Adequacy Test

There is a plethora of model adequacy tests that the user can employ to decide whether the assumed mathematical model is indeed adequate. Generally speaking these tests are based on the comparison of the experimental error variance estimated by the model to that obtained experimentally or through other means. 

Let us now consider models that have only more than one measured variable (w>l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

An independent estimate of COV(s), t, that is required for the adequacy tests can be obtained by performing N/t repeated experiments as... 

Although all the underlying assumptions (local linearity, statistical independence, etc.) are rarely satisfied, Bartlett s jf-test procedure has been found adequate in both simulated and experimental applications (Dumez et al., 1977 Froment, 1975). However, it should be emphasized that only the x2-test and the F-test are true model adequacy tests. Consequently, they may eliminate all rival models if none of them is truly adequate. On the other hand, Bartlett s x2-test does not guarantee that the retained model is truly adequate. It simply suggests that it is the best one among a set of inadequate models ... 

Step 6. Perform the appropriate model adequacy test (x2-test, F-test or Bartlett s -/2-test) for all rival models (just in case one of the models... 

In Tables 12.7 and 12.8 the results of the x2-adequacy test (here oe is assumed to be known) and Bartlett s x2-adequacy test (here oe is assumed to be unknown) are shown for both designed experiments (constant inputs and square wave for cF(t)). 

Table 12.7 Chemostat Kinetics Results from Model Adequacy Tests Assuming af. is Known (yftest) Performed at a=0.0I Level of Significance...

With this book the reader can expect to learn how to formulate and solve parameter estimation problems, compute the statistical properties of the parameters, perform model adequacy tests, and design experiments for parameter estimation or model discrimination. 

The adequacy test and the assignment of confidence intervals using a D-optimal design (Table 3.38) are accomplished along the same lines, as in the simplex-lattice method. The variation of with composition, are given in the reference literature [12], In constructing the fourth-order polynomial for the ternary system, the design will be D-optimal at ... 

Here, we want to emphasize that one is able to calculate the fraction of the experimental error only if replicate measurements (at least at one point x ) have been taken. It is then possible to compare model and experimental errors and to test the sources of residual errors. Then, in addition to the GOF test one can perform the test of lack of fit, LOF, and the test of adequacy, ADE, (commonly used in experimental design). In the lack of fit test the model error is tested against the experimental error and in the adequacy test the residual error is compared with the experimental error. 

The measurement of adequacy is burdensome, labor intensive, and prone to multiple measurement errors. Although dialysis center nursing staff and patients may collect the necessary samples required for adequacy testing with the utmost diligence, the very complexity and number of measurements taken wfll cause systematic errors. Nevertheless, in the opinion of the authors of the NKF-K/DOQI Clinical Practice Guidefines for PD Adequacy, when properly performed, these measures are reproducible enough to be useful in routine clinical practice. However, an alternative, simpler, method for defining dialysis adequacy would be very useful in practice. 

Addresses practical issues such as cost, feasibility, model adequacy-testing, and forecasting... 

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