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I am currently working on a report about the frequency of hand types in Texas Hold'em and whether or not the frequencies change when there are multiple players involved. For example, is there a change to the frequency of straights when there are five players compared to when there is one player? With each player having two whole cards and five community cards used by each player?

I am trying to answer this question through simulation. I wrote a program to simulate as many hands as needed with $n$ number of players ($n$ is user inputted). The program performs a $\chi^2$ test with the observed number of each hand type and the expected number of each hand type for any set of seven random cards.

What I am unsure about is how I should use this $\chi^2$ data? My current approach is to perform $10-20$ tests each ($10$ million hands per test) for $2$ through $10$ players, then plot the median $\chi^2$ value vs. number of players to see if there is any correlation between the number of players and the difference in hand frequency from the expected value.

Is it appropriate/meaningful to use $\chi^2$ data in this way? Or should I use a different approach to make a conclusion about my original question?

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  • $\begingroup$ Changing the number of players should not change the frequency of particular hands dealt for an individual. But what will change with the number of other players is betting and folding strategy, so there may be a change in the number of times an individual sees all the community cards while still playing with particular hands. $\endgroup$
    – Henry
    Sep 28, 2023 at 23:12

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I think one of the issues here is that you have potentially several categories here with the number of players involved, which makes interpretation with the chi-square difficult to ascertain. If you run a $\chi^2$ test on something with 10 categories against some other categorical variable (this is specifically a chi-square test of independence), you have to remember that this functions like an omnibus test...sure it will tell you there is some general difference in hand x player associations, but it won't tell you specific differences other than some vague inspection of your counts. What's more, because it only tests the overall association, you are more likely to have a flagged test with many heterogenous groups, but what this means in practice is vague without something quantifiable like phi-coefficients, which are still generalizations that are not going to tell you exactly the relationship between the categories here.

What may be a lot more illustrative is running the simulations but replacing it with a regression model instead. One option may be an ordinal logistic regression, with hand type as your outcome and number of players as your predictor. Number of players could be treated as continuous or categorical. I imagine it would be much more illustrative to use a continuous analysis, as it will give you a better idea of the overall trend (I doubt for example you have any specific hypotheses about $n_{players} = 6$). From there you could obtain probabilities from the model and make an approximation of what predictions your model should illicit.

You may be interested in the book Analysis of Categorical Data in R. R is particularly well suited for both simulation and statistical analysis, so this book may be a good compliment for your use case, and I believe that Chapter 3 covers multinomial cases such as these (one example I recall is a wheat kernel analysis where they treated the kernel type as response and several predictors of each).

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