Statistics measurement scales

There are many reasons why it is important to understand which type of measurement scale is being used to describe system inputs and outputs. One reason is that most statistical techniques are not applicable to data arising from all four types of measurement scales the majority of techniques are applicable to data from interval or ratio scales. 

Methodological and statistical problems in the construction of composite measurement scales a survey of six medical and epidemiological journals. Statistics in Medicine 14 331-345... 

Once the data have been recorded, there are a host of statistical measures that provide criteria for evaluating the quality of the data. These serve as important guides to possible weaknesses in the data set, give estimates of the useful resolution of the data, and yield quantitative measures of precision. These are normally provided as a matter of course by the data-processing programs that correct, scale, and merge the measured intensities into a comprehensive set. 

The precision of the analytical method depends on the stability of the instrument response for a given amount of analyte. In principle, a random dispersion of instrmnent signal at a given concentration transforms into dispersion on the measurement scale as schematically shown (Figure 14-2). The detailed statistical aspects of calibration are rather complex,but in the following, some approximate rela-... 

If the measurement scale is not linear, arithmetic means may. give a false value. For example, if three media had pH. values 6, 7 and 8, the appropriate mean pH is not 7 because the pH scale is logarithmic. The definition of pH is —logi0[H], where IH] is expressed in mol L"1 ( molar ) therefore, to obtain the true mean, convert data into [H] values (i.e. put them on,a linear scale) by calculating 10( pHvalue) as shown. Now calculate the mean of these values and convert the answer back into pH units. Thus, the appropriate answer is pH 6.43 rather than 7. Note that a similar procedure is necessary when calculating statistics of dispersion in " such cases, so you will find these almost certainly asymmetric about the mean. 

We assume that the errors in the measurements are statistically independent, scaled by the weights, u , in such a way that they have equal variance ((T ) and come from a Gaussian distribution. In case of these reasonable assumptions weighted least squares coincides with the maximum likelihood estimate. The (weighted) experimental errors of the measurements are given by Y 0)y as in (6.2). This means that the covariance matrix of the experimental errors is given by ... 

Many texts and monographs are devoted to careful discussions of turbulent flow fields, but only those aspe of turbulent flow that bear on the transport of particular species are discussed here. Davies book is a particularly good presentation of heat, mass, and momentum transport in tuibulent fluids. The turbulent motion of a fluid is characterized by relatively rapid, irregular changes in velocity at any one point in the fluid and by limited correlation between the velocities at different spatial positions at any time. The random nature of the turbulent fluctuations imposed on the flow requires description using various statistical measures. Two characteristics of the flow field are the intensity" and the scale" of turbulence. The intensity refers to the magnitude of the velocity changes at any particular poim with respect to the mean velocity the scale of turbulence refers to the correlation between velocities measured some distance apart or to die correlation between velocities at the same point at two different times. 

How to select the optimum measurement scale L, reflecting a balance between counting statistics, A[Pg.1608]

Once again, it is desirable to have a more statistical measure of a sample s potential to be an outher than simple visual inspection. For score clusters, it is possible to use a measure of the Mahalanobis distance (Figs. 16 and 17). This is calculated as the distance of the potential outlier sample point as measured from the mean of all the remaining points in the cluster. The distance is scaled for the range of variation in the cluster in all dimensions and then assigns a probabihty weight to the sample in terms of standard deviation. Any sample that Ues outside of 3 standard deviations from the mean can be considered suspicious. 

Finally, the hit quality index does not provide any absolute measure of the probability that the sample actually is the same as the library sample. The arbitrary scale of the hit quality values (0-1) does not give a very good statistical measure of the similarity of the spectra. In short, using only a single training spectrum to represent all possible samples in the future does not give the analyst any statistical assurance that the spectra are truly the same or different. It provides only a relative measure for all the library samples. For anyone who has tried simple library search techniques for spectrally similar samples, this result is all too obvious. 

The Scaled Deviance (SD) is defined as the likelihood ratio test statistic measuring twice the difference between the log likelihoods of the studied model and the full or saturated model. The full model has as maity parameters as there are observations so that the model fits the data perfectly. Therefore, the full model, which possesses the maximum log likelihood achievable under the given data, provides a baseline for assessing the goodness-of-fit of an intermediate model with p parameters [MCC 89],... 

Validity refers to whether the smvey instrument measures what it claims to measure. There are three basic types of validity for a measurement scale, each with particular experimental and statistical methodologies for evaluation. These are content validity (do relevant experts agree that the survey appears to measure what it is supposed to measme ), criterion validity (can scores from the survey be used to predict individual behavior or performance ), and construct validity (are the relationships found with the survey consistent with relevant theory and research ). 

Flavor Intensity. In most sensory tests, a person is asked to associate a name or a number with his perceptions of a substance he sniffed or tasted. The set from which these names or numbers are chosen is called a scale. The four general types of scales are nominal, ordinal, interval, and ratio (17). Each has different properties and allowable statistics (4,14). The measurement of flavor intensity, unlike the evaluation of quaUty, requires an ordered scale, the simplest of which is an ordinal scale. 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

Measures of potency are log normally distributed. Only p-scale values (i.e., pEC50) should be used for statistical tests. 

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