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NOTE: The pro projections below are first posted on Thursday, then updated on Saturday afternoon. On Thursday, commentary will be limited to early impressions. A more-detailed discussion of specific players will be added on Saturday afternoon.

Welcome back for Week 4 of Footballguys' new feature that uses a statistical model to project the roster percentages of "professional" tournament players on Draftkings. If you're curious about the full methodological details of the model, click here. The most important detail to keep in mind, however, is that the "professionals" in question were identified using a clustering algorithm and can be characterized by the following statement: Compared to the rest of the population, these 205 pros play more, they win more, and they do both consistently.

So let's examine some data about how well the model has performed through the first three weeks of the season.

Below is a table of correlations (by position) showing how well or poorly the pro projections model has predicted actual roster percentages based on the actual lineups pros have entered in the low buy-in featured tournaments this author also entered (as well as the Millionaire Maker).

As an example of how to read the table, across 98 projections made so far at running back, the model has demonstrated a 0.918 correlation with what the pros' actual roster percentages ended up being. But what does 0.918 mean? Well, for a correlation, the closer it is to 1.000, the stronger the relationship between the two sets of numbers. In this particular context, what's desirable is a correlation greater than or equal to .707. Therefore, the 0.918 figure for running backs implies the model has done an excellent job the first three weeks. It turns out the same can be said for tight ends, where a nearly equivalent correlation exists.

The lineup position at which the model hasn't done well is defense, where the correlation is only 0.646. This lower level of accuracy can be attributed to three double-digit errors in the first two weeks of the season. Specifically, in Week 1, the model projected Baltimore's defense to appear in 25.6% of pro lineups, but they ended up in only 15.5%. An even worse error in Week 1 involved the Patriots' defense, which the model projected at 5.7%, but ended up on 21.2% of pro lineups. The third double-digit error was discussed last week in this space: Houston in Week 2. If one were to ignore these three errors (out of 58 total projections), the correlation would increase all the way to 0.756.

Of course, that's not how the world works. We can't just wish our problems away, after all. Instead, the task is to figure out whether these large errors were the result of something knowable and predictable or the result of sheer randomness -- and then change the model or not depending on which one of those it is.

So now that you know how well the model is performing through three weeks, let's see what it has to say regarding pro roster percentages for Week 4.

Quarterbacks

Below is the table for quarterbacks. In this and all subsequent tables, "Overall" is Steve Buzzard's projection of roster percentage. "Pros" is the model's projection of roster percentage for Draftkings pros only.

Thursday Early Impressions

If you're experiencing a bit of déjà vu right now, you're not alone. Last week, this article recommended Andy Dalton and Matt Ryan. It also projected that the Falcons game would be popular among pros. And indeed it was, as evidenced by colleague Steve Buzzard having four players from that game in his million dollar-winning lineup on FanDuel.

As of right now, both Dalton and Ryan should be rostered more frequently because their wide receiver corps are projected as the top two highest-rostered of the week and their own individual probabilities for achieving 4x value sit near the top of the quarterback rankings.

SATURDAY UPDATE: Turns out that not much has changed since Thursday regarding Dalton and Ryan, so let's focus on two other quarterbacks.

Eli Manning ($5,600, 4.3% Overall Roster Percentage, 6.3% Pro Roster Percentage)

Manning no doubt benefits from facing a Saints defense projected to appear in less than one percent of tournament lineups. However, the main driver of his increased pro projection is the total roster percentage of his wide receiver corps. Sterling Shepard (24.6%) and Odell Beckham, Jr. (18.5%), combine for over 40 percent rostering, which is a full 7.0% higher than second-ranked Cincinnati (36.1%). As is said every week in this space, if Manning's wide receivers are going to perform well enough to justify their roster percentages, then Manning himself should perform well enough to be rostered more frequently than his overall projection suggests.

Tom Brady ($6,700, 4.6%, 3.2%)

Brady's wide receiver corps, on the other hand, is nowhere near the top of the roster percentage list. Specifically, at 3.8%, New England's wide receivers have the fifth-lowest mark of the week. The factor having an even bigger effect on Brady's pro projection is his probability of achieving 4x value, which at 17.3% ranks dead last among quarterbacks in the table.

