# Questions tagged [likelihood-ratio]

The likelihood ratio is the ratio of the likelihoods of two models (or a null and alternative parameter value within a single model), which may be used to compare or test the models. If either model is not fully specified then its maximum likelihood over all free parameters is used - this is sometimes called a generalized likelihood ratio.

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### A basketball probability question using Neyman–Pearson lemma

It is known that the probability of a basketball player to make his first shot is $p=0.6$ A player argues that it does not matter if he made the previous shot or not his odds stays the same. We say if ...
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### REML in non-mixed models

I want to know if the concept of Restricted Maximum Likelihood (REML) applies to non-mixed models. For example, suppose we want to perform a test equivalent of the one-sample t-test, it may be ...
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### Comparing groups, distinctly fitted with Weibull models, via the likelihood ratio test

I'm working with some data on the fatigue (i.e. time to failure under cyclic loads) of a certain alloy. I have four groups of data, each with the 'treatment' of a slightly different manufacturing ...
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### Optimizing cutoff for the negative likelihood ratio and negative predictive value

Supposing you're trying to create a test that does it best to exclude an outcome, for example miles driven on a car and chance of the car breaking down, or age and death or hospitalization, is it ...
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### LRT and Manually Finding Significance when Wilks Theorem isn't Valid

Hello and thank you for taking the time. I'm performing an LRT for a likelihood distribution which violates the regularity conditions for Wilks theorem and wald intervals. I'm running a monte-carlo ...
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### Decision boundary for logistic regression in Hurdle (two-part) GLM

I am reading . The authors use a two-part generalized linear model to detect differentially expressed genes in single-cell RNASeq data. Context Single-cell RNASeq is a technique for detecting ...
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### Showing a minimal sufficient statistic

If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$ with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample. Then how can we show that the ...
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### Hypotheses for likelihood-ratio confidence intervals

I'm working on a physics research project, and in the process I've managed to both firm up my understanding of statistical significance (I hope) and confuse myself somewhat. The question I'm looking ...
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### Manually calculating false positive risk (using Likelihood ratio and Bayes analysis)

The question is with reference to this paper: https://arxiv.org/pdf/1802.04888.pdf In the real life example on page 17-18, it is advised that the ...
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### Chi square approximation of the likelihood test ratio

I wasn't able to find any satisfying answer about that topic. I hope someone who understand correctly the subject could enlighten this shadow. This is not very important, just for the sake of ...
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### Derivation in the MLE calculation [duplicate]

In the context of the likelihood ratio test, I was told to use the following formula (1): $$\sum^n_{i=1}(X_i-\mu_0)^2=\sum^n_{i=1}(X_i-\bar{X})^2 + n(\bar{X}-{\mu}_0)^2 \qquad(1)$$ in order to ...
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### Likelihood equations returning different values in R

I made mechanistic adjustments to a negative binomial likelihood equation to account for the distribution of whole network sizes, Y generated from branching ...
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### likelihood ratio tests and t-tests

There are some derivations showing that likelihood ratio tests and t-tests (and other common hypothesis tests) are the same (e.g., page 1, page 2), but computed p-values are consistently different (e....
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### Can $G^2$ statistic in log-linear model for contingency tables be negative?

Can $G^2$ statistic of log-linear (unsaturated) model in contingency tables be negative? Since saturated model with perfect fit has $G^2=0$ I don't think the unsaturated models can get negative $G^2$. ...
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### How can I show that the LRT of $H_0:\theta_0=\theta_1=…=\theta_k$ is given by the F test under one-way ANOVA assumptions?

