Brazil has been FiveThirtyEight’s frontrunner to win the World Cup since the beginning of the tournament. But A Seleção will have to do it without its star, Neymar, who is out of the tournament after fracturing a vertebra in Brazil’s quarterfinal win against Colombia. In the semifinal against Germany on Tuesday, Brazil will also be without defender Thiago Silva, who was suspended after accumulating two yellow cards (Silva will return if Brazil reaches the final).

Our forecast, which gives Brazil a 54 percent chance of treating its home fans to a title, doesn’t account for these player absences. This article will attempt to measure their impact and recalculate the numbers.

We have some reasonably good news for Brazil. Even without Neymar and Silva, the team remains the leading contender to win the World Cup in our estimation. You may or may not agree with the math, but the intuition behind it is this: Soccer is a team sport, and Brazil is a very deep team. Whether Brazil is better than Germany without Neymar and Silva is up for debate. However, Brazil will play at home, where the national team hasn’t lost a competitive match since 1975.

Our rating system

Our World Cup forecasts are based on ESPN’s Soccer Power Index (SPI). SPI is essentially two ratings systems rolled into one. It measures a national team’s performance, placing more weight on its most competitive matches, and evaluates the talent on its roster based on players’ performance in national and club matches.

The player ratings work by means of a plus-minus system that assigns or subtracts points from players based on goals scored and allowed at the time they’re on the pitch. They include data from national team matches, along with the top-flight club leagues in Spain, England, Italy, Germany and France, and matches from the Champions League.

In principle, the system operates something like this. The average English Premier League team scores and allows about 1.4 goals per game. But let’s say Arsenal wins a game 3-1 instead against an average EPL opponent. Its three goals scored are 1.6 goals above the league average, so we have 1.6 goals worth of extra credit to apportion out to the players who were on the pitch at the time. Arsenal also allowed 0.4 goals less than average, so there’s 0.4 goals worth of defensive credit to split up, too.

In theory, this allows us to estimate the impact of any one player. But the plus-minus works better for some players than others. The statistics it accounts for are goals scored, bookings and the starting lineups and substitutions of a match, along with the players’ positions. This works fine for strikers like Neymar, whose main objective is to score goals and set them up, but it isn’t as helpful for defenders like Silva. SPI will give about the right overall amount of credit to the back line and midfielders on a team — for instance, to Silva’s Paris Saint-Germain after an impressive victory. But if Silva is better than the other defenders, it will give him too little credit and his teammates too much.

So we’ll use the SPI method to account for Neymar’s absence, and then an alternative method that might more accurately measure the loss of Silva.

The impact of Neymar’s absence

For forwards and strikers, the SPI plus-minus ratings line up reasonably well with subjective perceptions. The following chart compares SPI plus-minus ratings for forwards, measured as the number of additional goals they help score and prevent per 90 minutes of play, as compared to The Guardian’s consensus list of the top 100 footballers in the world.

The correlation isn’t perfect, but it’s reasonably good. The amazing Lionel Messi, along with Portugal’s Cristiano Ronaldo, rate as the best forwards in the world by far in both systems.

Based on data from his FC Barcelona games over the past two years, along with his recent play for Argentina, for instance, we estimate that Messi contributes about 0.65 goals per game relative to an average international footballer. (Almost all of that impact comes from Messi helping his team get more goals — only a little bit is defense or goal prevention.)

How big is that impact? The SPI match predictor estimates that if you put Messi on an average team, it would go from winning 50 percent of its games (assuming a knockout-style format with no draws) to around 65 percent instead. That’s impressive — not far from the effect you might see by replacing an average NBA player with LeBron James. But James plays a 5-on-5 game, while Messi and Ronaldo play one with 11 players to a side.

Neymar isn’t quite at Messi’s or Ronaldo’s level — nor is anyone else. Instead, SPI estimates his impact at about 0.4 goals per 90 minutes. Almost all of this (0.37 goals) comes from increasing his team’s scoring, while just a fraction (0.03 goals) comes from improving its goal prevention.

