Questions tagged [binomial]

The binomial distribution gives the frequencies of "successes" in a fixed number of independent "trials". Use this tag for questions about data that might be binomially distributed or for questions about the theory of this distribution.

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Clopper-Pearson Exact Confidence Interval for Overall Diagnostic Accuracy: Single Diagnostic Test Given a Disease with Low Prevalence

I am trying to understand how exactly to calculate the CP CI for the overall diagnostic accuracy of a single diagnostic test given a disease with low prevalence. Here is an example: Prevalence: 2.0% ...
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Which probability result is greater? Using hypergeometric distribution or binomial distribution?

Suppose I have $N$ samples, and 1/4 of them are bad. I draw $n$ samples $(n<N)$, and I want to know what's the probability that less than 1/3 are bad. I know it should use hypergeometric ...
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31 views

Binomial glm help!

I have been given a dataset with 750 participants, with information on Age, Location, Gender and the year it was obtained (predictor variables although responses may vary by location so this is a ...
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121 views

How to estimate the parameters of data with greater tail (seems as negative binomial)

I have a data with only 70 observations. The data contain one discrete variable. This variable contains the number of patient with flu (count). So, we have 70 days ...
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Comparing Begg and Gray method and MCMC for multinomial models

I am looking to build a multinomial model and I am considering using the "Begg and Gray" method: https://cran.r-project.org/web/packages/mlogitBMA/vignettes/conversion.pdf And I am also ...
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Linear Regression: Treat binomial dependent variable as normal due to large n

Say I wish to do a linear model and my dependent variable $Y$ follows a binomial distribution: $Y_i \sim Bin(n_i,p_i), i = 1,...,N$. Usually I would use a generalized linear model and perform a ...
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31 views

Statistical test to show if there is a difference between A and B (count data)

I conducted 20 tests (in parallel) where there was a choice between A and B and after a defined period of time, I counted the number of individuals that chose A or B (count for A and B is always ...
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39 views

When to use hypergeometric vs binomial

I'm having some trouble understanding if I need to use a hypergeometric distribution. I have a set of components, a small proportion of which are faulty. I now want to know if a feature of that ...
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General method to find the “best” binomial test confidence interval

I'm trying to implement a function while programming that finds an exact binomial test confidence given a probability of success, p, number of trials, k, and value of alpha. So for example given p=0....
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Binomial distribution mixed with geometric distribution

Suppose X is the number of heads in 10 tosses of a fair coin. Given X=5, what is the probability that the first head occurred in the third toss? We can assume X to a Binomial distribution with p=1/2 ...
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Interaction term alters degrees of freedom for main effects in glmmTMB model

I am using the glmmTMB package to test whether the probability of a seed remaining dormant in the soil (binary variable) depends on the population it comes from (fixed effect, 8 levels of population), ...
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2answers
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Conditional expectation of product of sums of bernoulli random variables

let's say we have $X_1,..,X_n$ i.i.d. Bernoulli random variables. For $l<m<n$, we want to calculate: $$ E \left[ \sum_{i=1}^{l}X_i \sum_{j=1}^{m}X_j \mid \sum_{k=1}^{n}X_k \right] $$ Does this ...
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Estimate binomials given monotonic probabilities

I am given $N$ ordered coins and for each coin $i=1,..,N$ some trials $X_i \sim Bin(n_i, p_i)$. The coins are ordered in the sense that I know a priori that $0\leq p_1\leq p_2 \leq ... \leq p_N \leq 1$...
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Does a multinomial regression model make sense for a low frequency?

I have data on job types with 3 different specialities: 100% of jobs have a primary speciality, 77% of jobs have a secondary speciality, and 1.6% of jobs have a tertiary speciality I want to model ...
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Which data visualization when running GLMM?

