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18 Mythbusters: Putting Fantasy Football Beliefs/Anecdotes to the Test
18.1 Getting Started
18.1.1 Load Packages
18.1.2 Specify Package Options
18.1.3 Load Data
We created the player_stats_weekly.RData and player_stats_seasonal.RData objects in Section 4.4.3.
18.2 Do Players Perform Better in their Contract Year?
Considerable speculation exists regarding whether players perform better in their last year of their contract (i.e., their “contract year”). Fantasy football talking heads and commentators frequently discuss the benefit of selecting players who are in their contract year, because it supposedly means that player has more motivation to perform well so they get a new contract and get paid more. To our knowledge, no peer-reviewed studies have examined this question for football players. One study found that National Basketball Association (NBA) players improved in field goal percentage, points, and player efficiency rating (but not other statistics: rebounds, assists, steals, or blocks) from their pre-contract year to their contract year, and that Major League Baseball (MLB) players improved in runs batted in (RBIs; but not other statistics: batting average, slugging percentage, on base percentage, home runs, fielding percentage) from their pre-contract year to their contract year (White & Sheldon, 2014). Other casual analyses have been examined contract-year performance of National Football League (NFL) players, including articles in 2012 [Bales (2012); archived here] and 2022 [Niles (2022); archived here].
Let’s examine the question empirically. In order to do that, we have to make some assumptions/constraints. In this example, we will make the following constraints:
- We will determine a player’s contract year programmatically based on the year the contract was signed. For instance, if a player signed a 3-year contract in 2015, their contract would expire in 2018, and thus their contract year would be 2017. Note: this is a coarse way of determining a player’s contract year because it could depend on when during the year the player’s contract is signed. If we were submitting this analysis as a paper to a scientific journal, it would be important to verify each player’s contract year.
- We will examine performance in all seasons since 2011, beginning when most data for player contracts are available.
- For maximum statistical power to detect an effect if a contract year effect exists, we will examine all seasons for a player (since 2011), not just their contract year and their pre-contract year.
- To ensure a more fair, apples-to-apples comparison of the games in which players played, we will examine per-game performance (except for yards per carry, which is based on \(\frac{\text{rushing yards}}{\text{carries}}\) from the entire season).
- We will examine regular season games only (no postseason).
- To ensure we do not make generalization about a player’s performance in a season from a small sample, the player has to play at least 5 games in a given season for that player–season combination to be included in analysis.
For analysis, the same player contributes multiple observations of performance (i.e., multiple seasons) due to the longitudinal nature of the data. Inclusion of multiple data points from the same player would violate the assumption of multiple regression that all observations are independent. Thus, we use mixed-effects models that allow nonindependent observations. In our mixed-effects models, we include a random intercept for each player, to allow our model to account for players’ differing level of performance. We examine two mixed-effects models for each outcome variable: one model that accounts for the effects of age and experience, and one model that does not.
The model that does not account for the effects of age and experience includes:
- random intercepts to allow the model to estimate a different starting point for each player
- a fixed effect for whether the player is in a contract year
The model that accounts for the effects of age and experience includes:
- random intercepts to allow the model to estimate a different starting point for each player
