Nonhierarchical models

Several examples of applicable formal methods were discussed. Nonhierarchical quantitative models can be applied several times based on plausible scenarios emerging from a qnalitative informed opinion. Expanding on the example above, a Monte Carlo simulation of pesticide ingestion rates may be conducted with and without consideration of water sources. Hierarchical Monte Carlo methods can be used in a similar manner. 

Examples of nonhierarchical clustering [22] methods include Gaussian mixture models, means, and fuzzy C means. They can be subdivided into hard and soft clustering methods. Hard classification methods such as means assign pixels to membership of only one cluster whereas soft classifications such as fuzzy C means assign degrees of fractional membership in each cluster. 

Stillwell WG, von Winterfeld D, John RS (1987) Comparing Hierarchical and Nonhierarchical Weighting Methods for Eliciting Multiattribute Value Models. Management Science 33 442-450 Stobaugh RB Jr (1969) Where in the world should we put that plant ... 

The methods reviewed above address primarily hierarchical models but an issue often arises concerning competing nonhierarchical models. That is, which model is the preferred These models are most often not independent. However, a test statistic can be used to discriminate between models, which is the difference of the minimized objective functions (log-UkeUhood differences, LLDs) for the two nonhierarchical models (18). In the next section the approach for obtaining the test statistic for comparing the two nonhierarchical models (18) is described. 

Note that the two likelihood ratio statistics LR XYX) and LR(XY,Y) are not independent. Therefore, testing whether two nonhierarchical models with equal degrees of freedom fit the data equally well is reduced to testing whether the noncentrality parameters of two independent distributions with equal degrees of freedom are identical. 

Therefore, 8 = Sy if and only if LR(X,F) = LR(Y,F). Testing this equality can be done by testing the equality of LR(X,F) and LR Y,F). The latter likelihood ratio statistics are the objective functions (i.e., -2 log-likehood of the data) of nonlinear mixed effects models. As a test statistic, the difference of objective functions (log-likelihood difference, LLD) of two nonhierarchical models can therefore be used. 

Until recently, no method of comparing nonhierarchical regression models has been available. The bootstrap has been proposed because it may estimate the distribution of a statistic under weaker conditions than do the traditional approaches. In general, for nonlinear mixed effects models that are not hierarchical, the preferred model has simply been selected as that with the lower objective function (2). A more rational approach has been proposed for comparing nonhierarchical models, which is an extension of Efron s method (2, 30). The test statistic is the difference between the objective functions (log-likelihood difference—LED) of the two nonhierarchical models. The method consists of constructing the confidence interval for the LLDs. 

To execute this, an estimate of the sample distribution of the LED under the null hypothesis must be derived to perform a test. The bootstrap method for estimating sample distribution of the difference of the objective function given the observations is used to solve the problem. This allows one to reject the null hypothesis of equal noncentrality parameters, that is, of equality of fit if zero is not contained in the confidence interval so derived. One thousand bootstrap pseudosamples were constructed, the nonhierarchical models of interest were applied, and the percentile method for computing the bootstrap confidence intervals was used. 

Ette, E.I. Comparing nonhierarchical models Application to nonlinear mixed effects modeling. Computers in Biology and Medicine 1996 6 505-512. 

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