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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Evaluating Block Bootstrap Confidence Interval

I have an estimation that is obtained by using MCMC. In order to calculate the standard error of the estimation, I use Block Bootstrap approach and using this standard error I create a 95% confidence ...
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Adding covariates in chi-squared test or proportions test?

I am having trouble figuring out what analysis is appropriate for my research question. I've been googling for the past few days but couldn't find the answer. I would greatly appreciate it if anyone ...
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Is it correct to compare two groups' delta G square to approximate an interaction effect?

My experimental design was Group (A vs. B) x Time (1 vs. 2). For group A, from time 1 to 2, ΔG2(1) = 3.97, p=.046 For group B, from time 1 to 2, ΔG2(1) = 1.16, p=.281 Is it correct to calculate a ...
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Likelihood ratio test for random intercept (MATLAB)

I need to program a likelihood ratio test for testing the significance of a random effect in a linear mixed model in MATLAB. But I am having some trouble finding out how to correctly specify the null ...
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Likelihood Ratio Test Equivalent with $t$ test: Difference of Two Means from Constant Variance Normal Distributions

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Suppose that independent random samples of sizes $n_1$ and $n_2$ are to be selected from normal ...
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Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of ...
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Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction ...
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Distinguishing 2 alternative hypotheses by relative likelihood

I have a system with a positive, Real parameter x. I can measure x directly, but the measurements are noisy. And probably has a small systematic bias. My priors are that x=a or x=b, where a & b ...
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For Poisson GLMs, when does the residual deviance follow a chi square distribution?

According to Generalized Linear Models by McCullagh and Nelder (I am looking at the 2nd edition, 1999), the deviance function is defined as $$D(y; \hat{\pi}) = 2[l(\tilde{\pi}; y)- l(\hat{\pi}; y)]$$ ...
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Using AIC vs Likelihood Ratio test for comparing Lognormal and Powerlaw distributions

I am interested in comparing whether a lognormal or a power law are a better fit for a given set of data. Both distributions have been fit using MLE, with $x_{min}$ determined using KS-minimization a ...
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Comparison between Nested and Non-nested Model: BIC and LRT

I would like to select the best model for predicting breast cancer risk, specifically, it is the comparisons between weight/BMI/height, as other covariates remain the same in all the models. But I got ...
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Significant coefficients but non-significant likelihood ratio test [closed]

Following a comment on this thread, I have a question about interpreting a logistic regression model with significant coefficients, but non significant likelihood ratio test. I have a super simple ...
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Where to find information on likelihood ratio tests [duplicate]

Can someone please recommend a good resource that explains the theory and logic behind likelihood ratio tests?
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ANOVA vs likelihood ratio test (different result)

I found the likelihood ratio test has a very similar setup as ANOVA: we are essentially testing if adding an additional variable would significantly increase the fitness. So I run both statistical ...
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What statistic am I actually looking for and how do I properly calculate it?

I think I am looking at odds or likelihood ratios in this scenario, but I'm having trouble figuring out how to actually calculate my numbers to get an answer to my questions. I have 309,508 total 3rd ...
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Does ( P(B|A) - P(B|~A) ) / P(B|A) have a name?

Without going into the details, which are unnecessary here, this morning I found uses for the quantity $$ S = \dfrac{P(B|A) - P(B|\overline A)}{P(B|A)} , $$ something like the amount of "...
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Do I have to drop a random slope if I drop the fixed effect in mixed-models for model comparison?

I'm performing a model comparison using the likelihood ratio test in R with two mixed models fitted with lme4. My model is: ...
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Mixed-models and PROC MI

I am running into a problem that I cannot seem to figure out. I am fitting a mixed-models logistic regression using PROC GLIMMIX using complex survey data from the Behavioral Risk Factor Surveillance ...
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GLR test for change in mean of Rician

I am trying to formulate the GLR test for the change in mean of a Rician distribution. I get the Rician pdf as $$ p(x) = \frac{x}{\sigma^2} \exp\left[ -\frac1{2\sigma^2}\left(x^2+\alpha^2\right) \...
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Interprete AIC as significance test and determine significance level for nested models

