Calculating a Football Team’s Strength of Schedule

Calculating Baseline Team Ratings

Instead, measuring the difficulty of a team’s schedule should start with making at least a half-decent effort to quantify each opponent’s strength. I will calculate a relatively crude measure of baseline team ratings using non-penalty goal difference per 90 (npG/90) and non-penalty expected goal difference per 90 (npxGD/90)³.

I will take a weighted combination of npxGD/90 (70%) and npGD/90 (30%) because previous work suggests this is the optimal way to combine the two (Torvaney 2021a).

This approach benefits from conceptual simplicity, making for a relatively straightforward interpretation.

Estimating Home Advantage

Rather than making wild assumptions about home advantage, I decided to do the responsible thing: measure the average home advantage across the big five leagues in recent seasons and use that number to weight home games.

I used the excellent worldfootballR package to grab match results data in the big five leagues from 2017/18 to 2022/23 from FBref focusing on actual and expected goals (because penalties are relevant when measuring home advantage). I have also dropped any rows where attendance is NA, assuming that NAs represent empty stadiums, particularly during the COVID-19 pandemic.

home_and_away <-
  worldfootballR::fb_match_results(
    country = c("ENG", "FRA", "ESP", "GER", "ITA"),
    season_end_year = 2018:2023,
    gender = "M"
  ) |>
  janitor::clean_names(
    replace = c("xG" = "xg")
  ) |>
  mutate(
    league = case_when(
      competition_name == "Fußball-Bundesliga" ~ "Bundesliga",
      .default = competition_name
    )
  ) |>
  select(league, ends_with("_goals"), ends_with("_xg"), attendance) |>
  tidyr::drop_na()

Table 1 shows the league average percentage difference in goals scored by home teams compared with away teams, and the average for the big five leagues is calculated, too.

home_and_away |>
  tidyr::drop_na() |>
  bind_rows(
    home_and_away |>
      summarise(
        across(ends_with("_goals") | ends_with("_xg"), mean)
      ) |>
      mutate("league" = "Big Five Leagues")
  ) |>
  summarise(
    across(ends_with("_goals") | ends_with("_xg"), mean),
    .by = c(league)
  ) |>
  mutate(
    goal_diff = (home_goals - away_goals) / away_goals,
    xg_diff = (home_xg - away_xg) / away_xg
  ) |>
  select(league, goal_diff, xg_diff) |>
  rename(
    "League" = "league",
    "Goals" = "goal_diff",
    "Expected Goals" = "xg_diff"
  ) |>
  gt() |>
  tab_spanner(
    label = "Difference (Home vs Away)",
    columns = 2:3
  ) |>
  tab_footnote("Excluding games played without fans in attendance.") |>
  tab_style(
    style = list(cell_text(weight = "bold")),
    locations = cells_body(
      columns = everything(),
      rows = League == "Big Five Leagues"
    )
  ) |> 
  tbl_theme(type = "pct", .round = 1)

The variance in the percentage difference in both goals and expected goals in Table 1 is pretty high, though that does appear to be driven by Serie A and La Liga being at the extreme ends. The Premier League, Bundesliga, and Ligue 1 variance is significantly lower. This is interesting, but I will avoid going down this specific rabbit hole right now because rabbit hole inception is a step too far.

I’m doing my duty as a Brexit-era Englishman focusing only on English leagues and pretending that football does not exist outside these fair shores. Taking the average across the big five leagues is acceptable for these purposes. Although the average is just above 25% for both goals and expected goals, I’m not going to be an insufferable bore⁵. I will treat the home advantage as 25% to keep things simple.

Team ratings will be weighted by home advantage by adding 25% of the team rating’s absolute value to the rating.

Putting it All Together

We should now be able to assemble everything to create an SOS model.

The first step is to calculate the opponents’ ratings (OR) and the opponents’ opponents’ ratings (OOR)⁶. I will do this by first calculating the weighted team rating for each fixture, meaning that when a team plays away from home OR will be 25% higher and when they are at home, OOR is 25% higher. I will then take the mean value of OR and OOR. This second value is more involved, but it is conceptually the same as the mean value of OR.

The average value of a team’s OR and OOR play slightly different roles in the model. OR is the model’s primary parameter, and OOR provides additional context, allowing us to weight OR by the quality of the opponents’ opposition.

I will take a weighted average of OR (66.6%) and OOR (33.3%) and sum the two values.

While this approach yields a defensible SOS metric, the interpretability of the results may be limited. I have concerns about the practical significance of SOS values measured on the same scale as goals per 90. What does it mean for a team to have a one-goal-per-90 advantage in SOS? Are unit increases and decreases in SOS linear across the range of values? I think using the same scale as goals per 90 may lead to oversimplification and potential misinterpretation of SOS values⁷.

