As I was reading the paper I was wondering, because it wasn\u2019t specified in the methods section, if the researchers had given any thought to whether their random effects were crossed or nested. Then I thought, maybe people aren\u2019t thinking about this and perhaps a short blog article with an example would be useful. So here we are\u2026<\/p>\n
For brevity, I’ll only present short snippets of code in the blog article. The entire code is available on my Github Page<\/a><\/strong><\/span>.\u00a0<\/em><\/p>\n Data<\/strong><\/span><\/p>\n We need to simulate some data to work with. To keep things simple, we will create a data set of a league that has 6 teams and 60 players. Each player will have a player value metric, the outcome of interest, for every team they played on during the season.<\/p>\n EDA<\/strong><\/span><\/p>\n How many unique players were on each team during the season?<\/p>\n Which players played on multiple teams?<\/p>\n Team Summary Stats<\/p>\n Crossed vs Nested Effects<\/strong><\/span><\/p>\n We have 14 players that played on multiple teams (Player 11 ended up playing for all 6 teams!) and, as a consequence, different teams had different numbers of individual players. For example, the Bears had 13 different players while the Pythons had 6 different players. This small detail is important to consider with respect to our random effects because crossed and nested effects behave differently with respect to how they partition variance.<\/p>\n When constructing mixed models we usually have a data hierarchy where observations are grouped or nested within specific categories. When the observations are nested the observations are exclusive to specific groups. Some examples:<\/p>\n By contrast, crossed effects can occur when observations move between groups during the same study\/observation period. A few examples of crossed effects are:<\/p>\n To help solidify this concept a bit more, I’ve created the below drawing.<\/p>\n We see that in the Completely Nested<\/strong> example there are individual players that only player for their respective teams. Contrast that to the Completely Crossed<\/strong> example, where every player ends up playing on each team. This would, obviously, be an incredibly weird scenario and not one we’d see in sport but probably one that you’d see in researched where there is a crossover of treatments. Finally, the Partially Nested & Crossed<\/strong> example has a few players playing for multiple teams (for example, Player 1 played for both Team 1 and Team 2) and some players that stay with a single team during the season. This version is the one that we most frequently encounter in team sport.<\/p>\n Now that we have a general understanding of the difference between crossed and random effects, let’s see how this looks when we build a model on our simulated data and (most importantly) let’s see how specifying the model to have crossed or nested random effects changes the model output.<\/p>\n Model Examples<\/strong><\/span><\/p>\n Crossed<\/em><\/p>\n Since we commonly deal with crossed (or partially nested\/crossed) data in team sport, I’ll start with modeling the data to have crossed random effects. As a side note, most of the time when I look at people’s models in sport science they are creating the model structure like this, by default.<\/p>\n We will build random intercept model (since we didn’t simulate any fixed effects) and allow a separate intercept variance for both team<\/strong><\/em> and player_id<\/strong><\/em>, keeping in mind that this specifies that there are players who are potentially playing on different teams (crossed observations) across the season.<\/p>\n Nested<\/em><\/p>\n Next, we fit the random intercept model but this time we treat our random effects of player_id<\/strong><\/em> and team<\/strong><\/em> as nested. That is to say that we are allowing the players to be nested within their respective teams. To review, this means that we are treating this data as if each player only played on a single team during the season and never changed teams.<\/p>\n Comparing the two models<\/strong><\/span><\/p>\n First, we notice that the fixed effects are slightly different between the two models, but the difference in small in this simulated data.<\/p>\n The random effects look very different between the two model structures. In the crossed effects model we get two different random effects — one for the player_id<\/strong><\/em> and one for the team<\/strong><\/em>. However, in the nested effects model we have a random effect for team<\/strong><\/em> but then we have a random effect for player_id<\/strong><\/em> nested within each team<\/strong><\/em>. So, in the nested model a player that played for multiple teams will have different random intercept for each of the teams they were nested in. Both models, of course, also have a residual in the random effects, however the values are very different.<\/p>\n Let’s compile the random effects side-by-side to get a better appreciation for the differences. We will start with the `team` random effect since it is present in both models.<\/p>\n There are pretty stark differences between the random intercepts of the nested and crossed models! Recall from our EDA that the Bears had the highest average value of all the teams and they also had the fewest number of different players (5).<\/p>\n The player random effects get a little wonky because players who played for multiple teams will end up having different random effects depending on which team we nested them in. Conversely, in the crossed model we get one random effect estimate per player.<\/p>\n To join the random effects from both models we have to do a bit of work on the nested model to extract the info. (NOTE: <\/strong>Since the table is really long I’ll only show the first 10 rows here but you can got to the Github page<\/a><\/span><\/strong> to see the full output.<\/em>)<\/p>\n Joining the two model random effects on player_id<\/strong><\/em> we see that player’s have some very different random effects between the two models. In particular, players that played on different teams can differ greatly between the models.<\/p>\n For example, let’s look at Player 21.<\/p>\n\r\nsuppressPackageStartupMessages({\r\n suppressMessages({\r\n suppressWarnings({\r\n library(tidyverse)\r\n library(lme4)\r\n library(arm)\r\n })\r\n })\r\n})\r\n\r\n\r\ntheme_set(theme_bw())\r\n\r\nset.seed(225544)\r\nteams <- sample(c("Sharks", "Bears", "Turtles", "Bisons", "Jaguars", "Pythons"), size = 60, replace = TRUE)\r\nplayers <- as.factor(sample(1:60, size = 60, replace = TRUE))\r\nintercept <- 50\r\nteam_effect <- case_when(teams == "Sharks" ~ rnorm(1, mean = 0, sd = 7), \r\n teams == "Bears" ~ rnorm(1, mean = 0, sd = 2),\r\n teams == "Turtles" ~ rnorm(1, mean = 0, sd = 1),\r\n teams == "Bisons" ~ rnorm(1, mean = 0, sd = 13),\r\n teams == "Jaguars" ~ rnorm(1, mean = 0, sd = 5),\r\n teams == "Pythons" ~ rnorm(1, mean = 0, sd = 3))\r\nplayer_effect <- rnorm(length(levels(players)), mean = 0, sd = 10)\r\nrandom_noise <- 2\r\n\r\n\r\n\r\ndat <- data.frame( team = teams, player_id = players ) %>%\r\n mutate(\r\n mu = intercept + team_effect + player_effect[players],\r\n player_value = round(mu + rnorm(n(), mean = 0, sd = random_noise), 1)\r\n ) %>%\r\n dplyr::select(-mu)\r\n\r\ndat %>%\r\n head()\r\n<\/pre>\n![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2025\/04\/Screenshot-2025-03-31-at-8.44.15\u202fPM.png\")
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<\/a><\/p>\n\r\nlme_crossed <- lmer(player_value ~ 1 + (1|team) + (1|player_id), data = dat)\r\n<\/pre>\n
\r\nlme_nested <- lmer(player_value ~ 1 + (1|team) + (1|team:player_id), data = dat)\r\n<\/pre>\n
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