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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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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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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What's the Issue with Modeling Admissions as Multinomial?
I am reading Chapter 4 (Testing and Confidence Regions: Basic Theory) of Mathematical Statistics by Bickel and Doksum. In Example 4.1.1, the authors use the study of sex bias in graduate admissions at ...
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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 ...
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How to test null hypothesis with partly confounded multinomial dependent variable
I have independent sample data consisting of 12 levels, where the dependent variable is multinomial with three possible outcomes corresponding to Strategy 1, Strategy 2, and Strategy 3. Each level is ...
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How to obtain the multinomial logistic regression via the framework of GLMs?
From wikipedia, the assumption we are making in multinomial logistic regression is that we take one class as "reference", say class $K$, and we assume that $\ln\left(\frac{\mathbb{P}(X=k_0)}{...
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What would be an appropriate way to analyze a dataset with nominal dependent variables with reapeated measures for some but not all subjects
I am trying to find the correlation between several clinical observations/demographic characteristics and the presence of certain electroencephalographic (EEG) features. The EEG features are my ...
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Infer a K-multinomial mixture model from "group-less" data
I want to infer a multinomial mixture model consisting of $K$ components where each component has the same number of trials, denoted as $N$. Such a multinomial mixture model can be written as:
$$
\...
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Expected number of multinomial samples to cover a multiset
Consider a multinomial distribution $[p_1, \ldots, p_n]$ and a collection of counts $[a_1, \ldots, a_n]$. I would like to know the expected number of multinomial samples needed until every element $i$ ...
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How does a fast Fourier transform yield the probabilities for a multinomial random variable?
Let $(A,B,C)$ follow a multinomial distribution:
$$(A,B,C) \sim \text{MultiNom}\left(n=100,p_1=p_2=p_3=\frac13\right).$$
Define $X$, a discrete random variable as
$$X = f(A,C) = 2A + 3B + 4C = 2A + 3(...
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Is there a non-brute-force solution to this single-valued random variable transformation problem?
Note: I'm happy to edit and rename the “question title” for this problem if someone proposes a better (and concise) way of asking about this problem.
In brief, the problem I'm presenting below was ...
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Metrics for multiclass classification
Background
Consider a clustering problem where a dataset of measurements $Z\triangleq\{z_i\}_{i=1}^m$ must be partioned in $n$ clusters, where $n$ is unknown and must be estimated.
Here the term "...
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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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How to interpret simultaneous confidence intervals for multinomial proportions?
I'm trying to learn on my own about confidence intervals for multinomial proportions, and I stumbled upon several resources on the subject talking about various methods (e.g. Goodman, or Sison-Glaz) . ...
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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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How do you estimate multiple poisson rates simultaneously?
I have data showing the number of cause-specific deaths in a cohort. I want to use Poisson regression to estimate the cause-specific rate by age. The complication is that I want the predicted cause-...
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Chi-Squared Test of Association Equivalent for 3 Categorical Variables and Observed Frequency [duplicate]
Originally asked on StackOverflow and now brought over here upon recommendation from another commenter.
I'm an undergrad writing a very simple lab report in which I'm analysing some data on observed ...
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Mean and variance of the G-test of goodness of fit?
Let $\mathcal{O}$ (for observed) and $\mathcal{E}$ (for expected) be two $D$-variate multinomial distributions, such that $\sum_{d=1}^{D}\mathcal{O}_{d}=1$ and $\sum_{d=1}^{D}\mathcal{E}_{d}=1$. The G-...
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Comparing emmeans values multinom model
I am performing quite some binomial/multinomial models for my thesis. After doing the emmeans statement, I used the contrast statement to compare the emmeans of the different groups. But, I ...
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What are the option for a post-hoc test after running a multinomial model?
Currently, I am trying to run the multinomial model Age = Gender to investigate the effect of Gender on the probability that my study animal belongs to a particular age class. I am not necessarily ...
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Help with rigorous derivation of multinomial distribution
$$
\newcommand{\pset}[1]{2^{#1}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbf{P}}
\newcommand{\OO}{\Omega}
\newcommand{\oo}{\omega}
\newcommand{\sal}{$\sigma$-algebra\...
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Show that two conditional distributions are the same
Let $\mathbf{Y}=\begin{pmatrix}
\mathbf{Y_1}\\\mathbf{Y_2}
\end{pmatrix}\sim N\left (\boldsymbol{\mu},\boldsymbol{\Sigma} \right ), $ $\boldsymbol{\mu}=\begin{pmatrix}
\boldsymbol{\mu_1}\\\...
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Is there a statistical test for mediation when the dependent variable is multinomial?
I am testing a mediation model and I want to test the direct and indirect effects. However, none of the commands or packages that I saw so far allow testing mediation when the dependent variable is ...
Ishwarya Balasubramanian
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Is the multinomial test the right way to test if the frequency of occurrence of two categories is different and is not just random chance?
