Ratio Scale Measures Research Articles

Distributional Equivalence and Subcompositional Coherence in the Analysis of Contingency Tables, Ratio-Scale Measurements and Compositional Data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to ?spectral mapping?, a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Properties of and Algorithms for Fitting Three-Way Component Models with Offset Terms.

Prior to a three-way component analysis of a three-way data set, it is customary to preprocess the data by centering and/or rescaling them. Harshman and Lundy (1984) considered that three-way data actually consist of a three-way model part, which in fact pertains to ratio scale measurements, as well as additive "offset" terms that turn the ratio scale measurements into interval scale measurements. They mentioned that such offset terms might be estimated by incorporating additional components in the model, but discarded this idea in favor of an approach to remove such terms from the model by means of centering. Then estimates for the three-way component model parameters are obtained by analyzing the centered data. In the present paper, the possibility of actually estimating the offset terms is taken up again. First, it is mentioned in which cases such offset terms can be estimated uniquely. Next, procedures are offered for estimating model parameters and offset parameters simultaneously, as well as successively (i.e., providing offset term estimates after the three-way model parameters have been estimated in the traditional way on the basis of the centered data). These procedures are provided for both the CANDECOMP/PARAFAC model and the Tucker3 model extended with offset terms. The successive and the simultaneous approaches for estimating model and offset parameters have been compared on the basis of simulated data. It was found that both procedures perform well when the fitted model captures at least all offset terms actually underlying the data. The simultaneous procedures performed slightly better than the successive procedures. If fewer offset terms are fitted than there are underlying the model, the results are considerably poorer, but in these cases the successive procedures performed better than the simultaneous ones. All in all, it can be concluded that the traditional approach for estimating model parameters can hardly be improved upon, and that offset terms can sufficiently well be estimated by the proposed successive approach, which is a simple extension of the traditional approach.

Ratio-Scale Measurement with Intransitivity or Incompleteness: The Homogeneous Case

In the homogeneous case of one-dimensional objects, we show that any relation that is positive and homothetic can be represented by a ratio-scale and a unique and constant biasing factor. This factor may favor or disfavor the preference for an object over another. In the first case, preferences are complete but not transitive and an object may be preferred even when its value is lower. In the second case, preferences are asymmetric and transitive but not negatively transitive and it may not be sufficient for an object to have a greater value to be preferred. In this manner, the biasing factor reflects the extent to which preferences may depart from a maximization process.

Rank reversals in multi-criteria decision analysis with statistical modelling of ratio-scale pairwise comparisons

In multi-criteria decision analysis, the overall performance of decision alternatives is evaluated with respect to several, generally conflicting decision criteria. One approach to perform the multi-criteria decision analysis is to use ratio-scale pairwise comparisons concerning the performance of decision alternatives and the importance of decision criteria. In this approach, a classical problem has been the phenomenon of rank reversals. In particular, when a new decision alternative is added to a decision problem, and while the assessments concerning the original decision alternatives remain unchanged, the new alternative may cause rank reversals between the utility estimates of the original decision alternatives. This paper studies the connections between rank reversals and the potential inconsistency of the utility assessments in the case of ratio-scale pairwise comparisons data. The analysis was carried out by recently developed statistical modelling techniques so that the inconsistency of the assessments was measured according to statistical estimation theory. Several type of decision problems were analysed and the results showed that rank reversals caused by inconsistency are natural and acceptable. On the other hand, rank reversals caused by the traditional arithmetic-mean aggregation rule are not in line with the ratio-scale measurement of utilities, whereas geometric-mean aggregation does not cause undesired rank reversals.

Rank-based modelling of preferences in multi-criteria decision making

In multi-criteria decision making, one approach to utilise mixed data consisting of both ordinal and ratio scale information is to consider ratio scale random samples fulfilling the existing ordinal data. However, some of the distributional assumptions proposed in the literature for the phase of transforming the ordinal data into ratio scale may lead to over-interpretation of the available information. In this paper, we propose a statistical model for overcoming these difficulties. In the cardinal part of the model, the ratio scale measurements are analysed by a regression model developed for pairwise comparisons data. The model for ordinal measurements is proposed by taking the regression model as a background model, and by assuming that instead of the ratio scale, only the ranks of the items are observed.

