004) ("Our court recognizes the known correlation between drug dealing and weapons, and accepts that they are closely and integrally related to the issue of possession of a firearm."); United States v. Hopkins, 310 F.3d 145, 151 (4th Cir. 2002) (holding that expert testimony that defendant's behavior was consistent with dealing crack was relevant and reliable in light of expert's law enforcement experience and analysis of germane facts in the case). Whether a particular opinion is relevant and reliable thus does not simply turn on whether the expert asserts a causal or correlative relationship, but is closely tied to the law and facts at issue in a given case. See Hodges, 474 F.3d at 195.

D.

Having reviewed the parties' arguments, O'Loughlin's testimony, and the district court's thorough examination of the proffered opinions, we cannot conclude that the court abused its discretion and committed manifest error in allowing O'Loughlin to testify about the tendency of defendants' false trade reports to affect the indices published by Inside FERC and NGI.

Regarding the Daubert factors, the district court recognized that the unprecedented nature of O'Loughlin's work made it impractical, if not impossible, to subject the methods to peer review and publication, and that there would be no general acceptance of the theory in the scientific or expert community. See 509 U.S. at 593, 113 S.Ct. 2786 (noting that publication "is not the sine qua non of admissibility; it does not necessarily correlate with reliability, and in some instances well-grounded but innovative theories will not have been published") (internal citations omitted). However, given the arithmetic underpinnings of O'Loughlin's analysis, one can test the theories and determine the rate of error. See id. Indeed, by noting that the volume-weighted averages matched the published indices approximately eighty percent of the time, O'Loughlin acknowledged that his opinion — that prices reports had the tendency to affect indices — was imperfect. The district court recognized that the fit between the data and O'Loughlin's theory was approximate, but reasoned this did not render the opinions altogether unreliable, and defendants could highlight any inconsistencies on cross-examination. In light of the district court's insightful consideration of, and fidelity to, the Daubert factors at this necessarily "flexible" stage of the trial, see id., we cannot say that the court abused its discretion in admitting O'Loughlin's testimony as sufficiently rigorous economic and statistical analysis. The record belies defendants' assertion that the court abdicated its duty to exclude irrelevant or spurious testimony from the courtroom. Moreover, the court did not abuse its discretion in declining to require that O'Loughlin be present at the Daubert hearing, especially in light of defendants' eleventh-hour motion to exclude his testimony.

Concerning relevance, it is important to bear in mind that in this case, defendants were charged with, and we now review convictions solely based upon, wire fraud. To prove wire fraud, the government must show a scheme to defraud, the use of wire communications in furtherance of the scheme, and the defendant's specific intent to participate in the scheme. United States v. Stalnaker, 571 F.3d 428, 436 (5th Cir. 2009) (citations omitted). The government must also prove that the communications were material. Neder v. United States, 527 U.S. 1, 25, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999). A "material" statement has "a natural tendency to influence, or is capable of influencing, the decision of the decision-making body to which it was addressed." See United States v. Lucas, 516 F.3d 316, 339 (5th Cir.) (quotation omitted), cert. denied, ___ U.S. ___, 129 S.Ct. 116, 172 L.Ed.2d 36 (2008); United States v. Philip Morris USA Inc., 566 F.3d 1095, 1122 (D.C. Cir. 2009) ("This materiality requirement is met if the matter at issue is of importance to a reasonable person in making a decision about a particular matter or transaction.") (citation and internal quotation omitted). Success of the scheme is not an element of the crime. See United States v. Loney, 959 F.2d 1332, 1337 (5th Cir. 1992).

Defendants agree that O'Loughlin's testimony concerned the materiality of defendants' false reports. For this element, it was incumbent on the government to show that the false reports had the tendency to influence the indices published by Inside FERC and NGI. It was not legally determinative whether a particular report actually changed a particular index price. O'Loughlin concluded, after evaluating the relationship between the data and the indices, that the information submitted by Valencia and Singleton, or at their direction, would be important to a reasonable person seeking to determine the price of natural gas at the trading points for which defendants submitted reports. This inference is largely consistent with the testimony of Doolan and Curran, who said they considered the bidweek reports when creating index prices. While it was a matter of editorial discretion whether a given trade would be included within a volume-weighted average, and ultimately, the index price, O'Loughlin demonstrated that this discretion was either infrequently invoked or played a comparatively minor role.

