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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.
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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,
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gradation 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
150:
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,
141:
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. Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including
132:
are 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
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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 in
163:, etc.), especially those developed for the analysis of ranked measurements. However, 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.
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In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, other percentiles (such as quartiles and deciles), and the quartile deviation. One-sample tests for ordinal data include the
1095:{\displaystyle 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 }}}}
123:
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 an
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can be used to show the relationship between an ordinal variable and a nominal or ordinal variable. 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.
1598:, 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.
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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
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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.
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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.
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There are several different models that can be used to describe the structure of ordinal data. Four major classes of model are described below, each defined for a random variable
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In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks.
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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.
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27:
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Alwin, Duane F. (2010). "Assessing the
Reliability and Validity of Survey Measures". In Marsden, Peter V.; Wright, James D. (eds.).
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40:
where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an
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There are variants of all the models that use different link functions, such as the probit link or the complementary log-log link.
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The non-ordered stereotype model has the same form as the ordered stereotype model, but without the ordering imposed on
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This model does not impose an ordering on the categories and so can be applied to nominal data as well as ordinal data.
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Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a
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The adjacent categories logit model can be thought of as a special case of the baseline category logit model, where
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have been proposed as the most appropriate procedures for inferential statistics involving ordinal data (e.g,
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are far apart. And if the values of the covariates change, then for that new data the fitted scores
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This is a more parsimonious, and more specialised, model than the baseline category logit model:
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by not having category widths that represent equal increments of the underlying attribute.
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The most commonly-used model for ordinal data is the proportional odds model, defined by
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Ataei, Younes; Mahmoudi, Amin; Feylizadeh, Mohammad Reza; Li, Deng-Feng (January 2020).
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Ordinal data can be visualized in several different ways. Common visualizations are the
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3517:
Good Charts: The HBR Guide to Making
Smarter, More Persuasive Data Visualizations
3417:(2nd ed.). Hillsdale, New Jersey: Lawrence Erlbaum Associates. p. 273.
3173:
3007:) using ordinal data has been recommended as a measure of statistical dominance.
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1418:{\displaystyle \log \left=\log \left=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} }
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3594:. Howard House, Wagon Lane, Bingley BD16 1WA, UK: Emerald House. p. 420.
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Psychological
Testing and Assessment: An Introduction to Tests and Measurement
1840:{\displaystyle \log \left=\mu _{k}+\phi _{k}\mathbf {\beta } ^{T}\mathbf {x} }
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is the model and c takes on the assigned levels of the categorical scale. In
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Applied
Multiple Regression/Correlation Analysis for the Behavioral Sciences
239:. Correlation measures appropriate for two ordinal-scaled variables include
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Example stacked bar plot of opinion on defense spending by political party
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2451:{\displaystyle \log \left=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} }
2051:, indicate how easy it is to distinguish between the different levels of
1712:{\displaystyle \log \left=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} }
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67:
3104:"Ordinal Priority Approach (OPA) in Multiple Attribute Decision-Making"
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are defined in advance, rather than being estimated based on the data.
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362:{\displaystyle \operatorname {logit} =\alpha +\beta _{1}c+\beta _{2}x}
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Example mosaic plot of opinion on defense spending by political party
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2569:{\displaystyle \log \left=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} }
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3662:"Dominance statistics: Ordinal analyses to answer ordinal questions"
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then that indicates that the current set of data for the covariates
1911:{\displaystyle 0=\phi _{1}\leq \phi _{2}\leq \dots \leq \phi _{q}=1}
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2942:
Example bump plot of opinion on defense spending by political party
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can be conducted with ordinal data in place of independent samples
3438:
Laird, Nan M. (1979). "A Note on
Classifying Ordinal-Scale Data".
3200:
Cohen, Ronald Jay; Swerdik, Mark E.; Phillips, Suzanne M. (1996).
2980:. Some common contexts for the collection of ordinal data include
3152:
Stevens, S. S. (1946). "On the Theory of Scales of
Measurement".
