Abstract
In a linear model relevance of a categorical predictor with ordered levels is typically tested by use of the standard F-test (known from statistical textbooks). Such a test can also be applied for testing whether the regression function is linear in the ordinal predictor’s class labels. In this paper we propose an alternative (restricted) likelihood ratio test for these hypotheses which is especially suited for ordinal predictors and is based on the mixed model formulation of penalized dummy coefficients. We show in simulation studies that the new test is more powerful than the standard F-test in many situations. The advantage of the new test is especially striking when the number of ordered levels is moderate or large. Using the relationship to mixed effect models and robust existent fitting software obtaining the test and its null distribution is very fast; a fast R implementation is provided.
Highlights
We consider an ordinal covariate x with levels k = 0, . . . , K
Penalties of higher order are possible, but we found these two penalties to be especially useful for ordinal predictors because: (1) they incorporate the ordinal scale level of x; and (2) penalize derivations from a model without predictor x, and a linear model based on the class labels, respectively
We proposed a new test for ordinal predictors in the classical linear model
Summary
In standard linear models the conditional expectation E(y|x) of y given an ordinal predictor x, is assumed to have a simple form. Whether methods for interval-level data (as model (1)) can be used for ordinal variables has long been debated, in particular in the social sciences; see, for example, Winship and Mare (1984), Labowitz (1970) and Mayer (1970, 1971). Using the mixed model formulation, we propose an alternative (restricted) likelihood ratio test for checking linearity – i.e., distinguish between model (1) and (2) – and checking relevance of an ordinal predictor (Section 5).
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