Ordinary Least Squares Regression of Ordered Categorical Data: Inferential Implications for Practice

Abstract

Ordered categorical responses are frequently encountered in many disciplines. Examples of interest in agriculture include quality assessments, such as for soil or food products, and evaluation of lesion severity, such as teat ends status in dairy cattle. Ordered categorical responses are characterized by multiple categories or levels recorded on a ranked scale that, while apprising relative order, are not informative of magnitude of or proportionality between levels. A number of statistically sound models for ordered categorical responses have been proposed, such as logistic regression and probit models, but these are commonly underutilized in practice. Instead, the ordinary least squares linear regression model is often employed with ordered categorical responses despite violation of basic model assumptions. In this study, the inferential implications of this approach are investigated using a simulation study that evaluates robustness based on realized Type I error rate and statistical power. The design of the simulation study is motivated by applied research cases reported in the literature. A variety of plausible scenarios were considered for simulation, including various shapes of the frequency distribution and different number of categories of the ordered categorical response. Using a real dataset on frequency of antimicrobial use in feedlots, I demonstrate the inferential performance of ordinary least squares linear regression on ordered categorical responses relative to a probit model.iii

Employment Status and Health in Spain Before and After the Great Recession

“…Liu and Agresti (2005) have shown that OLS regression is a frequent choice for the analysis of ordered categorical outcomes in social sciences. Moreover, recent research suggests that OLS-based inferences can be considered robust to the violation of model assumptions as regards Type I error and statistical power (Larrabee, Scott, and Bello 2014). Therefore, three models were fitted to data using OLS regression.…”

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“…The use of an ordinary LM was motivated by Larrabee et al. 17 They showed that it is not a big problem to naively treat ordinal scales as continuous to fit ordinary LMs when there is one covariate. By treating ordinal scales as continuous, the score parameter estimates might be more informative than the equally spaced scores.…”

“…It could be explained by the finding of Larrabee et al. 17 that fitting an ordinary LM for ordinal data might not be a problem when there is one covariate in the model. We notice that a binary predictor does not affect the result too much, as shown in Scenarios 4 and 6.…”

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“…Normal linear approximations, as implemented by linear models, have turned out to be surprisingly robust (or at least adaptable) to departures from basic statistical assumptions, due in no small part to the auspice of the Central Limit Theorem and to the fact that linear models often provide reasonable local approximations to a problem. Indeed, reasonably hardy inference can often be observed even under conditions of outright violations of statistical assumptions (Larrabee et al, 2014). It is then perhaps not surprising that oftentimes statistical assumptions receive little attention or are overlooked during a modeling exercise, despite at-times-serious detrimental consequences.…”