- random linear slopes (i.e., random effect of linear age) to allow the model to estimate a different form of change for each player
- a fixed quadratic effect of age to allow for curvilinear effects
- a fixed effect of experience
- a fixed effect for whether the player is in a contract year
Code
# Subset to remove players without a year signed
nfl_playerContracts_subset <- nfl_playerContracts %>%
dplyr::filter(!is.na(year_signed) & year_signed != 0)
# Determine the contract year for a given contract
nfl_playerContracts_subset$contractYear <- nfl_playerContracts_subset$year_signed + nfl_playerContracts_subset$years - 1
# Arrange contracts by player and year_signed
nfl_playerContracts_subset <- nfl_playerContracts_subset %>%
dplyr::group_by(player, position) %>%
dplyr::arrange(player, position, -year_signed) %>%
dplyr::ungroup()
# Determine if the player played in the original contract year
nfl_playerContracts_subset <- nfl_playerContracts_subset %>%
dplyr::group_by(player, position) %>%
dplyr::mutate(
next_contract_start = lag(year_signed)) %>%
dplyr::ungroup() %>%
dplyr::mutate(
played_in_contract_year = ifelse(
is.na(next_contract_start) | contractYear < next_contract_start,
TRUE,
FALSE))
# Check individual players
#nfl_playerContracts_subset %>%
# dplyr::filter(player == "Aaron Rodgers") %>%
# dplyr::select(player:years, contractYear, next_contract_start, played_in_contract_year)
#
#nfl_playerContracts_subset %>%
# dplyr::filter(player %in% c("Jared Allen", "Aaron Rodgers")) %>%
# dplyr::select(player:years, contractYear, next_contract_start, played_in_contract_year)
# Subset data
nfl_playerContractYears <- nfl_playerContracts_subset %>%
dplyr::filter(played_in_contract_year == TRUE) %>%
dplyr::filter(position %in% c("QB","RB","WR","TE")) %>%
dplyr::select(player, position, team, contractYear) %>%
dplyr::mutate(merge_name = nflreadr::clean_player_names(player, lowercase = TRUE)) %>%
dplyr::rename(season = contractYear) %>%
dplyr::mutate(contractYear = 1)
# Merge with weekly and seasonal stats data
player_stats_weekly_offense <- player_stats_weekly %>%
dplyr::filter(position_group %in% c("QB","RB","WR","TE")) %>%
dplyr::mutate(merge_name = nflreadr::clean_player_names(player_display_name, lowercase = TRUE))
#nfl_actualStats_offense_seasonal <- nfl_actualStats_offense_seasonal %>%
# mutate(merge_name = nflreadr::clean_player_names(player_display_name, lowercase = TRUE))
player_statsContracts_offense_weekly <- dplyr::full_join(
player_stats_weekly_offense,
nfl_playerContractYears,
by = c("merge_name", "position_group" = "position", "season")
) %>%
dplyr::filter(position_group %in% c("QB","RB","WR","TE"))
#player_statsContracts_offense_seasonal <- full_join(
# player_stats_seasonal_offense,
# nfl_playerContractYears,
# by = c("merge_name", "position_group" = "position", "season")
#) %>%
# filter(position_group %in% c("QB","RB","WR","TE"))
player_statsContracts_offense_weekly$contractYear[which(is.na(player_statsContracts_offense_weekly$contractYear))] <- 0
#player_statsContracts_offense_seasonal$contractYear[which(is.na(player_statsContracts_offense_seasonal$contractYear))] <- 0
#player_statsContracts_offense_weekly$contractYear <- factor(
# player_statsContracts_offense_weekly$contractYear,
# levels = c(0, 1),
# labels = c("no", "yes"))
#player_statsContracts_offense_seasonal$contractYear <- factor(
# player_statsContracts_offense_seasonal$contractYear,
# levels = c(0, 1),
# labels = c("no", "yes"))
player_statsContracts_offense_weekly <- player_statsContracts_offense_weekly %>%
dplyr::arrange(merge_name, season, season_type, week)
#player_statsContracts_offense_seasonal <- player_statsContracts_offense_seasonal %>%
# arrange(merge_name, season)
player_statsContractsSubset_offense_weekly <- player_statsContracts_offense_weekly %>%
dplyr::filter(season_type == "REG")
#table(nfl_playerContracts$year_signed) # most contract data is available beginning in 2011
# Calculate Per Game Totals
player_statsContracts_seasonal <- player_statsContractsSubset_offense_weekly %>%
dplyr::group_by(player_id, season) %>%
dplyr::summarise(
player_display_name = petersenlab::Mode(player_display_name),
position_group = petersenlab::Mode(position_group),
