Why can't least squares be used when the dependent variable is nominal or ordinal?

Why can't least squares be used when the dependent variable is nominal or ordinal?

• A. It violates regression assumptions
• B. It requires normal distribution
• C. It cannot handle linear relationships
• D. It overfits

Answer: (A)

The basic idea being tested is that ordinary least squares relies on the dependent variable being measured on a continuous, quantitative scale so that we can meaningfully minimize squared distances between observed and predicted values. When the outcome is nominal (categories with no order) or ordinal (ordered categories with potentially unequal spacing), those distances don’t have the same meaningful interpretation and the spacing between categories isn’t guaranteed to be equal. That means the residuals and the error structure assumed by least squares aren’t well-defined, so the method isn’t appropriate for predicting or inferring relationships with such outcomes. In practice, you’d use models designed for categorical data instead: logistic or probit regression for a binary outcome, multinomial logistic regression for a nominal outcome with more than two categories, or ordinal logistic regression when the order of categories matters. These approaches respect the measurement level of the dependent variable and provide well-defined predictions and inferences. So the reason it’s not suitable is that the least-squares framework relies on a continuous, quantitatively scaled dependent variable, and nominal or ordinal outcomes violate that requirement.

The basic idea being tested is that ordinary least squares relies on the dependent variable being measured on a continuous, quantitative scale so that we can meaningfully minimize squared distances between observed and predicted values. When the outcome is nominal (categories with no order) or ordinal (ordered categories with potentially unequal spacing), those distances don’t have the same meaningful interpretation and the spacing between categories isn’t guaranteed to be equal. That means the residuals and the error structure assumed by least squares aren’t well-defined, so the method isn’t appropriate for predicting or inferring relationships with such outcomes.

In practice, you’d use models designed for categorical data instead: logistic or probit regression for a binary outcome, multinomial logistic regression for a nominal outcome with more than two categories, or ordinal logistic regression when the order of categories matters. These approaches respect the measurement level of the dependent variable and provide well-defined predictions and inferences.

So the reason it’s not suitable is that the least-squares framework relies on a continuous, quantitatively scaled dependent variable, and nominal or ordinal outcomes violate that requirement.