Independency of Measurements

A basic requirement, in order that the above results mean and standard deviation are truly representative of the sampled population, is that the individual measurements should be independent of each other. Two general cases must be distinguished  

Samples are taken for classical off-line processing, e.g., a 10 ml aliquot is withdrawn from a reaction vessel every hour and measurements are conducted thereupon. 

The sensor is immersed in the reaction medium and continuously transmits values. 

In case (1) the different samples must be individually prepared. In the strictest interpretation of this rule, every factor that could conceivably contribute to the result needs to be checked for bias, i.e., solvents, reagents, calibrations, and instruments. That this is impractical is immediately apparent, especially because many potential influences are eliminated by careful exper- 

In other words, those factors and operations that contribute the most toward the total variance (see additivity of variances, next section) need to be individually repeated for two measurements on the same sample to be independent. Provided the two samples are taken with a sufficiently long delay between them, they can be regarded as giving independent information on the examined system. 

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The problem of determining K independently of measurement of Hq is the principal issue to be faced in establishing the Hq scale for a series of acidic solutions. What is done is to measure K for some base in aqueous solution where Hq pH. This base can than be used to find the Hq of a somewhat more acidic solution. The K of a second, somewhat weaker base is then determined in the more acidic solution. This second base can then be used to extend Hq into a still more acidic solution. The process is continued by using a... 

Note The concentration of cholesterol in the placebo group (n = 1893) was independent of measured compliance (Lipid Research Clinics Coronary Primary Prevention Trial, 1984). 

For instance, when more than one chain of links is maintained between the value in a sample and an SI unit, it might be possible to reduce the uncertainty even below that of the least uncertain single-path chain. That reduction would have to be carefully evaluated based not only on the uncertainties of all the links in the dual path, but especially on the independence of measurements in the alternative paths. Such a complex consideration is not recommended for protocols as here discussed... 

In the multivariate distribution, it is assumed that measurements of objects or aliquots of material (e.g., a single trial) produces vectors, x having a multivariate normal distribution. The measurements of the p variables in a single object, such as xT = x,xi>1,. .., x J will usually be correlated. In fact, this is expected to be the case. The measurements from different objects, however, are assumed to be independent. The independence of measurements from object to object or from trial to trial may not hold when an instrument drifts over time, as with sets of p wavelengths in a spectrum. Violation of the tentative assumption of independence can have a serious impact on the quality of statistical inferences. As a consequence of these assumptions, we can make the following statements about data sets that meet the above criteria ... 

Whereas most theorems of geometry are concerned with the concepts distance, angle and congruence, a smaller number of others are only concerned with the incidence of points and straight lines. This distinction differentiates between the common metrical properties of geometry and those, which are independent of measurement, and which reflect the characteristics of what became known as projective geometry. 

Irrespective of the measured positions of points A and B on the line, the correct condition of balance will always be described by assigning coordinates (fi, 2) to point P, where AB = ti + <2- Alternatively AB = fiti + /J,t2, with coordinates /iti, fj,t2) for point P. The coordinates of point P is independent of measure and conveniently formulated as ... 

If measurement uncertainties can be assumed to be bounded, then application of the triangle inequality may yield thresholds for parameter variations of ARR residuals that are independent of measurement uncertainties. For instance, let z be the predicted part of an output variable z of the diagnostic bond graph that controls a modulated sink of the incremental bond and let Az < be the bounded measurement uncertainty. Furthermore, let Ar be the variation of an ARR residual r that depends on z and its derivative. Then... 

The result is a threshold thr t) > thr t) that is independent of measurement uncertainties but that depends on the sampling rate of the measurement. In [14], Touati et al. have accounted for measurement uncertainties by adding modulated sources to an uncertain bond graph. 

Using the formula (11.2.S1) and the independence of measurement errors at r and at times up to t. i, the terms due to r. i-dt., thus to Gk., (11.2.53) will occur additively in the covariance matrix of at t. An assiduous reader can compute them explicitly. Clearly, also the variances computed for the estimated (integrals of the) mass flowrates will remain bounded and wrill not increase dramatically. 

For somewhat different reasons, Ramsey [1997] has argued that molecular shape does not exist independently of measurement and time-scale considerations. Multiple processes take place simultaneously on very different time scales. Shape is a response property rather than an intrinsic property.Microconstituents of substances are explanatory, but not intrinsic. What we see depends on how we look. 

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