I'm analysing correct response rate and response time of participants on a task. For the first one, I ran a mixed effect logistic regression: ...
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binomial test with multiple samples, whether to pool or not

I have a dataset consisting of multiple samples on two categories. ID A B 1 2 9 2 1 5 3 2 5 4 3 8 5 1 6 I want to test ...
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1answer
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finding PDF of Y, given Y|X [closed]

$$Y|X\sim Bin(X,n)$$ $$X\sim U([0,1])$$ How can I find the PDF of Y? I know that: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
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Interpretation of binomial GLMM with interaction fitted with glmer

I have a glmer model from the R package lme4 with a binomial distribution and I was wondering whether I am interpreting the ...
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Is there any hypothesis test for two binomial distribution without normal approximation?

Let's say we are running an A/B testing and each data point has a binary response. We would like to test whether the ratio of true are different between A and B. (e.g. ask a yes/no question to both ...
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56 views

Is a sum of two binomial distributions with different $p$ also binomial? [duplicate]

I have two independent random variables which follow binomial distributions $X \sim B (n_1, p_1)$ and $Y \sim B (n_2, p_2)$. Can we say that $Z = X + Y$ is also binomially distributed $Z \sim B (n_1+...
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1answer
48 views

Develop granularity-invariant criteria for comparison of logistic (binomial) models

I have a model with logistic (binomial) likelihood, with number of successes and failures as a response variable. I am comparing various models, which can be of different granularity. Different ...
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How to properly calculate the exact p-value for binomial hypothesis testing (e.g. in Excel)?

I am confused with calculation of the p-value for binomial hypothesis testing when not using the normal approximation and its Z-statistic. My hypothesis testing is for the situation when H₀: π = π₀ ...
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Wilson confidence interval - question on conventions

Is there a convention when using the Wilson interval and all observations are successes (or failures) to artificially increase or decrease the lower bound of your CI estimate to encompass the ...
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1answer
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Sampling distribution of sample means question (Newbold)

6.3 from Newbold (8th ed.): A population contains 6 million 0s and 4 million 1s. What is the approximate sampling distribution of the sample mean in each of the following cases? The sample size is n ...
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Why marginal impact of an observation to the posterior decreases in Bayesian inference?

When we toss a coin with an unknown Heads probability $p$, we can use Bayesian inference to estimate the unknown value $p$. Say, we start with a Beta prior distribution with parameters $(a,b)$ and ...
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31 views

Interval Estimation for a Binomial Proportion Given a Specific Test Outcome

Imagine that I had a coin, I tossed it 10 times (n) and it came up heads each time (x). What proportion heads I would get if I tossed it infinity times? A point estimate is 100%. I can get the ...
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1answer
31 views

Closed form of pairing probability

This situation happened to me at home today, and now it is bugging me. I didn't put all my clothes in the washing machine, and I retrieved 7 sockets, 2 pairs of socks and 3 unpaired ones. I wanted to ...
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24 views

Gaussian or Negative Binomial in glmmTMB

The Background: I'm working with a dataset that had randomization error at baseline; the methodology was solid, the PI was just unlucky. It's a three-level model (time within individual responses ...
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Empirical Bayes Estimation: Different transformation, fully different results

I would be grateful for some advice regarding the following problems: As I reviewed some empirical papers, I learned that most of them use a relatively small number of observations for performing ...
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Binomial Regression and Grouping Data

I'm currently trying to model Data where the dependent variable is either 0 or 1. Binary outcome. This would be the same as to say the data comes from a binomial distribution B(1, p). So I could use ...
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Bernoulli process with nonstationary probability

Say we have a process $X_t\vert P_t\sim \mathrm{Bin}(n,P_t)$ where $X_t$ is observable but $P_t$ is not. Also, the success probability $P_t$ might vary over time and I don't assume some ...
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Intercept significant in glmer and lmer

I'm quite new on this with binomial data tests, confusing when doing the analysis in R using glmer and lmer I am doing an ...
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Estimating the denominator of a binomial probability

Let $X \sim \mathrm{Binomial}\left(n, \frac{a^*}{b^*} \right)$, where $a^*$ and $b^*$ are coprime integers and $0 < a^* < b^* < 1000$. Here $n$ is known, but $a^*$ and $b^*$ are unknown. What'...
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May I use a grade of service problem to calculate how many doctors are needed in a hospital?

