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Understanding odds ratios in binary logistic regression

1. Contrived example, odds ratio of 2

Below we have a data file with information about families containing the husband's income (in thousands of dollars) ranging from 10,000 to 12,000, and whether the wife works, 1 if the wife does work, and 0 if the wife does not work.
clear
input inc wifework
10 0 
10 1 
10 1 
11 0 
11 1 
11 1 
11 1 
11 1 
12 0 
12 1 
12 1 
12 1 
12 1 
12 1 
12 1 
12 1 
12 1 
end
You might notice that for families earning $10,000, there are 2 wives who work and 1 who does not, for families earning $11,000 there are 4 wives who work, and 1 who does not, and for families earning $12,000 there are 8 wives who work, and 1 who does not.  We can confirm this using tabulate. 
tabulate inc wifework 
           |       wifework
       inc |         0          1 |     Total
-----------+----------------------+----------
        10 |         1          2 |         3 
        11 |         1          4 |         5 
        12 |         1          8 |         9 
-----------+----------------------+----------
     Total |         3         14 |        17
Let's run a logistic regression predicting wifework from inc. You can see below that the Odds Ratio predicting wifework from inc is 2. But what does this mean? The definition of an odds ratio tells us that for every unit increase in inc, the odds of the wife working increases by a factor of 2.
logistic wifework inc 
 
Logit estimates                                   Number of obs   =         17
                                                  LR chi2(1)      =       0.74
                                                  Prob > chi2     =     0.3891
Log likelihood = -7.5510435                       Pseudo R2       =     0.0468

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |          2   1.614483      0.859   0.391       .4110596    9.730949
------------------------------------------------------------------------------
Let us explore what this means. At the heart of this is the odds ratio, but let's first start with looking at the odds of the wife working at each level of inc, as shown below.
         Number   Number not  Odds
Income  Working  Working     of Working   
10      2        1           2 / 1 = 2   
11      4        1           4 / 1 = 4   
12      8        1           8 / 1 = 8
Suppose we compare the odds of working for those earning $10k (2) with those earning $11k (4). If we divide the odds for those earning $11k by the odds for those earning $10k, we get 4 / 2 = 2. Likewise, if we divide the odds of working for those earning $12k by the odds of working for those earning $11k, we get 8 / 4 = 2. Notice that when income increased by 1 unit ($1000) the odds of working increased by a factor of 2. This is what an odds ratio is. In this example, when we increase income by 1 unit, the odds of the wife working increases by a factor of 2.

We can ask Stata to compute the predicted odds of working broken down by income.

adjust , by(inc) exp 
-------------------------------------------------------------------------------
Dependent variable: wifework     Command: logistic
-------------------------------------------------------------------------------

----------+-----------
      inc |    exp(xb)
----------+-----------
       10 |          2
       11 |          4
       12 |          8
----------+-----------
Key:  exp(xb)    =  exp(xb)
Another way to compute odds is by using probabilities. For example, families that earn $10k have a probability of .666 of the wife working (1 / 3), and a probability of .333 of the wife NOT working. If we divide the probability of working by the probability of not working, we get the same result as we got before, an odds of 2. This is illustrated in the table below. 
                                Odds
Income  P(work)    P(not work)  of Working   
10      2/3=.666   1/3=.333     .666 / .333 = 2
11      4/5=.800   1/5=.200     .800 / .200 = 4
12      8/9=.888   1/9=.111     .888 / .111 = 8
We could ask Stata to compute the predicted probability of working by income.
adjust , by(inc) pr 
-------------------------------------------------------------------------------
Dependent variable: wifework     Command: logistic
-------------------------------------------------------------------------------

----------+-----------
      inc |         pr
----------+-----------
       10 |    .666667
       11 |         .8
       12 |    .888889
----------+-----------
Key:  pr         =  Probability
Note that we get the same odds whether we used the number working or the prob(working). The second method is the more traditional method, and the one we will use from this point forward.

