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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Permutation test for exponential null hypothesis: really bad?
Having found nice formulas for testing the null hypothesis under exponentially-distributed samples, I wanted to see how well permutation tests could do the job. And the answer, assuming no mistakes, ...
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When and why is a likelihood ratio preferable to a difference as a test statistic?
We want to know whether two sample sets {x} and {y} were drawn from the same distribution. The null hypothesis $H_0$ is that they are. As statisticians we test the hypothesis by calculating the p-...
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What is a *likelihood ratio test* for a specific distribution, and how does it relate to hypothesis tests?
I'm just now being introduced to likelihood-ratio tests (LRT), and I am having trouble following the concept and terminology.
For example, I posed a question about determining whether two samples {x} ...
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Likelihood ratio exponential family under permutation of parameters
I'm reading "ASYMPTOTIC NORMALITY OF MAXIMUM LIKELIHOOD AND ITS VARIATIONAL APPROXIMATION FOR STOCHASTIC BLOCKMODELS" Bickel et al. 2013. In their proof of Lemma 3, they claim a result and I ...
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How to interpret DESeq results with LRT vs. with Wald's test
I am new to the field of RNA-Seq and wanted to ask for advice concerning the proper use of the two DESeq() test options (LRT vs. Wald test).
Briefly, my ...
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UMP two sided tests for exponential families
Consider a random variable $X$ with density $$f(x : θ) = C(θ)e^{η(θ)T(x)}h(x), θ ∈ Θ$$.
Assume that $η(θ)$ is strictly increasing in $θ$ and that the family is full rank. Show that there will not be ...
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What are the degrees of freedom to consider for a G-test when some cells have expected values of 0?
Let's say I conduct a survey where people can mention their favorite
color among four options (red, green, blue, yellow).
After collecting the data, I create a contingency table crossing
gender with ...
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Likelihood ratio as minimal sufficient statistics in infinite parameter space
I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first:
Consider a family of density functions $f(x|\theta)$ where the ...
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How to use the likelihood ratio test (LRT) to test whether a three-way interaction is significant?
Our hypothesis is that there is a 3-way interaction between A, B, and C.
I have defined a model as follows:
Y=A+B+C+AB+AC+BC+ABC+error
I aim to use the likelihood ratio test (LRT) to determine if the ...
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Likelihood Ratio Testing for Binomial Distributions
I have a feeling this is a silly question. I am working on a research paper, at some point in it we perform a likelihood ratio test. The first guess would be to apply Wilks's theorem. However, if we ...
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Residual likelihood ratio test for fixed effects in a linear mixed model
I know (but now I have doubts) that "Comparing models that are fitted with REML and differ in their fixed effects never makes sense," just as @BenBolker explains in this answer.
I've been ...
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Does the "log-likelihood" measure cover all details about model fit, like covariance structure, adjustments, robust variance estimator, etc?
Just a general statistical question: when any statistical software returns log-likelihood of some model, does it account for all details in it?
For example, when we employ generalized least square ...
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Can be any example of testing contrasts using Wald's approach reproduced with Likelihood Ratio testing?
For illustration I will use R, but the question is general statistical question, totally not R related.
Assume I have a numerical variable and categorical variable with 3 levels, like A, B and C, for ...
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Understanding residuals vs. fitted plot for a linear mixed model
I am modeling body mass (y var) according to indices of dysregulation for different physiologic systems (x vars). I did a likelihood ratio test, which supported using a linear mixed model, with a ...
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Comparison of multilevel models via deviance test
I have a question regarding the comparison of the following two multilevel models:
Null model: outcome.nullmodel <- lmer(outcome ~ 1 + (1 | ID), data=multileveldata)
Random slopes model: outcome....
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Likelihood ratio tests vs. ANOVA for interactions in linear mixed model
I am analyzing a longitudinal study where patients received either treatment 1, treatment 2 or no treatment (placebo) using linear mixed models (LMM) in R. I have a baseline measure that is related to ...
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When using a likelihood ratio test to test for significance of a main effect, should I use the most maximal or minimal model as a base model?
Lets suppose I have a set of n covariates, and I want to test for the significance of the main effect of covariate i. I want to do this using a likelihood ratio test; fitting a model with covariate i ...
