Ordinal scale data

Again measurement is involved, but the characteristic being assessed is often more subjective in nature. It is all well and good to measure nice neat objective things like blood pressure or temperature, but it is also a good idea to get the patient s angle on 

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Figure 1.2 Ordinal scale data - scores for patient responses to treatment...

In Chapter 5 we saw how the calculation of the 95 per cent Cl for the mean can lead to nonsensical results if the data deviate severely from a normal distribution. This requirement for a normal distribution also applies to the t- tests, analyses of variance and correlation that we met in Chapters 6-14. These procedures are termed parametric methods and are quite robust, so moderate non-normality does little damage, but in more extreme cases, some pretty dumb conclusions can emerge. This chapter looks at steps that can be taken to allow the analysis of seriously non-normal data and also of ordinal scale data. 

It is theoretically possible for ordinal scale data to approximate a normal distribution, but marked non-normality is all too common. 

There is no absolute case that parametric methods cannot be used with ordinal scale data. If the scoring system allows a reasonably wide range of possible values and if these happen to approximate a normal distribution, parametric methods could be used. However the reality of working with ordinal data is ... 

An example of dealing with ordinal scale data - applying the Mann-Whitney test to the effectiveness of an analgesic... 

An application of Eq. (19) is shown in Fig. 4, which gives the solubility of solid naphthalene in compressed ethylene at three temperatures slightly above the critical temperature of ethylene. The curves were calculated from the equilibrium relation given in Eq. (12). Also shown are the experimental solubility data of Diepen and Scheffer (D4, D5) and calculated results based on the ideal-gas assumption (ordinate scale is logarithmic and it is evident that very large errors are incurred when corrections for gas-phase nonideality are neglected. 

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 ... 

Relative primary productivity, POC fluxes at 105 and 3000 m, and POC sediment accumulation rates versus latitude in the central equatorial Pacific Ocean. Data are normalized to the maximum value in each transect. Survey 1 was conducted during February-March 1992 under El Nino conditions and Survey 2 from August to September 1992 under non-El Nino conditions at longitudes ranging from 135 to 140°W. Ordinate scale is reset to 1.0 at each maximum, and the absolute magnitude (mmolCm ij-i) of each parameter is given next to its maximum. Source-. From Flernes, P. J., et al. (2001). Deep-Sea Research I 48, 1999-2023. 

Figure 6. ROA and Raman spectnim of (+H3 )-methylcyclohexanone in the skeletal region, as a neat liquid. The ROA spectnim has been redrawn on a linear ordinate scale from the data in ref. II. (Reproduced with permission from ref. 77. Copyright 1984 American Chemical Society.)...

Fig. 2 Same as Fig. 1 but for the C-H... O contact. In descending order of peak height acetylenic CH (wavy line because of too few data), aromatic CH, aliphatic CH. Note the large difference between ordinate scales in this figure and in Fig. 1... 

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). 

Figure 14. Photoionization cross section of NH3 and apparent photoionization cross section for NH produced by reaction of NH3+ with NH3. Variation of reaction cross section as function of vibrational state of reactant NH3+ ion was determined by comparing relative step heights of curves for NH3+ and NH after ordinate scales of both curves were adjusted so that data points of first plateau at about 10.2 eV coincide. Ratio of a pair of corresponding step heights is then proportional to ratio of cross section for reaction of vibrationally excited NH3+ to that for NH3+ in its ground vibrational state. Step heights used to determine relative cross section for reaction of NH3+ with v = 5 are shown. Step ratio NH//NH3+ decreases with increasing e.85 ...

Figure 14. Potentials for He (2 S)+He derived from data of Fig. 13. Apparent discontinuity at 50 meV results from change of ordinate scale. Dashed lines are GVB ab initio results due to Guberman and Goddard,84 Collision energies are given at right. Potentials are tabulated in Table III,...

Figure 10 presents the concn-time profiles at 212°C of three products for a constant initial wt of RDX in reactors of different volume. Figure 11 illustrates the data for the N02 in Fig 10 on an absolute basis and on an expanded ordinate scale. The profiles of the other gaseous products are not shown for the sake of clarity, but all follow the same general pattern as evidenced by N20 and C02... 

Figure 6.3 Absolute noble gas concentrations in selected mantle-derived samples. (Data are from Table 6.1. Air data are from Table 1.3.) The ordinate scale is in cm3 STP/g.

Fig. 22. Plot of log kq vs. AG° for the electron transfer quenching of Cr(bpy)3+ ( , ), Ru(bpy>3+ ( ) and Ir(5,6-Me2phen)2Cl2 ( , °) by aromatic quenchers (full points), and aliphatic amines (empty points) 54X The insert reports the data for the quenching of Ru(bpy)3+ and Ru(4,4 -Me2bpy)3 by Cr(bpy) +, Ru(bpy) +, Os(bpy) + and Ru(4,4 -Me2bpy)3+ 87 Note the expanded ordinate scale in the insert...