Understanding coefficients

In addition to getting odds ratios, you can also get coefficients. The coefficients are the estimates from the regression equation predicting logits. We can get the estimates using the logit command in Stata.
logit 
Logit estimates                                   Number of obs   =         17
                                                  LR chi2(1)      =       0.74
                                                  Prob > chi2     =     0.3891
Log likelihood = -7.5510435                       Pseudo R2       =     0.0468

------------------------------------------------------------------------------
wifework |      Coef.   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   .6931472   .8072415      0.859   0.391       -.889017    2.275311
   _cons |  -6.238325   8.979481     -0.695   0.487      -23.83778    11.36113
------------------------------------------------------------------------------
The equation shown obtains the predicted log(odds of wife working) = -6.2383 + inc * .6931 Let's predict the log(odds of wife working) for income of $10k.
display -6.2383 + 10 * .6931 
 .6927
We can take the exponential of this to convert the log odds to odds. Taking the exponential of 6927 yields 1.999 or 2. This was the odds we found for a wife working in a family earning $10k.
display exp(.6927) 
 1.9991058
We can convert the odds to a probability. The formula for converting an odds to probability is probability = odds / (1 + odds). We see the predicted probability of a wife working when the family earns $10k is .666 .
display 2 / (1 + 2) 
 .66666667
By the way, if we take the exp of a coefficient, it is the odds ratio.
display exp( _b[inc] )  
 2

Contrived example, odds ratio of 1.1

Below we explore another example, except in this case the odds ratio is 1.1 .  Like before, there is a variable called inc that represents the income of the family, and wifework that is 1 if the wife works, 0 if she does not.  Below we use the file.
use oddsrat2 , clear
Below we use tabulate to look at the number of wives who work (and don't work) for each level of income.  For example, there were 233 families earning $13,000, of which 133 had working wives and 100 had non-working wives.
tabulate inc wifework  
           |       wifework
       inc |         0          1 |     Total
-----------+----------------------+----------
        10 |       100        100 |       200 
        11 |       100        110 |       210 
        12 |       100        121 |       221 
        13 |       100        133 |       233 
        14 |       100        146 |       246 
        15 |       100        161 |       261 
        16 |       100        177 |       277 
        17 |       100        195 |       295 
        18 |       100        214 |       314 
        19 |       100        236 |       336 
-----------+----------------------+----------
     Total |      1000       1593 |      2593
Let's perform a logistic regression predicting wifework from inc.
logistic wifework inc 
 
Logit estimates                                   Number of obs   =       2593
                                                  LR chi2(1)      =      45.23
                                                  Prob > chi2     =     0.0000
Log likelihood = -1706.3066                       Pseudo R2       =     0.0131

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   1.100029   .0156951      6.682   0.000       1.069693    1.131225
------------------------------------------------------------------------------
This time we get an odds ratio of 1.1 . Let's see how we would interpret this. Let's use the adjust command to get the odds of the wife working by income.
adjust , by(inc) exp 
-------------------------------------------------------------------------------
Dependent variable: wifework     Command: logistic
-------------------------------------------------------------------------------

----------+-----------
      inc |    exp(xb)
----------+-----------
       10 |    .999386
       11 |    1.09935
       12 |    1.20932
       13 |    1.33029
       14 |    1.46335
       15 |    1.60973
       16 |    1.77075
       17 |    1.94788
       18 |    2.14272
       19 |    2.35705
----------+-----------
Key:  exp(xb)    =  exp(xb)
We see that the odds of the wife working for inc of 10 is .999 (let's say 1.0). The odds ratio of 1.1 tells us that the odds of the wife working should go up by a factor of 1.1 for ever unit increase in inc. Let's see how this works. If the family makes $11,000, the odds of the wife working will be 1.1 times greater or 1.1. If the family makes $12,000 the odds will again be 1.1 times greater or 1.1 * 1.1 or 1.21. If a family makes $13,000 the odds will again be 1.1 times greater or 1.1* 1.1*1.1 = 1.331.

Say that we wanted to know the odds of the wife working if we increased income by an additional 5 units ($5,000) to be $18,000. The odds would go up by 1.15 = 1.61 times.  So we would multiple the odds at $13,000 (1.33) by 1.61 = 2.14. So the odds of a wife working if the husband earns $18,000 is predicted to be 1.61, just as shown in the table above.

This shows that you can interpret the odds ratio in a couple of ways. 
1. For a one unit change in the predictor, the odds of a wife working increases by the odds ratio. 
2. For an x unit change in the predictor, the odds of a wife working increases by the odds ratio to the x power, odds-ratiox.