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Generalized likelihood ratio test for a left-truncated exponential distribution [duplicate]
I am doing self study in statistical inference and am rather confused about how to approach generalized likelihood ratio test (GLRT) problems. I am trying the traditional approach by definition and ...
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How to perform likelihood ratio test for bayesian neural network?
I am building a Bayesian neural network with Poisson likelihood and 50 features for time series prediction. Parameters of the model are learned using variational inference. I am trying to see whether ...
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p-value ratio for Likelihood Ratio Test with multicollinear data
I have two datasets where my independent variables (of which I have 6) are highly correlated. In one dataset I know for certain that the dependent variable should only depend on 1 independent variable ...
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Likelihood ratio test vs p value for Poisson regression
I have a Poisson regression model, from its summary table, I could see the p-value for a certain variable, e.g. gender. Since the p-value is testing the hypothesis whether the coefficient of gender ...
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Exact Likelihood ratio statistic for discrete distribution
Suppose that the random variables in a sample $Y_1, Y_2, \ldots, Y_n$ are iid with values in $[0,1]$, and that an investigator knows that the underlying probability density $f_Y(y)$ has the form
$f_Y(...
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Using likelihood ratios instead of p-values
I am analysing data from an experiment consisting of 4 treatments, and I am interested in treatment differences of DNA damage caused by a toxicant. I have consulted a statistician to discuss some of ...
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Is it valid to compare nested models even if proportional hazards assumption is violated in Cox models
I am trying to understand when are Cox models still informative and useful even when the proportional hazards (HR) assumption is violated and came across this interesting answer. It includes a link to ...
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UMP test of $H_0:\theta=1$ vs $H_A:\theta\neq1$ for Beta($\theta,\theta$)
Let $X_1,X_2,\dots,X_n$ be iid Beta($\theta,\theta$) samples. Is there a UMP level $\alpha$ test of $H_0:\theta=1$ vs $H_A:\theta\neq1$?
We first test $H_0:\theta=1$ vs $H_A:\theta<1$. The ...
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Why can a likelihood ratio not give evidence for the null since it is a model comparison?
I am curious as to why a likelihood ratio cannot give positive evidence for the null, since it is a model comparison.
Indeed, this is more confusing given the fact that Bayes Factors are similar ...
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Comparison of variances: symmetric F-test is likelihood ratio test?
Suppose $X_i\stackrel{IID}\sim N(\mu,\sigma^2)$ for $i=1,...,n$, where $\mu$ is known. We want to apply the likelihood ratio test to decide between the hypotheses
$$
H_0: \sigma=\sigma_0 \\
H_1: \...
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Why doesn't the G-test's Chi-squared "threshold" scale with sample size?
In the popular likelihood ratio test of goodness-of-fit (also known as the G-test: see, e.g., here), the test statistic is calculated as
$$G(\mathbf{O},\mathbf{E})=2\sum_{i=1}^{M}O_{i}\log\frac{O_{i}}{...
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Likelihood Ratio vs Modified Frequentist Approach (CLs)
I'm a physicist trying to finally get a hold on practical statistics for particle physics and am having problem with the following -- I apologize for the lack of formality below.
Suppose the number of ...
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Generalized Likelihood Ratio Test - Why is the denominator a union
For GLRT, the ratio is:
$$
\Lambda^* = \frac{\max_{\theta \in \omega_0} L(\theta)}{\max_{\theta \in \omega_1}L(\theta)}
$$
but we instead use:
$$
\Lambda = \frac{\max_{\theta \in \omega_0}L(\theta)}{\...
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Show a composite test is the most powerful after deriving a similar most powerful simple test
Let $X$ be a real-valued random variable with density $f(x) = (2\theta x + 1 - \theta) \mathbb{1}(x \in [0,1])$ where $1$ here is the indicator function and $-1 < \theta < 1$. I am trying to ...
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application of the Savage - Dickey ratio
The Savage - Dickey ratio is an equivalent, but more useable, form for the Bayes factor for two model, model $M_{0}$ and model $M_{1}$.
The definition for the Bayes Factor $BF_{ij}$is as follows:
$BF_{...
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How to compare 2 estimates from a model fitted by MLE when the covariance of the estimate can't be always estimated?
