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Questions tagged [binomial-distribution]

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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Overdispersion in a binomial GLMER model

I'm having trouble accounting for overdispersion in a binomial GLMER (lme4 package) - I'd read through other posts on the topic but haven't found anything that solves my problem. I tried adding an ...
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Sample size determination contradiction

I'm struggling with a contradiction when trying to determine a sample size when proportion is small. If I use binomial formula: $n = p(1-p)z^2/ME^2$ then the smaller the proportion - the smaller ...
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Hierarchical model: question on frequentist estimation

I am interested in understanding the differences between Bayesian and Frequentist estimation in the context of hierarchical models. Consider $n$ subjects, where for subject $i$ there are $k_i$ ...
Giancarlo's user avatar
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Interaction plot between categorical and quadratic continuous variable

I ran a GLMM model with a binomial response to analyse bear presence at feeding sites (0 = absent, 1 = present) within two years. My code is: ...
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Detecting outliers in percentages

My dataset looks like below - Total Success Percentage 100 65 65% 50 25 50% 30 20 66.6% 50 40 80% Plot - Each row is ...
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Including seasons and months into GLMM: should they be crossed or nested effects?

I have collected data from five consecutive fishing seasons (five factor levels). Each fishing season has five months within (five factor levels). Considering that I have a temporal correlation in my ...
Ignacio Gianelli's user avatar
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Linear mixed effect model with fractional/proportional outcome: choosing between binomial and beta

I'm looking to run a linear mixed effect model using lme4, where my dependent variable one_syllable_words / total_words_generated is a proportion and my random ...
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Use Chi-Squared or Binomial Test if Distribution is not Known?

Suppose you have a set of data (eg. [a, b, a, a, b, b, etc.]), and you have the suspicion that the set of data follows the binomial distribution. Your Null Hypothesis is: The probability of success ...
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Hypothesis testing for binomial distribution: a simple real-life case

I'm refreshing (or maybe just acquiring) some stats skills, and I have a real-life situation, probably very simple since it's quite close to a typical example from stats courses, that I would like to ...
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The marginals of a truncated multinomial are truncated binomials?

The multinomial distribution for a vector $\vec x$ of non-negative integers assigns a probability of: $$f(\vec x) = n!\prod_i p_i^{x_i}/x_i!$$ to every vector $\vec x$ of non-negative integers in ...
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Put constraint on max (or min) predicted value (mgcv)

I want to fit my data using a logistic GAM model with cubic regression splines. I know for sure that in reality my estimated probability should not go above 0.5 (due to mislabeling). So I thought ...
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Theoretical justification of choice for confidence interval exact method for the success probability parameter of negative binomial variable?

I have a computer experiment that runs the Bernoulli series with unknown probability $p$ of success. The experiment terminates when $m$ failures are observed. So, the unknown parameter $p$ has the ...
Piotr Semenov's user avatar
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Simple $\chi^2$ test question

I have the following question which seems extremely easy, but the way the data are set up is causing me some uncertainty: I plan to solve this problem through finding the maximum likelihood estimate ...
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Joint distribution of column sums when row sums are fixed

Suppose I have an $m$ by $n$ table $X_{ij} \in \{0,1\}$, where in each row, $r$ randomly chosen entries are set to 1 (the rest are 0), i.e. $\sum_j X_{ij}=r$. I know that e.g. the column sum $\sum_i ...
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Binomial mid-p value

I've been under the impression that the mid-$p$ values generally control the Type I error, and consequently confidence intervals based on mid-$p$ values control the coverage. However I have checked ...
Stéphane Laurent's user avatar
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Is the mean of the left-truncated binomial distribution convex in p?

The expectation of the binomial distribution of successes in $G$ trials, left-truncated at $R$, with success probability $p$, is $$ E[X|p] = \frac{\sum_{l=R}^Gl\phi(l)}{\sum_{l=R}^G\phi(l)} $$ where $$...
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Bernoulli random variables and correlation coefficient

Let's consider two random variables $X$ and $Y$ following a Bernoulli distribution such that: $$ P(X=1) = p\\ P(Y=1) = q $$ The correlation coefficient $\rho$ is given and my goal is to compute $P(X \...
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Modeling binomial outcomes with repeated measures

I'm looking at patterns of a particular injury within individuals and how they vary by age and sex. For each of 1365 individuals I have four locations each of which may be positive for this injury. ...
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Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
chickenNinja123's user avatar
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binomial glm where number of trials is also a predictor

I am modeling the probability of success $p_i$ under a binomial framework. In fact I am actually modeling $x_i \sim Bin\left( n_i, p_i\right)$ being the number of trial varying along each observation. ...
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How Negative Binomial Distribution and negative bionomial regression can be used to sales forecast?