These player ratings can be carried forward to the team level. For instance, if you removed Neymar from the pitch and replaced him with an average international footballer, Brazil’s scoring would drop by 0.37 goals per 90 minutes, we estimate. Brazil would also allow an additional 0.03 goals per 90 minutes of play.

But the players replacing Neymar are going to be well above average. An average international soccer player, according to SPI, is something like a starter on Israel or Libya or Uzbekistan. A player on Brazil’s second or third string would be a star on one of those sides.

It’s not clear exactly who will replace Neymar. In fact, it may wind up being a combination of players, given that his injury will have knockoff effects on Brazil’s depth. But Neymar’s playing time will likely be distributed principally among three players: the forwards Bernard and Jo, and the winger Willian.

Of the three, we have the best data on Willian, since he plays his club football for Chelsea in the EPL — one of the leagues we track. SPI estimates he’s worth about 0.2 goals per 90 minutes relative to an average footballer. If Neymar is worth about 0.4 goals per 90 minutes, Willian is 0.2 goals per 90 minutes worse. (Some numbers in the tables below may look slightly off due to rounding.)

For Bernard, who plays for Shakhtar Donetsk in Ukraine, we have data only from his performance with Brazil, plus a smattering of Champions League matches. And for Jo, who plays for Atlético in Brazil, we have the national team data only. But they rate similarly to Willian, at about +0.2 goals per 90 minutes.

Certainly these players lack Neymar’s explosiveness and creativity. But this isn’t a total disaster for Brazil. Ratings in the range of +0.2 goals per 90 minutes imply that Willian, Bernard and Jo would belong somewhere in the top 200 or 250 footballers in the world, and would start for most of the teams that made the World Cup.

The impact of Silva’s absence

Brazil will also be without Thiago Silva for the Germany match, and coach Luiz Felipe Scolari has a number of options to replace him. Brazil played Dani Alves in its first four World Cup matches, then replaced him with Maicon for its game against Colombia. Scolari could choose to play them both instead, though Brazil would need to rotate its formation, as they both typically play right back rather than center. Scolari could also go with Dante, who plays center defense for Bayern Munich and who has the advantage of being familiar with some of Germany’s stars. Henrique, who plays as a center back or defensive midfielder, would be another option.

The SPI plus-minus ratings suggest that there’s hardly any difference between Silva and some of these options — and what impact it does find comes on offense, such as on set pieces.

But this conclusion may be misguided. As I mentioned, the SPI ratings are a crude tool for evaluating defensive players. They do get us somewhere: SPI knows that Silva plays for Brazil and Paris Saint-Germain, two excellent teams, and gives him credit for being in those lineups. But it will have trouble distinguishing truly world-class defensive players from merely very good ones.

There’s a more aggressive way to approach the problem, however, which is to approximate Silva’s defensive value by using his position on The Guardian’s top 100 ranking. Silva was No. 17 on that list.

As I mentioned, the SPI plus-minus should be pretty solid for forwards and strikers. So we can calibrate the value of Silva by estimating how much a striker rated as the No. 17 player in the world would be worth. This value turns out to be about 0.37 goals per 90 minutes based on the regression line I drew in the chart above. By comparison, the raw SPI numbers put Silva’s impact at 0.19 goals per 90 minutes, considering both offensive and defensive contributions.

So let’s assume that The Guardian’s panelists have it right and that Silva is in fact worth 0.37 goals per match. We’ll further assume that SPI is missing Silva’s defense but that it gauges his offensive contributions about right. That implies having Silva on the pitch reduces an opponent’s goal-scoring by about 0.28 goals per 90 minutes relative to an average player, while he adds 0.09 goals per match with his offense.

If we make this fix for Silva, however, we should do so for Dante and Dani Alves, who also rank in The Guardian’s top 100. (We won’t make any adjustment for Henrique and Maicon, who are outside the top 100.)

Averaging the figures for Silva’s potential replacements yields a combination of players that rate at +0.22 goals per 90 minutes. That’s about 0.15 goals per match worse than Silva. More precisely, we estimate Brazil’s goal-scoring to decline by 0.03 goals per 90 minutes without Silva, and for it to allow an additional 0.11 goals in that time.

