Testing whether process that generates coins is biased

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Let $X_k$ be the number of heads observed when tossing coin $k$ $n_k$ times. Each coin has it's own independent heads probability $p_k$:



$$
X_k sim textBinomial(n_k, p_k)
$$



Each $p_k$ is drawn from some distribution $P$. I am interested in conducting a hypothesis test to test whether $P$ has mean $frac12$. Is there an appropriate testing framework to use?



Does the choice of $P$ matter (a lot?). $P$ will be a reasonably nice distribution (eg continuous, unimodal).



I know this question may not be fully precisely stated - the question is motivated by the design of a biological experiment. Any ideas or references would be appreciated.



Let's take $P$ to be distributed as $G/(G+H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10,100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a,1−a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$.




The spirit of the question is really to understand the relationship between $n$ and $k$. In order to have a powerful test, should I flip more coins, or should I flip each coin more often?





probability binomial-distribution hypothesis-testing




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  • 1




    Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
    – Math1000
    Jul 23 at 22:04










  • Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
    – user35546
    Jul 24 at 2:27















up vote
2
down vote

favorite












Let $X_k$ be the number of heads observed when tossing coin $k$ $n_k$ times. Each coin has it's own independent heads probability $p_k$:



$$
X_k sim textBinomial(n_k, p_k)
$$



Each $p_k$ is drawn from some distribution $P$. I am interested in conducting a hypothesis test to test whether $P$ has mean $frac12$. Is there an appropriate testing framework to use?



Does the choice of $P$ matter (a lot?). $P$ will be a reasonably nice distribution (eg continuous, unimodal).



I know this question may not be fully precisely stated - the question is motivated by the design of a biological experiment. Any ideas or references would be appreciated.



Let's take $P$ to be distributed as $G/(G+H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10,100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a,1−a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$.




The spirit of the question is really to understand the relationship between $n$ and $k$. In order to have a powerful test, should I flip more coins, or should I flip each coin more often?





probability binomial-distribution hypothesis-testing




share|cite|improve this question

















  • 1




    Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
    – Math1000
    Jul 23 at 22:04










  • Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
    – user35546
    Jul 24 at 2:27













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Let $X_k$ be the number of heads observed when tossing coin $k$ $n_k$ times. Each coin has it's own independent heads probability $p_k$:



$$
X_k sim textBinomial(n_k, p_k)
$$



Each $p_k$ is drawn from some distribution $P$. I am interested in conducting a hypothesis test to test whether $P$ has mean $frac12$. Is there an appropriate testing framework to use?



Does the choice of $P$ matter (a lot?). $P$ will be a reasonably nice distribution (eg continuous, unimodal).



I know this question may not be fully precisely stated - the question is motivated by the design of a biological experiment. Any ideas or references would be appreciated.



Let's take $P$ to be distributed as $G/(G+H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10,100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a,1−a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$.




The spirit of the question is really to understand the relationship between $n$ and $k$. In order to have a powerful test, should I flip more coins, or should I flip each coin more often?





probability binomial-distribution hypothesis-testing




share|cite|improve this question













Let $X_k$ be the number of heads observed when tossing coin $k$ $n_k$ times. Each coin has it's own independent heads probability $p_k$:



$$
X_k sim textBinomial(n_k, p_k)
$$



Each $p_k$ is drawn from some distribution $P$. I am interested in conducting a hypothesis test to test whether $P$ has mean $frac12$. Is there an appropriate testing framework to use?



Does the choice of $P$ matter (a lot?). $P$ will be a reasonably nice distribution (eg continuous, unimodal).



I know this question may not be fully precisely stated - the question is motivated by the design of a biological experiment. Any ideas or references would be appreciated.



Let's take $P$ to be distributed as $G/(G+H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10,100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a,1−a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$.




