Likelihood ratio tests

Mendal et al. (1993) compared eight tests of normality to detect a mixture consisting of two normally distributed components with different means but equal variances. Fisher s skewness statistic was preferable when one component comprised less than 15% of the total distribution. When the two components comprised more nearly equal proportions (35-65%) of the total distribution, the Engelman and Hartigan test (1969) was preferable. For other mixing proportions, the maximum likelihood ratio test was best. Thus, the maximum likelihood ratio test appears to perform very well, with only small loss from optimality, even when it is not the best procedure. 

Survival and failure times often follow the exponential distribution. If such a model can be assumed, a more powerful alternative to the Log-Rank Test is the Likelihood Ratio Test. 

Mendell, N.R., Finch, S.J. and Thode, H.C., Jr. (1993). Where is the likelihood ratio test powerful for detecting two component normal mixtures Biometrics 49 907-915. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

For the model in Exercise 3, test the hypothesis that X = 0 using a Wald test, a likelihood ratio test, and a Lagrange multiplier test. Note, the restricted model is the Cobb-Douglas, log-linear model. 

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variances, we refer %2 = 401n.58333 - 201n.847071 - 201n.320506 to the chi-squared table. (Under the null hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, %2 = 4.5164, which is roughly equal to the LM statistic and leads once again to rejection of the null hypothesis. 

The t-ratio for testing the hypothesis is. 15964/.202 =. 79. The chi-squared for the likelihood ratio test is 1.057. Neither is large enough to lead to rejection of the hypothesis. 

The log-likelihood function at the maximum likelihood estimates is -28.993171. For the model with only a constant term, the value is -31.19884. The t statistic for testing the hypothesis that (3 equals zero is 5.16577/2.51307 = 2.056. This is a bit larger than the critical value of 1.96, though our use of the asymptotic distribution for a sample of 10 observations might be a bit optimistic. The chi squared value for the likelihood ratio test is 4.411, which is larger than the 95% critical value of 3.84, so the hypothesis that 3 equals zero is rejected on the basis of these two tests. 

Suppose that the following sample is drawn from a nonnal distribution with mean u and standard deviation ct y = 3.1, -.1,. 3, 1.4, 2.9,. 3, 2.2, 1.5, 4.2,. 4. Test the hypothesis that the mean of the distribution which produced these data is the same as that which produced the data in Exercise 1. Test the hypothesis assuming that the variances are the same. Test the hypothesis that the variances are the same using an F test and using a likelihood ratio test. (Do not assume that the means are the same.)... 

The likelihood ratio test is based on the test statistic 7. = -2(lnZ - In/.,). The log-likelihood for the joint sample of 20 observations is the sum of the two separate log-likelihoods if the samples are a ssumed to be independent. A useful shortcut for computing the log-likelihood arises when the maximum likelihood... 

The selection of the appropriate population pharmacokinetic base model was guided by the following criteria a significant reduction in the objective function value (p < 0.01,6.64 points) as assessed by the Likelihood Ratio Test the Akaike Information Criterion (AIC) a decrease in the residual error a decrease in the standard error of the model parameters randomness of the distribution of individual weighted residuals versus the predicted concentration and versus time post start of cetuximab administration randomness of the distribution of the observed concentration versus individual predicted concentration values around the line of identity in a respective plot. 

Key words Frequent pattern mining, Graphs, Strings, Likelihood-ratio test, Polypharmacology,... 

Likelihood Ratio Test with Logistic Regression... 

Self, S. G., Liang, K.-Y., Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions,/. Am. Statist. Assoc. 1987, 82 605-610. 

Liao (2000) derived a test statistic for single dispersion effects in 2" k designs. He applied the generalized likelihood ratio test for a normal model to the residuals after fitting a location model, which results in Bartlett s (1937) classical test for comparing variances in one-way layouts. The test is then applied, in turn, to compare the variances at the two levels of each of the k experimental factors. We caution that the test statistic (equation (3) in Liao) is written incorrectly. 

Tebbs, J. M. and Swallow, W. H. (2003b). More powerful likelihood ratio tests for isotonic binomial proportions. Biometrical Journal, 45, 618-630. 

The relevance of CLCr for clearance is tested using the likelihood ratio test (Beal et al. 1992). The so called full model (alternative hypothesis) given in equation 3 is tested against the reduced model with 0 clCr = 0 (null hypothesis) characterised by equation 2. 

The OF (objective function negative log of probability, -21n(Prob)), calculated by NONMEM, is a measure for the deviation between the model prediction and the observed data. It enters into the likelihood ratio test as follows if the OF of the full model minus the OF of the reduced model is smaller than -3.84, than the full model can be accepted at a significance level of p < 0.05 (Beal et al. 1992). 

The volume given in Equation 5 as a full model (A) changes with 0 Sex = 0 to a reduced model (B). To perform the likelihood ratio test, both models were fitted with NONMEM and the OF of the full model (A) was 6.39 points lower than the OF obtained for the reduced model (B). This difference is highly significant, so the full model (A) is preferred when compared to a reduced model (B). 

In all formulations that have appeared in the literature thus far, a generalization of the CA reference concept was performed to statistically test for deviations from CA. This means that a function describing interaction is incorporated in the CA model such that if the interaction parameter is 0, the interaction function disappears from the function. This nested structure allows testing whether its appearance in the model improves the description of the data significantly by applying the likelihood ratio test. The various nonlinear response surface approaches do differ in the way this deviation function is formulated. 

Given the background and target space characterizations, a test is applied to each pixel to determine the likelihood that it contains the effects of the target gas in question. The test used is the Generalized Likelihood Ratio Test, GLRT The GLRT as formulated here is based on the matched subspace detector, MSD, written as... 

The generalized likelihood ratio test (GLRT) for the model 3 was derived in, given... 

A common form of model selection is to maximize the likelihood that the data arose under the model. For non-Bayesian analysis this is the basis of the likelihood ratio test, where the difference of two -2LL (where LL denotes the log-Ukelihood) for nested models is assumed to be approximately asymptotically chi-squared distributed. A Bayesian approach— see also the Schwarz criterion (36)—is based on computation of the Bayesian information criterion (BIC), which minimizes the KuUback-Leibler KL) information (37). The KL information relates to the ratio of the distribution of the data given the model and parameters to the underlying true distribution of the data. The similarity of the KL information expression (Eq. (5.24)) and Bayes s formula (Eq. (5.1)) is easily seen ... 

Goodness-of-Fit. It is implied in steps 2 to 6 above that diagnostic plots (e.g., weighted residual versus time, weighted residual versus predicted observations, population observed versus predicted concentrations, individual observed versus predicted concentrations) and a test statistic such as the likelihood ratio test would be used in arriving at the base model (see Section 8.6.1.1 for goodness of fit). Once the base model (with optimized structural and variance models) has been obtained, the next step in the PM model identification process is the development of the population model. 

Kowalski and Hutmacher (17) have proposed using the Wald approximation to the likelihood ratio test in conjunction with Schwarz s Bayesian criterion (SBC) to determine the covariates for inclusion in a population PM model. In this approach all possible models (with or without each of the covariate parameters in the model) are tested. The process proceeds as follows ... 

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