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I have a question about a claim in Richard.A.Jhonson's multivariate statistical analysis Chapter 1, page 13. So following the example in the picture, we have two variables $X_1$, $X_2$ which represents total salary of the baseball team and the ratio of win matches over the total number of played games respectively. According to the data, it seems high payroll have a higher success ratio. But it says that

" this cause-effect relationship cannot be substantiated, because the experiment did not include a random assignment of payrolls. "

I don't understand what it means. Does this mean that the data about payroll is not a randomly selected data or that each player's salary in a team is not random? Why can we not say that the wage and the success ratio is somewhat correlated?

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    $\begingroup$ The context of the Sports Illustrated article is probably the beginning of free agency in Major League Baseball in 1976. Before then, teams could pay their own players whatever they wanted, and since then, teams can openly bid on players from other teams. $\endgroup$
    – Matthew Leingang
    May 15, 2019 at 14:46

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The idea is that you do not know what would happen if you actually change the salary of a given team. Looking at the correlation, it would seem like increasing the payroll would lead to an increase in the number of wins, but how can you be sure? The answer is: change the salary and measure again the win/loose ratio.

  • If the correlation stays, then yes, there's a good chance that increasing payroll will impact the success ratio of the team

  • If the correlation breaks, then no. Increasing payroll probably has nothing to do with it

The problem is that just by looking at the table, it is impossible to decide. You would need to (randomly) change the salaries and measure again.

I do not have access to the book, and don't know whether the author will explore this effect in more detail, but think about this: usually good players have higher salaries, and the number of good players is definitely a strong indicative of the strength of a team. All this to say, it makes sense that a team with a high payroll will succeed more often (because it has probably a large number of exceptional players). But it does not make sense that just by paying more to a lot of unskilled players, the number of wins of the team increases

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    $\begingroup$ It seems kind of specious for the author to reject the article's claim on that basis. After all, “buying a championship” doesn't mean paying your current players a lot more to make them better. It means acquiring the best players by outbidding all other competing teams. (not a comment to you, or OP, but to Mr. Johnson) $\endgroup$
    – Matthew Leingang
    May 15, 2019 at 14:42

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