Ordinal data

Ordinal datais a categorical,statistical data typewhere the variables have natural, ordered categories and the distances between the categories are not known.^[1]^: 2These data exist on anordinal scale,one of fourlevels of measurementdescribed byS. S. Stevensin 1946. The ordinal scale is distinguished from thenominal scaleby having aranking.^[2]It also differs from theinterval scaleandratio scaleby not having category widths that represent equal increments of the underlying attribute.^[3]

Examples of ordinal data

A well-known example of ordinal data is theLikert scale.An example of a Likert scale is:^[4]^: 685

Like	Like Somewhat	Neutral	Dislike Somewhat	Dislike
1	2	3	4	5

Examples of ordinal data are often found in questionnaires: for example, the survey question "Is your general health poor, reasonable, good, or excellent?" may have those answers coded respectively as 1, 2, 3, and 4. Sometimes data on aninterval scaleorratio scaleare grouped onto an ordinal scale: for example, individuals whose income is known might be grouped into the income categories $0–$19,999, $20,000–$39,999, $40,000–$59,999,..., which then might be coded as 1, 2, 3, 4,.... Other examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework.^[5]

Ways to analyse ordinal data

Ordinal data analysis requires a different set of analyses than other qualitative variables. These methods incorporate the natural ordering of the variables in order to avoid loss of power.^[1]^: 88Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including the median or mode, are generally more appropriate.^[6]

General

Stevens (1946) argued that, because the assumption of equal distance between categories does not hold for ordinal data, the use of means and standard deviations for description of ordinal distributions and of inferential statistics based on means and standard deviations was not appropriate. Instead, positional measures like the median and percentiles, in addition to descriptive statistics appropriate for nominal data (number of cases, mode, contingency correlation), should be used.^[3]^: 678Nonparametric methodshave been proposed as the most appropriate procedures for inferential statistics involving ordinal data (e.g,Kendall's W,Spearman's rank correlation coefficient,etc.), especially those developed for the analysis of ranked measurements.^[5]^: 25–28However, the use of parametric statistics for ordinal data may be permissible with certain caveats to take advantage of the greater range of available statistical procedures.^[7]^[8]^[4]^: 90

Univariate statistics

In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median,^[9]^: 59–61other percentiles (such as quartiles and deciles),^[9]^: 71and the quartile deviation.^[9]^: 77One-sample tests for ordinal data include theKolmogorov-Smirnov one-sample test,^[5]^: 51–55theone-sample runs test,^[5]^: 58–64and the change-point test.^[5]^: 64–71

Bivariate statistics

In lieu of testing differences in means witht-tests,differences in distributions of ordinal data from two independent samples can be tested withMann-Whitney,^[9]^{: 259–264}runs,^[9]^{: 253–259}Smirnov,^[9]^{: 266–269}andsigned-ranks^[9]^{: 269–273}tests. Test for two related or matched samples include thesign test^[5]^: 80–87and theWilcoxon signed ranks test.^[5]^: 87–95Analysis of variance with ranks^[9]^{: 367–369}and theJonckheere test for ordered alternatives^[5]^{: 216–222}can be conducted with ordinal data in place of independent samplesANOVA.Tests for more than two related samples includes theFriedman two-way analysis of variance by ranks^[5]^{: 174–183}and thePage test for ordered alternatives.^[5]^{: 184–188}Correlation measures appropriate for two ordinal-scaled variables includeKendall's tau,^[9]^{: 436–439}gamma,^[9]^{: 442–443}r_s,^[9]^{: 434–436}andd_yx/d_xy.^[9]^: 443

Regression applications

Ordinal data can be considered as a quantitative variable. Inlogistic regression,the equation

\operatorname {logit} [P(Y=1)]=\ Alpha +\beta _{1}c+\beta _{2}x

is the model and c takes on the assigned levels of the categorical scale.^[1]^: 189Inregression analysis,outcomes (dependent variables) that are ordinal variables can be predicted using a variant ofordinal regression,such asordered logitorordered probit.

