Number of parameters in the model

Classic parameter estimation techniques involve using experimental data to estimate all parameters at once. This allows an estimate of central tendency and a confidence interval for each parameter, but it also allows determination of a matrix of covariances between parameters. To determine parameters and confidence intervals at some level, the requirements for data increase more than proportionally with the number of parameters in the model. Above some number of parameters, simultaneous estimation becomes impractical, and the experiments required to generate the data become impossible or unethical. For models at this level of complexity parameters and covariances can be estimated for each subsection of the model. This assumes that the covariance between parameters in different subsections is zero. This is unsatisfactory to some practitioners, and this (and the complexity of such models and the difficulty and cost of building them) has been a criticism of highly parameterized PBPK and PBPD models. An alternate view assumes that decisions will be made that should be informed by as much information about the system as possible, that the assumption of zero covariance between parameters in differ-... 

Table I lists the values of the rate coefficients used to simulate the transient response experiments shown in Figs. 3 through 8. These values were obtained in the following manner (29). Starting from a set of initial guesses, the values of k were varied systematically to obtain a fit between the predicted product responses and those obtained from experiments in which H2 was added suddenly to a flow of NO. These experiments while not described here were identical to that presented in Fig. 9, with the exception that only l NO was used. Because of the large number of parameters in the model, only a rough agreement could be achieved between experiment and theory even after 500 iterations of the optimization routine (30). The parameter values obtained at this point were now used to calculate the responses expected during the reduction of adsorbed NO. These computations produced responses similar to those observed experimentally (i.e., Fig. 3) but the appearance of the product peaks in time did not coincide with those observed. To correct for this, the values of kg, ky, and kg were adjusted in an empirical manner.

We started by generating a data base of inner-shell correlation contributions for some 130 molecules that cover the first two rows of the periodic table. In order to reduce the number of parameters in the model to be fitted, we introduced a Mulliken-type approximation for the parameters Dab (Da+Db)/2. Furthermore we did retain different parameters for single and multiple bonds, but assumed Da=b (3/2)Da=b-... 

For models containing a P term, the sum of squares due to the factors has p - I degrees of freedom associated with it, where p is the number of parameters in the model. For models that do not contain a Pq term, SSf has p degrees of freedom. 

We emphasize that if the lack of fit of a model is to be tested, f-p (the degrees of freedom associated with 55, f) and n - p (the degrees of freedom associated with 55pj) must each be greater than zero that is, the number of factor combinations must be greater than the number of parameters in the model, and there should be replication to provide an estimate of the variance due to purely experimental uncertainty. 

A second feature of the book is its emphasis on degrees of freedom. We have tried to remove the magic associated with knowing the source of these numbers by using the symbols n (the total number of experiments in a set), p (the number of parameters in the model), and / (the number of distinctly different factor combinations in the experimental design). Combinations of these symbols appear on the tree to show the degrees of freedom associated with various sums of squares (e.g., n-/for SS ). 

The number of parameters in the model is large. One could suspect that only a few parameters are critical, but we cannot know ( em a priori which. Rather than trying to determine this at an early state we concentrate on the determination of reasonable values for all parameters. When this has been done, we can test the model, against independent experimental data and we can perform a proper sensitivity analysis for the input parameters. We then backtrack and concentrate on the determination of accurate values... 

The coefficient b0 is very different. In die current calculation it represents die predicted response in the centre of the design, where the coded levels of the diree factors are (0, 0, 0). In the calculation in Section 2.2.3 it represents die predicted response at 0 pH units, 0 °C and 0 mM, conditions that cannot be reached experimentally. Note also that this approximates to die mean of the entire dataset (21.518) and is close to the average over the six replicates in die central point (17.275). For a perfect fit, with no error, it wdl equal the mean of the entire dataset, as it will for designs centred on the point (0, 0, 0) in which the number of experiments equals the number of parameters in the model such as a factorial designs discussed in Section 2.6. 

Calculate the matrix (D D) l. This will be a square matrix with dimensions equal to the number of parameters in the model. 

