Errors, standard

When we use experimental data to estimate parameters in response surface models, we will carry on the experimental error to the estimated model parameters. This will be manifested in such a way that the values of the model parameters will not be precisely known, i.e. they will have a probability distribution. To illustrate this, a simple example with only one experimental variable will first be discussed. It will then be obvious how the principles can be applied to models with several variables. 

Example A common routine task in the laboratory is to establish a calibration curve, e.g. to calibrate the signal from a chromatographic detector to known concentrations of the sample. When the concentration range is not too wide, often a straight line will give a good calibration, i.e. 

It is a common experience, however, that if we repeat the calibration experiment, the estimated values of the intercept, b, and the slope, hj, will not be identical. If we then repeat the calibration experiment several times, we will find that the estimated values of the parameters will be distributed around an average value, hj, sometimes denoted (beta hat). The situation is depicted in Fig. 3.10. 

From a series of n calibration experiments it will be possible to compute both the average 

This standard deviation is called the standard error of the estimated parameter. If the estimated average value of the parameter and its standard error are known it will be possible to assess the probability that new estimates of the model parameters will measure the same properties, i.e. they should not be outside the expected range of random variation of the previously known estimates. This can be tested as follows  

Supposing a large number of random samples, each containing n individuals, were taken from a population with a normal or near normal distribution then, the standard deviation of this set is termed the standard error of the mean (S.E.) and the following relationship holds  

Nominal mean — Standard mean Standard error 

Student s t can be calculated and compared with values in published Statistical Tables [3,4] at the 5% and 1% levels of probability and, if the calculated t exceeds the published value at the 1 % level, then a real difference exists and appropriate action must be taken. 

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The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

In Eq. (12), SE is the standard error, c is the number of selected variables, p is the total number of variables (which can differ from c), and d is a smoothing parameter to be set by the user. As was mentioned above, there is a certain threshold beyond which an increase in the number of variables results in some decrease in the quality of modeling. In fact, the smoothing parameter reflects the user s guess of how much detail is to be modeled in the training set. 

In Eq. (14) "est stands for the calculated (estimated) response, exp" for the experimental one, and n and m are the numbers of objects in the training set and the test set respectively. CoSE stands for Compound Standard Error. As an option, one can employ several test sets, if needed. 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

SEM = standard error of the means. IGF-I = insulin-like growth factor I. 

This gives the standard errors in the fitted parameters a. 

Example 1 Sample Quantity for Composition Quality Control Testing An example is sampling for quality control of a 1,000 metric ton (VFg) trainload of-Ks in (9.4 mm) nominal top-size bentonite. The specification requires silica to be determined with an accuracy of plus or minus three percent for two standard errors (s.e.). With one s.e. of 1.5 percent, V is 0.000225 (one s.e. weight fraction of 0.015 squared). The problem to be solved is thus calculating weight of sample to determine sihca with the specified error variance. 

Example 2 Calculation of Error with Doubled Sample Weight Repeated measurements from a lot of anhydrous alumina for loss on ignition established test standard error of 0.15 percent for sample weight of 500 grams, noting V is the square of s.e. Calculation of variance V and s.e. for a 1000 gram sample is... 

Example 3 Calculating Sample Weight for Screen-Size Measurement Weight W of bulk sample for screen analysis is calculated by the Gayle model for percent retained on a specified screen with relative standard error s.e. in percent... 

Sample weight estimated in this example is for two standard errors of 2.5 percent (resulting in V of 1.56) for testing iron ore (hematite) retained on a f4-in screen. Estimate of G is 5.5 for 94.5 percent of weight passing. Particle weight... 

The same results were gained for Cr analysis in high alloyed steels. The error of linear calibration in this case is 0.28%. The application of theoretical corrections decreases this error to 0.07%. The standard error of the linear calibration on the base of the analytical pai ameter Ici.j,yipj,j,p is 0.23% and the application of the theoretical corrections in this case gives error 0.04%. 

The classification as a carcinogen need not apply to fibres with a length weighted geometric mean diameter less two standard errors greater than 6 pm Sodium dichromate Sodium dichromate dihydrate... 

Another reason for investing in error reduction is to conform with regulatory standards. Error reduction yields regulatory relief. 

Standard-abweichung, /. standard deviation, -einheit, /. standard unit, -fehler, m. standard error, -losung, /. standard solution, -wert, m. standard value. 

B. Standard Error-Back-Propagation Training Routine... 

The standard deviation gives the accuracy of prediction. If Y is related to one or more predictor variables, the error of prediction is reduced to the standard error of estimate S, (the standard deviation of the errors), where... 

FIGURE 8.15 Confirmation of initial hits in the HTS. Top panel shows the distribution of values from a single test concentration of a high-throughput screen. The criteria for activity and subsequent retest is all values >3 standard error units away from the mean (dotted line). The process of retesting will generate another distribution of values, half of which will be below the original criteria for activity. 

The estimate of variability for a sample mean is the standard error of the mean ... 

There are instances where deviations, as measured by the standard error, are scaled to the magnitude of the mean to yield the coefficient of variation. This is calculated by... 

So how does one infer that two samples come from different populations when only small samples are available The key is the discovery of the t-distribution by Gosset in 1908 (publishing under the pseudonym of Student) and development of the concept by Fisher in 1926. This revolutionary concept enables the estimation of ct ( standard deviation of the population) from values of standard errors of the mean and thus to estimate... 

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