Variability and measurement errors

The data fluctuate because of uncontrolled variables and measurement error. Suppose we want the best single value of the yield, where best means the yield that we can expect in future runs at the same conditions. We could calculate the sample mean X=58.4, the median m=60 and the mode 63. But perhaps the 32% yield was a run involving some error of which we are unaware. We cannot arbitrarily drop the run without knowing the cause of the low value, but the mean places undue weight on it. 

According to the population approach, the analysis of collected data requires an explicit mathematical model, including parameters quantifying population mean profiles, interindividual variability, and residual variability including intraindividual variability and measurement error [460]. 

Nonlinear mixed-effects modeling methods as applied to pharmacokinetic-dynamic data are operational tools able to perform population analyses [461]. In the basic formulation of the model, it is recognized that the overall variability in the measured response in a sample of individuals, which cannot be explained by the pharmacokinetic-dynamic model, reflects both interindividual dispersion in kinetics and residual variation, the latter including intraindividual variability and measurement error. The observed response of an individual within the framework of a population nonlinear mixed-effects regression model can be described as... 

Fornell, C. and Larcker, D. F. (1981], "Evaluating structural equation models with unobservable variables and measurement error," Journal of Marketing Research, 18 (1], 39-50. 

The objective of sediment and water sampling is to obtain reliable information about the behavior of agrochemicals applied to paddy fields. Errors or variability of results can occur randomly or be due to bias. The two major sources of variability are sediment body or water body variability and measurement variability . For the former, a statistical approach is required the latter can be divided into sampling variability, handling, shipping and preparation variability, subsampling variability, laboratory analysis variability, and between-batch variability. ... 

Accordingly, we have for the estimate of the variables, the measurement errors, and the error estimate covariance... 

Note that the measurements and estimates include both measured state variables and measured input variables. The inclusion of the input variables among those to be estimated establishes the error-in-variable nature of the data reconciliation problem. 

In the error-in-variable method, measurement errors in all variables are treated in the calculation of the parameters. Thus, EVM provides both parameter estimates and reconciled data estimates that are consistent with respect to the model. 

In addition to the somewhat empirical and difficult development of NIR applications, thorough documentation must be produced. NIR methods have to comply with the current good manufacturing practice (cGMP) requirements used in the pharmaceutical industry. Various regulatory aspects have to be carefully considered. For example, NIR applications in classification, identification, or quantification require extensive model development and validation, a study of the risk impact of possible errors, a definition of model variables and measurement parameters, and... 

In practice, random fluctuations in process variables and random errors of measurement are always present. If our measurements are sufficiently sensitive, we will pick up these random fluctuations, and the variance of the measurements will not be zero. 

Input variables are controllable, uncontrollable and disturbance variables. Controllable variables or factors X1 X2,..., X are variables, that can be directed or that can affect the research subject in order to change the response. They can be numerical (example temperature) or categorical (example raw material supplier). Uncontrollable variables Z1 Z2,..., Zp are measured and controlled during the experiment but they cannot be changed at our wish. They can be a major cause for variability in the responses. Other sources of variability are deviations around the set points of the controllable factors, plus sampling and measurement error. Furthermore, the system itself may be composed of parts that also exhibit variability. Disturbance, non controlled variables Wi, W2,..., Wq are immeasurable and their values are randomly changed in time. 

This comparison never shows a perfect correspondence between models and experiments because of modeling and measurement errors. In fact, even if the presence of systematic experimental errors can be excluded, systematic errors generated by the inadequacy of the model must be added to random experimental errors for each measured variable (m = 1,..., Am) and each experimental time (j = 1,..., Ad), the errors generated by the model are defined as... 

The parameters of the mixing rules are estimated using the least square method. In the calculations "Y" (P) is the output variable and the errors of the experimental measurements are considered to be normally distributed. Next, the fit is performed by minimizing the following objective function with respect to the vector of parameters ... 

Population pharmacokinetic parameters quantify population mean kineticS/ between-subject variability (intersubject variability)/ and residual variability. Residual variability includes within-subject variability/ model misspecification/ and measurement error. This information is necessary to design a dosage regimen for a drug. If all patients were identical/ the same dose would be appropriate for all. However/ since... 