Let's say we have two nested models. The smaller one (corresponds to $H_0$) has $p_0$ parameters, so its AIC is given by $$AIC_0=-2\log L_0+2p_0$$ The larger model has $p_1:=p_0+d$ parameters of which ...
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Testing the ratio of SLR coefficients

Consider applying simple linear regression (SLR) to the data $\{(X_{i},Y_{i})\}_{i=1}^{n}$. Furthermore, let's assume that the errors are conditionally normal, i.e $$Y_{i} = \beta_{0} + \beta_{1}X+e_{...
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Which to Use: Likelihood Ratio Test or Uniformly Most Powerful Test?

I've recently been learning about MPTs (most powerful tests), UMPTs (uniformly most powerful tests) and LRTs (likelihood ratio tests), and do not totally understand in which context the different ...
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Convergence rate of log likelihood ratio

I have come across the following statement in the textbook A course on Large Sample Theory by Ferguson - Chapter 17. Strong Consistency of the Maximum Likelihood Estimates. The likelihood ratio, $L_n(...
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Testing for equal proportions when sample sizes are very small

Suppose I observe binary data for two samples (hopefully the notation below is obvious) and I wish to test the hypotheses: $$H_0: p_1 = p_2$$ $$H_A: p_1 \neq p_2$$ I know there is a $z$-test for doing ...
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How does the simplification step work during the derivation of the likelihood ratio for normally distributed data?

For $H_0: \mu = \mu_0$, $H_1: \mu \neq \mu_0$ where I have a random sample of size n for normally distributed data with a known variance $\sigma^2$. Whilst researching online, there are these two ...
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Asymptotic chi-squared distribution of likelihood ratio statistic in regression problem

There is a famous result, going back to Wilks (1938) "The large-sample distribution of the likelihood ratio for testing composite hypotheses" (Ann. Math. Stat., 9, 60-62) that states that ...
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What measure to use to compare stratified and non stratified Cox proportional hazards in R

I am doing past papers studying for an exam on medical statistics and am unsure about one of the questions. The question reads as follows: "Consider now a second model where we stratify by sex ...
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Likelihood Ratio for two-sample Exponential distribution with constant ratio and confidence interval

Let $X$ and $Y$ be two independent random variables with respective pdfs: $$f \left(x;\theta_i \right) = \begin{cases} \frac{1}{\theta_i} e^{-x/ {\theta_i}} \quad 0<x<\infty, 0<\theta_i< \...
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Degree of freedom in Wilk's Theorem

The data: Y_i is poisson(lambda), and and let xi be corresponding values of an explanatory variable x. I have hypothesis: H0: lambda_i = lambda, H_full: lambda_i free to be different for i = 1,....,n, ...
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How to combine two independent likelihood ratio tests?

Let us know that a patient has one of disease A or B. Suppose that we run an experiment to find that the patient has disease A or disease B. The null hypothesis is that the patient has disease A and ...
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Mixture of Chi Square for crossed mixed models

I have been reading the book "Linear Mixed Models for Longitudinal Data" by Geert Verbeke and Geert Molenberghs (Section 6.3.4) and they suggest that you can do marginal testing for the need ...
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Testing on uniform distribution with large sample?

Let $$X_1,X_2,\dotsc,X_n$$ be a random sample from U(0, a) and $$ Y_1,\dotsc,Y_n$$ be a random sample from U(-a,a) a is natural number. Let $$H_0$$ a is even and $$H_1$$ a is odd. On the basis of $$...
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How can I interpret the substantive size of a log likelihood or increase in log likelihood

If I run a linear regression and get an R-squared, it has a substantive interpretation: this is how much of the variance in the sample the model explained. If I add a term and see an increase in R-...
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Applying Wilks' theorem to uniform distribution

Suppose $X_i $ ~ $ U(0,b)$, for $i=1,2...n$ and we want to test the null hypothesis that $b=1$. Assume $H_0$. Then from Wilks' theorem, as $n \rightarrow \infty $, $ 2\ln(\frac{L_x(H_1)}{L_x(H_0)})$ ...
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Likelihood ratio test for two-parameter exponential distribution