The change to Brazil’s odds

The next step is to see what impact this has on Brazil’s overall chances of victory. Brazil, at full strength, has an SPI offensive rating of 3.12 (placing it No. 1 in the world) and an SPI defensive rating of 0.50 (putting it at No. 3 — at the team level, lower defensive ratings are better).

Without Neymar and Silva, Brazil’s offensive rating declines to 2.90 while its defensive rating worsens to 0.61. Both ratings are almost identical to Germany’s, which checks in with a 2.92 offensive rating and a 0.63 defensive rating.

But Brazil still has the home-country advantage. If we run the new numbers through the SPI match predictor, we come up with a 65 percent chance of Brazil prevailing against Germany. This is as opposed to a 73 percent chance of doing so with Silva and Neymar in the lineup.

Let me pause here to ruminate on the nature of home-country advantage in soccer. Brazil has developed a reputation for “winning ugly” so far in this World Cup. The team was helped by a dubious penalty in its opening match against Croatia. It needed a penalty shootout to get by Chile. And Brazil’s match against Colombia was marred by rough play, including the foul that knocked Neymar out.

But winning ugly is part of what home-field advantage is all about. Home teams benefit significantly from officiating decisions — it’s much harder for most referees to hand out a red card or to award a penalty against the home team’s hero. Home teams may also have some advantage in shootouts. Since 2005 — and counting Brazil’s win against Chile — home teams are 7-0 in penalty shootouts in major international competitions. That’s not to say that Brazil has been in its best form (in fact, its SPI rating has declined slightly over the course of the World Cup). But in Brazil, being in pretty good form may be good enough.

Should Brazil make it to the final — with Silva returning but still without Neymar — we estimate that it would have a 67 percent chance of beating Argentina (down from 72 percent with Neymar in the lineup). Almost all of that comes from home-field advantage — the teams would be almost level at a neutral site. Brazil would have a 73 percent chance against the Netherlands without Neymar, we estimate, down from 77 percent with him.

We can recalculate Brazil’s overall odds of winning the World Cup as well: They’re 45 percent, as opposed to 54 percent with the team at full strength. The other teams benefit from Brazil’s limitations: Argentina’s odds of winning the World Cup rise to 23 percent from 20 percent; Germany’s to 18 percent from 14 percent. The Netherlands gets the least help and goes to 14 percent from 12.5 percent.

Betting markets see things a bit differently. The odds available as of early Sunday afternoon have Germany, Brazil and Argentina as co-favorites, each with about a 27 percent chance of winning the World Cup, and the Netherlands just slightly behind at 20 percent.

It could be that the markets see a greater impact from the losses of Neymar and Silva than the ones we’ve estimated here. But I suspect that’s only part of the issue — from the start of the tournament, betting markets have consistently been lower on Brazil than SPI and another computer rating system, Elo, have.

So far, SPI and Elo have performed well compared to the betting markets. In general, markets are fairly tough to beat, so this could just reflect good luck. International football is particularly hard to rate, whether using subjective or objective methods. The main problems are that the teams don’t play very many competitive matches against one another, and that the composition of the rosters is always changing. SPI tries to address some of those problems, but it isn’t perfect.

But there could also be a pro-European bias in the markets. Even though Europe has performed rather poorly in this World Cup, it has the highest-profile club leagues, and it has a higher concentration of wealthy people who can afford to bet on soccer. FIFA’s ranking of international teams, with methodological flaws that may result in overrating European teams, also has some currency in shaping perceptions about international soccer. That may help to explain why betting markets give the European teams a 47 percent chance of winning the World Cup when SPI gives them a 32 percent chance (even after accounting for Brazil’s player absences). It may also help to explain why Brazil’s ugly wins against Colombia and Chile are given less credit than if they had come against, say, Portugal and France.

In a sense, the World Cup has come full circle. At the start of the tournament, SPI’s major contentions were that South America was underrated as compared with Europe — and that Brazil’s home-field advantage shouldn’t be overlooked. Brazil still has those things working in its favor, even without Neymar.