age = min(age, na.rm = TRUE),
years_of_experience = min(years_of_experience, na.rm = TRUE),
rushing_yards = sum(rushing_yards, na.rm = TRUE), # season total
carries = sum(carries, na.rm = TRUE), # season total
rushing_epa = mean(rushing_epa, na.rm = TRUE),
receiving_yards = mean(receiving_yards, na.rm = TRUE),
receiving_epa = mean(receiving_epa, na.rm = TRUE),
fantasyPoints = sum(fantasyPoints, na.rm = TRUE), # season total
contractYear = mean(contractYear, na.rm = TRUE),
games = n(),
.groups = "drop_last"
) %>%
dplyr::mutate(
player_id = as.factor(player_id),
ypc = rushing_yards / carries,
contractYear = factor(
contractYear,
levels = c(0, 1),
labels = c("no", "yes")
))
player_statsContracts_seasonal[sapply(player_statsContracts_seasonal, is.infinite)] <- NA
player_statsContracts_seasonal$ageCentered20 <- player_statsContracts_seasonal$age - 20
player_statsContracts_seasonal$ageCentered20Quadratic <- player_statsContracts_seasonal$ageCentered20 ^ 2
# Merge with seasonal fantasy points data18.2.1 QB
First, we prepare the data by merging and performing additional processing:
Code
# Merge with QBR data
nfl_espnQBR_weekly$merge_name <- paste(nfl_espnQBR_weekly$name_first, nfl_espnQBR_weekly$name_last, sep = " ") %>%
nflreadr::clean_player_names(., lowercase = TRUE)
nfl_contractYearQBR_weekly <- nfl_playerContractYears %>%
dplyr::filter(position == "QB") %>%
dplyr::full_join(
.,
nfl_espnQBR_weekly,
by = c("merge_name","team","season")
)
nfl_contractYearQBR_weekly$contractYear[which(is.na(nfl_contractYearQBR_weekly$contractYear))] <- 0
#nfl_contractYearQBR_weekly$contractYear <- factor(
# nfl_contractYearQBR_weekly$contractYear,
# levels = c(0, 1),
# labels = c("no", "yes"))
nfl_contractYearQBR_weekly <- nfl_contractYearQBR_weekly %>%
dplyr::arrange(merge_name, season, season_type, game_week)
nfl_contractYearQBRsubset_weekly <- nfl_contractYearQBR_weekly %>%
dplyr::filter(season_type == "Regular") %>%
dplyr::arrange(merge_name, season, season_type, game_week) %>%
mutate(
player = coalesce(player, name_display),
position = "QB") %>%
group_by(merge_name, player_id) %>%
fill(player, .direction = "downup")
# Merge with age and experience
nfl_contractYearQBRsubset_weekly <- player_statsContractsSubset_offense_weekly %>%
dplyr::filter(position == "QB") %>%
dplyr::select(merge_name, season, week, age, years_of_experience, fantasyPoints) %>%
full_join(
nfl_contractYearQBRsubset_weekly,
by = c("merge_name","season", c("week" = "game_week"))
) %>% select(player_id, season, week, player, everything()) %>%
arrange(player_id, season, week)
#hist(nfl_contractYearQBRsubset_weekly$qb_plays) # players have at least 20 dropbacks per game
# Calculate Per Game Totals
nfl_contractYearQBR_seasonal <- nfl_contractYearQBRsubset_weekly %>%
dplyr::group_by(merge_name, season) %>%
dplyr::summarise(
age = min(age, na.rm = TRUE),
years_of_experience = min(years_of_experience, na.rm = TRUE),
qbr = mean(qbr_total, na.rm = TRUE),
pts_added = mean(pts_added, na.rm = TRUE),
epa_pass = mean(pass, na.rm = TRUE),
qb_plays = sum(qb_plays, na.rm = TRUE), # season total
fantasyPoints = sum(fantasyPoints, na.rm = TRUE), # season total
contractYear = mean(contractYear, na.rm = TRUE),
games = n(),
.groups = "drop_last"
) %>%
dplyr::mutate(
contractYear = factor(
contractYear,
levels = c(0, 1),
labels = c("no", "yes")
))
nfl_contractYearQBR_seasonal[sapply(nfl_contractYearQBR_seasonal, is.infinite)] <- NA
nfl_contractYearQBR_seasonal$ageCentered20 <- nfl_contractYearQBR_seasonal$age - 20
nfl_contractYearQBR_seasonal$ageCentered20Quadratic <- nfl_contractYearQBR_seasonal$ageCentered20 ^ 2
nfl_contractYearQBR_seasonal <- nfl_contractYearQBR_seasonal %>%
group_by(merge_name) %>%
mutate(player_id = as.factor(as.character(cur_group_id())))
nfl_contractYearQBRsubset_seasonal <- nfl_contractYearQBR_seasonal %>%
dplyr::filter(
games >= 5, # keep only player-season combinations in which QBs played at least 5 games
season >= 2011) # keep only seasons since 2011 (when most contract data are available)Then, we analyze the data. Below is a mixed model that examines whether a player has a higher QBR per game when they are in a contract year compared to when they are not in a contract year.
Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: qbr ~ contractYear + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 9440.1
Scaled residuals:
Min 1Q Median 3Q Max
-3.2451 -0.5403 0.0767 0.5700 3.2399
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 110.5 10.51
Residual 199.0 14.11
Number of obs: 1127, groups: player_id, 262
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 44.6156 0.8526 237.4894 52.328 <2e-16 ***
contractYearyes -0.1715 1.1601 1008.9925 -0.148 0.883
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.239 R2m R2c
[1,] 1.478002e-05 0.3571011 contractYear emmean SE df lower.CL upper.CL
no 44.6 0.853 272 42.9 46.3
yes 44.4 1.270 774 42.0 46.9
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: qbr ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 9367.1
Scaled residuals:
Min 1Q Median 3Q Max
-3.367 -0.513 0.084 0.549 3.266
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 126.9109 11.2655
ageCentered20 0.3855 0.6209 -0.32
Residual 191.2250 13.8284
Number of obs: 1119, groups: player_id, 258
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 39.41240 2.25893 194.07184 17.447 < 2e-16 ***
contractYearyes 0.01999 1.20153 1000.46464 0.017 0.98673
ageCentered20 1.52956 0.64905 288.74817 2.357 0.01911 *
ageCentered20Quadratic -0.07463 0.02254 106.04139 -3.311 0.00127 **
years_of_experience -0.19545 0.53974 323.76549 -0.362 0.71750
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.059
ageCentrd20 -0.790 -0.071
agCntrd20Qd 0.736 0.049 -0.616
yrs_f_xprnc 0.238 -0.023 -0.693 -0.094 R2m R2c
[1,] 0.0124524 0.4032476 contractYear emmean SE df lower.CL upper.CL
no 44.2 0.905 251 42.5 46.0
yes 44.3 1.300 718 41.7 46.8
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: pts_added ~ contractYear + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 5132.9
Scaled residuals:
Min 1Q Median 3Q Max
-4.7312 -0.4973 0.0857 0.5453 4.2750
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 2.574 1.604
Residual 4.276 2.068
Number of obs: 1127, groups: player_id, 262
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.7982 0.1282 223.7371 -6.226 2.33e-09 ***
contractYearyes -0.1170 0.1705 993.6872 -0.686 0.493
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.233 R2m R2c
[1,] 0.0003108757 0.3759314 contractYear emmean SE df lower.CL upper.CL
no -0.798 0.128 272 -1.05 -0.546
yes -0.915 0.188 763 -1.28 -0.546
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: pts_added ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 5103.5
Scaled residuals:
Min 1Q Median 3Q Max
-4.8882 -0.4961 0.0894 0.5301 4.2771
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 3.57383 1.8905
ageCentered20 0.01126 0.1061 -0.52
Residual 4.09959 2.0247
Number of obs: 1119, groups: player_id, 258
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -1.579520 0.343782 185.015762 -4.595 8.01e-06 ***
contractYearyes -0.103404 0.176353 984.230548 -0.586 0.55778
ageCentered20 0.197941 0.097544 286.031947 2.029 0.04336 *
ageCentered20Quadratic -0.010740 0.003374 106.514781 -3.183 0.00191 **
years_of_experience 0.009662 0.080376 306.571919 0.120 0.90440
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.059
ageCentrd20 -0.790 -0.072
agCntrd20Qd 0.731 0.054 -0.624
yrs_f_xprnc 0.244 -0.023 -0.696 -0.081 R2m R2c
[1,] 0.01001632 0.4159062 contractYear emmean SE df lower.CL upper.CL
no -0.823 0.134 254 -1.09 -0.559
yes -0.927 0.192 714 -1.30 -0.550
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: epa_pass ~ contractYear + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 4784.9
Scaled residuals:
Min 1Q Median 3Q Max
-3.0638 -0.5098 0.0388 0.5482 4.4079
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 2.51 1.584
Residual 2.98 1.726
Number of obs: 1127, groups: player_id, 262
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 1.1290 0.1198 242.6476 9.426 < 2e-16 ***
contractYearyes 0.3946 0.1436 977.3789 2.747 0.00612 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.208 R2m R2c
[1,] 0.004393483 0.4595603 contractYear emmean SE df lower.CL upper.CL
no 1.13 0.120 272 0.893 1.36
yes 1.52 0.167 709 1.196 1.85
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
mixedModelAge_epaPass <- lmerTest::lmer(
epa_pass ~ contractYear + ageCentered20 + ageCentered20Quadratic + years_of_experience + (1 | player_id), # removed random slopes to address convergence issue
data = nfl_contractYearQBR_seasonal,
control = lmerControl(optimizer = "bobyqa")
)
summary(mixedModelAge_epaPass)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: epa_pass ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 4755.3
Scaled residuals:
Min 1Q Median 3Q Max
-3.1995 -0.4944 0.0481 0.5399 4.3277
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 2.560 1.600
Residual 2.933 1.713
Number of obs: 1119, groups: player_id, 258
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 3.318e-01 2.812e-01 9.569e+02 1.180 0.2383
contractYearyes 2.359e-01 1.490e-01 9.964e+02 1.583 0.1137
ageCentered20 1.374e-01 8.191e-02 7.390e+02 1.677 0.0939 .
ageCentered20Quadratic -4.524e-03 2.587e-03 1.057e+03 -1.749 0.0806 .