Just to give context, I never approached to HP testing with categorical variables, but for training and educational purpose I am now dealing with this topic and I was trying to understand the ...
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"Opposite" of chi-square test of goodness of fit (akin to a linear regression between predicted and observed values)
Suppose I have observed count data $x_1,...,x_k$. I have a hypothesis that this data was drawn from a multinomial distribution (derived from theory) with event probabilities $p_1,...,p_k$.
It is ...
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Premium bonds prize probabilities
The UK have an initiative called Premium Bonds. You can buy a £1 bond, and each month there is a 1 in 24,000 chance of winning a cash prize from that bond. The number of prizes given per month for ...
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Standard errors when transforming proportions and removing a category
I have multinomial distribution with three categories, and I've estimated the proportion of each category with some standard error. Now I want to remove one category and renormalize the other two to ...
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Relationship between the distribution of weighted sum of a multinomial distribution and other well-known distributions
Following is the explanation of multinomial distribution from a source of lecture notes (by Yen-Chi Chen).
The multinomial distribution is a common distribution for
characterizing categorical ...
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find the distribution of sum of a multinomial distribution
(Background: I am researching a economics problem and for the research, I need to know the general theory of multinomial distribution problems. This question is just an example to understand the ...
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Making sense of this section on likelihood of order statistics
I am currently studying the textbook In All Likelihood by Yudi Pawitan. In chapter 2 Elements of likelihood inference, the author presents the following example:
Example 2.4: Suppose $x$ is a sample ...
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Should I expect changes in my predicted probabilities when increasing the number of categories I am predicting membership in?
I calculated the predicted probability of membership in group A and not A with binary logistic regression. Then I broke the not A category into two distinct buckets of which it is comprised - B , C. ...
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I performed a multinomial model with bootstraping to predict the probability of the cows perform different behaviours over time period
I found an answer to model my data in order to predict behaviours over time.
The question answered was here I have zero inflated data, with discrete variables. Is it possible to use zero inflated ...
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Alternative to multinomial logistic regression that doesn't use a reference/baseline?
I am on a project developing a model to predict which of 3 groups customers will belong to.
I am currently using multinomial logistic regression. The three customer groups are the Loyals, who ...
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Hypothesis testing for multinomial distribution with a value for every possible categoric outcome
Suppose that i have a multinomial distribution of an outcome of a game such as [Win, Draw, Loss] with
P(Win)+P(Draw)+P(Loss) = 1
Furthermore I assign to every win a point, to every draw half a point ...
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How to derive bias in the Heckman Model
Regression regression:
$$
y_i=x'_i\beta+u_{1i}
$$
Selection equation:
$$
z'_i\gamma+u_{2i}>0
$$
The error terms are correlated:
\begin{equation}
\begin{pmatrix}
u_1 \\
u_2
\end{pmatrix} \sim N\...
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How to test whether one cohort is a mixture of two other cohorts for a set of binary variables?
I'll illustrate my question with an example. Suppose I have 25 observations for 3 binary variables. The observations belong to 3 pre-labelled cohorts 1, 2, and 3. Let's say:
...
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Compute error in GLMM with multinomial dependent variable
I am a novice at statistics doing psychophysics experiment. and I was trying to fit mixed models with random effect (participant) in R where there are two independent variables. 1) Categorical with ...
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Most likely counts of outcomes of a multinomial after N trials (mode of multinomial)
Given a categorical random variable $X = \{x_1,x_2,...,x_k\}$ with corresponding probabilities of each outcome $P = \{p_1, p_2,...,p_k\}$ (i.e. a multinomial):
What are the most likely counts of ...
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What are the relevant null distributions to infer the results of a simulation that renders a multinomial distribution with three possible results?
I have simulated 10 000 occurrences of a soccer match, by generating random numbers from 0 to 1, and then passing these, and the adjusted expected goals by team for the match, through an inverse ...
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How to simulate artifical data for multinomial regression? [duplicate]
I want to use some predictors (for example, $x_1$ and $x_2$ below) to simulate the multinomial outcome (for example, with 3 levels).
Below are the codes I found in another question regarding ...
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Multinomial mixed effect regression?
I have data regarding level of damage in tissue samples, using a Likert scale. Our main goal is to understand if there is a difference in the levels between two groups - control and treated.
An ...
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Response variable (Before,during, after) is it ordinal or multinomial?
Hi I am looking at elemental concentrations in pectoral fin rays the year before spawn, year of spawn, and the year after spawn. I will be creating a logistic regression model to see if I can predict ...
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Which test to compare the difference between the amount of job categories for men and women
I have a dataset with men and women (independent variable) and want to know if the job spectrum (dependent variable) of the men is broader than the job spectrum of the women. Their jobs are ...
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Multinomial regression
When I run my multinomial regression (SPSS) I get errors that there are too many empty cells in the sub populations. I have merged categories in my IVs to the extent possible. My DV has 4 categories....
