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I want to run a regression and I have a scale dependent variable that goes from 0 (extremely pessimistic) to 10 (extremely optimistic). My independent variables are only categorical variables such as age groups (15-24, 25-34, 35-44 etc.), gender, education level (Primary, Secondary, Tertiary etc.). My purpose is to find which demographic group exhibits the most pessimistic attitude. In other words, I am trying to explain the respondents' score (scale dependent variable) as a function of their demographic characteristic (independent variable). Which type of regression could help me to do so? I was thinking about using OLS, even though my dependent variable is clearly not continuous.

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It sounds like your response variable is using a type of Likert scale, which is an ordinal scale. Rather than using OLS, it is generally better to model this type of variable using some kind of ordinal logistic regression, since this latter model treats the response variable as an ordinal variable, and is invariant to renumbering of the values in the same order. The problem with OLS in this case is that it is affected by the numbering of the response values, and if you alter the intervals between the numbers, this changes the OLS estimates in the model (even when the ordinal ranking is preserved). People sometimes treat values on the Likert scale as if they were continuous variables, but it is best to avoid this if you can.

You can implement ordinal logistic regression in R using the polr function in the MASS package (this function implements "proportional odds" logistic regression). If you want to add random effects to the model you can use the clmm2 function in the ordinal package. In either case you use the standard syntax for specifying the regression relationship in the model. In particular, you can implement categorical explanatory variables in this model in the usual method, by specifying these as factor variables.

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I think that Ordinal Logistic Regression is fine for your problem.

A logistic regression model can be used to describe the average effect of independent variables (a.k.a. explanatory variables) on a binary dependent variable (i.e., the response). Although it cannot be used for non-binary problems, it can still be adopted when the dependent variable is ordinal, in the sense that its values follow some order and assuming that the difference between each value is equal, for example, Low-Medium-High, where the difference between Low and Medium is considered the same as Medium and High on that scale.

Therefore, ordinal logistic regression can be used to predict an ordinal dependent variable given one or more independent variables for a non-binary problem.

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