We acknowledge that O'Loughlin's conclusions were somewhat in tension with the statements of Doolan and Curran. However, O'Loughlin did not simply urge the jury to disregard their testimony. O'Loughlin's conclusion was well grounded in his thorough analysis of the data, not a metaphysical disagreement with the editors' assertions. We therefore reject defendants' argument that O'Loughlin's opinion was an unsubstantiated and impermissible attack on the editors' credibility. Because O'Loughlin's testimony helped show that the bidweek reports were important to Doolan and Curran, the testimony was relevant to the element of materiality. See Philip Morris, 566 F.3d at 1123 ("The question, however, is not whether a reasonable person would have believed Defendants' false statements, but only whether a reasonable person would have considered the issue of importance. . . .") (quotation marks omitted).

As for reliability, O'Loughlin carefully explained the nature of his conclusions so that he remained within the scope of his properly admitted opinions. Contrary to defendants' representations, O'Loughlin did not state that the false reports changed the published indices. Rather, he stated that volume-weighted averages and index prices often moved in tandem, and that volume-weighted averages were an "explanatory variable" for index prices. Based upon this observation, it was possible, or in some cases, likely that the false reports would have an effect on the volume-weighed averages, and hence, the indices. The statements cited by defendants as evidence of unsubstantiated causation testimony are either couched in terms of changes to volume-weighted averages or are answers to hypothetical questions. In isolated instances, O'Loughlin stated that false reports changed the indices. However, if counsel objected, the district court sustained the objection, and O'Loughlin then corrected himself. Such slip-ups are not prejudicial in light of the fact that O'Loughlin meticulously maintained the distinction between volume-weighted averages and final index prices. Moreover, on cross-examination, O'Loughlin stated that he did not assume, nor conclude after his analysis, that any one-cent change in a volume-weighted average would lead to a corresponding one-cent change in an index price. Therefore, O'Loughlin's opinions were both relevant and reliable.

We disagree with defendants' assertion that the testimony should have been excluded because O'Loughlin did not employ multiple regression analysis. Multiple regression analysis is a tool for understanding the relationship between a dependent and an explanatory variable. See generally Daniel L. Rubinfeld, Reference Guide on Multiple Regression, in REFERENCE MANUAL ON SCIENTIFIC EVIDENCE 179 (2d ed. 2000). Defendants are correct to point out that this is a powerful tool when the trier of fact must determine whether a causal link exists. Nevertheless, regression analysis is not a mandatory feature in all applications of economics or statistics. See Adams v. Ameritech Servs., Inc., 231 F.3d 414, 425 (7th Cir. 2000) ("[W]e are not prepared to hold as a matter of law that nothing but regression analyses can produce evidence that passes the Daubert and Kumho Tire thresholds. Statisticians might have good reasons to look at data in different ways.").

In cases where a causal relationship is not an essential fact of consequence, an expert need not eliminate all confounding variables or potential contributory factors in order to present an opinion that is both relevant and reliable. See Int'l Bhd. of Teamsters v. United States, 431 U.S. 324, 340, 97 S.Ct. 1843, 52 L.Ed.2d 396 (1977) (noting that the usefulness of statistics "depends on all of the surrounding facts and circumstances"); cf. Mathis v. Exxon Corp., 302 F.3d 448, 461 (5th Cir. 2002) ("Although Pulliam may not have isolated the precise effect Exxon's pricing had on each station, that was not the purpose of his testimony. The `subject of his testimony,' as listed by plaintiffs, was whether Exxon had set a commercially reasonable price in an economic sense."). In this case, to show that defendants' statements were material, the government had to prove that they were important to Doolan and Curran, not that the statements caused Doolan and Curran to change the indices. See Philip Morris, 566 F.3d at 1123. O'Loughlin reached his conclusions after applying statistical methods in a reliable manner. In this case, the lack of regression analysis affected the weight which the jury assigned to the expert's testimony, not its legal admissibility.