Another important feature of mathematical modeling techniques is the nature of the response data that they are capable of handling. Some methods are designed to work with data that are measured on a nominal or ordinal scale this means the results are divided into two or more classes that may bear some relation to one another. Male and female, dead and alive, and aromatic and nonaromatic, are all classifications (dichotomous in this case) based on a nominal scale. Toxic, slightly toxic, and non-toxic are classifications based on an ordinal scale since they can be written as toxic > slightly toxic > non-toxic. The rest of this section is divided into three parts methods that deal with classified responses, methods that handle continuous data, and artificial neural networks that can be used for both. 

Fig. 12 Calculated (solid line) site-site goo(r), pair correlation function of water (many body contribution included) in comparison with the corresponding experimental (dashed line) data. The ordinate scale refers to Run 1. Each of the remaining plots have been shifted upwards by an arbitrary unit, (a) Ref. 59.

Ordinal scale - measurements on a scale without defined intervals. Data are discontinuous. 

A common view is that, when planning any experiment where data will be collected on an ordinal scale, we may as well reconcile ourselves to the use of non-parametric methods from the outset. 

Figure 6. Compilation of TG/DTA responses for the crystallization of the amorphous alloy PdjjZr which was prepared by the melt-spinning technique. The red data were obtained in hydrogen, the blue data in oxygen. The responses in hydrogen are enlarged by a factor of 10, the enlarged weight curve by a factor of 100 relative top the ordinate scales. A SEIKO instrument was used and gas flows of lOOmlmin-1 were adjusted for sample masses of ca. 4 mg.

Of special interest are the data for the magic analyzer orientation —30° suppressing the isotropic scattering, measured in the same experimental runs and for identical conditions as the data of Fig. 6a. The results are presented in Fig. 6b. The ordinate scale refers to the same units as in Fig. 6a, while the time scale of the abscissa is stretched. The measured transient consists of two features a pronounced signal overshoot around to = 0 due to nonresonant scattering and an exponential tail, obviously representing the anisotropic contribution. The data represent novel evidence for the exponential time dependence of the reorientational autocorrelation function or [see Equation (3)] ... 

An experimental demonstration is depicted in Fig. 8 for the v mode of neat bromochloromethane (45). Part of the data of Fig. 5b are replotted in Fig. 8a with an enlarged ordinate scale. We recall that the anisotropic component (dotted line in Fig. 5b) is negligible for the special case here,... 

Figure 17 Transient spectra of a 1.2 M ethanol and CCU mixture taken at room temperature and excitation within the CH-stretching modes at 2974 cm-1. The data are shown at three different delay times of 0 ps (a), 4 ps (b), and 8 ps (c). The conventional absorption of the sample is shown for comparison in (c), right-hand ordinate scale and dash-dotted line. Measured data of the parallel signal calculated lines.

One solution is to plot the logarithms of particle diameter on the abscissa instead of the diameters themselves. This spreads out the presentation of distribution data so that a much broader range of particle sizes can be visualized. However, to maintain the relationship that the area between two particle size intervals is proportional to the total number of particles present, the ordinate scale must be altered. This is done by dividing the number of particles in each interval by the difference in the logarithms of the largest and smallest particle sizes of that interval, or, in mathematical terms,... 

Even superficial inspection of the three sets of data reported in Fig. 11 (note that the ordinate scales of parts (a) and (c) of Fig. 11 differ by one order of magnitude) demonstrates the dramatic decrease of the surface coverages of both hydride and hydroxyl groups formed upon H2 adsorption on samples having decreasing surface areas. The reduction of the density of surface defects as the surface area decreases is thus evident. On the smoke MgO sample, only the bands characterizing molecularly... 

Fig. 2. Titration curve of /3-lactoglobulin at ionic strength 0.15 and 25°C. The alkaline branch is time-dependent (cf. Fig. 12), and the figure shows data extrapolated to the time of mixing t = 0) and to infinite time. The figure also shows how the curve is divided into acid, neutral, and alkaline regions. Three ordinate scales with different reference points are given. (Data of Y. Nozaki.)...

Figure 5.11. Comparison of rain and fog analyses. The data are from samples taken in the proximity of Zurich, Switzerland. Note the difference in ordinate scale. Rain is characterized by 0.05-0.5 meq liter , while the concentration of ions in fog (lower liquid water content than with rain) is larger by one to two orders of magnitude. (From Sigg and Stumm, 1994.)...

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