How to interpret odds ratio for variables that range from 0 to 1

Question

How can I interpret the odds ratio of a variable in my logistic regression model that is a "rate," that is, a value between 0 and 1? What would a unit be in this case?

For example, suppose I am modeling basketball shot data, and my response variable is equal to 1 if the player makes the shot and 0 if they miss. Additionally, I have an explanatory variable that is the team's shooting accuracy rate up to that point, and this variable ranges from 0 to 1. So, how can I interpret the odds ratio of the coefficient for this variable? Because I cannot say it will increase by 1 unit, as the variable cannot exceed values greater than 1. Any tips on how I can interpret this, please?

Answer 1

Another approach would be to change the units. This is often useful for interpretation and, for most changes, it doesn't affect the meaning of the results.

Here, you could change "rate" (0 to 1) to "percentage" (0 to 100). Then the "one unit" is easy to think about. In your particular case, the independent variable not only has absolute limits of 0 and 1, but, except for very early in the game, has some practical limits. Certainly for pro or high level college basketball, the team shooting percentage is going to vary in a much smaller range. I think this adds to the appropriateness of using %.

One last reason is that field goals accuracy is usually reported as %. So, that unit will be more familiar to fans.

Answer 2

There are two issues here. First, forgetting about odds ratios for a moment, in any regression model, the coefficient (or odds ratio, or incidence risk ratio or whatever) always tells you "what happens to" the dependent variable when the independent variable it is attached to increases by "1." Now, as you note, sometimes "increased by 1" is not a realistic scenario for a particular dependent variable, although in your case it at least makes mathematical sense: it means going from a rate of 0 to a rate of 1. But the model doesn't care either way. It's up to you to decide what the "effect" of a "one unit increase" means in practical terms. Here, since going from 0 to 1 is the maximal theoretical amount that shooting rate could increase, so the coefficient or odds ratio gives you an "upper bound" on the size of the relationship. This is also how coefficients work for binary/dummy independent variables (like "player was drafted in first round"). "Increasing by 1" means "going from 0 to 1" ("being drafted in the first round vs not").

Everything I just said also applies to odds ratios - they tell you the "effect" of a one unit increase in the independent variable, whatever that means. However, odds ratios are very commonly misunderstood, so you should be aware that the odds ratio doesn't tell you anything about how the probability of making a shot changes: an odds ratio of 1.25 doesn't mean that the expected probability of making the shot increases by 25%. It tells you how the odds of making a shot goes up by 25%. So be sure you understand that difference before you try to interpret an odds ratio.

Answer 3

When dealing with a variable in a logistic regression model that is a rate (i.e., a value between 0 and 1), interpreting the odds ratio can be a bit different due to the nature of the variable. Here’s how to interpret the odds ratio for this type of variable.

Example Context

Suppose you are modeling basketball shooting data and have:

• Response variable: 1 if the player makes the shot and 0 if they miss.
• Explanatory variable: The team’s shooting percentage up to that point (ranging from 0 to 1).

Interpreting the Odds Ratio

1. Understanding the Coefficient:

   • The coefficient for the explanatory variable in a logistic regression model represents the natural log of the odds ratio associated with a one-unit change in the explanatory variable.
   • In your case, "one unit" refers to a change of 1 in the explanatory variable. However, since the variable is a rate and ranges from 0 to 1, a one-unit change is not realistic (as you cannot go from a rate of 0 to 1 in one step).

2. Interpreting the Odds Ratio:

   • To interpret the effect of a change in the rate variable, you should consider realistic incremental changes in the variable. For example, instead of thinking in terms of a one-unit increase, you might consider smaller increments, such as 0.1 units.
   • If the coefficient for the explanatory variable is, say, 0.5, then the odds ratio for a one-unit increase in the team’s shooting percentage is exp(0.5) ≈ 1.65. This means that the odds of making the shot increase by approximately 65% for a one-unit increase in the shooting percentage.

3. For Incremental Changes:

   • If you want to interpret smaller changes, such as an increase of 0.1 units, you should calculate the odds ratio for that specific change.
   • If the coefficient for the explanatory variable is β, for an increase of 0.1 units, you can calculate the odds ratio as exp(β × 0.1). This gives you a more practical understanding of the variable’s impact.