Valencia also contends that O'Loughlin relied on inadmissible hearsay to fill a "gap" in the government's case. Specifically, Jeffrey Hornback testified that Dynegy used NGI's indices to price California contracts. O'Loughlin relied on voice broker records to show that Dynegy also used Inside FERCs indices. Valencia says this is significant because the false reports she sent were mostly directed to Inside FERC. Valencia complains that O'Loughlin did not disclose in his Rule 16 report that he would rely on voice broker data, and that O'Loughlin directly conveyed inadmissible documents to the jury. At trial, the district court overruled Valencia's hearsay-based objections to Government Exhibit 686, which purported to show, inter alia, that Dynegy used Inside FERCs indices for contracts in California.

We do not believe that the court abused its discretion in allowing O'Loughlin to present this conclusion to the jury. O'Loughlin did not directly convey voice broker data to the jury. Rather, he analyzed the data, concluded that Dynegy used Inside FERC's indices to price transactions, and conveyed this conclusion to the jury in the form of an exhibit he created. Valencia has not shown that the court abused its discretion in treating the voice broker data as business records, nor that the court erred in holding that O'Loughlin was entitled to rely on the records in forming his expert opinion. See FED.R.EVID. 703 ("If of a type reasonably relied upon by experts in the particular field in forming opinions or inferences upon the subject, the facts or data need not be admissible in evidence in order for the opinion or inference to be admitted."); United States v. Avants, 367 F.3d 433, 447 (5th Cir. 2004). Because Valencia did not object at trial that the testimony was beyond the scope of O'Loughlin's Rule 16 expert report, we may only review for plain error. See Avants, 367 F.3d at 446. Government Exhibit 431, a Dynegy contract using an Inside FERC index to price a gas transaction in California, was submitted into evidence without objection. Because independent evidence corroborates O'Loughlin's conclusions, Valencia's substantial rights were not affected. See Mares, 402 F.3d at 520.

Singleton maintains that O'Loughlin's materiality opinion is inadmissible because the report Singleton submitted on July 31, 2000 could not have changed NGI's August 2000 Southern California Border Average index. O'Loughlin did state that the false trades reported by Singleton had no effect on the volume-weighted average. During cross-examination, O'Loughlin again acknowledged that the volume-weighted average remained the same whether Singleton's false trades were included or excluded. Singleton asserts that the inability of the report to change the volume-weighted average undermines O'Loughlin's opinions. This argument does not help Singleton for two reasons. First, it is the tendency of a communication to affect a decision maker which is relevant, not whether the communication actually had an effect. O'Loughlin posited such a tendency after demonstrating the propensity of reported trades to affect volume-weighted averages, and the propensity of volume-weighted averages to affect indices. Second, if the expert has used reliable methods to assess the data, isolated computational or arithmetic discrepancies affect the weight of the testimony, not its admissibility. See Daubert, 509 U.S. at 595, 113 S.Ct. 2786 (noting that trial courts must focus on experts' "principles and methodology, not on the conclusions that they generate"). O'Loughlin freely noted that Singleton's report of July 31, 2000 would not have changed the volume-weighted average. Nevertheless, the jury was free to continue to rely on O'Loughlin's opinion that the false reports had the tendency or capability to affect the indices.

In sum, the district court properly exercised its duties under Rule 702 and Daubert, and did not abuse its discretion in admitting the testimony of Matthew O'Loughlin. Defendants are not entitled to a new trial.

V.

Valencia says the numerous evidentiary errors in this case deprived her of a fair trial and, cumulatively, mandate a new trial. The government has not responded to this argument.

We have previously recognized the so-called cumulative error doctrine under which "an aggregation of non-reversible errors (i.e., plain errors failing to necessitate reversal and harmless errors) can yield a denial of the constitutional right to a fair trial." United States v. Munoz, 150 F.3d 401, 418 (5th Cir. 1998); see United States v. Sepulveda, 15 F.3d 1161, 1195-96 (1st Cir. 1993) (explaining the cumulative error doctrine).

United States v. Williams, 264 F.3d 561, 572 (5th Cir. 2001). A claim of cumulative error is " sui generis;" we evaluate the number and gravity of the errors in the context of the case as a whole. See Sepulveda, 15 F.3d at 1196.

