# OLS with ordinal dependent variable - do the coefficients mean anything?

I currently read a paper in which the author has asked people 3 different questions regarding their life satisfaction, all of which are to be rated on a four point scale: 1) very low, 2) low, 3) high, 4) very high. The author then takes the average of the answers to the three questions for each individual and then uses this individual average as dependent variable in an OLS regression with binary and continuous explanatory variables.

This does not make sense to me from an interpretation point of view. What does $\beta = 0.12$ tell me in this case given the nature of the dependent variable?

So here are my other questions:

1. Is OLS even unbiased and consistent for such outcome variables?
2. Would it be possible to first standardize the answers into the unit interval and then take the average to form a measure of life satisfaction?

For the second question I thought it might make sense to standardize the answers $j$ for individual $i$ as $$\tilde{X}_{i} = \frac{X_{ij}-X_{min}}{X_{max}-X_{min}}$$ and then take the average of that, such that $$\overline{\tilde{X}}_{i} = \frac{1}{N}\sum^N_{i=1}\tilde{X}_{i}$$ could be used as the dependent variable. Given that this measure of life satisfaction is between 0 and 1 this should give more interpretable OLS parameters, right?

• I deleted my answer because I don't think it really answered your question at all; apologies for misinterpreting. I hope someone comes along who can better answer your question. Mar 24, 2015 at 15:52
• I changed the title to make it more clear. Feel free to roll it back if this is not what you wanted.
– Andy
Mar 24, 2015 at 16:06
• Could you interpret it as if the life satisfaction were a continuous latent variable? So B = 0.12 indicates an increase of 0.12 units on this latent continuous QoL scale? The obvious problem is that the distance between low and high QoL is very unlikely to be equivalent to the difference between high and very high so it's quite crude Jun 11, 2021 at 9:57

Interpretive issues for the OLS estimator notwithstanding, the real issue here is in the treatment of an ordinal variable as if it were a variable on the ratio scale. By using standard linear regression analysis, the researchers are essentially treating the ordinal response as if it were a continuous quantity. By averaging three ratings they are also implicitly treating these life satisfaction measures as continuous measures of equal weighting in a continuous aggregated measure. This involves a lot of potentially dubious assumptions about the nature of the rating scale, so you could reasonably be skeptical of the legitimacy of this measure. At a minimum, such a treatment obscures a great deal of information in the specific effects of the explanatory variables on the ordinal categories in the individual response measures.

In any case, if we let $$\bar{Y}$$ denote the response variable in this case (i.e., the average of the three ratings for life satisfaction) then we have a model of the form:

$$\bar{Y}_i = u(\boldsymbol{\beta}, \mathbf{x}_i) + \varepsilon_i,$$

where the true regression function has the linear form:

$$u(\boldsymbol{\beta}, \mathbf{x}_i) = \beta_0 + \beta_1 x_{i,1} + \cdots + \beta_K x_{i,k}.$$

As usual, each slope coefficient $$\beta_k$$ (with $$k=1,...,K$$) is the rate-of-change of the conditional expected response with respect to the corresponding explanatory variable:

$$\beta_k = \frac{\partial u}{\partial x_{i,k}} (\boldsymbol{\beta}, \mathbf{x}_i).$$

As you can see, the coefficient values in the regression look at rates-of-change of the conditional expected value of the averaged life-satisfaction rating, which you may or may not regard as a dubious measure. The fact that all individual life-satisfaction ratings are ordinal integer values means that the averaged value is restricted to the support $$\{ 1, \tfrac{4}{3}, \tfrac{5}{3}, \cdots , \tfrac{11}{3}, 4 \}$$, and so the expected value is a convex combination of these possible values.

With regard to your follow-on questions: (1) the OLS estimator is unbiased and consistent (under broad limiting conditions on the explanatory variables) for the true coefficient values in the model, which in this case may be of dubious meaning to begin with; and (2) standardisation of the response values will merely transform them via a linear transformation, which will alter all the slope coefficients by the corresponding linear transformation; it does not fundamentally change the information coming out of the model.

Q1: The OLS is consistent and unbiased by standard theory. There are assumptions for this, but these don't seem more problematic for the outcome variables you have here than for "standard" applications of linear regression. These assumptions in particular do not say anything about how the quantitative outcome variable is obtained (and do not require it to be normally distributed).

Q2: As far as I can see, what you propose here is just a linear transformation of the outcome variables. Due to its affine equivariance (*), linear regression using this will be technically equivalent to using the original data, which, if I see it correctly, are scaled between 1 and 4 (I'm assuming here that you use $$X_{min}=1,\ X_{max}=4$$; equivalence may not hold if you use the minimum and maximum achieved in the data, which may not generally be 1 and 4). Regression coefficients tell you, as always, the (estimated) expected change of the response variable if the value of an explanatory variable is changed by 1. I don't see much difference between whether this has to be interpreted on a $$[1,4]$$-scale or on a $$[0,1]$$-scale, but if you feel more comfortable with the latter, nobody would stop you from using it. As said before, technically it's equivalent (for example, $$\hat\beta=0.12$$ on the $$[1,4]$$-scale with range 3 should change to $$0.12/3=0.04$$ on the $$[0,1]$$-scale).

(*) Affine equivariance means roughly that if the data are linearly transformed, the estimated regression parameters will change in the appropriate way implied by this transformation, so that they have the same meaning after transformation.

Addendum: To what extent it is appropriate to use ordinal responses in this way as if they were meaningful quantitative numbers is a controversial issue that may be worth some thought but for which there is no generally accepted true answer. In any case it doesn't have implications on your questions (other than knowing that background knowledge about how measurements were obtained is generally valuable for assessing model assumptions such as independence, and interpretative meaning of the results, but this is not specific to these data).