Ordinal data

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]^: 678 Nonparametric methods have 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]^: 25–28 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.[7][8][4]^: 90

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

In lieu of testing differences in means with t-tests, differences in distributions of ordinal data from two independent samples can be tested with Mann-Whitney,[9]^{: 259–264} runs,[9]^{: 253–259} Smirnov,[9]^{: 266–269} and signed-ranks[9]^{: 269–273} tests. Test for two related or matched samples include the sign test[5]^: 80–87 and the Wilcoxon signed ranks test.[5]^: 87–95 Analysis of variance with ranks[9]^{: 367–369} and the Jonckheere test for ordered alternatives[5]^{: 216–222} can be conducted with ordinal data in place of independent samples ANOVA. Tests for more than two related samples includes the Friedman two-way analysis of variance by ranks[5]^{: 174–183} and the Page test for ordered alternatives.[5]^{: 184–188} Correlation measures appropriate for two ordinal-scaled variables include Kendall's tau,[9]^{: 436–439} gamma,[9]^{: 442–443} r_s,[9]^{: 434–436} and d_yx/d_xy.[9]^: 443

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

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

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

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

Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a contingency tables. A correlation r is found between the variables where r lies between -1 and 1. To test the trend, a test statistic:

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

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

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

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

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 information theory.[11]

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

This model can be generalized by defining the model using ${\displaystyle \mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} }$ instead of ${\displaystyle \mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} }$, 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.

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

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

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

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

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

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

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

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

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 ${\displaystyle k}$, since ${\displaystyle Y\leq k}$ is compared to ${\displaystyle Y>k}$, whereas in the other models the size of the reference category remains fixed, as ${\displaystyle Y=k}$ is compared to ${\displaystyle Y=1}$ or ${\displaystyle Y=k+1}$.

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