# Questions tagged [multinomial-distribution]

A multivariate, discrete probability distribution used to describe the results of a random experiment where each of $n$ outcomes are placed into one of $k$ nominal categories.

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### Trend test for multinomial variables

Which test should be used to estimate a p-value for trend when the outcome (dependent) variable is multinomial (with multiple unordered categories like race) and the independent variable is an ordered ...
1 vote
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### Finding the expectation of a categorical variable times a random amount

Say we have $J$ trading cards and each have a dollar value of $u_{j}$ and I am allowed to make 1 draw. $$u_{j} \sim N(\mu,1)$$ Where the value of each trading card is independent and identically ...
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### Bias in simulation samples low frequency event at higher rates

I am trying to simulate a dataset of x trials and y subjects, where every trial can be one of 3 categorical vars, associated with a unique probability. For every subject and trial, I also want to add ...
1 vote
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### Necessary condition for constrained optimization

Suppose $X=(X_1,\cdots,X_k)$ follows the multinomial distribution with a known size $n$ and an unknown probability vector $(p_1,\cdots,p_k)$. Find the necessary conditions for the solution to the ...
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### Modification of Pearson's Chi-square test

At first glance, Pearson's chi-squared test seems flawed in a major way. Can you help me identify the error in my logic? I have a multinomial distribution with $k$ outcomes, and $p_i$ denotes the ...
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### R - multinomial logistic regression with relative frequencies as response variable

My colleagues observed in an experiment involving categorical and continuous independent variables, how the species composition changes. Approximately equal numbers of microbes were used in the ...
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1 vote
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### Probability of at least one element for each class in a multinomial distribution

Given a multinomial distribution $$X \sim \mathcal{M}(n, p_1, \ldots , p_K)$$ If I know $p_1 \ldots p_n$ I can easily obtain the following probability by repeatedly sampling from a multinomial ...
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### Testing the hypothesis that $\sum_i^N p_i^2 =(N-1)/2N$ given $\sum_i^N p_i=1$

In ecology there is a simple index for measuring diversity known as the Simpson index. In it simplest formula, it takes a series of proportions, representing the relative abundance (number of ...
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### Multiple goodness of fit tests for multinomial proportions of unbalanced classes

I am considering a case where I have a nominal variable with $L$ levels (let us say $L=2$ for simplicity) and a bunch of continuous features. I would like to test if the most centrally located point ...
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### What's the relation between the output of a neural network and a Multinomial distribution?

I am reading this paper, which has the following paragraph - "The gold standard for deep neural nets is to use the softmax operator to convert the continuous activations of the output layer to ...
1 vote
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### How to interpret multinomial logistic regression results in r with ordered independent variable?

If there is a independent variable X that is a ordered factor (low, medium, high) - high being the base class, the results of the multinomial regression for each class in X should be interpreted as ...
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### Assessing differences in effect size of an independent variable: multinomial logistic regression or one-vs-rest binary logistic regression

My question pertains to this Cross Validated post, but I specifically aim to compare the effect size of an independent variable between categories of the dependent variable. In practice, I have a non-...
1 vote
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### The test point’s standard deviation from the origin (ESL’s Exercise)

This is Ex.2.4 from The Elements of Statistical Learning. I don’t understand the sentence that I underline in the image. I know that $\sqrt{10}$ is approximately equal to 3.1, but I don’t know how to ...
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1 vote
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### Which parameters optimise the weighted cross-entropy loss for a pre-specified categorical distribution?

Question: Given a categorical distribution $C_q$ with parameters $q_1, \ldots, q_K$ with $K > 2$, $\sum_k q_k = 1$, which (new) categorical distribution $C_p$ with parameters $p_1, \ldots, p_K$ ...
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### Testing two proportions of a multinomial distribution

I am wondering if there is a statistical procedure to test if two proportions from a multinomial are equal. For instance: I take a sample of animals in a reserve in South-Africa and count the number ...
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### Generalized Linear Mixed Modeling questions

I have a dataset with assessments taken at different visits: baseline (visit=1), and then various post-baseline visits (2, 3, 4), and the following variables: DISEASE - the outcome, is an ordinal ...
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### Closed form formula for distance of multinomial sample from underlying distribution

Suppose that I have a probability vector $p$ e.g. of size 10, and that I draw a multinomial sample of size $n$ from $p$. Does there exist a closed form formula to compute the expected total variation ...
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### Multinomial gam diagnostic help

I have created a Generalized additive model of a multinomial distribution using the mgcv package, I have used check.gam to do some diagnostic tests but they look ...
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### Determine unknown probabilities from sample of multinomial distribution

Let there be a bag with arbitrarily many balls in k colours (presume we know k). There is a set (but unknown) probability p1, p2... pk of drawing a ball of a given colour. I take a sample of g balls (...
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### What kind of relationship is there, if any, between simultaneous confidence intervals for multinomial proportions and chi-squared residuals?

I'm trying to understand if/how the chi-square test relates to simultaneous confidence intervals for multinomial proportions. It doesn't relate to a practical problem to solve, so this is just a ...
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### Distribution of vector of $n$ Bernoulli trials

I have a random variable $X\in\{-1, 1\}^n$ that is just a vector of length $n$ containing either $-1$ or $+1$. I know that $\rho$ is the probability of the event $X = +1$. I would like to describe the ...
1 vote
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### Right way to compute pairwise significance for a multiple choice survey question?

Say I ask people via survey, "Pick your one favorite option: A, B, C, or D." What is the proper way to compute pairwise significance between the answer options? Can we say whether there is a ...
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### Obtaining the chi-squared test statistic via geometry

I’m trying to informally derive the chi-squared test statistic using a combination of basic geometry and algebra. I’m successfully able to obtain a system of equations that contain Karl Pearson’s chi-...
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