In the section on nonparametric tests in this book, each of the tests are 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 test hypotheses analogous to
those in the previous section. But also, models can be specified to handle
more complex data and situations.

Two disadvantages to using these models is that 1) your audience may not familiar with them, and 2) their results can be somewhat tricky to interpret or explain.

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 *4* > *3*
> *2* > *1*.

• Independent variables can be factors, ordered factors, or interval/ratio variables.

• The general interpretation for significant results of these models isn’t that there is a difference among medians, but 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
*lsmeans* package. An optional approach for post-hoc tests is to use pairwise
ordinal tests of groups, or with pairwise ordinal tests for paired data. These
can be conducted with my custom functions *pairwiseOrdinalTest* and *pairwiseOrdinalPairedTest*.

• 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, but 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. In
other types of models an analogous approach called *analysis of variance*
(*ANOVA*) is used.

#### Model assumptions for CLM

Most statistical models have some assumptions about the underlying data. In order for the model to be valid, these assumptions have to be met.

For CLM, the assumption of concern is called the *proportional
odds assumption*. An explanation of this assumption can be found in the
Wikipedia or IDRE articles cited below.

The ordinal package can test for the proportional odds
assumption with the *nominal_test* and

*scale_test* functions (Christensen 2015b). If any
independent variable fails these tests (that is, a significant *p*-value
is returned), that variable can be handled differently in the model using the *nominal*
and *scale* options in the *clm* function.

### References

For more information on these models and the *ordinal*
package, see:

• Christensen, H.R.B. 2015a. *Analysis of ordinal data with
cumulative link models—estimation with the R-package ordinal*. cran.r-project.org/web/packages/ordinal/vignettes/clm_intro.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. www.ats.ucla.edu/stat/r/dae/ologit.htm.

• Christensen, R.H.B. 2015b. *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.