Contrived example with odds ratio of 1.5

Here is another example like the ones above, except that the odds ratio is 1.5. 
clear
use oddsrat3 , clear
Here we show the number of wives who work, and don't work at each level of income.
tabulate inc wifework  
 
           |       wifework
       inc |         0          1 |     Total
-----------+----------------------+----------
        10 |       100        100 |       200 
        11 |       100        150 |       250 
        12 |       100        225 |       325 
        13 |       100        338 |       438 
        14 |       100        506 |       606 
        15 |       100        759 |       859 
        16 |       100       1139 |      1239 
        17 |       100       1709 |      1809 
        18 |       100       2563 |      2663 
        19 |       100       3844 |      3944 
-----------+----------------------+----------
     Total |      1000      11333 |     12333
Below we perform a logistic regression. We see that the odds ratio is 1.5. 
logistic wifework inc 
 
Logit estimates                                   Number of obs   =      12333
                                                  LR chi2(1)      =    1041.24
                                                  Prob > chi2     =     0.0000
Log likelihood = -2949.9768                       Pseudo R2       =     0.1500

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   1.499958   .0191732     31.718   0.000       1.462846    1.538012
------------------------------------------------------------------------------
We can use the adjust command with the exp option to get the predicted odds of the wife working at each level of income. We can see that for every unit increase in inc, the odds of the wife working increases by a factor of 1.5.  Try taking any of the odds ratios and multiplying it by 1.5 and you will get the odds ratio for the next level of income, e.g. taking the odds for income of 11 is 1.5, and multiplying that by 1.5 gives 2.25, which is the odds of working for an income of 12.
adjust , by(inc) exp 
-------------------------------------------------------------------------------
Dependent variable: wifework     Command: logistic
-------------------------------------------------------------------------------

----------+-----------
      inc |    exp(xb)
----------+-----------
       10 |    1.00019
       11 |    1.50025
       12 |    2.25031
       13 |    3.37537
       14 |    5.06291
       15 |    7.59415
       16 |    11.3909
       17 |    17.0859
       18 |    25.6281
       19 |    38.4411
----------+-----------
Key:  exp(xb)    =  exp(xb)

Contrived example, odds ratio of .66667

All the examples we have looked at so far have had odds ratios that are greater than one.  When the odds ratio is over 1, the odds of, say the wife working, increases as the predictor increases.  On the other hand, if the odds ratio is less than one, the odds of the wife working decreases as the predictor increases.
use oddsrat4 , clear 
tabulate inc wifework  
 
           |       wifework
       inc |         0          1 |     Total
-----------+----------------------+----------
        10 |       100       3844 |      3944 
        11 |       100       2563 |      2663 
        12 |       100       1709 |      1809 
        13 |       100       1139 |      1239 
        14 |       100        759 |       859 
        15 |       100        506 |       606 
        16 |       100        338 |       438 
        17 |       100        225 |       325 
        18 |       100        150 |       250 
        19 |       100        100 |       200 
-----------+----------------------+----------
     Total |      1000      11333 |     12333
We indeed see that the odds ratio is .666.
logistic wifework inc 
 Logit estimates                                   Number of obs   =      12333
                                                  LR chi2(1)      =    1041.24
                                                  Prob > chi2     =     0.0000
Log likelihood = -2949.9768                       Pseudo R2       =     0.1500

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   .6666852   .0085219    -31.718   0.000       .6501901    .6835989
------------------------------------------------------------------------------
We can get the odds of the wife working using the adjust command. You can see that the odds of the wife working go down as income increases.  In fact, the income goes down by a factor of .666.
adjust , by(inc) exp 
-------------------------------------------------------------------------------
Dependent variable: wifework     Command: logistic
-------------------------------------------------------------------------------

----------+-----------
      inc |    exp(xb)
----------+-----------
       10 |    38.4411
       11 |    25.6281
       12 |    17.0859
       13 |    11.3909
       14 |    7.59415
       15 |    5.06291
       16 |    3.37537
       17 |    2.25031
       18 |    1.50025
       19 |    1.00019
----------+-----------
Key:  exp(xb)    =  exp(xb)
For an income of 10, the odds of the wife working are 38.4411. If we multiply this by the odds ratio of .6666 we get get 25.62, which is the odds of a wife working when the husband earns 11.

When the odds ratio for inc is more than 1, an increase in inc increased the odds of the wife working. When the odds ratio for inc is less than one, an increase in inc leads to a decreased odds of the wife working. If the odds ratio for inc is exactly 1, the odds of the wife working would not change when income changes.