To recapitulate our holdings, we have concluded that the government committed non-reversible error in reading the letter of Jeffrey Hornback during opening statements, and non-reversible error in failing to timely and fully notify defense of its fee agreement with Glenn Labhart. We have rejected defendants' arguments concerning the propriety of the district court's evidentiary and admissibility rulings, as well as other matters concerning the fairness of their trial. We do not condone the government's missteps. But, we recognize that no trial is perfect, and the shortcomings here did not deprive defendants of a fair trial. Moreover, the district court assiduously and scrupulously weighed all objections which were timely brought to its attention. In the context of the vast amount of evidence presented over the course of four weeks, and in light of the weight of the evidence supporting defendants' wire fraud convictions, we reject Valencia's assertion that cumulative error mandates retrial in this case. See id.

VI.

Both Valencia and Singleton argue that the evidence presented was not sufficient to sustain their wire fraud convictions. Valencia says the evidence was insufficient to show that she submitted false trades, to show her mens rea, and to prove materiality. She says Labhart's testimony did not establish falsity because he did not show that the "false" trades listed in the trade reports did not actually occur. Valencia posits that not all trades may have been recorded, that trade data may have been lost, that there may have been transcription errors, and that Labhart's search may have been unreliable or incomplete. Regarding mens rea, Valencia says the jury would have to infer intent from scant evidence that her reports did not perfectly match actual bidweek trades. Thus, "the jury could only speculate impermissibly whether the mere presence of some incongruity was the result of intentionally fraudulent misreporting" or an honest error. Finally, Valencia insists materiality was not demonstrated because the editors of Inside FERC and NGI said they used editorial discretion to set index prices. Singleton similarly argues that there is insufficient evidence of the materiality of the false statement he sent on July 31, 2000. Alternatively, Singleton says the evidence preponderated against his guilt, such that the district court should have ordered a new trial.

Additionally, defendants have submitted pursuant to Federal Rule of Appellate Procedure 28(j) asserting that the government needed to prove that the false information provided to Inside FERC and NGI "affected and tended to affect index prices." This language was alleged in the CEA counts of both indictments, and was "adopted, realleged and incorporated" into the wire fraud counts as well. Defendants insinuate that unless the government has shown beyond a reasonable doubt that the false reports affected index prices, the materiality element of wire fraud has not been met. Alternatively, defendants suggest that failure to prove at trial that their communications "affected and tended to affect index prices" constitutes a material variance.

The government emphasizes that evidence in emails, spreadsheets, and other documents, as well as in taped phone conversations, shows that Valencia and Singleton engaged in the scheme with the intent to defraud, and that their statements were both false and material. The government says that even if the witnesses' statements do not always align, the jury was entitled to credit the theory supporting guilt. The government also argues that the allegation that defendants' actions "affected and tended to affect index prices" was mere surplusage and not a material variance from the proof at trial.

A.

We review de novo the denials of defendants' motions for judgment of acquittal under Federal Rule of Criminal Procedure 29(a). See United States v. Myers, 104 F.3d 76, 78 (5th Cir. 1997). We must affirm the verdict if "a reasonable trier of fact could conclude from the evidence that the elements of the offense were established beyond a reasonable doubt, viewing the evidence in the light most favorable to the verdict and drawing all reasonable inferences from the evidence to support the verdict." Id. The evidence "need not exclude every reasonable hypothesis of innocence or be wholly inconsistent with every conclusion except that of guilt." Id. at 79 (quotations omitted). The denial of a motion for a new trial under Rule 33(a) is reviewed for an abuse of discretion. See United States v. Sipe, 388 F.3d 471, 492-93 (5th Cir. 2004). To prove wire fraud, the government had to show a scheme to defraud, the use of wire communications in furtherance of the scheme, defendants' specific intent to participate in the scheme, and materiality of defendants' communications. See 18 U.S.C. § 1343; Neder, 527 U.S. at 25, 119 S.Ct. 1827; Stalnaker, 571 F.3d at 436; Lucas, 516 F.3d at 339.

B.