years_of_experience 1.751e-02 7.151e-02 4.131e+02 0.245 0.8067
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.066
ageCentrd20 -0.770 -0.068
agCntrd20Qd 0.704 0.054 -0.562
yrs_f_xprnc 0.278 -0.028 -0.737 -0.108 R2m R2c
[1,] 0.01881249 0.4760832 contractYear emmean SE df lower.CL upper.CL
no 1.21 0.123 271 0.965 1.45
yes 1.44 0.169 694 1.112 1.78
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: fantasyPoints ~ contractYear + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 13300
Scaled residuals:
Min 1Q Median 3Q Max
-3.7938 -0.5628 -0.0854 0.6325 2.7421
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 6298 79.36
Residual 5477 74.01
Number of obs: 1127, groups: player_id, 262
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 111.342 5.749 299.090 19.367 < 2e-16 ***
contractYearyes -29.589 6.205 989.601 -4.769 2.13e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.185 R2m R2c
[1,] 0.01143801 0.5401623 contractYear emmean SE df lower.CL upper.CL
no 111.3 5.75 271 100.0 122.7
yes 81.8 7.64 650 66.7 96.8
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
mixedModelAge_fantasyPtsPass <- lmerTest::lmer(
fantasyPoints ~ contractYear + ageCentered20 + ageCentered20Quadratic + years_of_experience + (1 | player_id), # removed random slopes to address convergence issue
data = nfl_contractYearQBR_seasonal,
control = lmerControl(optimizer = "bobyqa")
)
summary(mixedModelAge_fantasyPtsPass)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
fantasyPoints ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 | player_id)
Data: nfl_contractYearQBR_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 13181.5
Scaled residuals:
Min 1Q Median 3Q Max
-3.9481 -0.5697 -0.0770 0.6226 2.6078
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 6574 81.08
Residual 5333 73.03
Number of obs: 1119, groups: player_id, 258
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 138.77595 12.60858 971.08217 11.006 < 2e-16 ***
contractYearyes -23.08409 6.42163 997.45929 -3.595 0.000341 ***
ageCentered20 -8.83704 3.72487 818.10907 -2.372 0.017901 *
ageCentered20Quadratic -0.09183 0.11208 1044.34333 -0.819 0.412796
years_of_experience 8.44017 3.34466 523.36645 2.523 0.011915 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.068
ageCentrd20 -0.754 -0.065
agCntrd20Qd 0.675 0.051 -0.526
yrs_f_xprnc 0.304 -0.026 -0.764 -0.111 R2m R2c
[1,] 0.03658633 0.5685138 contractYear emmean SE df lower.CL upper.CL
no 110.3 5.96 271 98.5 122
yes 87.2 7.76 627 71.9 102
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 18.2.2 RB
Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: ypc ~ contractYear + (1 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 6361.6
Scaled residuals:
Min 1Q Median 3Q Max
-7.9762 -0.4002 0.0051 0.4031 14.8072
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 0.5084 0.713
Residual 1.8718 1.368
Number of obs: 1744, groups: player_id, 531
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 3.913e+00 5.135e-02 5.086e+02 76.202 <2e-16 ***
contractYearyes 8.829e-03 8.155e-02 1.677e+03 0.108 0.914
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.374 R2m R2c
[1,] 6.33599e-06 0.2135924 contractYear emmean SE df lower.CL upper.CL
no 3.91 0.0514 631 3.81 4.01
yes 3.92 0.0786 1257 3.77 4.08
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: ypc ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 6346.4
Scaled residuals:
Min 1Q Median 3Q Max
-7.7936 -0.3853 -0.0070 0.3843 14.2145
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 0.39621 0.6295
ageCentered20 0.01043 0.1021 -0.28
Residual 1.79738 1.3407
Number of obs: 1742, groups: player_id, 529
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 4.223e+00 1.705e-01 7.173e+02 24.772 <2e-16 ***
contractYearyes 1.093e-01 8.800e-02 1.602e+03 1.243 0.214
ageCentered20 -5.148e-02 5.996e-02 8.149e+02 -0.859 0.391
ageCentered20Quadratic -2.817e-03 4.225e-03 4.257e+02 -0.667 0.505
years_of_experience 1.444e-02 3.780e-02 4.721e+02 0.382 0.703