Deriving ratio-scale measures of sound quality from preference judgments

Deriving ratio-scale measures of sound quality from preference judgments

Measuring behavioral momentum

The metaphor of behavioral momentum proposes that when ongoing operant behavior is disrupted, changes in response rate are directly related to a force-like aspect of the disruptor and inversely proportional to behavioral mass. Several data sets suggest that differential resistance to change between the components of a multiple schedule satisfies the requirements of a ratio scale and is additive when different disruptors and different dimensions of reinforcement are combined. Differential resistance also provides a basis for scaling force in relation to rate of food presentation between components as a disruptor, and for scaling mass in relation to food rate within a component as a reinforcer. Preference in concurrent chains with terminal links identical to multiple-schedule components also meets the requirements of ratio-scale measurement, is additive when different dimensions of reinforcement are combined, and provides convergent measurement of behavioral mass.

An improved method for the rapid assessment of forest understorey light environments

Summary 1. The high spatial and temporal variability of forest understorey light environments requires lengthy and/or extensive sampling in order to characterize it by direct measurement. As this is often impractical, a number of surrogate measures have been developed that estimate light availability from assessments of forest canopy structure. 2. The subjective crown illumination index developed by Clark & Clark (1992) was compared with Garrison's (1949) moosehorn and two new methods: (i) the crown illumination ellipses method, which compares the size of canopy gaps with a series of standard area ellipses printed on a transparent screen; and (ii) the canopy‐scope that, like the moosehorn, uses an array of 25 dots printed on a transparent screen to assess canopy openness, but is more robust and portable, measuring the largest canopy gap visible from the point of measurement rather than canopy openness overhead. 3. The new measures were more highly correlated with canopy openness in the range 0–30%, measured from hemispherical photographs, than the crown illumination index, and showed lower levels of between‐observer variability. 4. The canopy‐scope has the potential to be widely used for the simple and rapid assessment of forest understorey light environments. It has the advantage of giving ratio scale measurements that can be used in parametric statistics. The crown illumination ellipses can be used to score the illumination of crowns that are above head height.

Seriation, stratigraphy, and index fossils: the backbone of archaeological dating

1. An Introduction to Time and Dating. Preliminary Considerations. Relative and Absolute Time. Continuous and Discontinuous Time. Direct and Indirect Dating. Scientific Dating. Time and Its Measurement. Nominal Scale Measurement. Ordinal Scale Measurement. Ordinal Scale Measurement. Interval Scale Measurement. Ratio Scale Measurement. Beyond Measurement Scale: Ideational and Empirical Units. 2. The Creation of Archaeological Types. Initial Considerations. Constructing Chronological Types. Chronological Types in Americanist Archaeology. Typological issues Begin to Take Shape. Typology in Retrospect. 3. Seriation I: Historical Continuity, Heritable Continuity, and Phyletic Seriation. What Is Seriation? Seriation in Americanist Archaeology. The Key Assumptions: Historical and Heritable Continuity. Continuity and the Study of Organisms. Tracing Lineages. Detecting Heritable Continuity. Historical Continuity, Heritable Continuity, and the Study of Artifacts. Culture Traits. W.M. Flinders Petrie and Artifacts from Egyptian Tombs. John Evans and Gold Coins from Britain. A. V. Kidder and Pottery from Pecos Pueblo. The Gladwin-Colton-Hargrave System. Projectile Point Evolution. 4. Seriation II: Frequency Seriation and Occurrence Seriation. The First Frequency Seriation. How Do Occurrence and Frequency Seriation Work? The Seriation Model. Requirements and Conditions of Seriation. Occurrence Seriation. Frequency Seriation. Meeting the Conditions of the Seriation Model. Temporal Resolution and Rates of Change. Absolute Seriation. A Final Note. 5. Superposition and Stratigraphy: Measuring Time Discontinuously. Strata, Stratigraphy, and Superposition. Stratigraphic Excavation. Stratigraphic Excavation in Historical Context. Early Stratigraphic Excavation. On the Eve of the 'Revolution'. The Real Revolution. What Was the Revolution? After the Revolution: Measuring Time with Strata. Measuring Time at Gatecliff Shelter, Nevada. The Final Proof Is in the Spade, But. 6. Cross Dating: The Use of Index Fossils. Folsom and Clovis Points. George C. Vaillant and the Mexican Formative. James A. Ford and the Lower Mississippi Valley. Measuring Time Discontinuously. 7. Final Thoughts on Archaeological Time: A Cash of Two Metaphysics. Measuring Time Continuously. Measuring Time Discontinuously. Concluding Remarks. References. Index.