To estimate the model parameters, it will be sufficient to study each variable at two levels only. This means that the number of necessary experiments will be of the same order as the number of parameters in the model, e.g. to fit a linear model with seven variables, eight experiments would be sufficient. 

In the past, the application of physiologically-based pharmacokinetics was limited by the complexity of the mathematics involved because of the large number of parameters in the models. In recent years, the advances in computer software have overcome this limitation. Thus, earlier this year, Clewell and Andersen (89) reported that by using the Advanced Continuous Simulation Language (ACSL), physiologically-based pharmacokinetic modelling may be carried out on personal computers with reasonably short turn-around times (i.e., execution time, 0.6-8 minutes) and in a user-friendly manner. 

For details of the log-normal distribution model and the appropriate activation law, the reader is referred to References 6 and 21. In these references details are provided as to the method of calculation. Three computer programs were used sequentially to fit the model to the experimental data in Figures 1 and 2. The total number of parameters in the model are five three (to, H, To) to fit the activation law for t, one to characterize the distribution width B, and oq the second moment. 

We have said that experimental design is to do with efficiency, and so we need to be able to measure a cost-benefit ratio. The cost is easy to measure, and is, essentially, the number of experiments. The "benefit" is more difficult to quantify. It may be considered as the amount of information from the design. The R-efficiency is the simplest measurement of a design s efficiency. It is the ratio of the number of parameters in the model (p) to the number of experiments (N) ... 

When the uncertainty in the parameter values becomes too large, the analyst should consider reducing the model. The correlation matrix between parameters can be useful in selecting the parameters that can be removed to make the model smaller. There are statistical criteria that can be used to select the better model. These include the Akaike Information Criterion (AIC) value and the F-test. The AIC value is calculated using the WSS, the number of parameters in the model, and the number of data points. The model with the lower AIC values is usually selected as the better model. The statistical F-test involves the calculation of an F value from the WSS and degrees of freedom from two analyses. The calculated F value is compared with the tabled values and a decision can be made whether the more complex model provides a significant improvement in the fit to the data. The analyst using a combination of subjective and objective criteria can make an educated decision about the best model. 

We have already seen that the number of degrees of freedom of the residual sum of squares is the total number of observed values minus the number of parameters in the model, Vr = (n - p). Subtracting from this the number of degrees of freedom corresponding to SSpe we will have the number of degrees of freedom for lack of fit ... 

Note Hi = number of replicates at the ith level m = number of distinct levels of the independent variables n = Su = total number of observations p = number of parameters in the model. 

The F tests demonstrated in Section 6.1 deal with a whole set of parameters. The selected significance level is therefore valid independent of the number of parameters in the model. However, one should bear in mind that, with those tests, by rejecting the null hypothesis, the parameters different from zero cannot be recognized individually. 

The minimum number of experiments that should be conducted is at least equal to the number of parameters in the model. A linear model has two parameters, /3o and (Strictly speaking, y = /Sixi is a linear model and y = Po+Pixi is nonlinear but this nuance will be disregarded in our discussion.) A quadratic model has three parameters (/3o,/3i, and P2) while there are four parameters in a cubic expression (see the relationship between specific heat and temperature above). The number of parameters for a simple polynomial is at most equal to m + 1, where m represents the order of the polynomial. 

Alternatively, criteria can be estimated for each model based on the principle of parsimony, that is, all else being equal, select the simplest model. The Akaike Information Criterion (AIC) is one of the most widely used information criterion that combines the model error sum of squares and the number of parameters in the model. 

When fitting models, the MLE is used to find the optimal fit to the dataset. However, maximizing the log likelihood often results in fitting noise and parameter estimates that are unstable, particularly when the data set is relatively small. This is because MLE trusts too much the observed trends in the, often limited, data (Moons et al., 2004). In order to avoid possible over-fitting, the Bayesian Information Criterion (BIC) was utilized (Schwarz, 1978). BIC is a criterion for model selection, and includes a penalty term for the number of parameters in the model. The BIC is given by the following equation ... 

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