The spirit of the question is really to understand the relationship between $n$ and $k$. In order to have a powerful test, should I flip more coins, or should I flip each coin more often?






probability binomial-distribution hypothesis-testing





share|cite|improve this question












share|cite|improve this question




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edited Jul 24 at 15:24









Strants

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asked Jul 23 at 17:50









user35546

31337




31337







  • 1




    Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
    – Math1000
    Jul 23 at 22:04










  • Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
    – user35546
    Jul 24 at 2:27













  • 1




    Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
    – Math1000
    Jul 23 at 22:04










  • Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
    – user35546
    Jul 24 at 2:27








1




1




Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
– Math1000
Jul 23 at 22:04




Yes, the choice of $P$ matters, and to get a precise answer, the question needs to be more precisely stated.
– Math1000
Jul 23 at 22:04












Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
– user35546
Jul 24 at 2:27





Let's take P to be distributed as $G / (G + H)$ where $G$ and $H$ are iid Poisson with parameter $lambda_k$. I'd be interested in the answer with $lambda in [10, 100]$. This may be intractable. I'd also be interested in the answer with a more simple $P$, say the hat distribution on $[a, 1-a]$. It would be great if there was an (approximate) answer that depends only on the moments of $P$. The spirit of the question is really to understand the relationship between $n$ and $k$. Ie, in order to have a powerful test, should I flip more coins, or should I flip each coin more often.
– user35546
Jul 24 at 2:27











1 Answer
1






active

oldest

votes

















up vote
1
down vote













You could structure this as a Bayesian hierarchical model and analyze
the posterior distribution of $ P $ conditioned on your "populations" which
are the coin toss outcomes under each $ p_k $. This is, of course, just one example.



The precise distribution of $ P $ will definitely depend on your choice
of prior on $ P $, but at the end of the day you can always numerically
approximate the posterior distribution of $ P $.






share|cite|improve this answer





















  • Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
    – user35546
    Jul 24 at 2:33










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1 Answer
1






active

oldest

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active

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active

oldest

votes








up vote
1
down vote













You could structure this as a Bayesian hierarchical model and analyze
the posterior distribution of $ P $ conditioned on your "populations" which
are the coin toss outcomes under each $ p_k $. This is, of course, just one example.



The precise distribution of $ P $ will definitely depend on your choice
of prior on $ P $, but at the end of the day you can always numerically
approximate the posterior distribution of $ P $.






share|cite|improve this answer





















  • Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
    – user35546
    Jul 24 at 2:33














up vote
1
down vote













You could structure this as a Bayesian hierarchical model and analyze
the posterior distribution of $ P $ conditioned on your "populations" which
are the coin toss outcomes under each $ p_k $. This is, of course, just one example.



The precise distribution of $ P $ will definitely depend on your choice
of prior on $ P $, but at the end of the day you can always numerically
approximate the posterior distribution of $ P $.






share|cite|improve this answer





















  • Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
    – user35546
    Jul 24 at 2:33












up vote
1
down vote










up vote
1
down vote









You could structure this as a Bayesian hierarchical model and analyze
the posterior distribution of $ P $ conditioned on your "populations" which
are the coin toss outcomes under each $ p_k $. This is, of course, just one example.



The precise distribution of $ P $ will definitely depend on your choice
of prior on $ P $, but at the end of the day you can always numerically
approximate the posterior distribution of $ P $.






share|cite|improve this answer













You could structure this as a Bayesian hierarchical model and analyze
the posterior distribution of $ P $ conditioned on your "populations" which
are the coin toss outcomes under each $ p_k $. This is, of course, just one example.



The precise distribution of $ P $ will definitely depend on your choice
of prior on $ P $, but at the end of the day you can always numerically
approximate the posterior distribution of $ P $.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 23 at 18:35









Kevin Li

22829




22829











  • Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
    – user35546
    Jul 24 at 2:33
















  • Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
    – user35546
    Jul 24 at 2:33















Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
– user35546
Jul 24 at 2:33




Yes, thank you for the suggestion. I agree that this is a reasonable approach. At the moment I'm most interested in a frequentist hypothesis test. This is because the design I've discussed is an alternative to another design which is well studied in the frequentist framework - and I'd like to be able to compare them
– user35546
Jul 24 at 2:33












 

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