Likelihood ratio

If we let H0 represent the hypothesis that organism A IS NOT present in the sample, and HA represent the hypothesis that organism A IS in the sample, then the likelihood ratio for H0 versus HA is given by the probability of the observed peak table under HA divided by the probability of observing the outcome under H0. Specifically,... 

Using the field model described in section 1, detection probabilities are to be computed for each grid point to find the breach probability. The optimal decision rule that maximizes the detection probability subject to a maximum allowable false alarm rate a is given by the Neyman-Pearson formulation [20]. Two hypotheses that represent the presence and absence of a target are set up. The Neyman-Pearson (NP) detector computes the likelihood ratio of the respective probability density functions, and compares it against a threshold which is designed such that a specified false alarm constraint is satisfied. 

Of the various available techniques, the most widely used are based on the Measurement Test (Mah and Tamhane, 1982). These are the Modified Iterative Measurement Test (MIMT) developed by Serth and Heenan (1986) and the Generalized Likelihood Ratio (GLR) method presented by Narasimhan and Mah (1987). The MIMT method uses a serial elimination strategy to detect and identify only biases in measuring instruments. The GLR method allows us to identify multiple gross errors of any type. It uses a serial compensation strategy. 

Narasimhan, S., and Mah, R. S. H. (1987). Generalised likelihood ratio method for gross error identification. AIChE J. 33, 1514-1521. 

As in the steady-state case, the implementation of the chi-square test is quite simple, but has its limitations. One can use a more sophisticated technique such as the Generalized Likelihood Ratio (GLR) test. An alternative formulation of the chi-square test is to consider the components of jk separately (this may be useful for failure isolation information). In this case we compute the innovation of the ith measurement as... 

Wilsky, A. S., and Jones, L. (1976). A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems. IEEE Trans. Autom. Control AC-21, 108-112. 

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. 

Thus the likelihood ratio is 0.048. The new state of knowledge concerning 0 is then expressed by the posterior odds ... 

Posterior Odds = Likelihood Ratio x Prior Odds = 54 x 2 = 108 1... 

The likelihood ratio (LR) is the ratio of the chance of a particular response (positive or negative) when a disease or condition is present to the chance of the same response when the disease or condition is absent. For example, a LR of 2.5 for a positive result for a disease indicates that a positive result is 2.5 times more likely in a patient with the disease than in one without it. For a positive result, LR is related to sensitivity and specificity as follows ... 

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 Wald statistic is far larger than the LM statistic. Since there are two restrictions, at significance levels of 95% or 99% with critical values of 5.99 or 9.21, the two tests lead to different conclusions. The likelihood ratio statistic based on the FGLS estimates is %2 = 301n(1396.162/30) - 101n(465.708/10)... = 6.42 which is between the previous two and between the 95% and 99% critical values. 

To use the likelihood ratio method to test the hypothesis, we will require the restricted maximum likelihood estimate. Under the hypothesis,the model is the one in Section 15.2.2. The restricted estimate is given in (15-12) and the equations which follow. To obtain them, we make a small modification in our algorithm above. We replace step (3) with... 

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... 

This is often termed the Bayes rule for minimum error. An associated concept of likelihood ratio, /r, to segment an observed profile into two classes (Fig. 8.4) is defined as follows ... 

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