Population parameter estimates

The random variable values 0 are more centered around the population parameter than the 02 ones (i.e. estimations). This means that the average error made in multiple population parameter estimation by means of 0 will be smaller than when we do the same for 02. The 0 estimation can be said to be more efficient. 

Naive Pooled Approach. The naive pooled approach, proposed by Sheiner and Beal, involves pooling all the data from all individuals as if they were from a single individual to obtain population parameter estimates.Generally, the naive pooled approach performs well in estimating population pharmacokinetic parameters from balanced pharmacokinetic data with small between-subject variations, but tends to confound individual differences and diverse sources of variability, and it generally performs poorly when dealing with imbalanced data. Additionally, caution is warranted when applying the naive pooled approach for PD data analysis because it may produce a distorted picture of the exposure-response relationship and thereby could have safety implications when applied to the treatment of individual patients. ... 

With this approach, individual parameters are estimated in the first stage by separately fitting each subject s data and combined in the second stage to obtain population parameter estimates. The salient features of the methods that constitute the two-stage approach will be discussed briefly. 

Population parameter estimates with between-subject variability, obtained from the analysis of the Phase 1 data, are presented in Table 13.3. Simulated plasma drug profiles in 50 subjects, following a single 10 mg oral administration, are plotted in Figure 13.9. Corresponding NONMEM control file and data set are in Appendix 13.6. 

M. C. KjeUsson, S. Jonsson, and M. O. Karlson, The back-step method for obtaining unbiased population parameter estimates for ordered categorical data. AAPS I 6(3) El-ElO (2004). 

Box (1976), in one of the most famous quotes reported in the pharmacokinetic literature, stated all models are wrong, some are useful. This adage is well accepted. The question then becomes how precise or of what value are the parameter estimates if the model or the model assumptions are wrong. A variety of simulation studies have indicated that population parameter estimates are surprisingly robust to all kinds of misspe-cifications, but that the variance components are far more sensitive and are often very biased when misspeci-fication of the model or when violations of the model assumptions occur. Some of the more conclusive studies examining the effect of model misspecification or model assumption violations on parameter estimation will now be discussed. 

As mentioned elsewhere in the chapter, EBEs are conditional estimates of an individual s model parameters given a model and the population parameter estimates. EBEs are useful for a variety of reasons, including ... 

It is shown that this test statistic approximately follows a chi-square distribution, with m - 1 - r degrees of freedom, where r is the number of population parameters estimated by... 

Finally, some results on optimal experiment design in a population parameter estimation context have also been recently presented [Hooker et al., 2003]. 

Data Acquisition and Parameter Estimation determines frequencies of the initiating events, component unavailability and probabilities of human actions were estimated from plant history. If insufficient, generic values were used including generic data from the nuclear industry (IAEA, 1988). In addition meteorological data and data on the population distribution around the plant were gathered and processed. 

The aim of the parameter estimation is to deduee the growth rate G, nuelea-tion rate agglomeration kernel /3aggi and disruption kernel /3disr from the experimental CSD. The CSD is deseribed mathematieally by the population balanee (Randolph and Larson, 1988)... 

One drag level (Cindiv) can be used with the means and standard deviation (SD) of population parameters (Ppop) as a priori knowledge for an individual parameter estimate using the Bayesian objective function. 

The standard way to proceed would be to fit the model to the data relative to each experimental unit, one at a time, thus obtaining a sample of parameter estimates, one for each experimental tumor observed. The sample mean and dispersion of these estimates would then constitute our estimate of the population mean and dispersion. By the same token, we could find the mean and dispersion in the Control and Treated subsamples. 

There is no restriction in the derivation of this relationship that would prevent its extension to cases where. YelR", EA-e9Tx", Y and Ae9 m, and Ae9lm n with m n. Then, Eye91mx" and equation (4.2.31) is still valid. Error propagation is achieved by replacing the population parameters by the value estimated by sampling, e.g., x for the sample mean... 

None of the population parameters x and y can be found since their determination would require the whole range of attainable values to be measured. The least-square criterion provides estimates x and of x and y, respectively, which also satisfy the model, i.e.,... 

For given data, the population parameters fij, X, and pj in Equation 5.2 have to be estimated. If the group sizes rij reflect the population groups sizes, the prior... 

The optimized values for and T, are determined by using mathematical approach without any significance attached to it for physiologic reasons. ° Generally, the re-sorting required to use a three-exponential term takes the estimates out of the population parameters or global minimum. 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

In general, bias refers to a tendency for parameter estimates to deviate systematically from the true parameter value, based on some measure of the central tendency of the sampling distribution. In other words, bias is imperfect accuracy. In statistics, what is most often meant is mean-unbiasedness. In this sense, an estimator is unbiased (UB) if the average value of estimates (averaging over the sampling distribution) is equal to the true value of the parameter. For example, the mean value of the sample mean (over the sampling distribution of the sample mean) equals the mean for the population. This chapter adheres to the statistical convention of using the term bias (without qualification) to mean mean-unbiasedness. 

Confidence interval The numerical interval constructed around a point estimate of a population parameter. It is combined with a probability statement linking it to the populations true parameter value, for example, a 90% confidence interval. If the same confidence interval construction technique and assumptions are used to calculate future intervals, they will include the unknown population parameter with the same specified probability. For example a 90% confidence interval around an arithmetic mean implies that 90% of the intervals calculated from repeated sampling of a population will include the unknown (true) arithmetic mean. 

Asymptotics take on a different meaning in the Bayesian estimation context, since parameter estimators do not converge to a population quantity. Nonetheless, in a Bayesian estimation setting, as the sample size increases, the likelihood function will dominate the posterior density. What does this imply about the Bayesian estimator when this occurs. 

The parameters A,k and b must be estimated from sr The general problem of parameter estimation is to estimate a parameter, 0, given a number of samples, x,-, drawn from a population that has a probability distribution P(x, 0). It can be shown that there is a minimum variance bound (MVB), known as the Cramer-Rao inequality, that limits the accuracy of any method of estimating 0 [55]. There are a number of methods that approach the MVB and give unbiased estimates of 0 for large sample sizes [55]. Among the more popular of these methods are maximum likelihood estimators (MLE) and least-squares estimation (LS). The MLE... 

To illustrate the difference between values of sample estimates and population parameters, consider the ten groups of five numbers each as shown in the table. The sample means and sample standard deviations have been calculated from appropriate formulas and tabulated. Usually we could calculate no more than that these val-... 

What we have done in the table is to take ten random samples from the infinite population of numbers from 0 to 9. In this case, we know the population parameters so that we can get an idea of the accuracy of our sample estimates. 

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