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20 Cluster Analysis
20.1 Getting Started
20.1.1 Load Packages
20.1.2 Load Data
20.1.3 Overview
Whereas factor analysis evaluates how variables do or do not hang together—in terms of their associations and non-associations, cluster analysis evaluates how people are or or not similar—in terms of their scores on one or more variables. The goal of cluster analysis is to identify distinguishable subgroups of people. The people within a subgroup are expected to be more similar to each other than they are to people in other subgroups. For instance, we might expect that there are distinguishable subtypes of Wide Receivers: possession, deep threats, and slot-type Wide Receivers. Possession Wide Receivers tend to be taller and heavier, with good hands who catch the ball at a high rate. Deep threat Wide Receivers tend to be fast. Slot-type Wide Receivers tend to be small, quick, and agile. In order to identify these clusters of Wide Receivers, we might conduct a cluster analysis with variables relating to the players’ height, weight, percent of (catchable) targets caught, air yards received, and various metrics from the National Football League (NFL) Combine, including their times in the 40-yard dash, 20-yard shuttle run, and three cone drill.
There are many approaches to cluster analysis, including model-based clustering, density-based clustering, centroid-based clustering, hierarchical clustering (aka connectivity-based clustering), etc. An overview of approaches to cluster analysis in R is provided by Kassambara (2017). In this chapter, we focus on examples using model-based clustering with the R package mclust (Fraley et al., 2024; Scrucca et al., 2023), which uses Gaussian finite mixture modeling. The various types of mclust models are provided here: https://mclust-org.github.io/mclust/reference/mclustModelNames.html.
20.1.4 Tiers of Prior Season Fantasy Points
20.1.4.1 Prepare Data
Code
[1] 2024Code
player_stats_seasonal_offense_recent <- player_stats_seasonal %>%
filter(season == recentSeason) %>%
filter(position_group %in% c("QB","RB","WR","TE"))
player_stats_seasonal_offense_recentQB <- player_stats_seasonal_offense_recent %>%
filter(position_group == "QB")
player_stats_seasonal_offense_recentRB <- player_stats_seasonal_offense_recent %>%
filter(position_group == "RB")
player_stats_seasonal_offense_recentWR <- player_stats_seasonal_offense_recent %>%
filter(position_group == "WR")
player_stats_seasonal_offense_recentTE <- player_stats_seasonal_offense_recent %>%
filter(position_group == "TE")20.1.4.2 Identify the Optimal Number of Tiers by Position
20.1.4.2.1 Quarterbacks
Code
Bayesian Information Criterion (BIC):
E V
1 -982.9038 -982.9038
2 -964.3518 -927.2534
3 -973.0978 -930.5327
4 -971.2212 -912.2067
5 -971.1002 -924.2192
6 -979.8174 -928.6330
7 -974.9956 -949.0362
8 -981.8676 -955.9257
9 -990.5409 -963.9506
Top 3 models based on the BIC criterion:
V,4 V,5 V,2
-912.2067 -924.2192 -927.2534 Best BIC values:
V,4 V,5 V,2
BIC -912.2067 -924.21918 -927.25337
BIC diff 0.0000 -12.01245 -15.04664
Code
Integrated Complete-data Likelihood (ICL) criterion:
E V
1 -982.9038 -982.9038
2 -972.1069 -933.7840
3 -1039.9236 -945.9954
4 -1040.3715 -927.2426
5 -1033.2208 -945.8315
6 -1061.5988 -935.2325
7 -1056.6193 -993.1199
8 -1065.2675 -976.8222
9 -1088.2374 -986.9286