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.151
ageCentrd20 -0.891 -0.173
agCntrd20Qd 0.795 0.127 -0.800
yrs_f_xprnc 0.057 -0.066 -0.355 -0.198 R2m R2c
[1,] 0.01717228 0.2674471 contractYear emmean SE df lower.CL upper.CL
no 3.86 0.0547 555 3.76 3.97
yes 3.97 0.0822 1231 3.81 4.13
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: rushing_epa ~ contractYear + (1 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 5057.4
Scaled residuals:
Min 1Q Median 3Q Max
-4.5544 -0.5021 0.0793 0.5799 3.4236
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 0.09945 0.3154
Residual 0.97437 0.9871
Number of obs: 1744, groups: player_id, 531
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.64551 0.03172 652.34985 -20.349 <2e-16 ***
contractYearyes 0.03646 0.05649 1741.99500 0.645 0.519
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.441 R2m R2c
[1,] 0.0002394901 0.09283116 contractYear emmean SE df lower.CL upper.CL
no -0.646 0.0317 650 -0.708 -0.583
yes -0.609 0.0512 1162 -0.710 -0.509
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: rushing_epa ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 5062.3
Scaled residuals:
Min 1Q Median 3Q Max
-4.6136 -0.4966 0.0707 0.5823 3.4224
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 0.173096 0.41605
ageCentered20 0.002046 0.04523 -0.67
Residual 0.958271 0.97891
Number of obs: 1742, groups: player_id, 529
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -6.602e-01 1.179e-01 4.018e+02 -5.597 4.04e-08 ***
contractYearyes 7.124e-02 5.994e-02 1.545e+03 1.189 0.23475
ageCentered20 5.052e-02 3.974e-02 4.083e+02 1.271 0.20434
ageCentered20Quadratic -1.233e-03 2.753e-03 2.098e+02 -0.448 0.65469
years_of_experience -5.986e-02 2.252e-02 4.869e+02 -2.658 0.00812 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.148
ageCentrd20 -0.902 -0.197
agCntrd20Qd 0.818 0.154 -0.838
yrs_f_xprnc 0.030 -0.053 -0.307 -0.191 R2m R2c
[1,] 0.008179426 0.1115709 contractYear emmean SE df lower.CL upper.CL
no -0.658 0.0326 581 -0.722 -0.594
yes -0.587 0.0532 1159 -0.691 -0.482
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: fantasyPoints ~ contractYear + (1 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 20809.1
Scaled residuals:
Min 1Q Median 3Q Max
-3.1340 -0.5057 -0.1729 0.4156 3.8988
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 3546 59.55
Residual 3042 55.16
Number of obs: 1844, groups: player_id, 548
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 81.495 3.055 654.865 26.672 < 2e-16 ***
contractYearyes -13.576 3.466 1589.483 -3.917 9.36e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.235 R2m R2c
[1,] 0.00528056 0.5406345 contractYear emmean SE df lower.CL upper.CL
no 81.5 3.06 597 75.5 87.5
yes 67.9 4.05 1246 60.0 75.9
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
fantasyPoints ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsRB_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 20671.4
Scaled residuals:
Min 1Q Median 3Q Max
-3.4371 -0.4920 -0.1588 0.4013 3.6457
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 6841.8 82.715
ageCentered20 54.6 7.389 -0.75
Residual 2641.7 51.398
Number of obs: 1842, groups: player_id, 546
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 70.0016 8.8203 729.4479 7.936 7.85e-15 ***
contractYearyes -9.3663 3.6074 1564.7556 -2.596 0.00951 **
ageCentered20 1.5209 3.0282 968.8867 0.502 0.61560
ageCentered20Quadratic -1.1212 0.1823 514.8215 -6.149 1.57e-09 ***
years_of_experience 12.0528 2.0883 659.2541 5.772 1.21e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.170
ageCentrd20 -0.868 -0.171
agCntrd20Qd 0.716 0.160 -0.740
yrs_f_xprnc 0.264 -0.065 -0.551 -0.086 R2m R2c
[1,] 0.06210342 0.6125754 contractYear emmean SE df lower.CL upper.CL
no 80.7 3.08 599 74.6 86.7
yes 71.3 4.04 1227 63.4 79.2
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 18.2.3 WR/TE
Code
Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: receiving_yards ~ contractYear + (1 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 32762.7
