Linearity is the absence of curvature.  The paper below or (to some degree) the abstract  describe useful methods for finding curvature in multiple regression.  If you find a curve you need to add it to the model.  If not you don't.  Various methods (treated in the paper) can be used but none are completely satisfactory. You might also look at Kutner's  book.  Further, Steven's measurement scales often apply nicely to psychometrics but not so nicely to statistics or science, in general. See the Wikipedia link and its references for useful ideas .

http://amstat.tandfonline.com/doi/abs/10.1080/00401706.1995.10484372

https://www.amazon.com/Applied-Linear-Statistical-Models-2013/dp/1259064743/ref=sr_1_1?ie=UTF8&qid=1489953688&sr=8-1&keywords=applied+linear+statistical+models+5th+edition

https://en.wikipedia.org/wiki/Level_of_measurement

Not sure how you will assess the linearity of the relationship between your response and an ordinal predictor. An ordinal predictor is not numeric, is categorical (however, with some information about the order of the categories). I am not aware of a regression technique that would be able to account for the ordering of the categories of an ordinal predictor.* If a predictor is ordinal, the oreder information is not used and the predictor is handled by the model as if it is simply categorical.

If you assign numerical values to the levels of the ordinal predictor, then you construct a new numerical variable and you use this variable as a predictor. This is not ordinal, this is numerical / metric and thus handled like any other numeric/metric predictor in the model.

NB: Statistics does not tell you what values you should assign to what category. The mapping should somehow makes sense in the research context. If you would only map the order of the categories (1, 2, 3, ...) without having something special in mind, you should probably better not create a numerical version at all.

*The orthogonal polynomial coding of the category-ranks can be used to asses a "trend",  that way the ordering information is used.

PS:

numerical predictors are handeled by models through a single column in the design matrix, whereas categorical predictors are handled through k-1 columns in the design matrix (k = the number of different categories); these columns usually contain only zeros and ones (0/1 coding), so that the coefficient of the first category is modelled as part of the intercept and the remaining k-1 coefficients will give the expected differences of the response in the respective category against the first (i.e. reference) category (see also http://stats.idre.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/, there also see "orthogonal polynomial coding").

Sylvia, what is your sample size?  I ask because 50 predictor variables sounds like an awfully large number.  And if you bear in mind that each of your ordinal predictors will have k-1 degrees of freedom (where k = number of categories), you eat up an even more than 50 df.  I am concerned that if the sample size is not quite large, you will be over-fitting your model.  HTH. 

http://www.cs.vu.nl/~eliens/sg/local/theory/overfitting.pdf

My sample size is 255 and it has just sufficient power (.80 for a medium effect) for the 32 variables I am regressing (plus another 12 dummy-coded). However, that's the not the issue. I have no problems with some of the ordinal variables forming a scale, as their numeric value result from the addition of the items. My main problem is how to test linearity with a bunch of 1-item 5-point Likert scales I have in the model.

I appreciate the contribution from everyone so far.  Your answers are very valuable.

Again: 1-item 5-point Likert values are not numeric. These are names or categories, and we only know that there is an ordering structure (a < b < c < d < e).

If you assign numerical values to the Likert values, you get a numeric variable that can be analyzed with statistical tools as any other random variable. As it is entirely arbitrary what values you assign (at least from the statistical point-of-view; retaining the ordering structure, however, should be ensured) it is completely arbitrary if the resulting variable is linearily related to some other variable. But as said above: after you have decided how to assign numerical values to the Likert values, you have a numerical variable that is treated/analyzed like any other numerical variable.

It is often missed to clearly state the rationale behind the value assignment (mapping Likert -> numbers) and to investigate or to state resonable assumptions about the distribution of the constructed random variable.

The random variable you get is actually discrete, but it might be treated as a continuous variable. This often allows to select a neat probability distribution model that will reflect relevant features of the distribution. But it can also be possible that there is no suitable distribution model, what considearbly limits the possible tools we have to analyze the variable.

One reasonable way to treat ordinal independent scale is to transform them through orthogonal polynomials ; that's about what R's lm does (see Venables & Ripley for explanations). Another way is to transform them as a set of increments to some latent variables, and to assess the increments, constraining them to monotonicity : see the R package brms for a Bayesian implementation of this idea, which can also be implemented in BUGS/JAGS or Stan, for Bayesian analysis, or (awkwardly) as elements of a likelihood finctioon, for direct optimization and frequentist inference, via (e. g.) R's optim function.

Emmanuel, many thanks for your very practical approach to answering my question. It is sometimes difficult to derive the right solution from knowledge provided not followed by a recommendation.

I would do a Principal Component Analysis on the 50 explanatory variables and create some composite variables, but also use this technique to remove some of the explanatory variables (i.e. not loading on any factors/components). PCA will tell you which ones are correlated and you can then create non-correlated (or partially correlated if you use oblique rotation after a principal factor analysis, which may be better again than PCA).

It is a good idea but as they study is designed, it is difficult to implement and if I create some composite variables, I will not be able to test my hypothesis. I am also conducting a hierarchical regression analysis. However, everything I am reading so far, inspires me to think about what other options I have to test linearity, which is my main concern.

is your data numerical ordinal or categorical ordinal?

Sylvia,

Testing the *linearity* of the link between an an *ordinal* predictor and a quantitative outcome has no meaning. The basic template for a *linear* relationship is that multiplication of (some value of) the predictor by a real has some meaning ; which is in direct contradiction with the hypothesis  that the predictor is ordinal (and only ordinal).

What can be (potentially) tested is the *monotonicity* of the relationship, i. e. do an increase in the predictor always lead to an increase (resp. decrease) in the outcome, or does thgis depend on the initial value of the predictor ?

A slightly different comparison (note that I did *not* write "test" !) is to compare  the likelihoods/posterior probabilities of

Note that the 3-4 relationship is equivalent to the test of relationship between a qualitative (unordered) predictor and a numeric outcome ; possible relevant tests are one-way ANOVA or Kruskal-Wallis.

The test of monotonicity would be (1 OR 2) vs 3, other comparisons may be usefiul (e. g. 1 vs 2).

Note that those comparisons, except 3 vs 4,  do NOT belong to the classical LR-based hypothesis testing framework, because the corresponding models are *NOT* nested. In a frequentist interpretation, one would have to devise a specific test.

The Bayesian framework *may* appeal by its ease of building a Bayes factor or analogous precedures (Carlin & Chib, RJMCMC, etc...). This is, however, an open problem.

The interpretation of the Bayes factor obtained by such modelisations is, of course, yet another open problem.... ISTR that Christian Robert and his team are working on the (different !) idea of using the weights of distincts model in a mixture to assess the relative importance of these models (look for a 2015 preprint of Kanary & al. in Arxiv).

HTH,

spearman rank correlation coefficients measures linearity