I am a bit curious as to how I can show this. I am aware that you can find this by using the union-intersection test, but it has been hinted that one can use the LRT test to find that LRT for $H_0$ is ...
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### How to construct the rejection region for a one-tailed Likelihood-ratio test

Suppose we have a likelihood-ratio test statistic for some parameter $\theta$, given by: $$\lambda_{\text{LR}} = -2 \left\{\ell(\theta_0) - \ell(\hat{\theta}) \right\}$$ where $\hat{\theta}$ is the ...
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### Find the Rejection Region using the Likelyhood Quotient

H0:= l=8, Ha:= l=10 from a random sample of N=9 where f(xi)=l*exp(-lxi) and a=0.05 Find the Rejection Region, please help I think i get confused when getting the likelyhood quotient because i ...
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### Local maximums in likelihood ratio test?

I am learning about likelihood ratio testing through inferring population means but am trying to generalize it in my head to estimating other parameters. Would it be possible for our likelihood ...
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### Interpreting Multilevel Logistic Regression?

I am working on a small project were I'm supposed to be using multilevel logistic regression. I am studying the likelihood of being unemployed for a specific population. At first, I inserted ...
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### Always 1 degree of freedom for hierarchical tests?

I'm trying to use the function lrtest() in r for two glm()-models. I want to do a hierarchical test for those two glm()-models, so that one of the glm()-models include all the variables and in the ...
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### Expressing z-test rejection region as LRT rejection region?

I'm trying to ensure that I understand the relationship between the rejection regions expressed in z-test and LRT. Given $h_0 \eq 0$ or ...
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### the meaning of likelihood ratio test

I just learned the likelihood ratio test (LRT) method for model selection and worked out some examples. However, I am still a bit confused with the meaning of it. Basically, for a family of model ...
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### Is it a valid likelihood ratio test?

We know a valid LRT should compare nested models. Suppose there are four models: ...
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### Highest probability set and density ratios equal to probability ratios

I came across a pretty result I had not seen before, and wondered if there were more examples For a random variable with an exponential distribution, if you want the highest probability set to ...
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### Liklihood ratio test and linear mixed effects regression

I have a data set which includes sex, age, and 5 polygenic scores as independent variables, with 16 dependent variables. I have constructed univariate linear mixed effects regression models and ...
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### “scale” in logistic regression

I am working on translating some R code into Python's statsmodels package, chiefly some logistic regression work that I've done, when I came across the following in ...
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### Interpreting LR test of model fit when additional variables are insignificant?

Suppose we estimate the following (made up) model: $$\tag{1} pizza = \beta_0 + \beta_1 price + \epsilon$$ where pizza is quantity of pizza purchased price is the price of a pizza Now suppose we ...
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### likelihood ratio test in Bernoulli logit model

Questions: Derive the likelihood ratio test (test statistic, its null distribution and rejection region) for testing $H_0 : β_1 = · · · = β_p$ in Bernoulli logit model. You should provide a clear and ...
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### Comparision of mixed effect models via likelihood ratio test and assumption of homogeneity of variance

I would like compare different models with increasing complexity, since I would like to check the impact of each predictor (and the combination of both predictor variables) on the independent variable ...
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### Meta-analyzing Likehood ratio or wald test outputs

Hope your day is going well! I've been reading a lot on the Wald test and likelihood ratio test recently because it is the output of my RNA sequencing data. I was interested in meta-analysis of ...
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### Ranking of Wald, LR and score statistic in the normal linear regression model

Consider the the partitioned linear regression model $$y=X_1\beta_{01}+X_2\beta_{02}+\epsilon,$$ where $y|X\sim\mathcal{N}(X\beta_0,\sigma^2I)$. We test \begin{equation}\label{hopartlinregrmod} H_0:\...
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### Likelihood ratio, Wald, and Score are equivalent?

In Foundations of Linear and Generalized Linear Models, Agresti makes a comment on page 131 about likelihood ratio, Wald, and Score testing of regression parameters. For the best-known GLM, the ...
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### Performance of TDA vs likelihood ratio test, AIC, BIC, $R^2$ for model selection

Topological data analysis (TDA) uses topological persistence to find which variables are important and to distinguish signal from noise. TDA can be used for variable selection. How does TDA perform ...