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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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Can positive and negative likelihood ratios be combined into one parameter?

Positive and negative likelihood ratios (PLR, NLR) are considered to be much better than sensitivity and specificity as parameters of usefulness of a test. Is it possible to combine PLR and NLR into ...
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Why is likelihood ratio test preferred over log rank and Wald in Cox model for small sample sizes?

It looks like a common consensus that likelihood ratio (LR) test is preferred over log rank and Wald in Cox model when sample size is small. I did some research and couldn't find any clear answer. My ...
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Why are Logrank, Wald and likelihood ratio test asymptotically equivalent?

I am trying survival analysis and it seems like a common consensus that Logrank, Wald and likelihood ratio are asymptotically equivalent I don't understand why they are asymptotically equivalent. As ...
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Likelihood ratio to quantify the similarity between one sample with two other matched samples

I conducted a study with 3 conditions and N subjects. All subjects performed all conditions once. I would like to know if the first condition is similar to the second or third condition. Formally, ...
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Can (log-)likelihood be used to compare a binomial model to its beta-binomial equivalent?

In this article the author talks about fitting beta-binomial models to data when the there data is over-dispersed relative to the assumptions of a model with binomial errors. Near the end they present ...
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Are the Peason's chi-square test and the -2-log likelihood ratio chi-square test expected to give different p-values?

I am trying to calculate a p-value after fitting a distribution to some data. In one way, I use the Pearson's chi-square test and get a p-value=0.369. I then use the log-likelihood ratio method and ...
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Summary Output for Nested Random Effect in Fixed Effect

I am trying to build a mixed model with random effect nested in fixed effect. Below is an example dataset. ...
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critical region of a binomial population

I have the following homework problem: The number of successes in $n$ trials is to be used to test the null hypothesis that the parameter $\theta$ of a binomial population equals 0.5 against the ...
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Likelihood ratio test of log normal distribution

$X_{1},X_{2}, … , X_{n}$ be a random sample from a $𝑁(\theta, 1)$ distribution. Instead of observing $X_{1},X_{2}, … , X_{n}$, $Y_{1},Y_{2}, … , Y_{n}$ was observed where $Y_{𝑖}= 𝑒^{X_{i}}$. Find ...
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Determining difference in vector of coefficients for a model according to likelihood ratio test in SPSS

I have performed logistic regression on my data set designed to explore significant independent variables (IVs) (Age, Gender, disease severity, co-morbidity, etc) associated with all-cause 30 day ...
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Uniformly most powerful unbiased test example

Question: We use the random sample $X=(X_{1},... ,X_{n})$ from Normal distribution $N(\theta ,1),-\infty <\theta <\infty $ . Define the test $\phi ^{*}=\begin{cases} & 1\text { if } \...
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Likelihood ratio test statistic for μ=σ²

I am trying to find the log likelihood ratio test statistic for the null hypothesis that $\mu=\sigma^2 where the observations are normally distributed and iid. I have figured that I need to maximise ...
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Likelihood ratio test and sample statistics

Given a sample $\mathbf X =(X_1,...,X_n)$ from a parent random variable $X$, Neyman-Pearson's test for two point hypotheses $H_0$ and $H_1$ is the one defined by the critical region $$C=\left\{\mathbf ...
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Non-analytical application of Neyman-Pearson lemma

I have a discrete random variable $N$, from which a random sample $N_1,\dots,N_{15}$ is drawn. I want to test two hypotheses about the distribution of $N$ which are represented by two histograms. In ...
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Likelihood modification in Metropolis Hastings ratio for transformed parameter

I want to use MH to get samples from $p(\theta \mid y) \approx p(y \mid \theta) p(\theta)$. Let's assume $\theta$ is heavily constrained and I transform $\theta$ to $f(\theta)$ so I can sample from ...
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Can the r-scale value (for Bayes Factor) be directly based on Cohen's d?