Instead, transforming SOS onto a standard scale, such as z-scores, offers a solution. Z-scores provide a standardised measure that facilitates precise comparative analysis across teams or seasons. This approach makes the results easier to interpret and allows for more meaningful comparisons within the league context.

SOS z-scores are calculated by taking the mean average and standard deviation of SOS for the entire league. For each unstandardised value of SOS, we subtract the mean and divide the result by the standard deviation.

An SOS that equals zero is a league-average schedule; scores greater than zero are harder than the league average, and scores less than zero are easier than the league average. If a team’s SOS z-score is +1, their SOS is one standard deviation above the average SOS for all teams. Similarly, a z-score of -1 indicates that the SOS is one standard deviation below the average. Z-scores allow us to compare different teams’ SOS values on a standard scale, making it easier to interpret and analyse their relative strengths of schedule.

Remaining Schedule

The first step is to filter the match results data to only include the remaining games we want to work with. We can leverage that worldfootballR’s fb_match_results() function also includes future games with undetermined details about the match, like home and away goals, set to NA.

remaining_games <-
  match_results_raw |>
  janitor::clean_names() |> 
  filter(is.na(home_goals) | is.na(away_goals)) |>
  mutate(
    team = home,
    opponent = away,
    home_or_away = "Home"
  ) |>
  bind_rows(
    match_results_raw |>
      janitor::clean_names() |> 
      filter(is.na(home_goals) | is.na(away_goals)) |>
      mutate(
        team = away,
        opponent = home,
        home_or_away = "Away"
      )
  ) |>
  select(team, home_or_away, opponent)

This creates a dataframe with every remaining fixture in the 2023/24 Premier League season repeated twice, with each team listed once in the team and opponent columns. Table 2, below, shows a sample of the data.

remaining_games |> 
  janitor::clean_names(case = "title") |> 
  slice_sample(n = 10) |> 
  gt() |> 
  tbl_theme(type = "num")

To me, this seems like the obvious way to structure the data, but it’s important to remember that I am an idiot, so there may well be a better way.

Baseline Team Ratings

The next step is to create the baseline ratings for each Premier League team using their npxGD/90 and npGD/90. As discussed in Section 1, calculating the team ratings is very simple. They are just a weighted average of npxGD/90 (70%) and npGD/90 (30%).

ratings <-
  team_stats_raw |>
  janitor::clean_names(
    replace = c(
      "Squad" = "team",
      "_Expected" = "",
      "G_minus_PK" = "npg",
      "npxG" = "npxg",
      "_Per_Minutes" = "90"
    )
  ) |>
  select(team, team_or_opponent, npxg90, npg90) |>
  mutate(team = stringr::str_remove(team, "vs ")) |>
  tidyr::pivot_wider(
    names_from = team_or_opponent,
    values_from = c(npg90, npxg90),
    names_glue = "{team_or_opponent}_{.value}"
  ) |>
  summarise(
    npxgd90 = team_npxg90 - opponent_npxg90,
    npgd90 = team_npg90 - opponent_npg90,
    rating = (npxgd90 * 0.7) + (npgd90 * 0.3),
    .by = team
  )

The ratings in Table 3 show each Premier League team’s npxGD/90, npGD/90, and baseline ratings, arranged from highest to lowest.

ratings |> 
  arrange(desc(rating)) |> 
  janitor::clean_names(case = "title") |> 
  rename(
    "npxGD/90" = Npxgd90,
    "npGD/90" = Npgd90
  ) |> 
  gt() |> 
  tbl_theme(type = "num")

According to Table 3, Arsenal are the best team in the league, which seems about right, followed by Manchester City and Liverpool, and then a giant chasm before we reach the rest of the league (which is topped by Tottenham). I am not compelled to argue with these ratings, but if you’re a Manchester United fan who doesn’t have the capacity for shame, you might?