The evidence presented at trial was sufficient to sustain all seven counts of wire fraud upon which Valencia was convicted, as well as the sole count of wire fraud on which Singleton was convicted. Jeffrey Hornback testified extensively about a scheme at Dynegy to misstate natural gas trades in order to benefit Dynegy's trading position. Hornback's testimony implicated Valencia in the scheme. Valencia's own recorded phone calls, while at times cryptic or oblique, presented evidence from which the jury could infer that Valencia knowingly and willingly participated in the scheme. Moreover, for each count on which Valencia was convicted, the government showed a communication transmitted via the wires by Valencia or at her direction to Inside FERC or NGI. The testimony of Glenn Labhart indicated that false information was contained in the bidweek reports. While Valencia surmises that Labhart's testimony might not have been wholly accurate, the jury was entitled to reject this hypothesis. The properly admitted testimony of Matthew O'Loughlin established that the communications were material, i.e., the communications were important to, and had the tendency to affect, the decision-makers to whom they were directed. See Philip Morris, 566 F.3d at 1124; Lucas, 516 F.3d at 339. The jury was entitled to believe that the false trades had the tendency to sway the indices, notwithstanding the testimony of Doolan and Curran that editorial discretion played a role in how the index prices were generated.

In Singleton's case, the evidence is likewise sufficient to sustain the guilty verdict based on Singleton's July 31, 2000 report to NGI. The jury could infer Singleton's participation in the scheme to misreport gas trades based upon a July 28, 2000 phone call with Valencia. Namely, Singleton and Valencia tentatively agree to a sale of gas, but Singleton tells Valencia: "If, if you buy it from me you don't have to report it. I won't report it." Valencia replies: "Ok, that's a cool thing." Alison Reitze, a trader with El Paso, testified that El Paso's records showed a fixed-price baseload trade between El Paso and Dynegy, entered into by Singleton and Valencia, respectively, on July 28, 2000. The trade met NGI's reporting criteria, but was not included on the report Singleton sent to NGI on July 31, 2001. Ronald Clay Sanders, the director of risk management and analysis for El Paso's marketing and trading company, likewise testified that many of the trades listed in Singleton's July 31, 2000 report to NGI were not listed in El Paso's risk management database. The foregoing facts adequately show Singleton's knowing participation in, and use of the wires in furtherance of, a scheme to misreport trades.

Singleton again suggests that the false trades were not material because NGI's Southern California Border index was the same whether his trades were included or excluded. He also states that O'Loughlin misrepresented the testimony of Mark Curran, chief editor of NGI, regarding how the index was calculated. Singleton emphasizes that Curran testified that he had no record of how he created the index, and that the index price was entirely at his discretion. As discussed above, even if the false trades reported by Singleton did not change the index, this does not mean the communication was immaterial. See Philip Morris, 566 F.3d at 1124; Lucas, 516 F.3d at 339. Success is not an element of the crime. See Loney, 959 F.2d at 1337. O'Loughlin's testimony also presented evidence which, if found credible by the jury, would support a well-grounded inference that trade reports from Singleton and other natural gas traders were important to Curran, regardless of his undisputed editorial discretion. At most, Singleton's arguments concern the weight the trier of fact could assign to various pieces of evidence. Because a reasonable jury could have found each element present beyond a reasonable doubt, Singleton's wire fraud conviction stands.

C.

The language in defendants' indictments is not defective or a material variance from the proof adduced at trial. We normally disregard arguments not briefed or raised for the first time at or after oral argument. See, e.g., United States v. Whitfield, 590 F.3d 325, 370 (5th Cir. 2009). At best, we may review for plain error. Id. at 371. Even so, defendants' arguments lack merit. We have long held that the government need only prove facts alleged in the indictment which meet the essential elements of the crime. United States v. Robinson, 974 F.2d 575, 578 (5th Cir. 1992); United States v. England, 480 F.2d 1266, 1269 (5th Cir. 1973). We treat the allegation of additional facts beyond those which comprise the elements of the crime as "mere surplusage." Robinson, 974 F.2d at 578. The materiality element of wire fraud does not require a showing that a false communication actually caused an intended consequence. Thus, the allegation in the indictment that Valencia and Singleton "affected and tended to affect index prices" is surplusage. See id.; United States v. Hughes, 766 F.2d 875, 879 (5th Cir. 1985).