Contrived example, 2 groups 1.1, and 1.5

Let us combine the data files from example 2 (where the odds ratio was 1.1) and example 3 (where the odds ratio was 1.5). Also, let's assume that example 2 was composed of families without children, and example 3 was from families with children. Below we combine the files, making child 0 for the data from example 2 and child 1 for the data from example 3.
use oddsrat2, clear
gen child = 0
append using oddsrat3
replace child = 1 if child == .  
(12333 real changes made)
We know from running the previous logistic regressions that the odds ratio was 1.1 for the group with children, and 1.5 for the families without children. Below we run a logistic regression and see that the odds ratio for inc is between 1.1 and 1.5 at about 1.32.
logistic wifework inc child 
Logit estimates                                   Number of obs   =      14926
                                                  LR chi2(2)      =    2187.87
                                                  Prob > chi2     =     0.0000
Log likelihood = -4785.5667                       Pseudo R2       =     0.1861

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   1.320337   .0128444     28.565   0.000       1.295401    1.345754
   child |   4.624184   .2583505     27.409   0.000       4.144565    5.159305
------------------------------------------------------------------------------
We know that the odds ratio of 1.32 is too high for those without children (who had an odds ratio of 1.1), and too low for those with children (who had an odds ratio of 1.5).

Below we create an interaction term by multiplying inc and child creating incchild.

generate incchild = inc*child
We now include incchild as a term in the regression.
logistic wifework inc child incchild 
Logit estimates                                   Number of obs   =      14926
                                                  LR chi2(3)      =    2446.43
                                                  Prob > chi2     =     0.0000
Log likelihood = -4656.2835                       Pseudo R2       =     0.2080

------------------------------------------------------------------------------
wifework | Odds Ratio   Std. Err.       z     P>|z|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     inc |   1.100029   .0156951      6.682   0.000       1.069693    1.131225
   child |   .0450401   .0130882    -10.669   0.000       .0254828    .0796069
incchild |   1.363563   .0261209     16.188   0.000       1.313316    1.415732
------------------------------------------------------------------------------
The odds ratio for inc of 1.1 is the same as the odds ratio for the group without children (when children=0). This tells us that for families with no children, every unit increase in income increases the odds of the wife working increases by a factor of 1.1.

The odds ratio for the term incchild is 1.36, which tells us that for families with children, for every unit increase in income the odds of the wife working increases by an additional factor of 1.36. So, for families with children, for a unit increase in income, the odds of the wife working increases by 1.1 times 1.36 which is 1.5 (1.496 rounds to 1.5).  This is as we saw above, that for families with children, the odds ratio was 1.5.

We can confirm the odds ratio by looking at the odds of women working separately for those with children, and without children. Let's use the prediction formula to confirm the results described above. We can compare the odds of the wife working for those earning $12,000 and $13,000 for those without children.
display exp( _b[_cons] + 12*_b[inc] + 0*_b[child] + 0 * _b[incchild] )  
1.2093207
display exp( _b[_cons] + 13*_b[inc] + 0*_b[child] + 0 * _b[incchild] )  
1.3302875
We see that this odds ratio is 1.1, as we expected. 
display 1.33 / 1.21 
1.0991736
Likewise, let's use the equation to make the predictions for those with children, comparing those earning $12,000 and those earning $13,000.
display exp( _b[_cons] + 12*_b[inc] + 1*_b[child] + 12 * _b[incchild] )  
2.2503079 
display exp( _b[_cons] + 13*_b[inc] + 1*_b[child] + 13 * _b[incchild] )  
3.3753679
We see that this odds ratio is 1.5, as we expected.
display 3.375 / 2.25 
1.5

Concluding comments

In these examples, we have tried to help make it easier to understand an interpret odds ratios.  We have fabricated data with certain odds ratios making data that fits perfectly.  If this were linear OLS regression, it would be like making up X and Y data and making up data that fits a line perfectly.  When you analyze your data, it will not fit perfectly so you won't see the kind of perfect relationships we have shown.  But, when you analyze your data the predicted values will be like the examples we have explored.  The difference is that in the examples we considered here, the data fit the predicted values exactly.  In your data, there will be discrepancies between the predicted and actual values.

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