In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks.^[10]

Linear trends

Linear trends are also used to find associations between ordinal data and other categorical variables, normally in acontingency tables.A correlationris found between the variables whererlies between -1 and 1. To test the trend, a test statistic:

M^{2}=(n-1)r^{2}

is used wherenis the sample size.^[1]^: 87

Rcan be found by letting $u_{1}\leq u_{2}\leq...\leq u_{I}$ be the row scores and $v_{1}\leq v_{2}\leq...\leq v_{I}$ be the column scores. Let ${\bar {u}}\ =\sum _{i}u_{i}p_{i+}$ be the mean of the row scores while ${\bar {v}}\ =\sum _{j}v_{j}p_{j+}.$ .Then $p_{i+}$ is the marginal row probability and $p_{+j}$ is the marginal column probability.Ris calculated by:

r={\frac {\sum _{i,j}\left(u_{i}-{\bar {u}}\ \right)\left(v_{j}-{\bar {v}}\ \right)p_{ij}}{\sqrt {\left\lbrack \sum _{i}(u_{i}-{\bar {u}}\ \right)^{2}p_{i+}\rbrack \lbrack \sum _{j}(v_{j}-{\bar {v}}\ )^{2}p_{+j}\rbrack }}}

Classification methods

Classification methods have also been developed for ordinal data. The data are divided into different categories such that each observation is similar to others. Dispersion is measured and minimized in each group to maximize classification results. The dispersion function is used ininformation theory.^[11]

Statistical models for ordinal data

There are several different models that can be used to describe the structure of ordinal data.^[12]Four major classes of model are described below, each defined for a random variable $Y$ ,with levels indexed by $k=1,2,\dots,q$ .

Note that in the model definitions below, the values of $\mu _{k}$ and $\mathbf {\beta }$ will not be the same for all the models for the same set of data, but the notation is used to compare the structure of the different models.

Proportional odds model

The most commonly-used model for ordinal data is the proportional odds model, defined by $\log \left[{\frac {\Pr(Y\leq k)}{Pr(Y>k)}}\right]=\log \left[{\frac {\Pr(Y\leq k)}{1-\Pr(Y\leq k)}}\right]=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x}$ where the parameters $\mu _{k}$ describe the base distribution of the ordinal data, $\mathbf {x}$ are the covariates and $\mathbf {\beta }$ are the coefficients describing the effects of the covariates.

This model can be generalized by defining the model using $\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x}$ instead of $\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x}$ ,and this would make the model suitable for nominal data (in which the categories have no natural ordering) as well as ordinal data. However, this generalization can make it much more difficult to fit the model to the data.

Baseline category logit model

The baseline category model is defined by $\log \left[{\frac {\Pr(Y=k)}{\Pr(Y=1)}}\right]=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x}$

This model does not impose an ordering on the categories and so can be applied to nominal data as well as ordinal data.

Ordered stereotype model

The ordered stereotype model is defined by $\log \left[{\frac {\Pr(Y=k)}{\Pr(Y=1)}}\right]=\mu _{k}+\phi _{k}\mathbf {\beta } ^{T}\mathbf {x}$ where the score parameters are constrained such that $0=\phi _{1}\leq \phi _{2}\leq \dots \leq \phi _{q}=1$ .

This is a more parsimonious, and more specialised, model than the baseline category logit model: $\phi _{k}\mathbf {\beta }$ can be thought of as similar to $\mathbf {\beta } _{k}$ .

The non-ordered stereotype model has the same form as the ordered stereotype model, but without the ordering imposed on $\phi _{k}$ .This model can be applied to nominal data.

Note that the fitted scores, ${\hat {\phi }}_{k}$ ,indicate how easy it is to distinguish between the different levels of $Y$ .If ${\hat {\phi }}_{k}\approx {\hat {\phi }}_{k-1}$ then that indicates that the current set of data for the covariates $\mathbf {x}$ do not provide much information to distinguish between levels $k$ and $k-1$ ,but that doesnotnecessarily imply that the actual values $k$ and $k-1$ are far apart. And if the values of the covariates change, then for that new data the fitted scores ${\hat {\phi }}_{k}$ and ${\hat {\phi }}_{k-1}$ might then be far apart.

Adjacent categories logit model

The adjacent categories model is defined by $\log \left[{\frac {\Pr(Y=k)}{\Pr(Y=k+1)}}\right]=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x}$ although the most common form, referred to inAgresti(2010)^[12]as the "proportional odds form" is defined by $\log \left[{\frac {\Pr(Y=k)}{\Pr(Y=k+1)}}\right]=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x}$

This model can only be applied to ordinal data, since modelling the probabilities of shifts from one category to the next category implies that an ordering of those categories exists.

The adjacent categories logit model can be thought of as a special case of the baseline category logit model, where $\mathbf {\beta } _{k}=\mathbf {\beta } (k-1)$ .The adjacent categories logit model can also be thought of as a special case of the ordered stereotype model, where $\phi _{k}\propto k-1$ ,i.e. the distances between the $\phi _{k}$ are defined in advance, rather than being estimated based on the data.

Comparisons between the models

The proportional odds model has a very different structure to the other three models, and also a different underlying meaning. Note that the size of the reference category in the proportional odds model varies with $k$ ,since $Y\leq k$ is compared to $Y>k$ ,whereas in the other models the size of the reference category remains fixed, as $Y=k$ is compared to $Y=1$ or $Y=k+1$ .