380:) that are ordinal variables can be predicted using a variant of
58:
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can also be useful for displaying ordinal data and frequencies.
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are the coefficients describing the effects of the covariates.
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do not provide much information to distinguish between levels
2128:{\displaystyle {\hat {\phi }}_{k}\approx {\hat {\phi }}_{k-1}}
1544:{\displaystyle \mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} }
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Ordinal data can be considered as a quantitative variable. In
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also use color or grayscale shading to display ordinal data.
2626:{\displaystyle \mathbf {\beta } _{k}=\mathbf {\beta } (k-1)}
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lies between -1 and 1. To test the trend, a test statistic:
211:
tests. Test for two related or matched samples include the
3567:
The
Truthful Art: Data, Charts, and Maps for Communication
3078:(3 ed.). Hoboken, New Jersey: John Wiley & Sons.
1591:{\displaystyle \mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} }
1499:
This model can be generalized by defining the model using
3101:
3569:(1st ed.). San Francisco: New Riders. p. 280.
1190:
Note that in the model definitions below, the values of
26:"Ordinal scale" redirects here. Not to be confused with
231:. Tests for more than two related samples includes the
3519:. Boston: Harvard Business Review Press. p. 228.
3206:(3rd ed.). Mountain View, CA: Mayfield. pp.
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3254:(2nd ed.). Boston: McGraw-Hill. pp. 25–26.
2838:
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2462:(2010) as the "proportional odds form" is defined by
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where the score parameters are constrained such that
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in 1946. The ordinal scale is distinguished from the
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Nonparametric
Statistics for the Behavioral Sciences
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3014:
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describe the base distribution of the ordinal data,
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the median or mode, are generally more appropriate.
3344:. New York: John Wiley & Sons. pp. 23–24.
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748:{\displaystyle {\bar {v}}\ =\sum _{j}v_{j}p_{j+}.}
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3632:Improving Survey Questions: Design and Evaluation
2702:
677:{\displaystyle {\bar {u}}\ =\sum _{i}u_{i}p_{i+}}
609:{\displaystyle v_{1}\leq v_{2}\leq ...\leq v_{I}}
544:{\displaystyle u_{1}\leq u_{2}\leq ...\leq u_{I}}
3723:
3327:"Measurement theory: Frequently asked questions"
2880:Differences in ordinal data can be tested using
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3250:Siegel, Sidney; Castellan, N. John Jr. (1988).
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2932:Example bar plot of opinion on defense spending
136:
2458:although the most common form, referred to in
233:Friedman two-way analysis of variance by ranks
2012:. This model can be applied to nominal data.
188:In lieu of testing differences in means with
3707:(2nd ed.). Hoboken, New Jersey: Wiley.
3480:(2nd ed.). Hoboken, New Jersey: Wiley.
3466:
3392:(Rev. 2nd ed.). New York: McGraw-Hill.
3003:Calculation of 'Effect Size' (Cliff's Delta
2887:
2339:The adjacent categories model is defined by
1086:
1019:
1016:
78:A well-known example of ordinal data is the
3544:(1st ed.). London: SAGE. p. 269.
3195:
3193:
3191:
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1729:The ordered stereotype model is defined by
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73:
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1946:{\displaystyle \phi _{k}\mathbf {\beta } }
1606:The baseline category model is defined by
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133:ranks, and letter grades for coursework.
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225:Jonckheere test for ordered alternatives
166:
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183:
161:Spearman's rank correlation coefficient
3724:
3628:
3413:Cohen, Jacob; Cohen, Patricia (1983).
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3659:
3589:
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1975:{\displaystyle \mathbf {\beta } _{k}}
411:is found between the variables where
3705:Analysis of Ordinal Categorical Data
3635:. Thousand Oaks, CA: Sage. pp.
3539:
3478:Analysis of Ordinal Categorical Data
3277:"Likert scales: how to (ab)use them"
3097:
3095:
3033:List of analyses of categorical data
2875:
2665:{\displaystyle \phi _{k}\propto k-1}
2324:{\displaystyle {\hat {\phi }}_{k-1}}
815:is the marginal column probability.