Scaled residuals:
Min 1Q Median 3Q Max
-4.8528 -0.5278 -0.1103 0.5063 4.5623
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 280.4 16.74
Residual 182.7 13.52
Number of obs: 3845, groups: player_id, 1087
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 25.1251 0.5908 1296.1433 42.528 < 2e-16 ***
contractYearyes -4.0728 0.5567 3257.3106 -7.316 3.2e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.234 R2m R2c
[1,] 0.00733895 0.6083696 contractYear emmean SE df lower.CL upper.CL
no 25.1 0.591 1189 24.0 26.3
yes 21.1 0.711 2075 19.7 22.4
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
mixedModelAge_receivingYards <- lmerTest::lmer(
receiving_yards ~ contractYear + ageCentered20 + ageCentered20Quadratic + years_of_experience + (1 + ageCentered20 | player_id),
data = player_statsContractsWRTE_seasonal,
control = lmerControl(optimizer = "bobyqa")
)
summary(mixedModelAge_receivingYards)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
receiving_yards ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 32324.7
Scaled residuals:
Min 1Q Median 3Q Max
-2.9759 -0.5212 -0.0962 0.4752 3.8093
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 546.051 23.368
ageCentered20 5.812 2.411 -0.69
Residual 136.886 11.700
Number of obs: 3843, groups: player_id, 1086
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 14.70081 1.57144 1529.88027 9.355 < 2e-16 ***
contractYearyes -3.03800 0.54699 3035.02757 -5.554 3.03e-08 ***
ageCentered20 2.45054 0.53041 2226.85462 4.620 4.05e-06 ***
ageCentered20Quadratic -0.43105 0.02688 1513.89053 -16.036 < 2e-16 ***
years_of_experience 3.34628 0.40668 1329.37210 8.228 4.48e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.122
ageCentrd20 -0.844 -0.150
agCntrd20Qd 0.659 0.071 -0.648
yrs_f_xprnc 0.336 0.030 -0.663 -0.067 R2m R2c
[1,] 0.1117034 0.744591 contractYear emmean SE df lower.CL upper.CL
no 24.0 0.622 1199 22.8 25.2
yes 20.9 0.713 1898 19.5 22.3
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: receiving_epa ~ contractYear + (1 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 12590.3
Scaled residuals:
Min 1Q Median 3Q Max
-5.5745 -0.5715 -0.0389 0.5376 3.8959
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 0.555 0.745
Residual 1.301 1.141
Number of obs: 3770, groups: player_id, 1070
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 0.65910 0.03347 1440.32757 19.692 < 2e-16 ***
contractYearyes -0.16319 0.04527 3585.01245 -3.605 0.000316 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.358 R2m R2c
[1,] 0.002954858 0.30102 contractYear emmean SE df lower.CL upper.CL
no 0.659 0.0335 1268 0.593 0.725
yes 0.496 0.0457 2454 0.406 0.585
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
receiving_epa ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 12573.5
Scaled residuals:
Min 1Q Median 3Q Max
-5.4226 -0.5652 -0.0324 0.5258 3.8586
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 0.9379 0.96844
ageCentered20 0.0055 0.07416 -0.66
Residual 1.2482 1.11724
Number of obs: 3769, groups: player_id, 1069
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 3.514e-01 1.040e-01 1.117e+03 3.379 0.000752 ***
contractYearyes -1.635e-01 4.791e-02 3.496e+03 -3.413 0.000651 ***
ageCentered20 7.729e-02 3.395e-02 1.207e+03 2.277 0.022982 *
ageCentered20Quadratic -9.348e-03 1.879e-03 4.726e+02 -4.974 9.18e-07 ***
years_of_experience 6.149e-02 2.301e-02 1.144e+03 2.672 0.007654 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.143
ageCentrd20 -0.871 -0.203
agCntrd20Qd 0.758 0.117 -0.741
yrs_f_xprnc 0.202 0.041 -0.524 -0.121 R2m R2c
[1,] 0.01285765 0.3377664 contractYear emmean SE df lower.CL upper.CL
no 0.657 0.0346 1214 0.589 0.725
yes 0.494 0.0467 2498 0.402 0.585
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: fantasyPoints ~ contractYear + (1 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 42562.6
Scaled residuals:
Min 1Q Median 3Q Max
-3.2503 -0.5270 -0.1438 0.4785 4.5956
Random effects:
Groups Name Variance Std.Dev.