My first question here. Due to the improper inventory management we seem to have dispersed sales, and the stores are unable to meet the demand because items are being out of stock. There are so much ...
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Multiple imputation of glm binomial size parameter

Suppose we have a generalized linear model with a binomial response $y_i\sim \mathrm{bin}(n_i,p_i)$ where $p_i$ is determined by the linear predictor in the usual way via some link function. Is there ...
Jarle Tufto's user avatar
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Election fraud detection: the statistics of Quick Count

I’m reading the book Quick Count and Election Observation (chapter 5). I’m interested in understanding the statistics used in Quick Counts. Quick Counts is a methodology for verifying official ...
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Bayesian inference with unequal sampling

I have a "two-column" data set, with a multi-class categorical variable A, and two-class variable B. It is assumed that each observation is independent. For each category of variable $A$, I want to ...
NaiveBayesian's user avatar
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How to calculate standard errors of a non-linear model prediction?

I'm trying to understand how to show the prediction error of a model fit in R using the non-linear least squares function nls. Although there is an argument ...
Marc in the box's user avatar
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189 views

Non-parametric estimators for time-varying binomial proportion

I have a bunch of count data associated with time intervals (potentially overlapping and of variable lengths), say $(s_i, t_i, n_i, N_i)$ where $N_i$ is a count of the total number of events ...
Matt's user avatar
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What is the distribution of independent binomial variables conditional upon the sum?

Suppose that we have independent binomial variates with differing sizes and probabilities $X_i \sim \operatorname{Binomial}(n_i,p_i)$, and $Z = \sum_iX_i$ is the sum. I understand that $Z$ is ...
tomriddle's user avatar
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How to analyse a continuous response having a bimodal distribution?

I am investigating unconscious racial prejudice as a predictor for guilty or not guilty judgements (Using SPSS). I have a continuous variable for unconscious racial prejudice (higher numbers equal ...
Kate's user avatar
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Generalized linear (mixed) model, binomial - help!

I work in biology and I´ve done an experiment exposing an invertebrate to a pesticide at different temperatures. One of my endpoints is hatching success of their eggs. The animals lay clutches of eggs,...
EcotoxicologyGirl's user avatar
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Unexpected post-hoc result using binomial GLM

I have a dataset that has a categorical factor and numerical response variables as proportions. A simplified deput() included at the end of the body. But, here is ...
scott.pilgrim.vs.r's user avatar
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1 answer
289 views

Probability of outcomes from overlapping samples of a random variable

I have a feeling there is a straight forward answer to my question, but I'm not sure what the appropriate term to search for is. I'm trying to get a sense of what the probability is of finding some ...
Max Miller's user avatar
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0 answers
228 views

Power calculation for binomial test

I have been asked to do a particular power calculation: We assume that the true probability of an event is 93%. We will do a binomial trial of size N, observe the outcome, and construct a confidence ...
Old_Mortality's user avatar
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How many orders will be placed on average until stock runs out, given a probability that the customer will find the item they want?

Let there be 500 units of stock $s$ of diverse items in a store. Customers arrive one by one and, for every given item, there is a likelihood $m$ of 5% that it is precisely the item that the customer ...
Pedro Schuller's user avatar
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Confidence intervals for binomial generalised linear model with cauchit link function

The correct way to calculate a confidence interval (CI) for a generalised linear model (GLM) and avoid the problems of normal approximation intervals has been adequately discussed by Gavin Simpson ...
Luka Seamus Wright's user avatar
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157 views

Counter-intuitive result from Cochran Armitage trend test and glm binomial

My question is motivated by the Q&A at How to test the increase of proportions. The question is how to test for a trend in proportion between events and non-events given an ordinal risk of 3 ...
dariober's user avatar
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3 votes
1 answer
116 views