785:is the marginal row probability and
684:be the mean of the row scores while
1119:Statistical models for ordinal data
82:. An example of a Likert scale is:
13:
3696:
2282:{\displaystyle {\hat {\phi }}_{k}}
2044:{\displaystyle {\hat {\phi }}_{k}}
237:Page test for ordered alternatives
174:Kolmogorov-Smirnov one-sample test
14:
3748:
3325:Sarle, Warren S. (Sep 14, 1997).
3275:Jamieson, Susan (December 2004).
3092:
2672:, i.e. the distances between the
1489:{\displaystyle \mathbf {\beta } }
1232:{\displaystyle \mathbf {\beta } }
3296:10.1111/j.1365-2929.2004.02012.x
3017:
2562:
2444:
2143:
1953:can be thought of as similar to
1833:
1705:
1584:
1537:
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1411:
466:{\displaystyle M^{2}=(n-1)r^{2}}
398:
3660:Cliff, Norman (November 1993).
3653:
3622:
3583:
3558:
3533:
3508:
3494:
3431:
3406:
3388:Blalock, Hubert M. Jr. (1979).
2970:
2335:Adjacent categories logit model
221:Analysis of variance with ranks
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2703:Comparisons between the models
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1180:{\displaystyle k=1,2,\dots ,q}
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3629:Fowler, Floyd J. Jr. (1995).
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2015:Note that the fitted scores,
1602:Baseline category logit model
180:, and the change-point test.
3174:10.1126/science.103.2684.677
2150:{\displaystyle \mathbf {x} }
1467:{\displaystyle \mathbf {x} }
137:Ways to analyse ordinal data
7:
3678:10.1037/0033-2909.114.3.494
3592:Handbook of Survey Research
3010:
62:. It also differs from the
10:
3753:
3342:Statistical Rules of Thumb
3340:van Belle, Gerald (2002).
3120:10.1016/j.asoc.2019.105893
616:be the column scores. Let
217:Wilcoxon signed ranks test
145:
25:
21:Ordinal data (programming)
18:
3737:Comparison (mathematical)
3076:Categorical Data Analysis
3038:Ordinal Priority Approach
2888:Visualization and display
2692:{\displaystyle \phi _{k}}
2331:might then be far apart.
2005:{\displaystyle \phi _{k}}
1143:, with levels indexed by
3614:: CS1 maint: location (
3515:Berinato, Scott (2016).
3440:Sociological Methodology
2868:Different link functions
1725:Ordered stereotype model
1445:{\displaystyle \mu _{k}}
1210:{\displaystyle \mu _{k}}
486:can be found by letting
74:Examples of ordinal data
19:Not to be confused with
3565:Cairo, Alberto (2016).
2747:{\displaystyle Y\leq k}
1474:are the covariates and
1243:Proportional odds model
273:Regression applications
3732:Statistical data types
3703:Agresti, Alan (2010).
3666:Psychological Bulletin
3476:Agresti, Alan (2010).
3108:Applied Soft Computing
3074:Agresti, Alan (2013).
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2773:{\displaystyle Y>k}
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808:{\displaystyle p_{+j}}
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778:{\displaystyle p_{i+}}
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551:be the row scores and
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3502:"Plotting Techniques"
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153:Nonparametric methods
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28:Ordinal Scale (movie)
16:Statistical data type
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184:Bivariate statistics
178:one-sample runs test
3540:Kirk, Andy (2016).
3166:1946Sci...103..677S
2825:{\displaystyle Y=1}
2799:{\displaystyle Y=k}
2442:
2246:{\displaystyle k-1}
2196:{\displaystyle k-1}
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378:dependent variables
374:regression analysis
279:logistic regression
3025:Mathematics portal
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405:contingency tables
382:ordinal regression
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36:is a categorical,
3601:978-1-84855-224-1
3390:Social Statistics
3290:(12): 1212–1218.
3284:Medical Education
3160:(2684): 677–680.
