I've recently started to participate in Tidy Tuesday, a weekly social data project in R. Each week they release a new dataset on their GitHub. It's a fun opportunity to explore new and interesting data sets. After you do a quick analysis and generate some figures, you're supposed to tweet it with #tidytuesday.

For my second #tidytuesday I show the power of centering your variables. The relationship b/w Goals & Penalty Minutes is made clearer when we center values on each player's career average. Variability within a player is driving this relationship! https://t.co/T2d6eN2bAO pic.twitter.com/d6GYCbmnA0

— Robin Sifre, Ph.D. (@SifreRobin) March 6, 2020

In this post, I walk through a bit of my code, but you can find the complete code on my GitHub.

This week's dataset comes from HockeyReference.com. I used 2 of the downloadable datasets:

1. Overall career goals.
2. Season level goals.

I thought that this multi-level data would be a fun way to explore the impact of how centering variables can clarify which level of analysis is driving the relationship.

The question

Is there a relationship between aggressive play (time spent in the penalty box) and scoring? On the one hand, more aggressive play could mean more goals. On the other hand, too much time in the penalty box takes you out of the game and could minimize your opportunity to score.

Disclaimer: I know nothing about Hockey, and most of my knowledge comes from the Mighty Ducks franchise. As someone who has lived in Minnesota for four years this is inexcusable, but here we are.

Mea culpa.

Why should I center my variables?

Let's imagine that there is a relationship between penalty minutes and goals scored. You can imagine this relationship work on two different levels: You can imagine that the relationship between penalty minutes and goals scored could work on 2 different levels:
1. Between-player effect. More aggressive players score more goals. A relationship between CAREER-AVERAGE penalty minutes and goals would suggest that the relationship is driven by between-player differences.
2. Within-player effect. When a given player plays more aggressively, they also score more goals. A relationship between SEASON-AVERAGE penalty minutes and goals would suggest that this relationship is driven by within-player differences.

However: when we examine season-averages we are “smushing together” variability driven by season-to-season changes within a player, and variability driven by differences between player skill. In other words, a top player may still score more goals even when they're having a poor season relative to a less skilled player.

Do aggressive players score more? Or do players score more when they play more aggressively? To better answer this question, we can center our variables.
If we take each player's season average (for penalty minutes & goals), and subtract it from their career average, we can isolate season-to-season variability.

Centering variables

So how did I center my variables? First, I created a variable that holds career-average penalty minutes and goals for each players. Let's call it player_stats.

player_stats = season_goals %>%
  dplyr::select(player, penalty_min, goals) %>%
  group_by(player) %>%
  summarize(ave_penalty = mean(penalty_min), #Career-average stats for player 
            ave_goals = mean(goals))

Then, I merged player_stats with season_goals, my data.frame that holds season-average goals and penalty minutes.

merged = merge(season_goals, player_stats, by = 'player')

Now creating the centered variables is a piece of cake.
We center each player's season-average penalty minutes (season_penalty) on their career-average penalty minutes (ave_penalty) to create the variable penalty_c. (We do the same thing for goals).

merged = merged %>%
  mutate(penalty_c = season_penalty - ave_penalty,
         goals_c = season_goals - ave_goals)

penalty_c now shows how a player's penalty minutes changes from season-to-season, holding ave_penalty constant.

Negative values of penalty_c indicate seasons where a player had lower than average minutes in the penalty box, and positive values indicate seasons where a player had higher than average minutes in the penalty box. In this case, average refers to their own personal average.

What does it buy us?

When we plot season-average penalty minutes against goals scores, it looks like there might be a relationship between these two variables:

But, it's a bit messy. In this figure we are smushing together variability between-players (e.g., high-scorers) and variability within a player (e.g., when someone is having a particularly good or bad season).

Now, let's plot the relationship between these two variables when we center them on a player's career average:

Here, the relationship becomes more clear. In a given season, when a player has more penalty minutes than their career average, they also score more goals than their career average.
Could this just be driven by how much play-time a player gets in a given season? Possibly! But this also illustrates how in this figure we are isolating change within a player, by holding their career averages constant.