Profile likelihood

Additionally, in a potential regulatory framework, a method of constructing 90% confidence interval for F must also be prespecified in order to control the Type I error. Several methods are possible. First, standard nonlinear regression packages provide asymptotic standard errors of parameter estimates. This readily leads to the construction of asymptotic confidence intervals, based on the normality assumption. Other methods include likelihood profile and bootstrapping. We considered these three methods briefly. 

We briefly explore later the potential of using likelihood profile and bootstrap as confidence interval construction methods through application examples with broncodUaion and broncoprovocation data. 

We also attempted the likelihood profile method to construct the 90% conhdence interval. However, the likelihood profile of F turned out to be extremely flat in this case. As a result, the 90% confidence interval included (0.1,10). This was considered unreasonable, given the similarity between the two formulations. Thus, the likelihood profile method did not seem suitable for confidence interval construction. 

Model selection is based on the likelihood ratio test with p < 0.001 and diagnostic plots. The difference in minus twice the log of the likelihood -ILL) between a fuU and a reduced model is asymptotically distributed with degrees of freedom equal to the difference in the number of parameters between two models. At p < 0.001, a decrease of more than 6.6 in -ILL is significant. Asymptotic standard errors are obtained from the asymptotic covariance matrix. Alternatively, confidence intervals on parameters can be computed for this very nonlinear situation from the likelihood profile plot (24). 

Some analysts, such as US EPA (2002) for the analysis of 1,3-butadiene carcinogenicity examined in more detail below, assumed a distribution shape (normal in that case, with a standard deviation calculated from the estimated MLE and confidence limits of the qi estimates). However, as will be shown below, such assumptions are approximate at best even in favorable cases. Zeise et al. (1991) demonstrated a procedure to generate the required distribution by tracing the likelihood profile of the linear term ( i) for a linearized multistage fit to dichotomous tumor data. This used a modified version (Zeise and Salmon 1991) of the MSTAGE program developed by Crouch (1985). More recently. Crouch (2006) developed a similar program that provides the likelihood distribution for estimates of qi when... 

Figure 7.13 Likelihood profile for clearance using the data in Table 7.4. Ninety percent CI is indicated by the arrows. The dotted—dashed line indicates the cut-off at which the change in objective function becomes statistically significant 2.70 in this case.

A diagnosis of possible damage should be made before beginning repairs with other construction measures [48,49]. There should be a checklist [48] of the important corrosion parameters and the types of corrosion effects to be expected. Of special importance are investigations of the quality of the concrete (strength, type of cement, water/cement ratio, cement content), the depth of carbonization, concentration profile of chloride ions, moisture distribution, and the situation regarding cracks and displacements. The extent of corrosion attack is determined visually. Later the likelihood of corrosion can be assessed using the above data. 

MSDS Material Safety Data Sheet - a document that provides pertinent information and a profile of a particular hazardous substance or mixture. An MSDS is normally developed by the manufacturer or formulator of the hazardous substance or mixture. The MSDS is required to be made available to employees and operators whenever there is the likelihood of the hazardous substance or mixture being introduced into the workplace. Some manufacturers prepare MSDS for products that are not considered to be hazardous to show that the product or substance is not hazardous. 

When cocaine is repeatedly administered, DAergic receptor numbers are altered.4041 The likelihood of hyperthermia may be increased with chronic cocaine abuse if the DAergic receptors involved in thermoregulation are undergoing adaptive changes with chronic cocaine exposure. In keeping with this hypothesis, cocaine delirium victims had a different profile of D2 receptor binding... 

It is imperative, however, to understand the probabilistic nature of such experiments a promising profile on pharmacogenomic and toxicogenomic screens will enhance the likelihood of having selected an ultimately successful compound, and will achieve this goal quicker than conventional animal experimentation, but will do so only with a certain likelihood of success. The less reductionist approach of the animal experiment will still be needed. It is to be anticipated, however, that such approaches will constitute an important, time- and resource-saving first evaluation or screening step that will help to focus and reduce the number of animal experiments that will ultimately need to be conducted. 

In view of the likelihood of wind-mixing, particularly on the Antarctic plateau, it is not surprising that the concentration variances are lower than in Greenland. Figure 3 shows the concentration profiles of Na+, Cl, N03—, and S042- measured on the 1977 firn core from South Pole. The N03— and S042- profiles have a very small variance, in marked contrast to the Camp Century Satellite profile shown in figure 2. Sea salt constituents Na and Cl-... 

Two samples from the soil profile were randomly separated into very small subsamples (a few millimetres across) and analysed after aqua regia digestion to assess the heterogeneous nature of the Au distribution. Replicate sub-samples were analysed to assess the likelihood of finding Au in other soil particles using LA-... 

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. 

By using this targeted approach, one can limit compound-profiling activities to areas of high likelihood of BMT and optimize the cost-effectiveness of such screening. 

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