Running Backs

Below is the table for running backs:

Thursday Early Impressions

Due to his low likelihood of achieving 4x value (9.1%) and his wide receiver teammates ranking sixth in aggregate roster percentage (18.9%), Jordan Howard figures to be in fewer pro lineups than the overall public this week. On the flip side, the pros are shaping up to be overweight on Ezekiel Elliott due to the Cowboys not having a single wide receiver projected to appear in at least one percent of tournament lineups.

SATURDAY UPDATE: Howard is still a "fade" in terms of being projected lower for pros than for the overall public, but now so is Elliott. The change for Elliott is due to the Lions defense going from an overall projection of less than one percent rostering on Thursday to 2.4% on Saturday.

Sony Michel ($4,500, 16.0%, 18.1%)

One of the biggest factors that goes into the pro projection model shows Michel to be in a position rarely enjoyed by a running back: His 33.8% likelihood of achieving 4x value ranks first this week, a full 10.0% ahead of second place! Now add in the fact that Miami's defense is projected to appear in less than one percent of lineups. And now add in the low rostering of Patriots wide receivers mentioned above in relation to Tom Brady. Granted, Michel isn't much of a receiver out of the backfield, but one could also interpret the low rostering of a team's wide receivers as saying "the offense will run the ball more often."

In other words, if the Patriots wide receivers aren't going to perform well enough to worthy high rostering, then either James White should be rostered more frequently because his receiving is going to fill the gap or Sony Michel should be rostered more frequently because his running is going to fill the gap. And as the table shows, White should not be rostered more frequently.

Tevin Coleman ($5,900, 12.3%, 8.0%)

That wide receiver factor is the main reason why Coleman qualifies as a "fade" according to the pro projection model. Whereas the Patriots receivers are projected to appear in a total of only 3.8% of lineups, the Falcons' wide receiver project to appear nearly 10 times as frequently (33.3%). This is more than enough to wash out his relatively high standing per other factors (i.e., 4x value probability and overall roster percentage for his opponent's defense).

Melvin Gordon III ($8,300, 18.3%, 13.7%)

and Saquon Barkley ($8,100, 17.6%, 11.8%)

Continuing the theme, Gordon and Barkley are also suffering from an acute case of receiver-itis this week. Just like Coleman, they have a 4x value probability ranked in the middle of the pack and are facing defenses projected to appear in less than one percent of lineups. And just like Coleman, the benefit of these stats is nullified by the projected roster percentage for their team's wide receivers. Barkley's wide receivers rank No. 1 at 43.1% aggregate rostering, while Gordon's wide receivers rank No. 4 at 24.4%.

Wide Receivers

Below is the table for wide receivers:

Thursday Early Impressions

Back to the Falcons-Bengals game. Tyler Boyd is a wide receiver to monitor throughout the rest of the week. Well, it's actually A.J. Green's health that needs monitoring, but you get the idea. If Green is out, Boyd is in a smash spot. But then again, even if Green plays, Boyd is probably still worth rostering more than the field simply due to his third-ranked likelihood of achieving 4x value. In short, $4,600 is far too low a salary for a wide receiver in a likely high-scoring game that's coming off back-to-back performances that easily achieved tournament value (an average of 24.9 points at an average of $3,450 salary, for an average value of 7.21x).

In the same salary stratum, Sterling Shepard is poised to be rostered more frequently by pros than the overall public thanks to his second-ranked 4x value probability (27.5%).

SATURDAY UPDATE: Dear readers, you're not going to believe this, but not much has changed since Thursday regarding Boyd and Shepard. They're still highly favored by the pro projection model. With respect to Boyd, this is despite A.J. Green's questionable availability ceasing to be questionable.

Julio Jones ($8,200, 12.8%, 8.5%)

Jones is a "fade" simply due to his probability of achieving 4x value. At 8.6%, he ranks 41st out of 50 wide receivers in the table. Keep reading and there's a pivot play at tight end you might like.