Top 3 models based on the ICL criterion:
V,4 V,2 V,6
-927.2426 -933.7840 -935.2325 Best ICL values:
V,4 V,2 V,6
ICL -927.2426 -933.78400 -935.232482
ICL diff 0.0000 -6.54137 -7.989849
Code
tiersQB_boostrap <- mclust::mclustBootstrapLRT(
data = player_stats_seasonal_offense_recentQB$fantasyPoints,
modelName = "V") # variable/unequal variance (for univariate data)
numTiersQB <- as.numeric(summary(tiersQB_boostrap)[,"Length"][1]) # or could specify the number of teams manually
tiersQB_boostrap-------------------------------------------------------------
Bootstrap sequential LRT for the number of mixture components
-------------------------------------------------------------
Model = V
Replications = 999
LRTS bootstrap p-value
1 vs 2 68.720575 0.001
2 vs 3 9.790787 0.032
3 vs 4 31.396105 0.001
4 vs 5 1.057678 0.624
20.1.4.2.2 Running Backs
Code
Bayesian Information Criterion (BIC):
E V
1 -1888.714 -1888.714
2 -1817.804 -1769.298
3 -1827.956 -1699.724
4 -1817.083 -1701.580
5 -1827.203 -1708.617
6 -1837.331 -1719.106
7 -1817.623 -1721.044
8 -1827.752 -1735.666
9 -1834.919 -1746.427
Top 3 models based on the BIC criterion:
V,3 V,4 V,5
-1699.724 -1701.580 -1708.617 Best BIC values:
V,3 V,4 V,5
BIC -1699.724 -1701.580264 -1708.616531
BIC diff 0.000 -1.855914 -8.892182
Code
Integrated Complete-data Likelihood (ICL) criterion:
E V
1 -1888.714 -1888.714
2 -1823.200 -1793.185
3 -1991.232 -1728.105
4 -1974.495 -1745.695
5 -2074.939 -1750.066
6 -2123.855 -1757.956
7 -2081.524 -1765.455
8 -2133.100 -1796.801
9 -2136.424 -1795.120
Top 3 models based on the ICL criterion:
V,3 V,4 V,5
-1728.105 -1745.695 -1750.066 Best ICL values:
V,3 V,4 V,5
ICL -1728.105 -1745.69534 -1750.06574
ICL diff 0.000 -17.58998 -21.96037
The model-based bootstrap clustering of Running Backs’ fantasy points is unable to run due to an error:
Thus, we cannot use the following code, which would otherwise summarize the model results, specify the number of tiers, and plot model comparisons:
20.1.4.2.3 Wide Receivers
Code
Bayesian Information Criterion (BIC):
E V
1 -2761.531 -2761.531
2 -2703.730 -2574.337
3 -2714.665 -2561.183
4 -2690.946 -2551.896
5 -2701.848 -2559.810
6 -2679.348 -2566.401
7 -2690.252 -2567.887
8 -2693.451 -2579.761
9 -2704.412 -2594.502
Top 3 models based on the BIC criterion:
V,4 V,5 V,3
-2551.896 -2559.810 -2561.183 Best BIC values:
V,4 V,5 V,3
BIC -2551.896 -2559.809568 -2561.182771
BIC diff 0.000 -7.913781 -9.286984
Code
Integrated Complete-data Likelihood (ICL) criterion:
E V
1 -2761.531 -2761.531
2 -2728.952 -2597.147
3 -2967.945 -2623.521
4 -2909.051 -2643.926
5 -3004.434 -2652.681
6 -2995.921 -2665.160
7 -3044.355 -2642.838
8 -3043.060 -2662.966
9 -3081.954 -2680.271
Top 3 models based on the ICL criterion:
V,2 V,3 V,7
-2597.147 -2623.521 -2642.838 Best ICL values:
V,2 V,3 V,7
ICL -2597.147 -2623.52084 -2642.83833
ICL diff 0.000 -26.37432 -45.69181
Code
tiersWR_boostrap <- mclust::mclustBootstrapLRT(
data = player_stats_seasonal_offense_recentWR$fantasyPoints,
modelName = "V") # variable/unequal variance (for univariate data)
numTiersWR <- as.numeric(summary(tiersWR_boostrap)[,"Length"][1]) # or could specify the number of teams manually
tiersWR_boostrap-------------------------------------------------------------
Bootstrap sequential LRT for the number of mixture components
-------------------------------------------------------------
Model = V
