![[banner]](images/banner.jpg "Cohansey River, Fairfield, New Jersey")
In the section on nonparametric tests in this book, each test is used for data from a specific situation or design, such as comparing groups from two-sample unpaired data, or two-sample paired data, or with an unreplicated complete block design.
Cumulative link models are a different approach to analyzing ordinal data. Models can be chosen to handle simple or more complex designs.
This approach is very flexible and might be considered the best approach for data with ordinal dependent variables in many cases.
However, a few disadvantages to using these models are that 1) your audience may not be familiar with them, 2) their results can be somewhat tricky to interpret or explain, and 3) some models won’t converge or model assumptions won’t be met for some data sets.
These models are also called ordinal regression models, or proportional odds models.
The ordinal package
These models and tests will use the ordinal package, and either of two functions, clm and clmm.
A few notes on using cumulative link models:
• The dependent variable must be an ordered factor variable. It does not need to have numerals for levels. For example it could have levels doctorate > masters > bachelors > associates > high.school. But also it could have the levels 5 > 4 > 3 > 2 > 1.
• Independent variables can be factors, ordered factors, or interval/ratio variables.
• The general interpretation for significant results of these models is that there is a significant effect of the independent variable on the dependent variable, or that there is a significant difference among groups.
• Post-hoc tests for factors or groups can be conducted with the emmeans package.
• The threshold = “equidistant” and threshold = “symmetric” options can be used to indicate to the software that levels of the response variable are equally spaced or symmetrically spaced, respectively. This is useful to indicate when these conditions are assumed to be true, and are also useful to try if the model procedure produces errors. Likert items using symmetrical language in the range of responses could be considered symmetric. Likert items with several numbered options with anchor terms only at the ends of scale might be considered equidistant.
Analysis of deviance
The significance of the effects of independent variables will be tested with an analysis of deviance (ANODE) approach. This is analogous to the analysis of variance (ANOVA) used in linear models.
Model assumptions for CLM
Most statistical models have some assumptions about the underlying data, or more properly, the populations from which the data are sampled. In order for the model to be valid, these assumptions should be met.
For CLM, the assumption of concern is called the proportional odds assumption. Some discussion of this can be found in the references below. The ordinal package, however, can also allow for partial proportional odds effects (“nominal” effects) and multiplicative effects (“scale” effects) (Christensen 2015). In addition, tests for the proportional odds assumption can be conducted with the nominal_test and scale_test functions. Any further discussion of proportional, partial proportional, or scale effects is beyond the scope of this book.
References
• Christensen, R.H.B. No date. Cumulative Link Models for Ordinal Regression with the R. cran.r-project.org/web/packages/ordinal/vignettes/clm_article.pdf.
• library(ordinal); help(package="ordinal")
• Wikipedia. 2015. “The model and the proportional odds assumption” in Ordered logit. en.wikipedia.org/wiki/Ordered_logit#The_model_and_the_proportional_odds_assumption.
• IDRE . 2015. R Data Analysis Examples: Ordinal Logistic Regression. UCLA. stats.oarc.ucla.edu/r/dae/ordinal-logistic-regression/.
• Christensen, R.H.B. 2015. Package ‘ordinal’. cran.r-project.org/web/packages/ordinal/ordinal.pdf.
• Hervé, M. 2014. “72. Analyser des notes” in Aide-mémoire de statistique appliquée à la biologie. cran.r-project.org/doc/contrib/Herve-Aide-memoire-statistique.pdf.