# Odds Ratio vs. Elasticity

I have a discrete choice model with multiple $\beta$ coefficient estimates. For its interpretation I consider two measures. While discrete choice models are somewhat different to regression models, I reckon that the concepts of these two measures are basically the same:

1. Odds Ratio: $OR = exp(\beta)$
Straight forward, everybody knows it.

2. Aggregated Elasticity:

First compute the disaggregate elasticity according to Train (2009, p. 63): $\epsilon_{Xn} = (1-P_n)*\beta*X_n$

where $P_n$ is the probability to chose the alternative and $X_n$ the value of the variable for observation $n$.

Then aggregate it according to Ben-Akiva (1985, p. 133): $\epsilon_X = \frac{ \sum^N (P_n * \epsilon_{Xn})}{\sum^N P_n}$

Here my thoughts to the measures:

As far as I understood, OR and elasticity both indicate how much the probability to observe the dependent variable changes if the variable is present or not.

The OR is derived directly from the $\beta$ coefficients and doesn't account for interactions between the variables in the sample.

The elasticities are calculated based on the sample itself and show the real percentage changes in probability for the variable given the sample.

The latter accounts better for correlations of variables, while the former is a more straight forward concept.

My questions:

• Did I get that right?
• Would you suggest me to use only one of the measures or both? If only one: which one?
• Also general thoughts about odds ratio vs. elasticities will help me! Doesn't have to be in a discrete choice context.

References:

• Ben-Akiva (1985): Discrete Choice Analysis: Theory and Application to Travel Demand. Link
• Train (2009): Discrete Choice Methods with Simulation. Link

1.Did I get that right?

As far as I understood, OR and elasticity both indicate how much the probability to observe the dependent variable changes if the variable is present or not.

Let's go back to the very basic definitions of odds/odds ratios and elasticity. An odds is the defined as the probability of an event occurring divided by the probability of that event not occurring, i.e.: $Odds = \frac{Pr}{1-Pr}$

An odds ratio is therefore the ratio of two odds, so it is technically not correct to say that it indicates the relative probability of the dependent variable changing in the presence of the variable (a ratio of two probabilities is commonly known as a risk ratio). The technically correct interpretation is to use "odds," but odds are not really intuitive or straightforward for lay people, so a common way of interpreting odds ratios in the epidemiology literature is to use "likely." For example, if the OR for the dichotomous variable $Female$ is 1.2, then you would interpret it as "Females are 1.2 times more likely than males to have the disease" or "the odds of disease for females is 1.2 times that of males." Odds ratios are commonly used in public health and the medical literature.

Elasticity is a common measure in economics and comes from economic theory. The basic definition of elasticity is the percent change in one variable corresponding to the percent change in another variable (Train, 2009). That is: $\epsilon=\frac{\% \Delta X}{\% \Delta Y}$

If you were fitting a regression model where the outcome is quantifiable (by which I mean you can fit an OLS or Poisson model), the interpretation is straightforward. For example, a 1% increase in price corresponds to a 0.2% decrease in doctor visits. In discrete choice models, the interpretation is not that straightforward, because we are dealing with likelihoods, instead of something more tangible. In a logistic regression output (for example, output from Stata's margins, eyex(*) postestimation commands), a straightforward interpretation of elasticity would be the percent change in probability of the outcome corresponding to a 1 percent change in the covariate of interest. For instance, if your output says that the elasticity of income was 0.122, the interpretation would be "a 1 % increase in income is associated with a 0.122% increase in probability of blablabla."

Note that both OR and elasticity show you the direction and magnitude of the effect under examination, and they are both measures of relative changes, rather than discrete changes. A third way of quantifying effect size also popular in economics, the marginal effect, measures discrete changes in predicted probability. That is, if variable $Female$ has a marginal effect of 0.302, the interpretation is that females are 30.2 percentage points more likely than males to have the disease.

2. Would you suggest me to use only one of the measures or both? If only one: which one?

It depends on your discipline. Use what the people in your field or your intended audience are familiar with, as they are both correct measures of effect. Do not use the measure with which your intended audience is not familiar. I dabble in both epidemiology and economics (field is public health/health policy) and have worked with both epidemiologists and economists at university, and none of the epidemiologists I worked with had heard of elasticity until they started working with me. On the other hand, my old econometrics professor really hated odds ratios ("it doesn't make sense") and exclusively used marginal effects and elasticities.

3. Also general thoughts about odds ratio vs. elasticities will help me! Doesn't have to be in a discrete choice context.

A parting note, although I probably shouldn't be dishing out this advice since it borders on academic dishonesty, is that sometimes people choose one over the other if it makes their results look "favorable." For example, the table below contains the same result from a simple logistic regression of using a certain health service on income using 4 different measures: the $\beta$ coefficient, OR, elasticity $\epsilon$, and the marginal effect (ME). The important thing to note is that all of these measures came from the same logistic regression model, conveying the same information in different ways.

$$\begin{array}{|c|c|c|c|} \hline & \beta & \text{OR} & \epsilon &\text{ME}\\ \hline \text{Income (USD)} & .226 & 1.25 &.009 & .036\\ \hline \end{array}$$

Of all the measures presented, one may be tempted to present the OR, because it has the largest effect size. That is, a .009% increase in probability or a 3.6 percentage point increase in probability of using a service are not as impressive-sounding as having someone be 1.25 times more likely to use a service as their income increases.

• Great post!,"my old econometrics professor really hated odds ratios ("it doesn't make sense") and exclusively used marginal effects and elasticities" My old professor of Epidemiology (also a statistics professor) once said "The Nobel prize of Economy is not a real Nobel prize". Sep 17, 2015 at 1:38
• @DeepNorth Haha, that's pretty funny. I think nobody wants to admit that we're all on the same team: Team Data. A couple of years ago, I was at a dissertation defense where the student had an economist and a biostatistician on her committee. The biostatistician was her methods person. The economist asked the economists' favorite question: "What about endogeneity?" The biostatistician shut him down with the quickness saying, "We assume everything was measured correctly." Luckily for the student, her chair is an applied person and didn't know any better and deferred to the biostatistician. Sep 17, 2015 at 6:23
• @marquisdecarabas: Thanks for the great explanation. My advisor deals with elasticities regularly, so that shouldn't be a problem. For a dichotomous variable, two questions arise for me: (1) is the elasticity the relative change when the variable is changed from 0 to 1? Since there are no % changes... (2) Is the elasticity of X for all observations $n$ where $X_n=0$ really 0? Deriving from Train's formula given in my question... Sep 17, 2015 at 7:59
• actually, I'll ask the first part in a new question since it may be interesting for other folk as well. The (2) question would really interest me in this context though! Sep 17, 2015 at 16:42