Although the fixed effects have been well estimated, it is also of interest to examine how closely the estimated standard deviations of the random effects reflect the true variability in the simulated data. The dp2 data frame includes values of the generated subject random effects, interoccasion random effects, and measurement errors, from which sample variances can be obtained and compared to the model estimates. The intersubject sample standard deviations of log(ic ), og AUC), and log(T) are 0.33,0.41, and 0.23, respectively. The corresponding model estimates are 0.36, 0.39, and 0.26. For the lO random effects, the sample SDs are 0.17, 0.22, and 0.17, while the corresponding values obtained in the model fit are 0.20, 0.19, and 0.18, respectively. The sample and model SD for measurement error are both equal to 0.1, indicating a good agreement overall between sample and model estimates. 

Instead, we mean here the use of experimental data that can be expected to lie on a smooth curve but fail to do so as the result of measurement uncertainties. Whenever the data are equidistant (i.e., taken at constant increments of the independent variable) and the errors are random and follow a single Gaussian distribution, the least-squares method is appropriate, convenient, and readily implemented on a spreadsheet. In section 3.3 we already encountered this procedure, which is based on least-squares fitting of the data to a polynomial, and uses so-called convoluting integers. This method is, in fact, quite old, and goes back to work by Sheppard (Proc. 5th... 

There are differences between regulatory authorities in procedures used to set milk withholding periods. USFDA/CVM requires use of at least 20 animals and analysis of milk samples for the marker residue in triplicate.If the product is authorized for mastitis treatment, it is assumed that no more than one-third of the milk is derived from treated cows. A regression line is fitted to the log residue concentration data for each cow, and then fitted lines are used to estimate the distribution of log residue concentrations at each sampling time. Between-animal variance and measurement error variability are estimated and used to calculate a tolerance limit at each time. The WhT is set as the first time at which the upper 95% confidence limit of the 99th percentile of residue concentrations is equal to or less than the MRL. 

Analytical redundancy relations are balance equations of effort or flow variables, in which unknown variables have been replaced by input variables and measured output variables and in which parameters are known. Evaluation of an ARR provides a residual that theoretically should be zero. In practice, however, the residual of an ARR is within certain error bounds as long as no faults occur during system operation. The value is not exactly zero over some time interval due to noise in measurement, parameter uncertainties, and numerical inaccuracies. If, however, the numerical value of a residual exceeds certain thresholds, then this is an indicator to a fault in one of the system s components. Noise in measured output variables may result in residual values indicating a fault that does not exist. Hence, measured data should pass appropriate filters before being used in ARRs. 

Functions on the sampled observations such as the sample mean, the sample standard deviation and many others (see Sect. 20.2.2) are called (sample) statistics. They are random variables, which have a probability distribution. Observations from subsequent samples vary due to at least three sources of variation deviations reflecting the variation in the population (e.g. variation of the true weights of the capsules in the batch), sampling error (deviations due to differences between subsequent samples) and measurement errors. 

At this point one has a compartmental model structure, a description of the measurement error, and a numerical value of the parameters together with the precision with which they can be estimated. It is now appropriate to address the optimal experiment design issue. The rationale of optimal experiment design is to act on design variables such as number of test input and outputs, form of test inputs, number of samples and sampling schedule, and measurement errors so as to maximize, according to some criterion, the precision with which the compartmental model parameters can be estimated [DiStefano, 1981 Carson et al, 1983 Landaw and DiStefano, 1984 Walter and Pronzato, 1990]. 

After an initial model has been developed, one should check the autocorrelation of the residuals (error between predicted variable and measured variable). For a good model, the autocorrelation of the residuals should be small. In addition, there should be no cross correlation between the process input and residuals. If either the autocorrelation or cross correlation of the residuals is not small, the model stmcture should be expanded, usually resulting in a more complex model. 

In Section F.l, we considered the probability of one or more events occurring. The same probability concepts are also applicable for random variables such as temperatures or chemical compositions. For example, the product composition of a process could exhibit random fluctuations for several reasons, including feed disturbances and measurement errors. A temperature measurement could exhibit random variations due to turbulence near the sensor. Probability analysis can provide useful characterizations of such random phenomena. 

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