Will Fuller V ($6,800, 12.9%, 8.9%)

and DeAndre Hopkins ($8,400, 8.8%, 5.4%)

One of Hopkins' issues is the same as Jones: His 8.6% likelihood of achieving 4x value ranks 42nd out of 50. Unlike Jones, however, Hopkins is also negatively affected in the model by Indianapolis' defense having the fourth-highest projected roster percentage for the overall public. Naturally, this factor also negatively affects Fuller. And indeed, it's the main problem for Fuller, as his 4x value probability of 16.6% ranks a respectable 19th.

Mike Evans ($7,900, 5.1%, 3.7%)

If you've read this far and for this many weeks, you know this author is enamored by themes. Humans are narrative beings, after all. And so, continuing the theme, Evans is Hopkins-esque this week: He has a 4x value probability of 9.1% ranks 38th, while the projected roster percentage for his opposing defense -- Chicago at 15.6% -- ranks No. 1 by far. If you're looking for exposure to the Buccaneers offense, the table suggests pivoting to Chris Godwin is a better bet than rostering Evans.

Tight Ends

Below is the table for tight ends:

Thursday Early Impressions

At the moment, Jared Cook and Dallas Goedert appear to be the pro plays for Week 4 due to their top-ranked 4x value probabilities and the infrequent rostering of their respective opponent defenses. However, the value probability side of the equation may change considerably because their point projections -- 12.5 for Cook and 8.3 for Goedert -- seem a bit high for players facing two of the five stingiest defenses against opposing tight ends.

SATURDAY UPDATE: Hey, whaddyknow! Cook and and Goedert are still favored by the pro projection model -- and this author is still skeptical. So here's another tight end the model likes and one that it doesn't.

Zach Ertz ($6,200, 9.0%, 6.7%)

It stands to reason that if pros are going to roster Goedert more frequently, then they'll roster Ertz less frequently. What's interesting from the model's perspective is that, being stablemates, one would expect both to be popular since the team-level factors affecting them are the same. The difference is due to the main individual-level factor, 4x value probability. Whereas Goedert ranks No. 1 among tight ends at 32.2%, Ertz ranks 14th (of 18) at 17.3%.

Austin Hooper ($2,900, 3.8%, 4.5%)

If you want exposure to the high-flying Falcons-Bengals game while getting relief with respect to both salary and chalk, Hooper is an ideal pivot play. He benefits from all of the factors his teammates do, primarily Cincinnati's defense being projected for less than one percent rostering, but his 26.4% probability of achieving 4x value is the highest of them all.

Defenses

Below is the table for defenses:

Thursday Early Impressions

There isn't much that stands out at defense. The Cowboys look intriguing, but their track record without Sean Lee is, um, not ideal, to put it politely. The Bears top the list, but Tampa Bay's high-octane offense presents the toughest challenge they've faced since Week 1.

Of course, for both of these defenses, the points just made can be characterized as conventional wisdom. So perhaps, based on contrarianism, Dallas and Chicago actually are good plays. Which one it is will be known on Saturday.

SATURDAY UPDATE: It's fitting that the last section of the article continues the theme of Thursday's early impressions holding steady come Saturday. So here's just about the only other defense in the top half of the table that's likely to be rostered by pros more frequently than by the overall public.

Detroit Lions ($2,500, 2.4%, 2.9%)

As mentioned earlier in the introduction, the projection model for defense is the least accurate of the five roster positions. One reason for this is that fewer factors make any difference above and beyond Steve Buzzard's overall projection. One of the only factors that matters is the opposing quarterback's roster percentage, and that's where the Lions get a boost this week. Namely, Dak Prescott's 1.9% roster percentage ranks 18th out of 23 quarterbacks. Importantly, aside from Indianapolis, the five defenses facing a quarterback projected to be rostered even less than Detroit's opponent are the at the top of the table: Bears, Chargers, Packers, Jaguars, and Eagles. At $2,500, the Lions are a bargain in comparison.

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