Replications = 999
LRTS bootstrap p-value
1 vs 2 203.573535 0.001
2 vs 3 29.532613 0.001
3 vs 4 25.665741 0.001
4 vs 5 8.464976 0.056
20.1.4.2.4 Tight Ends
Code
Bayesian Information Criterion (BIC):
E V
1 -1416.311 -1416.311
2 -1382.530 -1330.306
3 -1392.221 -1305.417
4 -1401.914 -1304.670
5 -1370.398 -1314.375
6 -1380.110 -1322.054
7 -1387.386 -1329.543
8 -1397.037 -1343.259
9 -1406.769 -1349.787
Top 3 models based on the BIC criterion:
V,4 V,3 V,5
-1304.670 -1305.417 -1314.375 Best BIC values:
V,4 V,3 V,5
BIC -1304.67 -1305.4171376 -1314.374518
BIC diff 0.00 -0.7472878 -9.704669
Code
Integrated Complete-data Likelihood (ICL) criterion:
E V
1 -1416.311 -1416.311
2 -1393.104 -1350.405
3 -1524.763 -1331.375
4 -1592.916 -1341.536
5 -1569.134 -1358.678
6 -1611.364 -1360.491
7 -1616.459 -1360.443
8 -1650.436 -1392.210
9 -1687.470 -1383.417
Top 3 models based on the ICL criterion:
V,3 V,4 V,2
-1331.375 -1341.536 -1350.405 Best ICL values:
V,3 V,4 V,2
ICL -1331.375 -1341.53615 -1350.40527
ICL diff 0.000 -10.16078 -19.02991
Code
tiersTE_boostrap <- mclust::mclustBootstrapLRT(
data = player_stats_seasonal_offense_recentTE$fantasyPoints,
modelName = "V") # variable/unequal variance (for univariate data)
numTiersTE <- as.numeric(summary(tiersTE_boostrap)[,"Length"][1]) # or could specify the number of teams manually
tiersTE_boostrap-------------------------------------------------------------
Bootstrap sequential LRT for the number of mixture components
-------------------------------------------------------------
Model = V
Replications = 999
LRTS bootstrap p-value
1 vs 2 100.537455 0.001
2 vs 3 39.421427 0.001
3 vs 4 15.279849 0.006
4 vs 5 4.827893 0.220
20.1.4.3 Fit the Cluster Model to the Optimal Number of Tiers
20.1.4.3.1 Quarterbacks
In our data, all of the following models are equivalent—i.e., they result in the same unequal variance model with a 4-cluster solution—but they arrive there in different ways.
Code
mclust::Mclust(
data = player_stats_seasonal_offense_recentQB$fantasyPoints,
G = numTiersQB,
)
mclust::Mclust(
data = player_stats_seasonal_offense_recentQB$fantasyPoints,
G = 4,
)
mclust::Mclust(
data = player_stats_seasonal_offense_recentQB$fantasyPoints,
)
mclust::Mclust(
data = player_stats_seasonal_offense_recentQB$fantasyPoints,
x = tiersQB_bic
)Let’s fit one of these:
Here are the number of players that are in each of the four clusters (i.e., tiers):
20.1.4.3.2 Running Backs
Here are the number of players that are in each of the four clusters (i.e., tiers):
20.1.4.3.3 Wide Receivers
Here are the number of players that are in each of the four clusters (i.e., tiers):
20.1.4.3.4 Tight Ends
Here are the number of players that are in each of the four clusters (i.e., tiers):
20.1.4.4 Plot the Tiers
We can merge the player’s classification into the dataset and plot each player’s classification.
20.1.4.4.1 Quarterbacks
Code
player_stats_seasonal_offense_recentQB$tier <- clusterModelQBs$classification
player_stats_seasonal_offense_recentQB <- player_stats_seasonal_offense_recentQB %>%
mutate(
tier = factor(max(tier, na.rm = TRUE) + 1 - tier)
)
player_stats_seasonal_offense_recentQB$position_rank <- rank(
player_stats_seasonal_offense_recentQB$fantasyPoints * -1,
na.last = "keep",
ties.method = "min")
plot_qbTiers <- ggplot2::ggplot(
data = player_stats_seasonal_offense_recentQB,
mapping = aes(
x = fantasyPoints,