3085:978-0-470-46363-5
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2721:{\displaystyle k}
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3492:
3491:
3473:
3464:
3463:
3435:
3429:
3428:
3410:
3404:
3403:
3385:
3356:
3355:
3337:
3331:
3330:
3322:
3316:
3315:
3281:
3272:
3266:
3265:
3247:
3222:
3221:
3197:
3186:
3185:
3149:
3140:
3139:
3099:
3090:
3089:
3071:
3027:
3022:
3021:
2924:
2923:
2863:
2861:
2860:
2855:
2831:
2829:
2828:
2823:
2805:
2803:
2802:
2797:
2779:
2777:
2776:
2771:
2753:
2751:
2750:
2745:
2727:
2725:
2724:
2719:
2698:
2696:
2695:
2690:
2688:
2687:
2671:
2669:
2668:
2663:
2649:
2648:
2632:
2630:
2629:
2624:
2607:
2599:
2598:
2593:
2575:
2573:
2572:
2567:
2565:
2560:
2559:
2554:
2545:
2544:
2532:
2528:
2526:
2500:
2480:
2457:
2455:
2454:
2449:
2447:
2441:
2436:
2431:
2422:
2421:
2409:
2405:
2403:
2377:
2357:
2330:
2328:
2327:
2322:
2320:
2319:
2308:
2307:
2299:
2288:
2286:
2285:
2280:
2278:
2277:
2272:
2271:
2263:
2252:
2250:
2249:
2244:
2226:
2224:
2223:
2218:
2203:, but that does
2202:
2200:
2199:
2194:
2176:
2174:
2173:
2168:
2156:
2154:
2153:
2148:
2146:
2134:
2132:
2131:
2126:
2124:
2123:
2112:
2111:
2103:
2096:
2095:
2090:
2089:
2081:
2070:
2068:
2067:
2062:
2050:
2048:
2047:
2042:
2040:
2039:
2034:
2033:
2025:
2011:
2009:
2008:
2003:
2001:
2000:
1981:
1979:
1978:
1973:
1971:
1970:
1965:
1952:
1950:
1949:
1944:
1942:
1937:
1936:
1917:
1915:
1914:
1909:
1901:
1900:
1882:
1881:
1869:
1868:
1846:
1844:
1843:
1838:
1836:
1831:
1830:
1825:
1819:
1818:
1806:
1805:
1793:
1789:
1787:
1767:
1747:
1718:
1716:
1715:
1710:
1708:
1702:
1697:
1692:
1683:
1682:
1670:
1666:
1664:
1644:
1624:
1597:
1595:
1594:
1589:
1587:
1582:
1581:
1576:
1567:
1566:
1550:
1548:
1547:
1542:
1540:
1534:
1529:
1524:
1515:
1514:
1495:
1493:
1492:
1487:
1485:
1473:
1471:
1470:
1465:
1463:
1451:
1449:
1448:
1443:
1441:
1440:
1424:
1422:
1421:
1416:
1414:
1409:
1408:
1403:
1394:
1393:
1381:
1377:
1375:
1349:
1329:
1314:
1310:
1308:
1285:
1265:
1238:
1236:
1235:
1230:
1228:
1216:
1214:
1213:
1208:
1206:
1205:
1186:
1184:
1183:
1178:
1142:
1140:
1139:
1134:
1101:
1099:
1098:
1093:
1091:
1085:
1084:
1072:
1071:
1060:
1059:
1058:
1050:
1044:
1043:
1030:
1015:
1014:
1002:
1001:
996:
992:
989:
988:
987:
979:
973:
972:
959:
944:
943:
942:
941:
929:
925:
922:
921:
920:
912:
906:
905:
891:
887:
884:
883:
882:
874:
868:
867:
852:
836:
814:
812:
811:
806:
804:
803:
784:
782:
781:
776:
774:
773:
754:
752:
751:
746:
741:
740:
728:
727:
717:
703:
702:
701:
693:
683:
681:
680:
675:
673:
672:
660:
659:
649:
635:
634:
633:
625:
615:
613:
612:
607:
605:
604:
580:
579:
567:
566:
550:
548:
547:
542:
540:
539:
515:
514:
502:
501:
472:
470:
469:
464:
462:
461:
434:
433:
407:. A correlation
368:
366:
365:
360:
355:
