Stata FAQ
How do I interpret odds ratios in logistic regression?
You may also want to check out,
FAQ: How do I
use odds ratio to interpret logistic regression?, on our General FAQ page.
Introduction
Let's begin with probability. Probabilities
range between 0 and 1. Let's say that the
probability of success is .8, thus
Then the probability of failure is
Odds are determined from probbailities and range between 0 and infinity.
Odds are defined as the ratio of the probability of success and the probability
of failure. The odds of success are
odds(success) = p/(1p) or
p/q = .8/.2 = 4,
that is, the odds of success are 4 to 1. The odds of failure would be
odds(failure) = q/p = .2/.8 = .25.
This looks a little strange but it is really saying that the odds of failure are 1 to 4. The odds of success and the odds of failure are just reciprocals of one another, i.e.,
1/4 = .25 and 1/.25 = 4. Next, we will add another variable to the equation so that we can compute an odds ratio.
Another example
This example is adapted from Pedhazur (1997). Suppose
that seven out of 10 males are admitted to an engineering school while three of 10 females
are admitted. The probabilities for admitting a male are,
p = 7/10 = .7 q = 1  .7 = .3
If you are male, the probability of being admitted is 0.7 and the probability
of not being admitted is 0.3.
Here are the same probabilities for females,
p = 3/10 = .3 q = 1  .3 = .7
If you are female it is just the opposite, the probability of being admitted
is 0.3 and the probability of not being admitted is 0.7.
Now we can use the probabilities to compute the odds of admission for both males and females,
odds(male) = .7/.3 = 2.33333
odds(female) = .3/.7 = .42857
Next, we compute the odds ratio for admission,
OR = 2.3333/.42857 = 5.44
Thus, for a male, the odds of being admitted are 5.44 times larger than the odds for a female
being admitted.
Logistic regression in Stata
Here are the Stata logistic regression commands and
output for the example above. In this example
admit is coded 1 for
yes and 0 for no
and
gender is coded 1 for male and 0 for female. In Stata, the
logistic
command produces results in terms of odds ratios while
logit produces results in
terms of coefficients scales in log odds.
input admit gender freq
1 1 7
1 0 3
0 1 3
0 0 7
end
This data represents a 2x2 table that looks like this:


Admission 


1 
0 
Gender 
1 
7 
3 
0 
3 
7 
logit admit gender [fweight=freq], nolog or
(frequency weights assumed)
Logistic regression Number of obs = 20
LR chi2(1) = 3.29
Prob > chi2 = 0.0696
Log likelihood = 12.217286 Pseudo R2 = 0.1187

admit  Odds Ratio Std. Err. z P>z [95% Conf. Interval]
+
gender  5.444444 5.313234 1.74 0.082 .8040183 36.86729

/* Note: the above command is equivalent to 
logistic admit gender [weight=freq], nolog */
logit admit gender [weight=freq], nolog
(frequency weights assumed)
Logistic regression Number of obs = 20
LR chi2(1) = 3.29
Prob > chi2 = 0.0696
Log likelihood = 12.217286 Pseudo R2 = 0.1187

admit  Coef. Std. Err. z P>z [95% Conf. Interval]
+
gender  1.694596 .9759001 1.74 0.082 .2181333 3.607325
_cons  .8472979 .6900656 1.23 0.220 2.199801 .5052058

Note that
z = 1.74 for the coefficient for
gender and for the odds ratio for gender.
About logits
There is a direct relationship between the
coefficients produced by
logit and the odds ratios produced by
logistic.
First, let's define what is meant by a logit: A logit is defined as the log
base e (log) of the odds. :
[1] logit(p) = log(odds) = log(p/q)
The range is negative infinity to positive infinity. In regression it is
easiest to model unbounded outcomes. Logistic regression is in reality an ordinary regression using the logit as
the response variable. The logit transofrmation allows for a linear relationship between the
response variable and the coefficients:
[2] logit(p) = a + bX
or
[3] log(p/q) = a + bX
This means that the coefficients in logistic regression are in terms of
the log odds, that is, the coefficient 1.694596 implies that a one unit change in gender
results in a 1.694596 unit change in the log of the odds. Equation [3] can be expressed in odds by getting rid of the
log. This is done by taking
e to the power for both sides of the equation.
[4] e^{log(p/q)} = e^{a + bX}
or
[5] p/q = e^{a + bX}
The end result of all the mathematical manipulations is that the odds
ratio can be computed by raising
e to the power of the logistic coefficient,
[6] OR = e^{b} = e^{1.694596} = 5.44
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