A simulation-based evaluation of the approximate and the exact eigenvector methods employed in AHP

The Analytic Hierarchy Process (AHP) is a popular multi-criteria decision making method. It provides ratio-scale measurements of the priorities of elements in various levels of a hierarchy. These priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level. Saaty has suggested two methods, viz., the approximate eigenvector method (AEV) and the exact eigenvector method (EEV) for calculating the priorities. Research on AHP has led to conflicting views on the use of these two methods. This paper presents a simulation analysis to evaluate them. The simulation analysis uses the concept of approximating a continuous pairwise comparison (CPC) matrix by its closest discretized pairwise comparison (DPC) matrix. This exercise supports Saaty's theoretical statements and empirically concludes that the EEV method is to be preferred over the AEV method for the calculation of priority vectors.

A comparison of pain measurement characteristics of mechanical visual analogue and simple numerical rating scales

Numerical rating scales and mechanical visual analogue scales (M-VAS) were compared for their capacity to provide ratio scale measures of experimental pain. Separate estimates of experimental pain sensation intensity and pain unpleasantness were obtained by each method, as were estimates of clinical pain. Orofacial pain patients made numerical scale and VAS ratings in response to noxious thermal stimuli (45–51°C) applied for 5 sec to the forearm by a contact thermode. The derived stimulus-response function was well fit as a power function only in the case of sensory M-VAS. The power function derived from sensory M-VAS ratings predicted temperatures chosen as twice as intense as standard temperatures of 47°C and 48°C, thereby providing evidence for ratio scale characteristics of M-VAS. The stimulus-response function derived from sensory numerical ratings differed from that obtained with M-VAS and did not provide accurate predictions of temperatures perceived as twice intense as 47°C or 48°C. Both M-VAS and numerical rating scales produced reliably different stimulus response functions for pain sensation intensity as compared to pain unpleasantness and both provided consistent measures of experimental and clinical pain intensity. Finally, both mechanical and pencil-and-paper VAS produced very similiar stimulus-response functions. The ratio scale properties of M-VAS combined with its ease of administration and scoring in clinical settings offer the possibility of a simple yet powerful pain measurement technology in both research and health care settings.

A Ratio Scale Measurement of Conformity

Since the seminal work on conformity, there has been a steady stream of research attempting to isolate individual and situational variables that influence its expression. Most previous studies have employed either proportions of participants showing conformity or have used Likert scales to measure changes in conformity as a function of different variables. The current presentation describes the use of a measurement of conformity on a ratio scale as well as several variables that are easily manipulated to assess normative and information factors involved. Two successful attempts at validating the procedure are reported.

Structural-equation models of current drug use: are appropriate models so simple(x)?

The simplex and common-factor models of drug use were compared using maximum-likelihood estimation of latent variable structural models in two samples: a sample of 226 high school students, using ratio-scale measures of current drug use, and a sample of 310 industrial workers and 811 college students, using ordinal-scale measures of current drug use. Latent variables of alcohol, marijuana, enhancer hard drugs, and dampener hard drugs were specified in a series of structural models. Contrary to previous findings with cumulative drug-use data, the common-factor model provided a more acceptable representation of the observed current-use data than did the simplex model in both samples. In addition, the similarity of results across both of these samples supports recent contentions by Huba and Bentler (1982) that quantitatively measured variables are not necessarily superior to qualitative, ordinal indicators in latent variable models of drug use.

Numerical Types and Inspectional Types: Evaluating Shape Characterization Procedures

Projectile point data are used in testing and evaluating various numerical shape classification procedures. Inspectional point types play the role of a control population while clustering algorithms, attribute section, and standardization techniques are tested. The most effective numerical procedure identified for the classification of projectile point form is a combination of k-means analysis, ratio scale measurements between attributes, and standardization of interval measures to a mean of zero and standard deviation of one.