We then have to calculate the mean rating for a team’s OR and OOR from these baseline ratings. The process for calculating each of these is similar. However, the home advantage weighting is reversed for the OOR, and a little extra work is required to get the opponents that a team’s opponents will face.

opponent_ratings <-
  remaining_games |>
  left_join(ratings, by = join_by(opponent == team)) |>
  mutate(
    rating =
      if_else(home_or_away == "Away", rating + (abs(rating) * 0.25), rating)
  ) |>
  summarise(opponent_rating = mean(rating), .by = team)

opponent_opponent_ratings <-
  remaining_games |>
  left_join(ratings, by = join_by(team)) |>
  mutate(
    rating =
      if_else(home_or_away == "Home", rating + (abs(rating) * 0.25), rating)
  ) |>
  summarise(rating = mean(rating), .by = opponent) |>
  full_join(remaining_games) |>
  summarise(opponent_opponent_rating = mean(rating), .by = team)

Table 4 shows the two sets of ratings joined by their corresponding team, which hopefully helps make sense of what is happening here. This part of the process is a little confusing because the steps are kind of stacked on top of each other. Fingers crossed, this makes it easier to follow.

remaining_games |>
  distinct(team) |>
  arrange(team) |> 
  left_join(opponent_ratings, by = join_by(team)) |>
  left_join(opponent_opponent_ratings, by = join_by(team)) |> 
  rename(
    "Team" = team,
    "Opponents' Rating" = opponent_rating,
    "Opponents' Oppponents Rating" = opponent_opponent_rating,
  ) |> 
  gt() |> 
  tbl_theme(type = "num")

Strength of Schedule

Having created all the components needed, we must piece it together like a dullard’s jigsaw puzzle.

strength_of_schedule <-
  remaining_games |>
  distinct(team) |>
  left_join(opponent_ratings, by = join_by(team)) |>
  left_join(opponent_opponent_ratings, by = join_by(team)) |>
  mutate(
    unstandardized = (2/3 * opponent_rating) + (1/3 * opponent_opponent_rating),
    sos = scale(unstandardized)
  )

We can start by looking at the Premier League title race, comparing Arsenal, Manchester City, and Liverpool’s schedules. Table 5 shows the results.

strength_of_schedule |>
  select(team, sos) |> 
  filter(team %in% c("Arsenal", "Manchester City", "Liverpool")) |> 
  arrange(sos) |> 
  rename(
    "Team" = team,
    "Strength of Schedule (Z-Score)" = sos
  ) |> 
  gt() |> 
  tab_footnote(
    footnote = 
      "z < 0 = Easier than league average; z > 0 = Harder than league average"
    ) |> 
  tbl_theme(type = "num")

As discussed in Section 1.3, z-scores are a measure of standard deviation from the mean. A z-score of -1.12 means that Liverpool’s schedule is 1.12 standard deviations lower than the mean league schedule.

All three teams’ schedules are easier than the league average, which is unsurprising because they have played each other, and many teams still have to play them. However, there is plenty of variance in schedule strength between the three title contenders. Arsenal’s schedule is very close to the league average, while Liverpool’s remaining schedule is one of the easiest in the Premier League.

Finally, we can plot the Strength of Schedule Z-Scores for all Premier League teams’ remaining games this season. The plot below orders teams by SOS, from the hardest schedules to the easiest.

strength_of_schedule |>
  mutate(team = forcats::fct_reorder(team, sos)) |>
  ggplot(aes(x = team, y = sos, fill = sos > 0)) +
  geom_col(colour = "grey20") +
  geom_hline(yintercept = 0, colour = "grey20") +
  geom_text(
    aes(
      label = round(sos, 2),
      colour = between(sos, -0.25, 0.25),
      hjust = case_when(
        between(sos, -0.25, 0) ~ 1.2,
        between(sos, 0, 0.25) ~ -0.4,
        sos < -0.25 ~ -0.2,
        sos > 0.25 ~ 1.2
        )
      ),
    size = 5, fontface = "bold", family = "Poppins"
    ) +
  coord_flip() +
  scale_fill_manual(values = c("#026E99", "#D93649"), guide = "none") +
  scale_colour_manual(values = c("white", "grey20"), guide = "none") +
  labs(
    title = "Premier League Teams' Strength of Schedule for Remaining Games",
    subtitle =
      stringr::str_wrap(
        glue::glue(
          "Remaining schedule difficulty in the Premier League, represented as ",
          "Z-Scores. Scores above zero are harder than league average, below ",
          "easier. Z-Scores are the standard deviation distance from the mean ",
          "(i.e, scores of +1 (or -1) are 1 SD above (or below) mean)."
          ),
        width = 95
        ),
    x = NULL, y = "Strength of Schedule (Z-Score)",
    caption = "Visualisation: Paul Johnson  |  Data: FBref Via {worldfootballR}"
    )

Aston Villa and Luton have much left to play for, so this is not great news for either side. Meanwhile, Newcastle doesn’t have to face any of the top three but still have to play Sheffield United and Burnley. Their schedule includes only three teams ranked in the top ten in the baseline team ratings, two of which they will face at St James’ Park. Awful slackers.