Statistics data points

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

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). 

Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products.Two types of control charts are commonly used in quality assurance a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line representing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measurement process. The position of the data points about the boundary lines determines whether the system is in statistical control. 

When a system is in statistical control, the data points should be randomly distributed about the center line. The presence of an unlikely pattern in the data is another indication that a system is no longer in statistical control. > Thus,... 

The degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

Statistical quaUty control charts of variables are plots of measurement data, preferably the average result of repHcate analyses, vs time (Fig. 2). Time is often represented by the sequence of batches or analyses. The average of all the data points and the upper and lower control limits are drawn on the chart. The control limits are closely approximated by the sum of the grand average plus for the upper control limit, or minus for the lower control limit, three times the standard deviation. 

Subtracting y —y) from y — T) gives (T — y) which is that portion of the deviation of any data point from the mean which is explained by the correlation. The coefficient of determination, y, a statistical parameter which varies from 0.0 to 1.0, is defined as ... 

Assumption 4 There is no systematic association of the random error for any one data point with the random error for any other data point. Statistically this is expressed as Correlation ( , 8 ) = 0. For u, V = 1, 2,. . . n, u V,... 

Equipment failure rate data points carry varying degrees of uncertainty expressed by two measures, confidence and tolerance. Confidence, the statistical measurement of uncertainty, expresses how well the experimentally measured parameter represents the actual parameter. Confidence in the data increases as the sample size is increased. 

However, the data that are contributed to a generic failure rate data base are rarely for identical equipment and may represent many different circumstances. Generic data must be chosen carefully because aggregating generic and plant-specific data may not improve the statistical uncertainty associated with the final data point, owing to change in tolerance. 

Aggregation The statistical combination of several data points to form a single data point and confidence interval. 

Data point A numerical estimate of equipment reliability as a mean or median value of a statistical distribution of the equipment s failure rate or probability. 

Figure 10.24a and the allosteric model in Figure 10.24b. The circled data points were changed very slightly to cause an F-test to prefer either model for each respective model, illustrating the fallacy of relying on computer fitting of data and statistical tests to determine molecular mechanism. As discussed in Chapter 7, what is required to delineate orthosteric versus allosteric...

FIGURE 11.13 A collection of 10 responses (ordinates) to a compound resulting from exposure of a biological preparation to 10 concentrations of the compound (abscissae, log scale). The dotted line indicates the mean total response of all of the concentrations. The sigmoidal curve indicates the best fit of a four-parameter logistic function to the data points. The data were fit to Emax = 5.2, n = 1, EC5o = 0.4 pM, and basal = 0.3. The value for F is 9.1, df=6, 10. This shows that the fit to the complex model is statistically preferred (the fit to the sigmoidal curve is indicated). 

SSqs (clfs) is number of data points minus the common max, common slope, and four fitted values for ECso- Thus, clfs = 24 — 6=18. The value for F for comparison of the simple model (common maximum and slope) to the complex model (individual maxima and slopes) for the data shown in Figure 11.14 is F = 2.4. To be significant at the 95% level of confidence (5% chance that this F actually is not significant), the value of F for df =12, 18 needs to be >2.6. Therefore, since F is less than this value there is no statistical validation for usage of the most complex model. The data should then be fit to a family of curves of common maximum and slope and the individual ECS0 values used to calculate values of DR. 

FIGURE 11.16 Control dose-response curve and curve obtained in the presence of a low concentration of antagonist. Panel a data points. Panel b data fit to a single dose-response curve. SSqs = 0.0377. Panel c data fit to two parallel dose-response curves of common maximum. SSqc = 0.0172. Calculation of F indicates that a statistically significant improvement in the fit was obtained by using the complex model (two curves F = 4.17, df=7, 9). Therefore, the data indicate that the antagonist had an effect at this concentration. 

We can see, in Figure 51, that the spectra form a spherical cloud in this 3-dimensional subset of the absorbance data space- In other words, this data is isotropic. No matter in which direction we look, we will see no significant (in the statistical sense of the word) difference in the distribution of the data points. If we were able to show the plot for all 10 dimensions, we would see a 10-dimensional hyperspherical cloud that is isotropic within the spherical distribution of points. 

Experience has shown that correlations of good precision are those for which SD/RMS. 1, where SD is the root mean square of the deviations and RMS is the root mean square of the data Pfs. SD is a measure equal to, or approaching in the limit, the standard deviation in parameter predetermined statistics, where a large number of data points determine a small number of parameters. In a few series, RMS is so small that even though SD appears acceptable, / values do exceed. 1. Such sets are of little significance pro or con. Evidence has been presented (2p) that this simple / measure of statistical precision is more trustworthy in measuring the precision of structure-reactivity correlations than is the more conventional correlation coefficient. 

For the 150 data points the squared correlation coefficient was 0.955 with a root mean square error of 0.0062. The graph of predicted versus actual observed kWh/lb along with the summary of fit statistics and parameter estimates is shown in Figure 16.8. 

Because the number of data points is low, many of the statistical techniques that are today being discussed in the literature caimot be used. While this is true for the vast majority of control work that is being done in industrial labs, where acceptability and ruggedness of an evaluation scheme are major concerns, this need not be so in R D situations or exploratory or optimization work, where statisticians could well be involved. For products going to clinical trials or the market, the liability question automatically enforces the tried-and-true sort of solution that can at least be made palatable to lawyers on account of the reams of precedents, even if they do not understand the math involved. 

For an example of a control chart see Fig. 1.31 and Sections 4.1 and 4.8. Control charts have a grave weakness the number of available data points must be relatively high in order to be able to claim statistical control . As is often the case in this age of increasingly shorter product life cyeles, decisions will have to be made on the basis of a few batch release measurements the link between them and the more numerous in-process controls is not necessarily straight-forward, especially if IPC uses simple tests (e.g. absorption, conductivity) and release tests are complex (e.g. HPLC, crystal size). 

Restrictions (1) and (2) are of no practical relevance, at least as far as the slope and the intercept are concerned, when all data points closely fit a straight line the other statistical indicators are influenced, however. 

Figure 3.9. Demonstration of ruggedness. Ten series of data points were simulated that all are statistically similar to those given in Table 4.5. (See program SIMILAR.) A quadratic parabola was fitted to each set and plotted. The width of the resulting band shows in what ar-range the regression is reliable, higher where the band is narrow, and lower where it is wide. The bars depict the data spread for the ten statistically similar synthetic data sets.

Each data point is assigned to the appropriate cell, where the following statistics are calculated number of values, Zmean, and s. ... 

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