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Is there a way to compare individual team member contributions when pitted against another team multiple times? This is in a sport like rowing where you take N number of individuals from a set A and M number of individuals against set B.

Is there a way to make estimations on-going with incomplete sets (not all unique combinations of N vs M exist) or with some interdependence where some in N or M are never changed out?

In a similar vein to this question How to extract an individual's (normalized) contribution from a group? I am interested if there are some recommended approaches or literature on this subject. The difference in my question is this limited to two groups with known team vs team data.

Just to make it clear, I'm trying to compare relative times between teams and then try to see individual performance. So for example here is some fake data between Team A and Team B with 9 members to choose 5 from with the teams results for that Event (note the times are the same for members because they are racing in the same boats).

TEAM A      Event1     Event2   Event3
N1          55.5       53.3     51.2
N2          ----       53.3     ----
N3          55.5       53.3     ----
N4          55.5       -----    51.2
N5          55.5       ----     51.2
N6          ----       53.3     51.2
N7          ----       53.3     ----
N8          55.5       ----     ----
N9          ----       ----     51.2

Vs TEAM B
TEAM B      EVENT1     Event2   Event3
M1          55.0       52.9     53.2
M2          ----       52.9     ----
M3          55.0       52.9     ----
M4          55.0       ----     53.2
M5          55.0       ----     53.2
M6          ----       52.9     53.2
M7          ----       52.9     ----
M8          55.0       ----     ----
M9          ----       ----     53.2

Each event occurs in different conditions so only the same events can be directly compared of Team A vs B. So for example Team A group at Event1 clocked 55.5 and Team B group won at 55.0.

What I've Tried

I've computed their relative time difference per event. Then I've taken each individuals average time difference and compare them. This works ok, but I don't think this is an accurate comparison and I don't know how to estimate cases where the data is limited or how to know what data still should be gathered.

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user6972
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  • How many events can you observe for comparison of two teams? Typically more than in the example, i.e. three? – g g Jul 08 '19 at 20:09
  • The sample size is typically very small, usually no more than 6 or 7. But I don't want to limit the question too much and discourage discussion. – user6972 Jul 08 '19 at 20:30
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    You need to come up with a mathematical model describing how the team's performance depends on the team members. For example, if you say each person $i$ is capable of rowing at a random speed $X_i$, normally distributed as $N(\mu_i, \sigma_i^2)$, and you say the speed of the boat is the sum of the speeds of the rowers, then the race time can be calculated as a function of which rowers are rowing. Then you can attempt to deduce the $\mu_i$ and $\sigma_i$ from the data. – Jack M Jul 08 '19 at 21:56

1 Answers1

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I would use a fixed effects regression, with a fixed effect for each team member, and the team time as a dependent variable. Formally, this is identical to an OLS regression with a dummy variable for each team member.

Let your dependent variable be $Outcome_{ts}$ for team $s$ at time $t$ - the outcome could be the time by team $s$ on day $t$ in the race. Let $Mi_{ts}$ be a dummy variable denoting team member $i$ on team $s$ at time $t$, i.e., $Mi_{ts}=1$ if $i$ was on team $s$ at time $t$, and $Mi_{ts}=0$ otherwise. The simplest fixed effects regression is then: $$Outcome_{ts}=a_1 Constant+a_2M2_{ts}+\ldots+a_nMn_{ts}+\varepsilon_{ts}.$$ Note we are omitting variable $M1$ and instead include a Constant in the regression; thus, member 1 is the "reference category" in the regression and all other team member estimates are relative to member 1.

The estimate $a_1$ for the constant then represents the average contribution of member 1; the remaining estimates $a_2,\ldots,a_n$ give you the average contributions of the other team members relative to member 1. Negative estimates mean that member is on average worse than member 1, positive estimates mean that member is on average better than member 1. To obtain the absolute average contribution of members $i=2,\ldots,n$, just compute $a_1+a_i$.

You can use additional variables in the regression if you have more data; for example, you could control for things like whether which might differ by $t$. In that case, the regression would be $$Outcome_{ts}=a_1 Constant+a_2M2_{ts}+\ldots+a_nMn_{ts}+\beta\cdot X_{ts}+\varepsilon_{ts},$$ where $X$ is a vector of these control variables and $\beta$ is a vector of their contributions to the team outcome.

This fixed effects approach has been used in labor economics before, where the effect of supervisors on team output was determined (supervisors were randomly allocated to teams at the beginning of each day) - I believe this was a study where Edward Lazear was one of the authors.

I think the important thing for you is to think about how each individual's performance maps into the team output. Does the average contribution matter, or the minimum contribution ("weakest link in the chain")? Depending on the answer, different estimation techniques might be suitable. The above FE approach would be suitable if the average contribution matters. You can see that in the above regressions, where the team outcome is modeled as a (weighted) sum of the individual team members' contributions.

Nameless
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  • The examples I see for FE show full data sets for each member. What would you recommend I do for cases with no data (member was not in the race)? – user6972 Jul 08 '19 at 21:52
  • You don't need all members to participate in all races, or even that all members participated in the same number of races. You can still estimate the fixed effects for all members that participated at least once. Just use the data from all races; this then a fixed effects estimation with an "unbalanced panel". – Nameless Jul 08 '19 at 23:25
  • How would I create a FE model for both individuals and crews? Should I just organize the data by crews instead? And should I consider a two way regression to allow for different conditions on difference race days? – user6972 Jul 09 '19 at 19:11
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    I added more detail on how to run such a regression, and also elaborated on how you can control for different conditions on different race days. This can be done in one regression.

    I am not sure what you mean by "a FE model for both individuals and crews". IF a crew is the set of team members active in a race, then you can estimate crew fixed effects by defining dummies for crew rather than team members. However, you cannot estimate both member and crew fixed effects in the same regression because of perfect collinearity. (This might be different if you understand "crew" differently.)

     – Nameless Jul 09 '19 at 20:46
  • Thanks this really helps me understand the method better and how collinearity would be a problem. – user6972 Jul 09 '19 at 22:03