I have one model, that I fit using MLE. In order to feel what could be a good initial guess for the fitting process, I try different arbitrary smartly chosen initial guess (not random) and report the ...
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Is it better to select variables before regression or performing regression and then performing tests to select the variables?
I'm working on a regression model that predicts age of clients. The problem is that there aren't many variables to work with. So my question, is it better to study correlations and contingency tables ...
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Generalized Likelihood Ratio for variance in Normal Hierarchical Models
We have a Linear Hierarchical Model where
$$Y_i | \theta_i \sim N(\theta_i,1)$$
$$\theta_i | A \sim N(0,A)$$
with
$$Y_i |A \sim N(0,A+1)$$
where $ i = 1,2,\ldots,k.$
I found the likelihood function ...
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Why does the Type-3 ANOVA using LRT via car::Anova() give different result than term-by-term LRT model comparison via anova() in R?
With this very simple data:
...
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Likelihood Ratio when true/false positive/negative rates are not available
In this paper (PMID 23123231; paywalled), the authors develop a logistic regression prediction model for Alzheimer's disease. In Table 3 the authors then present disease prediction results after ...
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Model Comparison for Overlapping Models (Choosing between the different measurements of the same skill)
I have a large dataset with 15 independent variables. I am interested in investigating the potential interactions between certain independent variables; however, some independent variables measure the ...
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Null model is a better fit for the data compared to experimental model?
I have built a generalised linear mixed effects model fitted to a gamma distribution. I am wanting to compare this experimental model to a nested null model to see whether it is a better fit for the ...
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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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Subspace test for multivariate normal distribution [duplicate]
Subspace test for multivariate normal distribution
Suppose $X_1, X_2,\ldots, X_n$ are i.i.d. observations from a multivariate normal distribution $N(\mu,\Sigma)$ where $\Sigma$ is known. Furthermore, ...
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What does positive likelihood ratio mean outside of medicine?
I am a meteorologist and I regularly hear of POD (we call it probability of detection) $ POD = \frac{TP}{TP+FN} $ as well as FAR (False Alarm Rate) $ FAR = \frac{FP}{FP+TN}$. But I recently had a ...
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Find the likelihood ratio test for to test $H_0$:$\log(1-\theta)=\eta_0$ against $H_1$:$\log(1-\theta)\neq\eta_0$
My approach is given below.
Let $X_1, \ldots, X_n \stackrel{\text{i.i.d}}{\sim} \text{Geometric}(\theta)$, so that
$$f_{\theta}(x) = \theta(1 - \theta)^x, \hspace{10mm} x \in \mathbb{N}.$$
Find the ...
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Detection limit for a drop in speed, due to stopping between measurements
Assume that a device is traveling at a constant expected speed $\mu$, subject to random variation with standard deviation $\sigma$, for a measurement period of duration $t$. Then the expected travel ...
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Likelihood ratio test for model specification with boundary Null
I am interested in understanding the asymptotic distribution of Likelihood ratio (LR) test statistic for model specification. I am focusing on the case in which the null hypothesis is of the form (i.e....
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Multiple Kolmogorov-Smirnov tests with likelihood ratio
I would like to be able to establish whether a sample (let's call it A) is more likely than another sample (let's call it B) to be drawn from the same distribution as a third sample (let's call it C).
...
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What regression should i perform in order to obtain an R-squared or pseudo R-squared with my data properties?
I've got a rather hard question concerning my regression.
My data has the following properties.
Dependent variable is count data and is overdispersed and consist of repeated measurements within ...
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Type-2 errors in the likelihood-ratio/G-test?
I am quite new to hypothesis testing, and I am currently trying to familiarize myself with the G-test for independence. In my research, I came up with a question for which I could not fit an answer ...
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Significance test on likelihood ratio of two completely specified models
Suppose that I collect some dataset $Y$ which is a sample from a multinomial distribution with unknown event probabilities $\theta_{truth}=(p_1,p_2,...,p_k)$ where $\sum{p}=1$.
I have two competing ...
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What is df of the Chi-squared distribution for Likelihood ratio test
I have got 2 sample points $x_i, i = 1,2$ from $ \text{poisson} \left( {\lambda}_i \right)$. and want to perform likelihood ratio test for $H_0 : \lambda_1=\lambda_2 $ vs. $H_0 : \lambda_1 \neq \...
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