I am considering a random sample $X_1, \ldots, X_m; \ m \geq 2$, from a 2-parameter exponential distribution with pdf $$f_X(x; \mu, \sigma) = \frac{1}{\sigma} \exp \left( -\frac{x-\mu}{\sigma} \right) ...
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How to perform likelihood ratio hypothesis test between two hidden markov models

I'm currently working on a problem posed as follows: given some data $\mathbf{x}$, what's the appropriate way to accept/reject between two hypotheses which are hidden markov models that could have ...
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Likelihood ratio of Hardy–Weinberg proportions

Consider a population with three kinds of individuals labeled $1, 2$, and $3$ occuring in the Hardy–Weinberg proportions $f(1,\theta)=\theta^2,f(2,\theta)=2\theta(1−\theta),f(3,\theta)=(1−\theta)^2$. ...
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compute likelihood statistic with permutation test

This question is about hypothesis testing, where we want to use the likelihood ratio statistic with permutations test. Suppose we sample $n$ observations from the distribution $F_{XY}$, which is the ...
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Should you still report or interpret odds ratios on levels of a factor if the likelihood ratio test is not significant?

I am running a logistic regression to study the association between a three-level factor and a binary outcome, after controlling for covariates. My sample is small to medium (~800). The likelihood ...
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Is it legitimate to compare likelihood ratios from different datasets?

I have two nonlinear models, $M_1$ and $M_2$, where $M_2$ has all parameters of $M_1$ and a few additional ones. Since $M_2$ is more expressive than $M_1$, it will always be at least as good as $M_1$ ...
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LR test for overlapping return data using bootstrap

I wish to test a null hypothesis as in Christoffersen (1998) to see whether a sequence of Value-at-Risk forecasts $Q_t(p) \in \mathcal{F}_t$ possesses correct conditional coverage. Here $p \in [0,1]$ ...
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What exactly is the difference between LMR-LRT, adjusted LMR-LRT, and VLMR-LRT?

I'm trying to understand the difference between these models and all of the citations I'm finding seem to interchange the names, or in some cases confuse which came first. I'm pretty sure it was LMR, ...
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Which statistical test to compare same model with different parameters?

I have two datasets on people buying apples based on weight and price. One dataset in 2019 the other in 2020. I estimate a logit model with Utility = betaWeight * weight + betaPrice * price. Training ...
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Converting Log-Likelihood to Chi-square

I'm using two different algorithms to get a periodogram. One outputs log-likelihood and the other outputs chi-squared test statistic, but I would like a way to convert from log-likelihood to $\chi^2$ ...
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Finding region of rejection with likelihood ratio test

Let $X_1,\ldots,X_n$ be i.i.d. from a Gamma distribution with p.d.f. $f(x;\theta) = \theta^{-2} x e^{-x/\theta}$ for $x>0$ where $\theta$ is an unknown parameter. I would like to test the ...
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Monotone Likelihood Ratio for simple null vs composite alternative

I was studying the monotone likelihood ratio property. I have a small query. We know that once a distribution has a MLR in $T(X)$ for $\theta$ then I can test one sided hypothesis of the form $H_0:\...
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Critical region for an uniform distribution

Let $X_1, X_2, ... , X_n$ be a random sample from the uniform distribution over $[0, \theta]$. Suppose we wish to test $H_0 : \theta = 5$ versus $H_A : \theta < 5$ at significance level $\alpha = 0....
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Non-centrality of likelihood ratio test statistic chi2 under alternate hypothesis

I am having trouble understanding how to determine the non-centrality parameter of the $\chi^2$ distribution symptotically followed by the likelihood ratio test statistic if the data follow the ...
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On the test of significance under penalized likelihood estimation

I start using brglm2 package to implement logistic regression under a perfect separation problem. Is there any way to test the significance of the parameters using ...
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Likelihood Ratio Test for Averaging Two Regressors

How would I go about in formulating a likelihood ratio test to compare the fit of two models, one of them containing an average of two columns? For example, I would like to compare these two models: $$...

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