Distribution of replicates

Figure 1.1 Normal distribution of replicate measurements about a mean. 

Replicates don t all have to be carried out at the center point. Using the experimental design of Figure 13.2 as a basis. Figures 13.9 and 13.10 show the effects of different distributions of replicates. 

Spann, TR, Moir, R.D., Goldman, A.E., Stick, R. and Goldman, R.D. (1997) Dismption of nuclear lamin organization alters the distribution of replication factors and inhibits DNA synthesis. J. Cell Biol. 136, 1201-1212. 

One possible option is to adopt a statistical description of the kinetic parameters and to ask how likely it is for the quasi-species to be localized about the wild type. This undertaking requires an analysis beyond the second order in perturbation theory since a distant mutant with a selective value very close to that of the wild type may jeopardize the stability of the latter in the population. We were however encouraged by the progress that had been made with a problem of similar difficulty in the very different area of electron or spin localization in disordered solids. Indeed, it turns out that an expression of the form of Eqn. (III.5) may be obtained, with an explicit expression for the superiority parameter Oq, dependent on the distribution of replication rates but not on any average involving population variables. 

In comparison with Eqn. (III.4) the result of the statistical analysis is to provide an expression for the effective superiority parameter er ff of the wild type in terms of the distribution of replication rates of its mutants. 

From experience with many determinations, we find that the distribution of replicate data from most quantitative analytical experiments approaches that of the Gaussian curve shown in Figure 6-2c. As an example, consider the data in the spreadsheet in Table 6-2 for the calibration of a 10-mL pipet. In this experiment a small flask and stopper were weighed. Ten milliliters of water were transferred to the flask with the pipet, and the flask was stoppered. The flask, the stopper, and the water were weighed again. The temperature of the water was also measured to determine its density. The mass of the water was then calculated by taking the difference between the two masses. The mass of water divided by its density is the volume delivered by the pipet. The experiment was repeated 50 times. 

Systematic errors have a deliniie value and an assignable cause and are of the same magnitude for replicate measurements made in the same way, Systematic errors lead to bias in measurement results. Bias is illustrated by the two curves in Hgure a -2, which show the frequency distribution of replicate results in the analysis of identical samples by two methods that have random errors of identical size. Method A has no bias so that the mean fiA 5> be true value. Method B has a bias that is given by... 

Y,i,k (Equation [8.26]) is the standard deviation of k values of Yj predicted by Equation [8.19a] for a known Xj and Vy i is the variance of the normal (Gaussian) distribution of replicate determinations assumed to underlie the small data sets usually obtained in analytical practice, as predicted from Equation [8.19a] for a chosen value of Xj. Clearly (sy j t) = Vy j = (sy j) in the limit k - 00, i.e. Vy i pjjji is the variance of the normal distribution assumed to describe the determinations of Yj. Later the quantity s(Yj) is used to denote a simple experimental determination (not prediction as for Sy j ) of the standard deviation of a set of replicate determinations of Yj for a fixed Xj in the calibration experiments (Equation [8.2] with Y replacing x). 

When an analyst performs a single analysis on a sample, the difference between the experimentally determined value and the expected value is influenced by three sources of error random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to determine the effect of random error on the analysis. The position of the distribution relative to the sample s true value, p, is determined both by systematic errors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. 

Exposure and latent image formation. The sensitized photoreceptor is exposed to a light and dark image pattern in the light areas the surface potential of the photoconductor is reduced due to a photoconductive discharge. Since current can only flow perpendiculai to the surface, this step produces an electrostatic-potential distribution which replicates the pattern of the image. 

When an experimental value is obtained numerous times, the individual values will symmetrically cluster around the mean value with a scatter that depends on the number of replications made. If a very large number of replications are made (i.e., >2,000), the distribution of the values will take on the form of a Gaussian curve. It is useful to examine some of the features of this curve since it forms the basis of a large portion of the statistical tools used in this chapter. The Gaussian curve for a particular population of N values (denoted x ) will be centered along the abscissal axis on the mean value where the mean (r ) is given by... 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

The standard deviation s is the square root of the variance graphically, it is the horizontal distance from the mean to the point of inflection of the distribution curve. The standard deviation is thus an experimental measure of precision the larger s is, the flatter the distribution curve, the greater the range of. replicate analytical results, and the Jess precise the method. In Figure 10-1, Method 1 is less precise but more nearly accurate than Method 2. In general, one hopes that a and. r will coincide, and that 5 will be small, but this happy state of affairs need not exist. 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

Kessler RC, Berglund P, Dernier O, et al. Lifetime prevalence and age-of-onset distributions of the DSM-IV disorders in the National Comorbidity Survey Replication. Arch Gen Psychiatry 2005 62 593-602. 

Random deviations (errors) of repeated measurements manifest themselves as a distribution of the results around the mean of the sample where the variation is randomly distributed to higher and lower values. The expected mean of all the deviations within a measuring series is zero. Random deviations characterize the reliability of measurements and therefore their precision. They are estimated from the results of replicates. If relevant, it is distinguished in repeatability and reproducibility (see Sect. 7.1)... 

In the given formula, o av is the quantile of the respective f-distribution (the degrees of freedom v relates to the number of replicates by which srepeat has been estimated)... 

Additional applications of BSOCOES and sulfo-BSOCOES include investigations of the cellular and subcellular distribution of the type II vasopressin receptor (Fenton et al., 2007), TNF-alpha (Grinberg et al., 2005), and studying mechanisms in the control of plasmid replication (Das et al., 2005). 

The analytical variance can be determined by carrying out replicate analysis of samples that are known to be homogeneous. You can then determine the total variance. To do this, take a minimum of seven laboratory samples and analyse each of them (note that Sample characterizes the uncertainty associated with producing the laboratory sample, whereas sanalysis w h take into account any sample treatment required in the laboratory to obtain the test sample). Calculate the variance of the results obtained. This represents stQtal as it includes the variation in results due to the analytical process, plus any additional variation due to the sampling procedures used to produce the laboratory samples and the distribution of the analyte in the bulk material. 

Bias is a measure of trueness . It tells us how close the mean of a set of measurement results is to an assumed true value. Precision, on the other hand, is a measure of the spread or dispersion of a set of results. Precision applies to a set of replicate measurements and tells us how the individual members of that set are distributed about the calculated mean value, regardless of where this mean value lies with respect to the true value. 

Factor the spec into model types. If some parts of the pre or postcondition seem to be particularly associated with types in the model, write them there as effects that you use in the system operation specs (Section 3.8.3, Effects factor common postconditions, on page 150). This helps avoid replication in the spec, and allows it to be polymorphic. Although the type model is not necessarily a blueprint for its design, this distribution of concerns obviously looks forward to the distribution of responsibilities in design. effect LibraryManagementSystem Book return () pre borroweronull post etc... 

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