y = position_rank,
color = tier
)) +
geom_point(
aes(
text = player_display_name # add player name for mouse over tooltip
)) +
scale_y_continuous(trans = "reverse") +
coord_cartesian(clip = "off") +
labs(
x = "Projected Points",
y = "Position Rank",
title = "Quarterback Fantasy Points by Tier",
color = "Tier") +
theme_classic() +
theme(legend.position = "top")
plotly::ggplotly(plot_qbTiers)20.1.4.4.2 Running Backs
Code
player_stats_seasonal_offense_recentRB$tier <- clusterModelRBs$classification
player_stats_seasonal_offense_recentRB <- player_stats_seasonal_offense_recentRB %>%
mutate(
tier = factor(max(tier, na.rm = TRUE) + 1 - tier)
)
player_stats_seasonal_offense_recentRB$position_rank <- rank(
player_stats_seasonal_offense_recentRB$fantasyPoints * -1,
na.last = "keep",
ties.method = "min")
plot_rbTiers <- ggplot2::ggplot(
data = player_stats_seasonal_offense_recentRB,
mapping = aes(
x = fantasyPoints,
y = position_rank,
color = tier
)) +
geom_point(
aes(
text = player_display_name # add player name for mouse over tooltip
)) +
scale_y_continuous(trans = "reverse") +
coord_cartesian(clip = "off") +
labs(
x = "Projected Points",
y = "Position Rank",
title = "Running Back Fantasy Points by Tier",
color = "Tier") +
theme_classic() +
theme(legend.position = "top")
plotly::ggplotly(plot_rbTiers)20.1.4.4.3 Wide Receivers
Code
player_stats_seasonal_offense_recentWR$tier <- clusterModelWRs$classification
player_stats_seasonal_offense_recentWR <- player_stats_seasonal_offense_recentWR %>%
mutate(
tier = factor(max(tier, na.rm = TRUE) + 1 - tier)
)
player_stats_seasonal_offense_recentWR$position_rank <- rank(
player_stats_seasonal_offense_recentWR$fantasyPoints * -1,
na.last = "keep",
ties.method = "min")
plot_wrTiers <- ggplot2::ggplot(
data = player_stats_seasonal_offense_recentWR,
mapping = aes(
x = fantasyPoints,
y = position_rank,
color = tier
)) +
geom_point(
aes(
text = player_display_name # add player name for mouse over tooltip
)) +
scale_y_continuous(trans = "reverse") +
coord_cartesian(clip = "off") +
labs(
x = "Projected Points",
y = "Position Rank",
title = "Wide Receiver Fantasy Points by Tier",
color = "Tier") +
theme_classic() +
theme(legend.position = "top")
plotly::ggplotly(plot_wrTiers)20.1.4.4.4 Tight Ends
Code
player_stats_seasonal_offense_recentTE$tier <- clusterModelTEs$classification
player_stats_seasonal_offense_recentTE <- player_stats_seasonal_offense_recentTE %>%
mutate(
tier = factor(max(tier, na.rm = TRUE) + 1 - tier)
)
player_stats_seasonal_offense_recentTE$position_rank <- rank(
player_stats_seasonal_offense_recentTE$fantasyPoints * -1,
na.last = "keep",
ties.method = "min")
plot_teTiers <- ggplot2::ggplot(
data = player_stats_seasonal_offense_recentTE,
mapping = aes(
x = fantasyPoints,
y = position_rank,
color = tier
)) +
geom_point(
aes(
text = player_display_name # add player name for mouse over tooltip
)) +
scale_y_continuous(trans = "reverse") +
coord_cartesian(clip = "off") +
labs(
x = "Projected Points",
y = "Position Rank",
title = "Tight End Fantasy Points by Tier",
color = "Tier") +
theme_classic() +
theme(legend.position = "top")
plotly::ggplotly(plot_teTiers)20.1.5 Types of Wide Receivers
Code
# Compute Advanced PFR Stats by Career
pfrVars <- nfl_advancedStatsPFR_seasonal %>%
select(pocket_time.pass:cmp_percent.def, g, gs) %>%
names()
weightedAverageVars <- c(
"pocket_time.pass",