I want to set an r-scale value for a Bayesian t-test (i.e. to calculate Bayes Factor likelihood ratio) based on previous results (i.e., from a posterior). But I simply cannot find a straightforward ...
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Understand a statement about likelihood function

I'm reading Agresti - Categorical Data Analysis and it says Consider two models, $M_0$ with fitted values $\hat{\mu}_0$ and $M_1$ with fitted values $\hat{\mu}_1$ with $M_0$ a special case of $M_1$....
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Do we maximize likelihood or likelihood ratio for ML estimation? [closed]

I was reading link. And I rewrite (3), here, in link to simplify notation as follows $$ \Lambda(X) = \frac{\mathcal{L}(\lambda_S | X)}{\mathcal{L}(X)} $$ Here $\lambda_S$, variance in presence of ...
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Is likelihood also defined as ratio of pdfs

My understanding of likelihood is that it is pdf except that it is a function of parameters rather than observations (as in link). I was reading link. Can likelihood be defined as ratio of ...
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Usefulness of Point Estimators: MVU vs. MLE

In a past class, two types of point estimators were introduced: minimum variance unbiased estimators (MVUs) and maximum likelihood estimators (MLEs). Supposedly, the MVU is optimal, unless an unbiased ...
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Likelihood ratio test, Wald test and LM test for variance of a normal distribution

Let y1, y2....yt follow a N(0,sigma^2) distribution. [Note that the mean is zero and you know that it is zero]. Derive the LR, LM and Wald test of hypothesis sigma^2 = 1. I have got the MLE, the ...
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ANOVA selects a model with autocorrelated residuals

I want to know which temperature dataset (Aa1, Bb, Cc, Dd) is/are the best predictor for laying date (medini). First, I used simple linear regression: median...
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Statistical test to compare the mean of two samples

There's a competition with 20 categories, and each category has 3 winners. I have an array of the winner's performance of the next year (60 values), and other array with the average of the competitors ...
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Select mixed model same degree of freedom

When I have a two mixed models (lme function) with different df then ANOVA summary shows the p-value of likelihood ratio test as following : ...
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How much bias am I risking by doing model goodness of fit comparison without accounting for clustering?

I am interested in testing whether an interaction term is statistically significant or not in a logistic regression. Data is large and observations are clustered by family and suffer from sparsity for ...
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Reproduce figure of “Computer Age Statistical Inference” from Efron and Hastie

The summarized version of my question (26th December 2018) I am trying to reproduce Figure 2.2 from Computer Age Statistical Inference by Efron and Hastie, but for some reason that I'm not able to ...
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confusing result from likelihood ratio test when covariate is factor

I got really confusing result from likelihood ratio test. I'm using mtcars data as an example. The following code works as expected. In ...
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Calculating the relative likelihood with AIC values

I'm using AIC for model selection, and would like to use a relative likelihood measure to quantify how many times a model with minimum AIC (AICmin) fits better than the alternative (with AICi). For ...
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Comparing nested models with orthogonal predictors using the t-statistic

Imagine we have a pair of nested models. Model A includes $n$ terms. Model B includes $n+m$ terms. To assess the "value add" of ...
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Calculating Log-Likelihood of Logistic Adaptive-Quadrature GLMM for Comparison with Fixed Model

Fitting a binary logistic GLMM here, with ungrouped data (all responses either 0 or 1). It says in this thread and in the documentation of anova.merMod that the ...
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Degrees of freedom of likelihood ratio test with equal dimension on the null and the parameter space?

Let $x_1,\dots,x_n$ be iid samples from a $N(\mu,\sigma^2)$ and consider the hypotesis $$H_0:\theta\in\Theta_0,\,\,\,\,\,\,vs\,\,\,\,\,\,H_1:\theta\in\Theta _{0}^c$$ where $\Theta _{0} = \{\mu>0,\...
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What should my critical region look like in this LR test for shifted exponential distribution with pdf $e^{-(x-\theta)}\mathbf1_{x>\theta}$?

I have a small confusion over describing the cutoff point for the critical region in a likelihood ratio test when the null hypothesis is composite. Take this exercise in particular: Let $(X_1,X_2,\...
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Basic likelihood ratio test

I should perform a likelihood ratio test on the following null hypothesis : α = Aψ where α and A (p * m) are known vectors. ψ (m*r) is a matrix of free parameters. The likelihood ratio test is ...
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Why is the false acceptance probability not improving with increasing sample size?