354:
339:
338:
85:
84:
3752:
3751:
3747:
3746:
3745:
3743:
3742:
3741:
3722:
3721:
3715:
3699:
3697:Further reading
3694:
3693:
3658:
3654:
3647:
3627:
3623:
3607:
3606:
3602:
3588:
3584:
3577:
3563:
3559:
3552:
3538:
3534:
3527:
3513:
3509:
3500:
3499:
3495:
3488:
3474:
3467:
3436:
3432:
3425:
3411:
3407:
3400:
3386:
3359:
3352:
3338:
3334:
3323:
3319:
3279:
3273:
3269:
3262:
3248:
3225:
3218:
3198:
3189:
3150:
3143:
3100:
3093:
3086:
3072:
3061:
3056:
3023:
3016:
3013:
2998:decision-making
2982:survey research
2973:
2919:Choropleth maps
2890:
2878:
2870:
2837:
2834:
2833:
2811:
2808:
2807:
2806:is compared to
2785:
2782:
2781:
2759:
2756:
2755:
2754:is compared to
2733:
2730:
2729:
2713:
2710:
2709:
2705:
2683:
2679:
2677:
2674:
2673:
2644:
2640:
2638:
2635:
2634:
2603:
2594:
2589:
2588:
2586:
2583:
2582:
2561:
2555:
2550:
2549:
2540:
2536:
2501:
2481:
2479:
2475:
2467:
2464:
2463:
2443:
2437:
2432:
2427:
2417:
2413:
2378:
2358:
2356:
2352:
2344:
2341:
2340:
2337:
2309:
2298:
2297:
2296:
2294:
2291:
2290:
2273:
2262:
2261:
2260:
2258:
2255:
2254:
2232:
2229:
2228:
2212:
2209:
2208:
2182:
2179:
2178:
2162:
2159:
2158:
2142:
2140:
2137:
2136:
2113:
2102:
2101:
2100:
2091:
2080:
2079:
2078:
2076:
2073:
2072:
2056:
2053:
2052:
2035:
2024:
2023:
2022:
2020:
2017:
2016:
1996:
1992:
1990:
1987:
1986:
1966:
1961:
1960:
1958:
1955:
1954:
1938:
1932:
1928:
1926:
1923:
1922:
1896:
1892:
1877:
1873:
1864:
1860:
1852:
1849:
1848:
1832:
1826:
1821:
1820:
1814:
1810:
1801:
1797:
1768:
1748:
1746:
1742:
1734:
1731:
1730:
1727:
1704:
1698:
1693:
1688:
1678:
1674:
1645:
1625:
1623:
1619:
1611:
1608:
1607:
1604:
1583:
1577:
1572:
1571:
1562:
1558:
1556:
1553:
1552:
1536:
1530:
1525:
1520:
1510:
1506:
1504:
1501:
1500:
1481:
1479:
1476:
1475:
1459:
1457:
1454:
1453:
1436:
1432:
1430:
1427:
1426:
1410:
1404:
1399:
1398:
1389:
1385:
1350:
1330:
1328:
1324:
1286:
1266:
1264:
1260:
1252:
1249:
1248:
1245:
1224:
1222:
1219:
1218:
1201:
1197:
1195:
1192:
1191:
1148:
1145:
1144:
1128:
1125:
1124:
1121:
1108:
1077:
1073:
1067:
1063:
1049:
1048:
1039:
1035:
1026:
1007:
1003:
997:
978:
977:
968:
964:
955:
950:
946:
945:
934:
930:
911:
910:
901:
897:
896:
892:
873:
872:
863:
859:
858:
854:
842:
837:
835:
827:
824:
823:
796:
792:
790:
787:
786:
766:
762:
760:
757:
756:
733:
729:
723:
719:
713:
692:
691:
689:
686:
685:
665:
661:
655:
651:
645:
624:
623:
621:
618:
617:
600:
596:
575:
571:
562:
558:
556:
553:
552:
535:
531:
510:
506:
497:
493:
491:
488:
487:
457:
453:
429:
425:
423:
420:
419:
401:
350:
346:
334:
330:
289:
286:
285:
281:, the equation
275:
266:
262:
253:
186:
169:
148:
139:
76:
31:
24:
17:
12:
11:
5:
3750:
3740:
3739:
3734:
3720:
3719:
3714:978-0470082898
3713:
3698:
3695:
3692:
3691:
3672:(3): 494–509.