"ybc_att.rush","yac_att.rush",
"ybc_r.rec","yac_r.rec","adot.rec","rat.rec",
"yds_cmp.def","yds_tgt.def","dadot.def","m_tkl_percent.def","rat.def"
)
recomputeVars <- c(
"drop_pct.pass", # drops.pass / pass_attempts.pass
"bad_throw_pct.pass", # bad_throws.pass / pass_attempts.pass
"on_tgt_pct.pass", # on_tgt_throws.pass / pass_attempts.pass
"pressure_pct.pass", # times_pressured.pass / pass_attempts.pass
"drop_percent.rec", # drop.rec / tgt.rec
"rec_br.rec", # rec.rec / brk_tkl.rec
"cmp_percent.def" # cmp.def / tgt.def
)
sumVars <- pfrVars[pfrVars %ni% c(
weightedAverageVars, recomputeVars,
"merge_name", "loaded.pass", "loaded.rush", "loaded.rec", "loaded.def")]
nfl_advancedStatsPFR_career <- nfl_advancedStatsPFR_seasonal %>%
group_by(pfr_id, merge_name) %>%
summarise(
across(all_of(weightedAverageVars), ~ weighted.mean(.x, w = g, na.rm = TRUE)),
across(all_of(sumVars), ~ sum(.x, na.rm = TRUE)),
.groups = "drop") %>%
mutate(
drop_pct.pass = drops.pass / pass_attempts.pass,
bad_throw_pct.pass = bad_throws.pass / pass_attempts.pass,
on_tgt_pct.pass = on_tgt_throws.pass / pass_attempts.pass,
pressure_pct.pass = times_pressured.pass / pass_attempts.pass,
drop_percent.rec = drop.rec / tgt.rec,
rec_br.rec = drop.rec / tgt.rec,
cmp_percent.def = cmp.def / tgt.def
)
uniqueCases <- nfl_advancedStatsPFR_seasonal %>% select(pfr_id, merge_name, gsis_id) %>% unique()
uniqueCases %>%
group_by(pfr_id) %>%
filter(n() > 1)Code
nfl_advancedStatsPFR_seasonal <- nfl_advancedStatsPFR_seasonal %>%
filter(pfr_id != "WillMa06" | merge_name != "MARCUSWILLIAMS" | !is.na(gsis_id))
nfl_advancedStatsPFR_career <- left_join(
nfl_advancedStatsPFR_career,
nfl_advancedStatsPFR_seasonal %>% select(pfr_id, merge_name, gsis_id) %>% unique(),
by = c("pfr_id", "merge_name")
)
# Compute Player Stats Per Season
player_stats_seasonal_careerWRs <- player_stats_seasonal %>%
filter(position == "WR") %>%
group_by(player_id) %>%
summarise(
across(all_of(c("targets", "receptions", "receiving_air_yards")), ~ weighted.mean(.x, w = games, na.rm = TRUE)),
.groups = "drop")
# Drop players with no receiving air yards
player_stats_seasonal_careerWRs <- player_stats_seasonal_careerWRs %>%
filter(receiving_air_yards != 0) %>%
rename(
targets_per_season = targets,
receptions_per_season = receptions,
receiving_air_yards_per_season = receiving_air_yards
)
# Merge
playerListToMerge <- list(
nfl_players %>% select(gsis_id, display_name, position, height, weight),
nfl_combine %>% select(gsis_id, vertical, forty, ht, wt),
player_stats_seasonal_careerWRs %>% select(player_id, targets_per_season, receptions_per_season, receiving_air_yards_per_season) %>%
rename(gsis_id = player_id),
nfl_actualStats_career_player_inclPost %>% select(player_id, receptions, targets, receiving_air_yards, air_yards_share, target_share) %>%
rename(gsis_id = player_id),
nfl_advancedStatsPFR_career %>% select(gsis_id, adot.rec, rec.rec, brk_tkl.rec, drop.rec, drop_percent.rec)
)
merged_data <- playerListToMerge %>%
reduce(
full_join,
by = c("gsis_id"),
na_matches = "never")Additional processing:
Code
merged_data <- merged_data %>%
mutate(
height_coalesced = coalesce(height, ht),
weight_coalesced = coalesce(weight, wt),
receptions_coalesced = pmax(receptions, rec.rec, na.rm = TRUE),
receiving_air_yards_per_rec = receiving_air_yards / receptions
)
merged_data$receiving_air_yards_per_rec[which(merged_data$receptions == 0)] <- 0
