# logit - interpreting coefficients as probabilities

I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called the logit scale. Therefore to interpret them, exp(coef) is taken and yields OR, the odds ratio.

If $$\beta_1 = 0.012$$ the interpretation is as follows: For one unit increase in the covariate $$X_1$$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.

Exponentiation yields that for one unit increase in the covariate $$X_1$$, the odds ratio is 1.012 ($$\exp(0.012)=1.012$$), or $$Y=1$$ is 1.012 more likely than $$Y=0$$.

But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:

The coefficients β can be exponentiated and treated as multiplicative effects."

Such that if β1=0.012, then "the expected multiplicative increase is exp(0.012)=1.012, or a 1.2% positive difference ...

However, according to my scripts

$$\text{ODDS} = \frac{p}{1-p}$$

and the inverse logit formula states

$$P=\frac{OR}{1+OR}=\frac{1.012}{2.012}= 0.502$$

Which i am tempted to interpret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which I assume is wrong, but I do not understand why.

How can logit coefficients be interpreted in terms of probabilities?

• (1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber
Aug 24 '18 at 16:34
• @whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also! Aug 24 '18 at 18:07

These odds ratios are the exponential of the corresponding regression coefficient:

$$\text{odds ratio} = e^{\hat\beta}$$

For example, if the logistic regression coefficient is $\hat\beta=0.25$ the odds ratio is $e^{0.25} = 1.28$.

The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 \times 1.28$.

Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.

In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.

The formula is:

$$\text{Percent Change in the Odds} = \left( \text{Odds Ratio} - 1 \right) \times 100$$

• +1: good explanation.
– whuber
Aug 24 '18 at 18:23
• @user1607 this makes sense. However, I don't see how it answers the question regarding if inverse logit to get probabilities is the correct way or not? Jul 13 '19 at 2:07

Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:

Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)

Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...

If you want to interpret in terms of the percentages, then you need the y-intercept ($\beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.

The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.

So when all $x=0$:

$p(Y=1) = \frac{e^{\beta_0}}{1+e^{\beta_0}}$

and if $x_1=1$ (and any other covariates are 0) then:

$p(Y=1) = \frac{ e^{(\beta_0 + \beta_1)}}{ 1+ e^{(\beta_0 + \beta_1)}}$

and those can be compared. But notice that the effect of $x_1$ is different depending on $\beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.

Also notice that your estimate of $\beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $\beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.