I created the following problem (based on grade of service problem of Intro to Prob by Tsitsiklis book): original problem: My analogy problem: A Hospital unit has 15 doctors to serve the needs of a ...
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How to derive this inverse standard deviation error bound for binomial random variable? [duplicate]

Is this error bound suggesting the normal approximation is good for binomial random variables? How was it derived? Why is the reciprocal standard deviation in the error bound?
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Performance measures for modelling proportions

Suppose that for each of $k$ individuals, I have a vector of covariates $x_i$, a total number of 'attempts' $n_i$ and a total number of 'successes' $y_i$. I model each individual's success ...
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Factorial moment bound for discrete Binomial distribution

I need to compute the upped bound for the tail (survivor) probability $P(X \ge t)$ for the discrete Binomial random variable $X$. I could use Chernoff bounds, however according to this paper [1] the ...
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percentage difference between two group when one group contains only positive cases

I have two groups $i=1,2$, with different total cases $n_1$ and $n_2$ on which the number of positive cases is $x_1<n_1$ and $x_2=n_2$. So there is no variability in the second group. I wonder how ...
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1answer
46 views

Test hypothesis of the probabilities being the same using using Wald test statistic [closed]

I am trying to understand how to use the Wald test to test the hypothesis that the probabilities of getting a faulty product are the same in two samples. The whole text of a task is: Two factories ...
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60 views

Sum of Bernoulli variables with known probabilities

Following the ideas from this post and, specially, this post, i was wondering if the a sum of two independent groups of Bernoulli distributed variables whose probabilities are know a priori is a ...
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1answer
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Intuition Behind Kurtosis of Binomial Distribution

I'm trying to get a good understanding of the higher moments of the Binomial Distribution; it's an important building block for more complex distributions so I want to get a strong intuition for this. ...
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Testing for potential discrimination

Apologies, I know this is incredibly simple but it's been a while since I was in school! I'd like to run a significance test on the (dummy) data in the table attached to ascertain whether there has ...
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Hypothesis test involving proportion of proportions

I have 100 different coins, each with probability $p_i$ of being heads - but these $p_i$s are unknown. I flip each coin $n_i$ times, and see that it lands on heads $x_i$ times. So the proportion of ...
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1answer
46 views

Given we observe $x$ successes in sample of $n$, what is the probability that $p>0.5$

I flip a coin $n$ times, and I find that for $x$ of these it lands on heads. Let $p$ be the true probability of the coin landing on heads, though this is currently unknown. I am interested in $P(p>...
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Variance is larger for hierarchical models

The following is from a book Suppose that $m_iy_i$ follows a binomial distribution with $m_i$ cases and success probability $p_i$. Suppose that the $p_i$ is itself a random variable, with mean $...
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How to calculate coverage probability from binomial confidence intervals?

For a given alpha I have been calculating various confidence intervals for a binomial distribution(Wald, Wilson, Agrest-Coulla etc.): ...
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Difference between cumulative binomial probability and discrete

I am new to this so I don't know if this question is too obvious or not. I have this question (its already solved I'm just trying to understand why). Say you want to prove that less than 10% of US ...
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1answer
79 views

Spatio-temporal autocorrelation

I have a huge data frame (300k + rows) on GPS animal positions. I want to model the probability of the presence of chamois taking into consideration as variables: distance (from a disturbance), ...
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1answer
29 views

Probability question - is this binomial?

Question from a Statistics textbook written in 1978 (ISBN 0-393-09076-0): It is claimed that a vitamin supplement helps kangaroos to learn to run a special maze. To test whether this is true, ...
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MLE for parameter n in Binomial [closed]

How is the MLE for n calculated if p and k are given? I've been stuck on this forever and cannot seem to even find a similar example of calculating MLE of n.

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