Statistics ordinal data

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. 

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. 

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... 

Another method of detecting a dose-response relationship is to fit the data to various models for dose-response curves. This method statistically determines whether or not a dose-response model (such as a Logistic function) fits the data points more accurately than simply the mean of the values this method is described fully in Chapter 12. The most simple model would be to assume no dose-response relationship and calculate the mean of the ordinate data as the response for each concentration of ligand (horizontal straight line parallel to the abscissal axis). A more complex model would be to fit the data to a sigmoidal dose-response function (Equation 11.2). A sum of squares can be calculated for the simple model (response — mean of all response) and then for a fit of the data set refit to the four parameter Logistic shown... 

A statistical test that is often appropriate for comparing two groups in terms of a quantitative outcome measure is the unpaired t-test. Other assumptions underpin the use of a t-test (see later) and it is therefore sometimes desirable to use one of the tests primarily intended for use on ordinal data even if the data are quantitative. 

Estimation is the use of the sample data to make inferences about the population that the sample represents . With qualitative data, we would usually be interested in estimating the proportion or percentage of individuals in the population having some outcome or characteristic with ordinal data we would probably wish to estimate the population median, and with quantitative data the population mean. Although percentages, medians and means are most often of interest, it is possible to use any sample statistic to estimate the corresponding population value thus in Sections 7.3.1.3.3 and 7.3.1.3.4 we were interested in whether a sample gl or g2 was consistent with ffie true or population values being zero. 

McCullagh P (1980) Regression models for ordinal data. Journal of the Royal Statistical Society B 42 109-142. 

Graphically, the most important details of descriptive statistics of data can be represented in a box-and-whisker plot or, for short, box plot (Figure 2.5). Along the variable axis, here the ordinate, a box is drawn, with the lower and upper quartile being the bottom and top of the box, respectively. The width of the box has no meaning. 

What is the significance of these different scales of measurement As was mentioned in Section 1.5, many of the well-known statistical methods are parametric, that is, they rely on assumptions concerning the distribution of the data. The computation of parametric tests involves arithmetic manipulation such as addition, multiplication, and division, and this should only be carried out on data measured on interval or ratio scales. When these procedures are used on data measured on other scales they introduce distortions into the data and thus cast doubt on any conclusions which may be drawn from the tests. Non-parametric or distribution-free methods, on the other hand, concentrate on an order or ranking of data and thus can be used with ordinal data. Some of the non-parametric techniques are also designed to operate with classified (nominal) data. Since interval and ratio scales of measurement have all the properties of ordinal scales it is possible to use non-parametric methods for data measured on these scales. Thus, the distribution-free techniques are the safest to use since they can be applied to most types of data. If, however, the data does conform to the distributional assumptions of the parametric techniques, these methods may well extract more information from the data. 

The choice of analysis is influenced by the type of data and its distribution. Survey data, such as that collected during anthropometric studies (typically interval data), may be analyzed using parametric statistical techniques, while questionnaires (typically opinion-based), are more appropriately tested using nonparametric tests (nominal or ordinal data). In the case of questionnaires, the wording of questions, the organization of response categories, the method to categorize or code, and the format to collect the data all may influence the analysis method which can be used. 

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. 

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. 

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). 

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... 

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. 

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. 

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). 

That categorical or discrete concepts are naturally mapped to entities that contain and ordinal concepts are naturally mapped to entities that connect seems to underlie the use and interpretation of bars and lines in graphs (Zacks and Tversky, 1999). Bars are container-like and lines connect. Zacks and Tversky (1999) distinguished two related uses. The use that is statistical lore, and sometimes explicit advice, is the bar-line data use display discrete data with bars and continuous data with lines. But we have already seen that the same data can be viewed in different ways. The more subtle use is the bar-line message use interpret/produce bars as discrete comparisons and lines as trends. 

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