Statistics ordinal

Cliff N. 1993. Dominance statistics Ordinal analyses to answer ordinal questions. Psychol Bull 114 494-509. FDA (Food and Drug Administration). 1996. Current good manufacturing practice, quality control procedures, quality factors, notification requirements, and records and reports, for the production of infant formula. Proposed rule. Fed Regist 61 36153—36219. 

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. 

FIGURE 11.13 A collection of 10 responses (ordinates) to a compound resulting from exposure of a biological preparation to 10 concentrations of the compound (abscissae, log scale). The dotted line indicates the mean total response of all of the concentrations. The sigmoidal curve indicates the best fit of a four-parameter logistic function to the data points. The data were fit to Emax = 5.2, n = 1, EC5o = 0.4 pM, and basal = 0.3. The value for F is 9.1, df=6, 10. This shows that the fit to the complex model is statistically preferred (the fit to the sigmoidal curve is indicated). 

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. 

Figure 2.15. The limit of detection LOD the minimum signal/noise-ratio necessary according to two models (ordinate) is plotted against log 0(n) under the assumption of evenly spaced calibration points. The three sets of curves are for p = 0.1 (A), 0.05 (B), and 0.02 (C). The correct statistical theory is given by the fine points, while the model presented here is depicted with coarser dots. The widely used S/N = 3. .. 6 models would be represented by horizontals at y = 3. .. 6.

Calculate New Data) generates a statistically similar ordinate value for each Xi by superimposing ND(0, s ) noise on the model or previous data this option can be repeatedly accessed. 

Here you can still use the Pearson chi-square test as shown in the 2x2 table example as long as your response variable is nominal and merely descriptive. If your response variable is ordinal, meaning that it is an ordered sequence, and you can use a parametric test, then you should use the Mantel-Haenszel test statistic for parametric tests of association. For instance, if in our previous example the variable called headache was coded as a 2 when the patient experienced extreme headache, a 1 if mild headache, and a 0 if no headache, then headache would be an ordinal variable. You can get the Mantel-Haenszel /pvalue by running the following SAS code ... 

EXELFS (the extended energy-loss fine structure) carries information about the bonding and co-ordination of the atoms contributing to the edge. However, the signal needs to be strong before statistically reliable information can be obtained. 

The behavior of the different amines depends on at least four factors basicity, nucleophilicity, steric hindrance and solvation. In the literature (16), 126 aliphatic and aromatic amines have been classified by a statistical analysis of the data for the following parameters molar mass (mm), refractive index (nD), density (d), boiling point (bp), molar volume, and pKa. On such a premise, a Cartesian co-ordinate graph places the amines in four quadrants (16). In our preliminary tests, amines representative of each quadrant have been investigated, and chosen by consideration of their toxicity, commercial availability and price (Table 1). 

Group comparison tests for proportions notoriously lack power. Trend tests, because of their use of prior information (dose levels) are much more powerful. Also, it is generally believed that the nature of true carcinogenicity (or toxicity for that matter), manifests itself as dose-response. Because of the above facts, evaluation of trend takes precedence over group comparisons. In order to achieve optimal test statistics, many people use ordinal dose levels (0,1,2..., etc.) instead of the true arithmetic dose levels to test for trend. However, such a decision should be made a priori. The following example demonstrates the weakness of homogeneity tests. 

It is important to appreciate that the statistical significance of the results is wholly dependent on the quality of the data obtained from the trial. Data that contain obvious gross errors should be removed prior to statistical analysis. It is essential that participants inform the trial co-ordinator of any gross error that they know has occurred during the analysis and also if any deviation from the method as written has taken place. The statistical parameters calculated and the outlier tests performed are those used in the internationally agreed Protocol for the Design, Conduct and Interpretation of Collaborative Studies.14... 

For PDF profiles in vivo, the peak co-ordinates are frequently used, because they are immediately read from the tabulated observations or from a corresponding plot. In statistical terms, Zmax is the modus of the distribution, i.e.,... 

Data belonging to distribution profiles may be compared either vertically along the release/response ordinate or horizontally along the time abscissa. The semi-invariants (moments) provide a complete set of metrics, representing both aspects in logical sequence AUC accounts (vertically) for the difference of the extent, the mean compares (horizontally) the rates, and higher-order moments and higher-order statistics (variance, etc.) characterize the shape aspect from coarse to finer. 

There is a hierarchy of usefulness of data, according to how well it can be statistically manipulated. The accepted order is continuous data > ordinal data > nominal data. 

One way to compare data to predicted fractionation laws is to plot the data on the three isotope plot in which 5 "Mg is the ordinate and 5 Mg is the abscissa, and examine how closely the data fall to the different curves defined by the exponent p. However, the differences between the different P values are often evident only with careful attention to the statistics of the data. Ideally, the values of P should be obtained by a best fit to the data. This is most easily accomplished if the problem can be rewritten so that P is the slope in a linear regression. 

When the statistieally sophisticated psychologists realized what I was doing, they had a field day pointing out my failings unjustified assumptions, violations of statistical theory and other mathematical crimes. They talked about ordinal scales versus ratio scales and scolded me for not using analysis of variance instead of Chi-square and Student s T tests of significance. 

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. 

D Z)-statistic extended to ordinal forms of two observation sets, defined by Eq.(l 1)... 

The Z)-statistic concept can be readily extended to the two-factor case, where (A, B,) are corresponding ranks, i.e. if A and B are the ordinal form of observation sets (the A7B notation is used instead of Lehman s (R.S) notation in order to avoid confusion between similar symbols). If rank positions are tied, they are replaced by their mid-rank, yielding modified distributions A and B. By a straightforward extension of Eq.(6), the modified Z)-statistic is computed as [17]... 

As we shall see later the data type to a large extent determines the class of statistical tests that we undertake. Commonly for continuous data we use the t-tests and their extensions analysis of variance and analysis of covariance. For binary, categorical and ordinal data we use the class of chi-square tests (Pearson chi-square for categorical data and the Mantel-Haenszel chi-square for ordinal data) and their extension, logistic regression. 

The calculation of mean and standard deviation only really makes sense when we are dealing with continuous, score or count data. These quantities have little relevance when we are looking at binary or ordinal data. In these situations we would tend to use proportions in the various categories as our summary statistics and population parameters of interest. 

There are various ways of normalizing the gaussian, in both abscissa and ordinate. In statistics, we often deal with... 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

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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