We have a normal distribution with zero mean. We have two hypotheses for the variance $\sigma^2$: $$H_0: \sigma^2=\sigma_0^2$$ $$H_1: \sigma^2=\sigma_1^2$$ We make $n$ independent observations $X_1, ...
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Are EFA and/or ESEM models on the same data set with different factor solutions nested models?

Are EFA and/or ESEM models on the same data set with different factor solutions nested models? If yes, how are they nested? Are EFA and/or ESEM models and more restrictive CFA models on the same data ...
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318 views

Different p-values for coefficients and LRT in univariate cox regression (coxph R)

I have used Cox PH to test the relationship between one predictor and survival for 6 patients. Cox PH was used since the predictor is continuous. Using the "cox.zph" function, there was no significant ...
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difference between Wald and LR in Anova(car) and p-values

I am having difficulty understanding the p-values of a Anova(glm) in the car package in terms of Type II error. Are they testing ...
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Why always AIC and BIC are used in mixture model than Vuong test

I am working with mixture models. I fitted more than one model to the data and then try to select the most appropriate model using different selection criteria, for example, AIC. My supervisors asked ...
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Philosophy behind “hypothesis” in Bayes factor

The posterior odds is the ratio of the Bayes factor $\times$ prior odds of the hypotheses. $\frac{p(H_0 | D)}{p(H_1 | D)}$ = $\frac{p(D | H_0)}{p(D | H_1)} \frac{p(H_0)}{p(H_1)}$ It is considered a ...
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How to obtain a power curve from a likelihood ratio test for linear regression as a function of varying a coefficient of the more complex model?

I am currently doing a test of model complexities for two linear regression models: First Model: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon$ vs. Second Model: $Y = \beta_0 + \beta_1X_1 + \...
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Joint probability of multivariate normal distributions with missing dimensions

Suppose I conduct two experiments, each measuring a subset of possible parameters. From experiment #1 I measure two parameters and estimate the multivariate normal distribution $$ \mathbf{X}_1=\left [...
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Is there a special meaning behind a hypothesis test that consistently report 0.5 as the p-value? [closed]

I have been conducting Likelihood Ratio Tests on Full and Reduced form linear models and there is a specific scenario where I consistently obtain p-values of 0.5. I am wondering if there was a special ...
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Comparison of nested regression models vs predictor's significance within model

I am building regression models to evaluate the effect of several characteristics of genetic variants (my predictors) on a handful of phenotypic parameters (continuous and binary - hence I am building ...
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Is this model “nested” and fit for LR Test?

i want to see if different dummy variables are capturing significanly different effects from one another so i am running LR tests in which two (of the 4) dummies independent variables are merged in ...
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Likelihood Ratio Testing For Model Selection

I have seen, in various places, the likelihood ratio test being used to compare models. From my understanding, the likelihood ratio statistic is very useful because Wilks Theorem shows that ...
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Reporting betareg outcome - how to compare non-nested models?

I am looking for advice how to gain and report results using beta regression for an ANCOVA-like model. My model is as follows: ...
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LRT and $\text{Uniform}(0, \theta)$

We have one observation of variable $X$ that is distributed uniformly on $[0, \theta]$. Null hypothesis is $H_0 \colon \theta = 1$ and alternative hypothesis is $H_1 \colon \theta = 2$. What is the ...
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Analysing Repeated Measures RCT study. emmeans / lsmeans estimate and back-transform problems. Approach doubts

Background I am writing a project on a big multicenter RCT study, where subjects are following different dietary patterns for 2 years. I have access to their dietary intakes and I am grouping them in ...
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When to use Neyman Pearson or Likelihood ratio

question about Neyman-Pearson lemma vs the likelyhood ratio. From my textbook it says that if you want to test: $H_0: \theta \in \Theta_0$ vs $H_1: \theta \in \Theta_0 ^ c$, then you can use a LRT $...
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LR test of merged dummy coefficients in a Heckman

i'm running a Heckman test and want to demonstrate if different dummy variables have a statistically similar effect. im performing an LR test by first running the full Heckman, storing the log-...