# How to handle multicollinearity in a linear regression with all dummy variables?

First, a little background:

I'm a college paintball coach and I'm trying to identify which of my players have the biggest impact on various statistics (e.g. winning percentage etc).

In order to do this I've put together a csv file with the following columns for each point that we play (you can use hockey as a mental model for how this works and each point as a "shift"):

• Win/Loss/Draw (-1 for loss, 0 for draw, 1 for win)
• Dummy variable for team A (1 if we are playing them, 0 if we are not)
• Dummy variables for each other team we play
• Dummy variable for each player on our side during that point

The dummy variable for each team is to that we can isolate better teams from the impact of each player.

For example, the headers would look like this

WinLoss,P_1,P_2,P_3,P_4,T_5,T_1,T_2,T_3


If Game 1 vs Team 1 had the following outcome:

Point 1: Players 1, 2 and 3 lost
Point 2: Players 1, 2 and 4 won
Point 3: Players 3, 4 and 5 lost

would look like this

WinLoss,P_1,P_2,P_3,P_4,T_5,T_1,T_2,T_3
0,1,1,1,0,0,1,0,0
1,1,1,0,1,0,1,0,0
0,0,0,1,1,1,1,0,0


In the actual data set the players are in groups of 5 but the above gives the general format. We try to keep players together on the same "lines" as we assume that helps build both team rapport and communication.

I then ran the below:

mydata <- read.csv('lines_data.csv')
attach(mydata)
wins2 = lm(WinLoss ~ P_1 + P_2 + P_3+ P_4 + P_5 + T_1 + T_2 + T_3 + T_4)
summary(wins2)


I noticed recently that if I change the order of the players, I get different co-efficient values for each player.

Some searching here on Cross Validated led me to this question.

It makes sense that there is a high degree of multicollinearity between the player dummy variables as the players are on the field in "lines"/"shifts" as mentioned above.

My question is to how to account for this when running the regression? Do I just need more data? Do I need more dummy variables?