player_id (Intercept) 2935 54.18
Residual 2473 49.73
Number of obs: 3845, groups: player_id, 1087
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 76.244 1.978 1334.194 38.543 < 2e-16 ***
contractYearyes -14.199 2.032 3340.122 -6.989 3.32e-12 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr)
contrctYrys -0.258 R2m R2c
[1,] 0.007634808 0.5462717 contractYear emmean SE df lower.CL upper.CL
no 76.2 1.98 1207 72.4 80.1
yes 62.0 2.44 2199 57.3 66.8
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 Code
mixedModelAge_fantasyPtsReceiving <- lmerTest::lmer(
fantasyPoints ~ contractYear + ageCentered20 + ageCentered20Quadratic + years_of_experience + (1 + ageCentered20 | player_id),
data = player_statsContractsWRTE_seasonal,
control = lmerControl(optimizer = "bobyqa")
)
summary(mixedModelAge_fantasyPtsReceiving)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
fantasyPoints ~ contractYear + ageCentered20 + ageCentered20Quadratic +
years_of_experience + (1 + ageCentered20 | player_id)
Data: player_statsContractsWRTE_seasonal
Control: lmerControl(optimizer = "bobyqa")
REML criterion at convergence: 42196.7
Scaled residuals:
Min 1Q Median 3Q Max
-3.0668 -0.4967 -0.1292 0.4563 4.9582
Random effects:
Groups Name Variance Std.Dev. Corr
player_id (Intercept) 5863.26 76.572
ageCentered20 60.76 7.795 -0.71
Residual 1960.17 44.274
Number of obs: 3843, groups: player_id, 1086
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 43.34064 5.49900 1469.85276 7.882 6.25e-15 ***
contractYearyes -10.48463 2.03309 3169.40708 -5.157 2.66e-07 ***
ageCentered20 7.30197 1.83894 2147.02160 3.971 7.40e-05 ***
ageCentered20Quadratic -1.37759 0.09638 1290.30852 -14.293 < 2e-16 ***
years_of_experience 11.34115 1.36400 1283.33031 8.315 2.32e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) cntrcY agCn20 agC20Q
contrctYrys 0.128
ageCentrd20 -0.852 -0.161
agCntrd20Qd 0.684 0.080 -0.675
yrs_f_xprnc 0.306 0.032 -0.629 -0.076 R2m R2c
[1,] 0.1034356 0.6793804 contractYear emmean SE df lower.CL upper.CL
no 72.5 2.07 1204 68.5 76.6
yes 62.1 2.45 2055 57.3 66.9
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 18.2.4 QB/RB/WR/TE
18.3 Conclusion
18.4 Session Info
R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
locale:
[1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8
[4] LC_COLLATE=C.UTF-8 LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8
[7] LC_PAPER=C.UTF-8 LC_NAME=C LC_ADDRESS=C
[10] LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
time zone: UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lubridate_1.9.4 forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4
[5] purrr_1.0.4 readr_2.1.5 tidyr_1.3.1 tibble_3.3.0
[9] ggplot2_3.5.2 tidyverse_2.0.0 emmeans_1.11.1 MuMIn_1.48.11
[13] lmerTest_3.1-3 lme4_1.1-37 Matrix_1.7-3 nflreadr_1.4.1
[17] petersenlab_1.1.5
loaded via a namespace (and not attached):
[1] tidyselect_1.2.1 psych_2.5.3 viridisLite_0.4.2
[4] farver_2.1.2 fastmap_1.2.0 TH.data_1.1-3
[7] digest_0.6.37 rpart_4.1.24 timechange_0.3.0
[10] estimability_1.5.1 lifecycle_1.0.4 cluster_2.1.8.1
[13] survival_3.8-3 magrittr_2.0.3 compiler_4.5.0
[16] rlang_1.1.6 Hmisc_5.2-3 tools_4.5.0
[19] yaml_2.3.10 data.table_1.17.4 knitr_1.50
[22] htmlwidgets_1.6.4 mnormt_2.1.1 plyr_1.8.9
[25] RColorBrewer_1.1-3 multcomp_1.4-28 withr_3.0.2
[28] foreign_0.8-90 numDeriv_2016.8-1.1 nnet_7.3-20
[31] grid_4.5.0 stats4_4.5.0 lavaan_0.6-19
[34] xtable_1.8-4 colorspace_2.1-1 scales_1.4.0
[37] MASS_7.3-65 cli_3.6.5 mvtnorm_1.3-3
[40] rmarkdown_2.29 reformulas_0.4.1 generics_0.1.4
[43] rstudioapi_0.17.1 tzdb_0.5.0 reshape2_1.4.4
[46] minqa_1.2.8 DBI_1.2.3 cachem_1.1.0
[49] splines_4.5.0 parallel_4.5.0 base64enc_0.1-3
[52] mitools_2.4 vctrs_0.6.5 sandwich_3.1-1
[55] boot_1.3-31 jsonlite_2.0.0 hms_1.1.3
[58] pbkrtest_0.5.4 Formula_1.2-5 htmlTable_2.4.3
[61] glue_1.8.0 nloptr_2.2.1 codetools_0.2-20
[64] stringi_1.8.7 gtable_0.3.6 quadprog_1.5-8
[67] pillar_1.10.2 htmltools_0.5.8.1 R6_2.6.1
[70] Rdpack_2.6.4 mix_1.0-13 evaluate_1.0.3
[73] pbivnorm_0.6.0 lattice_0.22-6 rbibutils_2.3
[76] backports_1.5.0 broom_1.0.8 memoise_2.0.1
[79] Rcpp_1.0.14 coda_0.19-4.1 gridExtra_2.3
[82] nlme_3.1-168 checkmate_2.3.2 xfun_0.52
[85] zoo_1.8-14 pkgconfig_2.0.3