Ordinal measurement scales

Figure 1.20. Monte Carlo simulation of 25 normally distributed measurements raw data are depicted in panel A, the derived means Xmean CL(Xmean) in B, and the standard deviation % + CL( t) in C. Notice that the mean and/or the standard deviation can be statistically different from the expected values, for instance in the range 23 < n < 25 in this example. The ordinates are scaled in units of la. 

Four types of measurement scale can be used for assigning values to varying amounts of a property associated with a system input or system output [Summers, Peters, and Armstrong (1977)]. In order of increasing informing power, they are nominal, ordinal, interval, and ratio scales. The characteristics at determine a measurement scale s level of sophistication are name, order, distance, and origin. The characteristics of the four types of measurement scale are shown in Table 1.3. The nominal scale possesses only one of these characteristics the ratio scale possess all four characteristics. 

Indicate which type of measurement scale (nominal, ordinal, interval, or ratio) is usually used for the following characteristics time, mass, library holdings, gender, type of heart attack, cholesterol level as measured by a clinical chemical laboratory, cholesterol level as reported by a doctor to a patient, pipet volume, and leaves on a plant. 

In Section 8.3.1 we introduced the idea of a simple ordinal rating scale such as the 4-point scale for a migraine headache absent, mild, moderate, severe. Another simple approach to measurement is the so-called visual analogue scale (VAS). The VAS is simply a line - normally 100 mm long - the ends of which are associated with descriptions of opposite extremes of the disease. 

In some diseases a simple ordinal scale or a VAS scale cannot describe the full spectrum of the disease. There are many examples of this including depression and erectile dysfunction. Measurement in such circumstances involves the use of multiple ordinal rating scales, often termed items. A patient is scored on each item and the summation of the scores on the individual items represents an overall assessment of the severity of the patient s disease status at the time of measurement. Considerable amoimts of work have to be done to ensure the vahdity of these complex scales, including investigations of their reprodu-cibihty and sensitivity to measuring treatment effects. It may also be important in international trials to assess to what extent there is cross-cultural imiformity in the use and imderstand-ing of the scales. Complex statistical techniques such as principal components analysis and factor analysis are used as part of this process and one of the issues that need to be addressed is whether the individual items should be given equal weighting. 

Back in Chapter 1, data were described as interval (measurements on a regular scale), ordinal (measurements on a scale of undefined steps) and nominal (classifications). We have dealt extensively with two of these, but ordinal data have thus far been ignored. 

Fig. 12, Liapunov exponents of the unstable periodic orbits labelled — + + +, where each period contains one slow Coulombic oscillation of the farther electron and n Coulombic oscillations of the closer electron in between two crossings of the line ri = T2 in coordinate space. The ordinate measures the products n2 on a linear scale and the abscissa measures n on a logarithmic scale. The solid dots show the results for s-wave helium (Eq. (16)) [55], the open circles show the corresponding results for collinear helium (Eq. (15)) [52]. The linear behaviour for large n illustrates the proportionality of /l to (log n)/n. (Results are for charge Z = 2 in the Hamiltonian, Eq. (14))...

This scale is best defined as one in which an ordering of values can be assigned. Examples of data from clinical studies measured on an ordinal scale include severity of an adverse event classified as mild, moderate, or severe age categorized as < 65, 65-70, 71-75, and > 75 years. The ordinal nature of the measurement scales means that we can say that a mild headache is less severe than a moderate headache, which is less severe than a severe headache. However, we cannot say that the difference between mild and moderate is the same as the difference between moderate and severe. 

The superpositioning of experimental and theoretical curves to evaluate a characteristic time is reminiscent of the time-tefnperature superpositioning described in Sec. 4.10. This parallel is even more apparent if the theoretical curve is drawn on a logarithmic scale, in which case the distance by which the curve has to be shifted measures log r. Note that the limiting values of the ordinate in Fig. 6.6 correspond to the limits described in Eqs. (6.46) and (6.47). Because this method effectively averages over both the buildup and the decay phases of radical concentration, it affords an experimentally less demanding method for the determination of r than alternative methods which utilize either the buildup or the decay portions of the non-stationary-state free-radical concentration. 

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. 

The conventional control chart is a graph having a time axis (abscissa) consisting of a simple raster, such as that provided by graph or ruled stationary paper, and a measurement axis (ordinate) scaled to provide six to eight standard deviations centered on the process mean. Overall standard deviations are used that include the variability of the process and the analytical uncertainty. (See Fig. 1.8.) Two limits are incorporated the outer set of limits corresponds to the process specifications and the inner one to warning or action levels for in-house use. Control charts are plotted for two types of data ... 

Figure 4.48. The calculated standard deviation and its upper CL. A series of 10 measurements was simulated, bottom panel), with the newest addition at each step given in bold. The corresponding SD is given by the thick line in the top panel, and the 80. .. 97.5% CLy by thin lines. Notice that point 5, which is high, drives the SD up from = 0.9 to = 1.5 (E(a) = 1) the 95% CL is at 2.38 cr, respectively 3.6. The ordinates are both scaled in units of a. This depiction, for just one level of p, is part of the display of program CONVERGE.

Fig. 5.—The partial degradation of decamethylene adipate polyester with small percentages of decamethylene glycol (experiments 8, 13, and 17), or with lauryl alcohol (experiment 19), at 109°C, catalyzed with 0.1 equivalent percent of p-toluenesulfonic acid. The fraction of added glycol, or alcohol, unassimilated has been calculated indirectly from melt viscosity measurements and is plotted on the logarithmic ordinate scale.2 ...

Table 32.1 describes 30 persons who have been observed to use one of four available therapeutic compounds for the treatment of one of three possible disorders. The four compounds in this measurement table are the benzodiazepine tranquillizers Clonazepam (C), Diazepam (D), Lorazepam (L) and Triazolam (T). The three disorders are anxiety (A), epilepsy (E) and sleep disturbance (S). In this example, both measurements (compounds and disorders) are defined on nominal scales. Measurements can also be defined on ordinal scales, or on interval and ratio scales in which case they need to be subdivided in discrete and non-overlapping categories. 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

Fig. 28. Time-resolved phosphorescence spectra of quinoxaline in durene host observed at 1.38 K and at (a) 30 msec, (b) 450 msec, and (c) 1500 msec after excitation cutoff. The ordinate scale is normalized with respect to the 0 - 0" band. The numbers shown in (c) represent the vibrational frequencies (in wavenumber unit) measured from the 0 - 0" band (21639 cm r). The arrows indicate the bands whose relative intensities are remarkably enhanced at later times after the excitation cutoff. (From Yamauchi and Azumi, Ref. >)...

Full factorial designs have been especially useful for describing the effects of qualitative factors, factors that are measured on nominal or ordinal scales. This environment of qualitative factors is where factorial designs originated. Because all possible factor combinations are investigated in a full design, the results using qualitative factors are essentially historical and have little, if any, predictive ability. 

Nonetheless, categorisation can be of benefit under some circumstances. In an earlier section we discussed the categorisation of number of cigarettes to a four-point ordinal scale, accepting that measures on the original scale may be subject to substantial error and misreporting the additional information contained in the number of cigarettes smoked is in a sense spurious precision. 

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

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