merged_dataWRs <- merged_data %>%
filter(position == "WR")
merged_dataWRs_cluster <- merged_dataWRs %>%
filter(receptions_coalesced >= 100) %>% # keep WRs with at least 100 receptions
select(gsis_id, display_name, vertical, forty, height_coalesced, weight_coalesced, adot.rec, drop_percent.rec, receiving_air_yards_per_rec, brk_tkl.rec, receptions_per_season) %>% #targets_per_season, receiving_air_yards_per_season, air_yards_share, target_share
na.omit()20.1.5.1 Identify the Number of WR Types
Code
Bayesian Information Criterion (BIC):
EII VII EEI VEI EVI VVI EEE
1 -8454.016 -8454.016 -5152.072 -5152.072 -5152.072 -5152.072 -5061.963
2 -8048.408 -8016.971 -5158.404 -5133.220 -5107.907 -5097.764 -5091.426
3 -7902.509 -7867.378 -5113.313 -5079.821 -4985.776 -5008.906 -5094.265
4 -7810.730 -7814.913 -5081.823 -5052.165 -5013.736 -5013.465 -5112.015
5 -7698.027 -7678.692 -5097.037 -5069.040 -5069.200 -5033.071 -5075.946
6 -7737.280 -7701.533 -5093.353 -5066.906 -5064.475 -5074.656 -5098.294
7 -7683.895 -7666.122 -5124.703 -5090.771 -5118.641 -5119.460 -5135.785
8 -7708.298 NA -5110.228 NA NA NA -5110.070
9 -7774.346 NA -5136.425 NA NA NA -5122.346
VEE EVE VVE EEV VEV EVV VVV
1 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963
2 -5007.572 -4976.592 -4968.094 -5076.718 -5128.228 -5085.093 -5108.094
3 -5034.293 -4920.517 -4971.412 -5142.228 -5204.047 -5200.756 -5207.984
4 -5024.710 -5018.231 -4986.038 -5310.765 -5372.021 -5362.652 -5384.626
5 -5037.345 -4998.579 -5012.025 -5411.801 -5455.265 -5479.829 -5557.317
6 NA NA NA -5526.068 -5513.935 NA NA
7 NA NA NA -5607.206 -5644.139 NA NA
8 NA NA NA -5827.743 NA NA NA
9 NA NA NA -6009.122 NA NA NA
Top 3 models based on the BIC criterion:
EVE,3 VVE,2 VVE,3
-4920.517 -4968.094 -4971.412 Best BIC values:
EVE,3 VVE,2 VVE,3
BIC -4920.517 -4968.09394 -4971.41195
BIC diff 0.000 -47.57727 -50.89529
Code
Integrated Complete-data Likelihood (ICL) criterion:
EII VII EEI VEI EVI VVI EEE
1 -8454.016 -8454.016 -5152.072 -5152.072 -5152.072 -5152.072 -5061.963
2 -8054.467 -8022.074 -5188.589 -5168.348 -5121.102 -5113.110 -5106.041
3 -7912.142 -7875.762 -5133.949 -5098.692 -5007.408 -5033.036 -5112.127
4 -7819.921 -7826.700 -5102.602 -5076.934 -5038.652 -5031.973 -5139.461
5 -7709.079 -7688.423 -5128.500 -5096.415 -5097.985 -5052.260 -5092.795
6 -7751.195 -7713.392 -5123.927 -5086.934 -5086.074 -5094.807 -5119.475
7 -7697.688 -7677.348 -5158.280 -5109.184 -5137.655 -5136.448 -5163.501
8 -7725.662 NA -5140.532 NA NA NA -5137.724
9 -7790.141 NA -5172.862 NA NA NA -5146.788
VEE EVE VVE EEV VEV EVV VVV
1 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963 -5061.963
2 -5008.372 -4986.265 -4980.919 -5079.017 -5142.868 -5087.834 -5113.593
3 -5045.216 -4932.619 -4990.999 -5147.506 -5211.583 -5205.964 -5215.238
4 -5037.849 -5035.341 -4996.453 -5322.424 -5383.665 -5368.753 -5390.619
5 -5047.991 -5008.817 -5026.191 -5415.840 -5459.559 -5486.313 -5563.007
6 NA NA NA -5532.165 -5518.187 NA NA
7 NA NA NA -5610.051 -5646.800 NA NA
8 NA NA NA -5830.894 NA NA NA
9 NA NA NA -6011.256 NA NA NA
Top 3 models based on the ICL criterion:
EVE,3 VVE,2 EVE,2
-4932.619 -4980.919 -4986.265 Best ICL values:
EVE,3 VVE,2 EVE,2
ICL -4932.619 -4980.91857 -4986.26523
ICL diff 0.000 -48.29982 -53.64648
Based on the cluster analyses, it appears that three clusters are the best fit to the data.