There was also no material variance between the indictment and the proof at trial. "A variance is material if it prejudices the defendant's substantial rights, either by surprising the defendant at trial or by placing the defendant at risk of double jeopardy." Robinson, 974 F.2d at 578 (citations and internal quotations omitted). A variance is immaterial if the nature of the charge remains the same. See id.; see also United States v. Millet, 123 F.3d 268, 272 (5th Cir. 1997) ("A constructive amendment to the indictment occurs when the jury is permitted to convict the defendant on a factual basis that effectively modifies an essential element of the offense charged in the indictment.") (citations omitted). In this case, the wire fraud scheme alleged in the indictment was the same as that demonstrated at trial, namely, that defendants submitted false natural gas trade reports to Inside FERC and NGI. The success of the schemes is not relevant to wire fraud. Therefore, defendants could not have been surprised or exposed to double jeopardy for these counts if the proof did not demonstrate an unequivocal causal effect between the reports and changes to the indices. See Robinson, 974 F.2d at 578. There was no plain error. See Millet, 123 F.3d at 272.

In sum, there was sufficient evidence to sustain all counts on which the jury returned verdicts of guilty. We reiterate that we have only reviewed the sufficiency of the counts of wire fraud. We have not considered the conspiracy counts, nor the substantive counts of violations of the Commodities Exchange Act, which were dismissed at the government's motion in light of the jury's partial verdict. We express no opinion regarding whether the evidence was sufficient to support such counts.

VII.

Finally, Valencia argues that the district court erred in failing to consider whether she was entitled to a downward departure based on her caregiver status. Sentencing took place on August 21, 2008, more than two years after trial. The Guidelines range was forty-six to fifty-seven months. Valencia had recently given birth to a daughter with Turner's syndrome, and requested a departure for this reason. The court said Valencia's decision to conceive and carry a child to term, despite her imminent incarceration, "was a personal choice. . . . I cannot change my sentence for that reason." The court sentenced Valencia to concurrent fifty-seven month terms of imprisonment, the top of the Guidelines range, followed by concurrent two-year terms of supervised release. Valencia avers that the court's statement, "I cannot change my sentence for that reason," reveals the district court's failure to comprehend its broad authority to grant a departure. See Gall v. United States, 552 U.S. 38, 49-50, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007). Valencia requests resentencing. The government counters that the court's statement reflects its understanding of its duties to uphold the law and apply the factors of 18 U.S.C. § 3553(a), and not a sense that it lacked discretion to grant a downward departure.

Review of a district court's sentence is governed by a two-step process, in which we first ask whether the district court committed a procedural error. United States v. Delgado-Martinez, 564 F.3d 750, 751 (5th Cir. 2009). If there is no error or the error is harmless, we review the substantive reasonableness of the sentence for an abuse of discretion. Id.; see Gall, 552 U.S. at 51, 128 S.Ct. 586. There is no indication, nor does Valencia argue, that the court committed a procedural error in calculating the Guidelines range. Moreover, the court acknowledged its authority and responsibilities, including those stated in Kimbrough v. United States, 552 U.S. 85, 128 S.Ct. 558, 169 L.Ed.2d 481 (2007), United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), and under § 3553(a). The court's statement does not indicate misapprehension of its authority to give Valencia the sentence it deemed proper, whether within, above, or below the Guidelines range. The court also stated to Valencia: "I'm going to give you the sentence that I think is the right sentence regardless of the Guidelines." The court thereafter explicitly acknowledged that the Guidelines were not binding, and that the court had the authority to deviate if it so chose. In context, the words "I cannot" are best understood as "I will not." This isolated comment does not indicate that the district court believed that the Guidelines range should presumptively apply. See United States v. Cisneros-Gutierrez, 517 F.3d 751, 766 (5th Cir. 2008). Valencia is not entitled to resentencing.

VIII.

We have exhaustively examined the vast trial record, defendants' arguments on appeal, and the relevant law. We conclude that defendants' convictions withstand each of the challenges raised. We therefore affirm the convictions and sentences of defendants Valencia and Singleton.

AFFIRMED.