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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Confidence interval of precision / recall and F1 score

To summarise the predictive power of a classifier for end users, I'm using some metrics. However, as the users input data themselves, the amount of data and class distribution varies a lot. So to ...
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1answer
108 views

Binomial confidence intervals - which is correct?

Background: I am working with a data set that requires a transformation. It's prevalence data so I have proportions to deal with, and as the proportions are quite low, I'm using the Freeman-Tukey ...
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137 views

Are there better estimators of misclassification error than the fraction of misclassified test points?

Assume we train a binary classification model using the training set. Also assume that the model returns an estimate of the probability of success $\hat f(x)$ for every feature vector $x$ and was ...
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873 views

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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437 views

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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406 views

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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204 views

Identifiability in generalized linear random effect model?

Suppose I observe binary $Y_{ij}$ for $i = 1, ..., N$ and $j = 1, ..., J$ and I want to model $$\Pr(Y_{ij} = 1 \mid \lambda_{i}) = \Phi(\lambda_{ij}), \qquad [Y_{ij} \perp Y_{ij'} \mid \lambda_i]$$ ...
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775 views

Chi-squared test and binomial distribution

In some circumstances the Chi-square test can be replaced by a test based on the binomial distribution. The common issue with using the binomial distribution is that it may quickly become tricky to ...
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240 views

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 $k_i$ ...
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2k views

How to get distribution of sum of dependent bernoulli variables

I have $N$ Bernoulli variables, $X_1,...,X_N$ and $X_i\sim B(1, \pi_i)$, $\pi$ is known for each $X_i$, and $Y=X_1+...+X_N$, now I need to get the destribution of $Y$. If $X_i$ and $X_j$ are ...
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165 views

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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128 views

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 ...
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2answers
338 views

How do I test difference in Percentages?

I have some survey data and know the number and percentage of "yes" responses. How do I determine if any of the percentage yes regional responses are statistically different from the national yes ...
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42 views

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 ...
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74 views

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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417 views

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 ...
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252 views

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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78 views

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 ...
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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 ...
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1answer
11k views

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 ...
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1answer
47 views

binomial responses in h2o gbm

I am modeling the probability of success in a dataset where I have a both the number of trials and the number of successes (and, obviously, I am modeling $p_i=\frac{total successes}{total trials}$). I ...
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107 views

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 $p_1$. ...
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48 views

What are models for count data where the variance is not related to the sample size?

For modeling count data that arises as the sum of Bernoulli random variables or according to an occurrence process with constant rate in some interval, there are many techniques, like using a Binomial ...
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78 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 ...
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39 views

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{...
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411 views

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 ...
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1k views

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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1answer
14 views

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) ...
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72 views

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 ...
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144 views

Does clustering lead to overdispersion?

TL;DR Clustering is often cited as a source of overdispersion in count data. However, I seem to arrive at the conclusion that clustering actually reduces the dispersion. Could someone confirm this ...
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886 views

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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134 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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116 views

Probability of a Blizzard - Poisson vs Binomial vs Bayesian

My goal is to model data I have collected to compute the probability of a severe blizzard occurring in a given set of states (region). Assume the US is broken into 3 regions that categorize the ...
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207 views

Limiting distribution of $\frac{\bar{x}-p}{\frac{pq}{n}}$ from mean of $Bin(1,n)$?

I found some difficulties in here. We know that if $X$ has Binomial distribution with $1$ trial and $p$ success, or what we called $X$~$Bin(1,p)$, we have $\mu=p$ and $\sigma^2=p(1-p)=pq$. From that, ...
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77 views

confidence interval for proportion

Suppose $k$ events were observed in $n$ trials. We then have $\hat{p} = k/n$ and the standard Wald confidence interval is $$\hat{p} \pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}$$ I wonder why the ...
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69 views

Test of strict difference in independent binomial probabilities

Suppose $X,Y$ are two independent binomial random variables with parameters $n_1,p_1$ and $n_2,p_2$ respectively. Suppose one wanted to test the hypothesis $p_1>p_2$. Conditional on $X+Y=s$, ...
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173 views

Variance of binomial vs. multinomial distribution in cross-validation

Suppose we have a dataset with $N=100$ observations. We do $K$-fold cross-validation with $K=10$ and $K=100$. In the first case, the classification decisions are sampled (can I say it like this?) ...
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193 views

Binomial data: Null Hypothesis $p = 0$ when all Sample Values are 0 - testing and power analysis

I am trying to investigate if the proportion of successes in my population can be shown to be larger than zero. Thus, $H_0: p = 0$ and $H_A: p>0$. Since $p=0$, the prop.test in R does not work, ...
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178 views

Determining preference using a binomial test. Is this the right way to go?

Let's say you are running a blind essay on preference of a new product. You make people taste two different drinks and record their response. They prefer A and you record it as a failure in a ...
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515 views

Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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506 views

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 ...
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413 views

Should I use a beta-binomial or binomial glmm?

I have several data sets on wildlife disease incidence. One of the issues with my dependent variable is that it represents only current infection status, therefore 0 (no disease) can represent either ...
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182 views

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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170 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 ...
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130 views

The probability that one bernoulli process has a higher p than another?

I have two data generating processes that are independent Bernoulli processes with probabilities of success $p_A$ and $p_B$. I am taking repeated samples from these two data generating processes, so ...
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392 views

Plotting fit for binomial lme

I've been asked by a reviewer on a manuscript to provide plots of a model fit for a binomial lme which is specified as follows: ...
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228 views

binomial test for testing significance of classification very sensitive to hits count

I'm doing a multi-class classification task and I wonder if I'm doing the binomial test correctly since it is very sensitive to the count of correctly classified trials (hits count). Say, there are ...
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1k views

Mixture of binomial distributions

I am experimenting with a mixture of binomial models. Consider a binary variable $y_i$. Furthermore, there are two sub-groups in the population (not known a priori and not observable): $z_i=0$ or $z_i=...
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577 views

Why may one use qbeta() to calculate CIs for binomial proportions that are actually based on the F-distribution?

I just barely grasped why large-sample confidence limits that assume approximately normal distributions are not the best approach when relative frequencies approach either one or zero - so that in ...