20.1.5.2 Fit the Cluster Model to the Optimal Number of WR Types
Code
----------------------------------------------------
Gaussian finite mixture model fitted by EM algorithm
----------------------------------------------------
Mclust EVE (ellipsoidal, equal volume and orientation) model with 3 components:
log-likelihood n df BIC ICL
-2242.626 126 90 -4920.517 -4932.619
Clustering table:
1 2 3
30 80 16 20.1.5.3 Plots of the Cluster Model
20.1.5.4 Interpreting the Clusters
1 2 3
30 80 16 Code
[,1] [,2] [,3]
type 1.00 2.00 3.00
vertical 36.32 36.12 35.75
forty 4.48 4.46 4.47
height_coalesced 73.07 72.54 73.19
weight_coalesced 206.77 197.65 204.62
adot.rec 10.11 10.58 12.12
drop_percent.rec 0.04 0.05 0.07
receiving_air_yards_per_rec 7.84 9.60 7.26
brk_tkl.rec 26.60 7.85 0.31
receptions_per_season 79.23 44.51 39.35Based on this analysis (and the variables included), there appear to be three types of Wide Receivers. Type 1 Wide Receivers includes the Elite WR1s who are strong possession receivers (note: not all players in a given cluster map on perfectly to the typology—i.e., not all Type 1 Wide Receivers are elite WR1s). They tend to have the lowest drop percentage, the shortest average depth of target, and the fewest receiving air yards per reception. They tend to have the most receptions per season and break the most tackles.
Type 2 Wide Receivers includes the consistent contributor, WR2 types. They had fewer receptions and fewer broken tackles than Type 1 Wide Receivers. Their average depth of target was longer than Type 1, and they had more receiving air yards per reception than Type 1.
Type 3 Wide Receivers includes the deep threats. They have the greatest average depth of target and the most receiving yards per reception. However, they also have the fewest receptions, the highest drop percentage, and the fewest broken tackles. Thus, they may be considered the boom-or-bust Wide Receivers.
The tiers were not particularly distinguishable based on their height, weight, vertical jump, or forty-yard dash time.
Type 1 (“Elite/WR1”) WRs:
Type 2 (“Consistent Contributor/WR2”) WRs:
Type 3 (“Deep Threat/Boom-or-Bust”) WRs:
20.2 Conclusion
20.3 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_3.2.1
[9] tidyverse_2.0.0 plotly_4.10.4 ggplot2_3.5.2 mclust_6.1.1
[13] nflreadr_1.4.1 petersenlab_1.1.5
loaded via a namespace (and not attached):
[1] gtable_0.3.6 xfun_0.52 htmlwidgets_1.6.4 psych_2.5.3
[5] lattice_0.22-6 tzdb_0.5.0 crosstalk_1.2.1 quadprog_1.5-8
[9] vctrs_0.6.5 tools_4.5.0 generics_0.1.4 stats4_4.5.0
[13] parallel_4.5.0 cluster_2.1.8.1 pkgconfig_2.0.3 data.table_1.17.2
[17] checkmate_2.3.2 RColorBrewer_1.1-3 lifecycle_1.0.4 compiler_4.5.0
[21] farver_2.1.2 mnormt_2.1.1 mitools_2.4 htmltools_0.5.8.1
[25] lazyeval_0.2.2 yaml_2.3.10 htmlTable_2.4.3 Formula_1.2-5
[29] pillar_1.10.2 cachem_1.1.0 Hmisc_5.2-3 rpart_4.1.24
[33] nlme_3.1-168 lavaan_0.6-19 tidyselect_1.2.1 digest_0.6.37
[37] mvtnorm_1.3-3 stringi_1.8.7 reshape2_1.4.4 labeling_0.4.3
[41] fastmap_1.2.0 grid_4.5.0 colorspace_2.1-1 cli_3.6.5
[45] magrittr_2.0.3 base64enc_0.1-3 pbivnorm_0.6.0 foreign_0.8-90
[49] withr_3.0.2 scales_1.4.0 backports_1.5.0 timechange_0.3.0
[53] httr_1.4.7 rmarkdown_2.29 nnet_7.3-20 gridExtra_2.3
[57] hms_1.1.3 memoise_2.0.1 evaluate_1.0.3 knitr_1.50
[61] mix_1.0-13 viridisLite_0.4.2 rlang_1.1.6 Rcpp_1.0.14
[65] xtable_1.8-4 glue_1.8.0 DBI_1.2.3 rstudioapi_0.17.1
[69] jsonlite_2.0.0 R6_2.6.1 plyr_1.8.9