3652:
3645:
3621:
3600:
3582:
3576:978-0321934079
3575:
3557:
3551:978-1473912144
3550:
3532:
3526:978-1633690707
3525:
3507:
3493:
3487:978-0470082898
3486:
3465:
3452:10.2307/270775
3430:
3423:
3405:
3398:
3357:
3350:
3332:
3317:
3267:
3260:
3223:
3216:
3187:
3156:. New Series.
3141:
3091:
3084:
3058:
3057:
3055:
3052:
3051:
3050:
3045:
3043:Ordinal number
3040:
3035:
3029:
3028:
3012:
3009:
2978:questionnaires
2972:
2969:
2966:
2965:
2955:
2945:
2935:
2889:
2886:
2877:
2874:
2869:
2866:
2853:
2850:
2847:
2844:
2841:
2821:
2818:
2815:
2795:
2792:
2789:
2769:
2766:
2763:
2743:
2740:
2737:
2717:
2704:
2701:
2686:
2682:
2661:
2658:
2655:
2652:
2647:
2643:
2622:
2619:
2616:
2613:
2610:
2606:
2602:
2597:
2592:
2564:
2558:
2553:
2548:
2543:
2539:
2535:
2531:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2499:
2496:
2493:
2490:
2487:
2484:
2478:
2474:
2471:
2446:
2440:
2435:
2430:
2425:
2420:
2416:
2412:
2408:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2376:
2373:
2370:
2367:
2364:
2361:
2355:
2351:
2348:
2336:
2333:
2318:
2315:
2312:
2305:
2302:
2276:
2269:
2266:
2242:
2239:
2236:
2216:
2192:
2189:
2186:
2166:
2145:
2122:
2119:
2116:
2109:
2106:
2099:
2094:
2087:
2084:
2060:
2038:
2031:
2028:
1999:
1995:
1969:
1964:
1941:
1935:
1931:
1907:
1904:
1899:
1895:
1891:
1888:
1885:
1880:
1876:
1872:
1867:
1863:
1859:
1856:
1835:
1829:
1824:
1817:
1813:
1809:
1804:
1800:
1796:
1792:
1786:
1783:
1780:
1777:
1774:
1771:
1766:
1763:
1760:
1757:
1754:
1751:
1745:
1741:
1738:
1726:
1723:
1707:
1701:
1696:
1691:
1686:
1681:
1677:
1673:
1669:
1663:
1660:
1657:
1654:
1651:
1648:
1643:
1640:
1637:
1634:
1631:
1628:
1622:
1618:
1615:
1603:
1600:
1586:
1580:
1575:
1570:
1565:
1561:
1539:
1533:
1528:
1523:
1518:
1513:
1509:
1484:
1462:
1439:
1435:
1413:
1407:
1402:
1397:
1392:
1388:
1384:
1380:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1348:
1345:
1342:
1339:
1336:
1333:
1327:
1323:
1320:
1317:
1313:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1284:
1281:
1278:
1275:
1272:
1269:
1263:
1259:
1256:
1244:
1241:
1227:
1204:
1200:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1132:
1120:
1117:
1107:
1104:
1103:
1102:
1088:
1083:
1080:
1076:
1070:
1066:
1056:
1053:
1047:
1042:
1038:
1034:
1029:
1025:
1021:
1018:
1013:
1010:
1006:
1000:
995:
985:
982:
976:
971:
967:
963:
958:
954:
949:
940:
937:
933:
928:
918:
915:
909:
904:
900:
895:
890:
880:
877:
871:
866:
862:
857:
851:
848:
845:
841:
834:
831:
802:
799:
795:
772:
769:
765:
744:
739:
736:
732:
726:
722:
716:
712:
708:
699:
696:
671:
668:
664:
658:
654:
648:
644:
640:
631:
628:
603:
599:
595:
592:
589:
586:
583:
578:
574:
570:
565:
561:
538:
534:
530:
527:
524:
521:
518:
513:
509:
505:
500:
496:
476:is used where
474:
473:
460:
456:
452:
449:
446:
443:
440:
437:
432:
428:
400:
397:
390:ordered probit
370:
369:
358:
353:
349:
345:
342:
337:
333:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
274:
271:
264:
260:
251:
185:
182:
168:
165:
147:
144:
138:
135:
126:interval scale
119:
118:
115:
112:
109:
106:
102:
101:
98:
95:
92:
91:Like Somewhat
89:
75:
72:
64:interval scale
44:, one of four
15:
9:
6:
4:
3:
2:
3749:
3738:
3735:
3733:
3730:
3729:
3727:
3716:
3710:
3706:
3701:
3700:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3656:
3648:
3646:0-8039-4583-3
3642:
3638:
3634:
3633:
3625:
3617:
3611:
3603:
3597:
3593:
3586:
3578:
3572:
3568:
3561:
3553:
3547:
3543:
3536:
3528:
3522:
3518:
3511:
3503:
3497:
3489:
3483:
3479:
3472:
3470:
3461:
3457:
3453:
3449:
3445:
3441:
3434:
3426:
3424:0-89859-268-2
3420:
3416:
3409:
3401:
3399:0-07-005752-4
3395:
3391:
3384:
3382:
3380:
3378:
3376:
3374:
3372:
3370:
3368:
3366:
3364:
3362:
3353:
3351:0-471-40227-3
3347:
3343:
3336:
3328:
3321:
3313:
3309:
3305:
3301:
3297:
3293:
3289:
3285:
3278:
3271:
3263:
3261:0-07-057357-3
3257:
3253:
3246:
3244:
3242:
3240:
3238:
3236:
3234:
3232:
3230:
3228:
3219:
3217:1-55934-427-X
3213:
3209:
3205:
3204:
3196:
3194:
3192:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3148:
3146:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3098:
3096:
3087:
3081:
3077:
3070:
3068:
3066:
3064:
3059:
3049:
3048:Ordinal space
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3030:
3026:
3020:
3015:
3008:
3006:
3001:
2999:
2995:
2991:
2987:
2983:
2979:
2960:
2956:
2950:
2946:
2940:
2936:
2930:
2926:
2925:
2922:
2920:
2915:
2910:
2907:
2903:
2899:
2895:
2885:
2883:
2873:
2865:
2851:
2848:
2845:
2842:
2839:
2819:
2816:
2813:
2793:
2790:
2787:
2767:
2764:
2761:
2741:
2738:
2735:
2715:
2700:
2684:
2680:
2659:
2656:
2653:
2650:
2645:
2641:
2617:
2614:
2611:
2604:
2600:
2595:
2590:
2579:
2576:
2556:
2551:
2546:
2541:
2537:
2533:
2529:
2520:
2517:
2514:
2511:
2508:
2494:
2491:
2488:
2476:
2472:
2469:
2461:
2438:
2433:
2428:
2423:
2418:
2414:
2410:
2406:
2397:
2394:
2391:
2388:
2385:
2371:
2368:
2365:
2353:
2349:
2346:
2332:
2316:
2313:
2310:
2300:
2274:
2264:
2240:
2237:
2234:
2214:
2206:
2190:
2187:
2184:
2164:
2120:
2117:
2114:
2104:
2097:
2092:
2082:
2058:
2036:
2026:
2013:
1997:
1993:
1983:
1967:
1962:
1939:
1933:
1929:
1919:
1905:
1902:
1897:
1893:
1889:
1886:
1883:
1878:
1874:
1870:
1865:
1861:
1857:
1854:
1827:
1822:
1815:
1811:
1807:
1802:
1798:
1794:
1790:
1781:
1778:
1775:
1761:
1758:
1755:
1743:
1739:
1736:
1722:
1719:
1699:
1694:
1689:
1684:
1679:
1675:
1671:
1667:
1658:
1655:
1652:
1638:
1635:
1632:
1620:
1616:
1613:
1599:
1578:
1573:
1568:
1563:
1559:
1531:
1526:
1521:
1516:
1511:
1507:
1497:
1482:
1437:
1433:
1405:
1400:
1395:
1390:
1386:
1382:
1378:
1369:
1366:
1363:
1354:
1351:
1343:
1340:
1337:
1325:
1321:
1318:
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1296:
1290:
1287:
1279:
1276:
1273:
1261:
1257:
1254:
1240:
1225:
1202:
1198:
1188:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1130:
1116:
1114:
1081:
1078:
1074:
1068:
1051:
1045:
1040:
1036:
1027:
1023:
1011:
1008:
1004:
998:
993:
980:
974:
969:
965:
956:
952:
947:
938:
935:
931:
926:
913:
907:
902:
898:
893:
888:
875:
869:
864:
860:
855:
849:
846:
843:
839:
832:
829:
822:
821:
820:
818:
800:
797:
793:
770:
767:
763:
742:
737:
734:
730:
724:
720:
714:
710:
706:
694:
669:
666:
662:
656:
652:
646:
642:
638:
626:
601:
597:
593:
590:
587:
584:
581:
576:
572:
568:
563:
559:
536:
532:
528:
525:
522:
519:
516:
511:
507:
503:
498:
494:
485:
481:
479:
458:
454:
447:
444:
441:
435:
430:
426:
418:
417:
416:
414:
410:
406:
399:Linear trends
396:
393:
391:
387:
386:ordered logit
383:
379:
375:
356:
351:
347:
343:
340:
335:
331:
327:
324:
321:
312:
309:
306:
300:
294:
291:
284:
283:
282:
280:
270:
268:
267:
255:
254:
246:
242:
241:Kendall's tau
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
192:
181:
179:
175:
164:
162:
158:
154:
143:
134:
131:
127:
116:
113:
110:
107:
104:
103:
99:
96:
93:
90:
87:
86:
83:
81:
71:
69:
65:
61:
60:
55:
54:nominal scale
51:
50:S. S. Stevens
48:described by
47:
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42:ordinal scale
39:
35:
29:
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3704:
3669:
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3111:
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3075:
3004:
3002:
2996:testing and
2986:intelligence
2974:
2971:Applications
2911:
2906:Mosaic plots
2891:
2879:
2871:
2706:
2580:
2577:
2338:
2204:
2014:
1984:
1920:
1728:
1720:
1605:
1498:
1246:
1189:
1122:
1109:
816:
483:
482:
477:
475:
412:
408:
402:
394:
376:, outcomes (
371:
276:
257:
248:
209:signed-ranks
197:Mann-Whitney
190:
187:
170:
149:
140:
122:
80:Likert scale
77:
57:
56:by having a
41:
34:Ordinal data
33:
32:
3446:: 303–310.
2994:personality
1551:instead of
157:Kendall's W
130:ratio scale
68:ratio scale
3726:Categories
3114:: 105893.
3054:References
2882:rank tests
384:, such as
3686:1939-1455
3610:cite book
3136:209928171
3128:1568-4946
2914:grayscale
2912:Color or
2898:pie chart
2894:bar chart
2739:≤
2681:ϕ
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235:and the
223:and the
215:and the
100:Dislike
94:Neutral
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3162:Bibcode
3154:Science
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193:-tests
176:, the
3456:JSTOR
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3280:(PDF)
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292:logit
245:gamma
229:ANOVA
88:Like
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3641:ISBN
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