Sequential Bayesian estimation of binary outcome

I am planning an experiment to determine the frequency of a binary variable (valued 1 or 0). Each day, there are ~ 10,000 new events taking place, thought this number may shift by ~ 100 day to day. ...
user6883405's user avatar
3 votes
1 answer
166 views

Logistic regression with binomial independent variable

I have a table of observations, with three columns --- (a) class labels (can be 0 or 1), (b) counts of successes (out of a certain number of Bernoulli trials) and, (c) numbers of Bernoulli trials. I ...
Mr K's user avatar
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Literature request: (in)appropriateness of negative binomial for count data with an upper bound

I conducted an analysis where I used binomial logistic regression to analyze x successes in n trials (where n varies between observations) in aggregate (using the R syntax ...
NatWH's user avatar
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1 answer
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Model/predict the number of malaria-infected cells

Background: In order to determine the severity of a Malaria infection, one takes a sample of red blood cells and determines, through a microscope, the number of cells infect by the malaria parasite. ...
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Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
rsmith49's user avatar
3 votes
0 answers
94 views

Confidence interval of proportion of a proportion

Let's say we have $N$ users. Some of them could be hackers and some not. Only for a random $p_u$ percent (e.g. 1%) of users have we actually investigated manually to know which ones are hackers and ...
Robert Lindsey's user avatar
3 votes
0 answers
2k views

Hypothesis testing for the binomial distribution - critical region

Consider the following problem: $H_0:p=0.2, H_1: p\neq 0.2$ $X\sim B(25,0.2)$ Find the critical region for a hypothesis test using a $5\%$ significance level. I have found $$\begin{...
A. Goodier's user avatar
3 votes
1 answer
42 views

Statistical test with different kinds of null hypothesis

Let $X_1$ and $X_2$ be two binomal random variable with respective parameters $n_1$, $p_1$, $n_2$ and $p_2$. Let $x_1$ and $x_2$ be observations of $X_1$ and $X_2$ respectively. I want to try ...
Anthony's user avatar
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0 answers
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Binomial GLMM for proportion (glmmTMB/lme4)

Hopefully this will be a straightforward question for anyone experienced with GLMMs. I'm trying to model aggregation of parasites on different locations on their host. I'd like to approach this by ...
Dylan's user avatar
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3 votes
0 answers
229 views

Confidence intervals and hypothesis tests for binary choice data with repeated measures

I have data with two between-subjects factors (each with two levels; bs1 has levels ‘hi’ and ‘lo’, and bs2 has levels ‘happy’ and ‘sad’) and one within-subjects factor (ws1, with levels ‘good’, ‘...
userLL's user avatar
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1 answer
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Summation of combinations up to $r-1$ terms

I am trying to come up with a simplified expression for $$\sum_{k=r}^{n}\binom{n}{k}$$ Choosing $x=y=1$ in Binomial theorem, I have $$2^n = \sum_{k=0}^{n}\binom{n}{k}$$ $$2^n = \sum_{k=0}^{r-1}\binom{...
Emon's user avatar
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0 answers
239 views

Infimum of Binomial Confidence Interval Coverage Probability

Let $X \sim \text{Binomial}(n,p)$. Let $\hat{p} = X/n$ be the estimator for $p$. Let the confidence interval for unknown $p$ be $$ \hat{p} \pm z_{\alpha/2}\ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. $$ I ...
Taylor's user avatar
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1 answer
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sampling order sum from different order size and probabilities

I want to analyze the probable outcomes of an order pipeline for a business. Say we have ten potential orders, each (business) order has a different size (OS), say 1, 20, 3, 4, 5, 6, 6, 8, 8, 9 (mio) ...
Jakn09ab's user avatar
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Differences from logistic regression and mixed effects logistic regression - rounding error or conceptual mistake?

I'm a bit confused. To my understanding, the standard logistic regression should be equivalent to a mixed effect logistic regression where the statistical unit is defined as random effect - but I ...
StupidQuestion's user avatar
3 votes
0 answers
290 views

Residuals in Negative Binomial Data

I'm tackling the problem of Anomaly Detection in a dataset that's comprised of call counts to a call-centre. The data exhibits daily and weekly seasonality and is known to be over-dispersed. ...
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