Different link functions

There are variants of all the models that use different link functions, such as the probit link or the complementary log-log link.

Statistical tests

Differences in ordinal data can be tested usingrank tests.

Visualization and display

Ordinal data can be visualized in several different ways. Common visualizations are thebar chartor apie chart.Tablescan also be useful for displaying ordinal data and frequencies.Mosaic plotscan be used to show the relationship between an ordinal variable and a nominal or ordinal variable.^[13]A bump chart—a line chart that shows the relative ranking of items from one time point to the next—is also appropriate for ordinal data.^[14]

Color orgrayscalegradation can be used to represent the ordered nature of the data. A single-direction scale, such as income ranges, can be represented with a bar chart where increasing (or decreasing) saturation or lightness of a single color indicates higher (or lower) income. The ordinal distribution of a variable measured on a dual-direction scale, such as a Likert scale, could also be illustrated with color in a stacked bar chart. A neutral color (white or gray) might be used for the middle (zero or neutral) point, with contrasting colors used in the opposing directions from the midpoint, where increasing saturation or darkness of the colors could indicate categories at increasing distance from the midpoint.^[15]Choropleth mapsalso use color or grayscale shading to display ordinal data.^[16]

Example bar plot of opinion on defense spending

Example bump plot of opinion on defense spending by political party

Example mosaic plot of opinion on defense spending by political party

Example stacked bar plot of opinion on defense spending by political party

Applications

The use of ordinal data can be found in most areas of research where categorical data are generated. Settings where ordinal data are often collected include the social and behavioral sciences and governmental and business settings where measurements are collected from persons by observation, testing, orquestionnaires.Some common contexts for the collection of ordinal data includesurvey research;^[17]^[18]andintelligence,aptitude,personalitytesting anddecision-making.^[2]^[4]^: 89–90

Calculation of 'Effect Size' (Cliff's Deltad) using ordinal data has been recommended as a measure of statistical dominance.^[19]

References

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^^a ^bAtaei, Younes; Mahmoudi, Amin; Feylizadeh, Mohammad Reza; Li, Deng-Feng (January 2020)."Ordinal Priority Approach (OPA) in Multiple Attribute Decision-Making".Applied Soft Computing.86:105893.doi:10.1016/j.asoc.2019.105893.ISSN 1568-4946.S2CID 209928171.
^^a ^bStevens, S. S. (1946). "On the Theory of Scales of Measurement".Science.New Series.103(2684): 677–680.Bibcode:1946Sci...103..677S.doi:10.1126/science.103.2684.677.PMID 17750512.
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^Jamieson, Susan (December 2004)."Likert scales: how to (ab)use them"(PDF).Medical Education.38(12): 1212–1218.doi:10.1111/j.1365-2929.2004.02012.x.PMID 15566531.S2CID 42509064.
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^Cohen, Jacob; Cohen, Patricia (1983).Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences(2nd ed.). Hillsdale, New Jersey: Lawrence Erlbaum Associates. p. 273.ISBN 0-89859-268-2.
^Laird, Nan M. (1979). "A Note on Classifying Ordinal-Scale Data".Sociological Methodology.10:303–310.doi:10.2307/270775.JSTOR 270775.
^^a ^bAgresti, Alan (2010).Analysis of Ordinal Categorical Data(2nd ed.). Hoboken, New Jersey: Wiley.ISBN 978-0470082898.
^"Plotting Techniques".
^Berinato, Scott (2016).Good Charts: The HBR Guide to Making Smarter, More Persuasive Data Visualizations.Boston: Harvard Business Review Press. p. 228.ISBN 978-1633690707.
^Kirk, Andy (2016).Data Visualisation: A Handbook for Data Driven Design(1st ed.). London: SAGE. p. 269.ISBN 978-1473912144.
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^Alwin, Duane F. (2010). "Assessing the Reliability and Validity of Survey Measures". In Marsden, Peter V.; Wright, James D. (eds.).Handbook of Survey Research.Howard House, Wagon Lane, Bingley BD16 1WA, UK: Emerald House. p. 420.ISBN 978-1-84855-224-1.{{cite book}}:CS1 maint: location (link)
^Fowler, Floyd J. Jr. (1995).Improving Survey Questions: Design and Evaluation.Thousand Oaks, CA: Sage. pp.156–165.ISBN 0-8039-4583-3.
^Cliff, Norman (November 1993)."Dominance statistics: Ordinal analyses to answer ordinal questions".Psychological Bulletin.114(3): 494–509.doi:10.1037/0033-2909.114.3.494.ISSN 1939-1455.