The validation of visual analogue scales as ratio scale measures for chronic and experimental pain

Visual analogue scales (VAS) of sensory intensity and affective magnitude were validated as ratio scale measures for both chronic and experimental pain. Chronic pain patients and healthy volunteers made VAS sensory and affective responses to 6 noxious thermal stimuli (43, 45, 47, 48, 49 and 51°C) applied for 5 sec to the forearm by a contact thermode. Sensory VAS and affective VAS responses to these temperatures yielded power functions with exponents 2.1 and 3.8, respectively; these functions were similar for pain patients and for volunteers. The power functions were predictive of estimated ratios of sensation or affect produced by pairs of standard temperatures (e.g. 47 and 49°C), thereby providing direct evidence for ratio scaling properties of VAS. VAS sensory intensity responses to experimental pain, VAS sensory intensity responses to different levels of chronic pain and direct temperature (experimental pain) matches to 3 levels of chronic pain were all internally consistent, thereby demonstrating the valid use of VAS for the measurement of and comparison between chronic pain and experimental heat pain.

Nontransitive measurable utility

A new theory of preferences under risk is presented that does not use the transitivity and independence axioms of the von Neumann-Morgenstern linear utility theory. Utilities in the new theory are unique up to a similarity transformation (ratio scale measurement). They key to this generalization of the traditional linear theory lies in its representation of binary preferences by a bivariate rather than univariate real valued function. Linear theory obtains a linear function u on a set P of probability measures for which u( p) > u( q) if and only if p is preferred to q. The new theory obtains a skew-symmetric bilinear function φ on P × P for which φ( p, q) > 0 if and only if p is preferred to q. Continuity, dominance, and symmetry axioms are shown to be necessary and sufficient for the new representation.

Łukasiewicz logic and the foundations of measurement

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable us to treat formally errors arising in nominal and ordinal measurements.

Social Consensus on Norms of Justice: Should the Punishment Fit the Crime?

Social consensus on norms of justice has long been of concern to sociologist. The present paper presents a model of justice norms as congnitions and tests the degree of cognitive consensus on the norm of just deserts (i.e., "letting the punishment fit the crime"). It is argued that consensus on justice norms should be tested using a combination of within-respondent and between-respondent techniques. Such tests can (1) simultaneously reveal the presence of consensus on the justice principle involved and on the evaluation of the specific social stimuli presented, (2) facilitate demographic comparisons on adhering to principle or agreeing on facts, and (3) reveal conflicts between these two versions of consensus. For testing the norm of just deserts, ratio scale measures of crime seriousness and punishment severity were employed, and a formula derived from both equity theory and psychophysics was utilized in model fitting. Results from a sample survey indicated dramatically strong use of the principle of just deserts by members of the public but less adherence to just deserts by demographically disadvantaged (low-income or black) respondents. A path model of the relation between aggregate and individual scores further demonstrated a fundamental tension between the two versions of normative consensus, in that the more respondents used the principle the more systematically they deviated from the group average response. The paper concludes by discussing the implications of both the substantive findings and the new methodologies employed for understanding normative consensus and the assessment of justice norms.

Causal Models with Nominal and Ordinal Data: Retrospective

The five previous chapters are part of the work started at a three-week seminar on causal models for nominal and ordinal data. The seminar took place in Seattle at the University of Washington from June 25 to July 13, 1973. Participants were Hubert M. Blalock, Jr., Phillip Bonacich, Roland K. Hawkes, Gudmund Hernes, James D. Laing, Robert K. Leik (director), Donald Ploch, and Aage S0renson. Funding was provided by The Center for Advanced Study in the Behavioral Sciences. The primary question around which the seminar was structured was how to reap the benefits of path analysis or its equivalent when data are not up to the standards of intervalor ratio-scale measurement. This problem requires us to consider the properties of regression and association at the interval level and compare them with possible analogues of regression for the

Ratio scale measurement of the perceived lengths of lines.

Ratio scale measurement of the perceived lengths of lines.